No title

Letters to the Editor

The Mathematical Jntelligencer

The Alternating Harmonic Series

encourages comments about the

Mengoli-Mercator Formula"

material in this issue. Letters to the

the definition of

editor should be sent to either of the editors-in-chief, Chandler Davis or

I enjoyed Friesecke and Wehrstedt's Note. ''An Elementary Proo f of th e Gregory­

( Mathematical lrztelligencer 28 (2006), no. 3. 4-Sl.

While reading it. I wondered if there was a geometric argument proceeding from

e r v = � between 1 and 2. A little tinkering showed me there was. It is simply that the J(\wer Riemann sum for � dx using

ln(2) as the area u

equal intervals of width

nd

(21

1

-;;•

.

1

I1+ n

Marjorie Senechal.

II

1

1

;�1

1

=I-+ .1 n II

;� 1 11

.\'

1.

•

1

is equal to the first 2n terms of the alternating harmonic series n

1

211

1

n

1

2n

1

11

;�1

I

;�1

I

;�1

I

;�1

211

( - 1 )i

. =II�-I�=I�-2II-= 21 + 1

;�1 11

1

;�1

ll

(equation (6) in Friesecke and Wehrstedt's note). Thus the alternating series must converge to ln(2).

Jim Henle Department of Mathematics Smith College Northampton, MA 01 063 USA e-mail: [email protected] du

Editor's

Note. Rob Uurckel informs us that H. G. Forder, in a half-page note in

Mathematical Gazette 12 0925), 390, gives a brief derivation of the formula, much

like Henle's. And Bruce Berndt points out that the idea of using Riemann sums for

the integral of� to sum series dates bJck at least to Ramanujan, who used the techx

nique to show first that

1

+ 2::-:�1

2

( 2 n) 3 - 2 n

=

2

ln (2) ("Ramanujan's Notebooks,'' Bruce Berndt, construct an identity for In (3)

Math. Mag. 1 5, 1 978) , and later to ( Ramanujan �' Notehook.s, Pm1 I. Bruce Berndt, pp.

26-27). Meanwhile, we have the following

4

THE MATH EMATICAL INT ELLIGENCER © 2007 Springer Science+Business Media, Inc.

from Down Under.

Still Shorter For n E N, let

dx.

1 +X Then

1 1l

+1

By induction

1 - - + - -+ · ··+ ( -1)"+ 1 1

1

3

2

Let rz ----> co I" ----> 0 and

1-l_+l_-···=Io= It's that simple.

2

3

{ o

1

x+1

1

n

=Io + ( -1)"+ 1 In.

dx=

J2 I

1 X

dx= In 2.

Michael D. Hirschhorn School of Mathematics and Statist ics University of New South Wales Sydney, NSW 2052 Australia e-mai l : m . hirsch [email protected] .au

© 2007 Springer Sc1ence+ Bus1ness Media, Inc., Volume 29, Number 2 , 2007

5

•:mt1

An othe r M otivat i o n fo r the Hype rbo l i c P l ane Segments Moving on the Line MARCOS SALVAI

n interval on the real line is said to be nontrivial if it has more than one point. Let us denote :J = (nontrivial hounded and closed

intervals of the real linel.

Notice that .'J is not a subset of a Eu­ clidean space (although its elements. the intervals. are so) . We ask ourselves how to define a notion of distance and of best path joining two elements of .'J. For example, x < y and x' < y' \Ve can take dist C[x, }'L[x' , v'D

� vrccx )2, ' ·-).,-2 -+----,- -J--., _ (J--:, ·':-_-x -

.

that is, copying the Euclidean distance from the open set ((x,y) E IR:2 :�: < yl by the obvious bijection. The best path joining these two intervals is then the curve y: [0, 1]--'> .'J given by y(t) = [(1- t) x + tx', (1 - t) y + (v'l. Now. this notion of distance may he not the most convenient one if we con­ sider the nature of the elements of :J. In a certain sense, it discriminates by the size of the segments. For instance, it gives dist ([0,10-'], [1,1 + 10-'1]) = v2 = dist ([0,10-:1], [1,1 + 10-"D. Nevertheless , considering the relative size of the intervals, the first two of them seem to be much closer to each other than the last two are (let us imag­ ine that we ask the interval [0,10-3] to move to the position of [1,1 + 10-.:l]; it will seem to it very far away ) . Let us seek a notion of distance ·which takes this into account . We will need some basic facts about the hyperbolic plane [1].

The Hyperbolic Plane

Let� = (( u,v) E IR:211'> Ol. The hyper­ bolic length of a differentiable curve y : [a , b] -'>�, y(t) = (u(t), l'(t)), is de­ fined by �ii'Y'(t)ll dt, where

J

lly'(t)ll = 1Cu'Ct), l''cm:/l'Ct)

6

THE MATHEMATICAL INTELLIGENCER © 2007 Spnnger Sc1ence-+-Business Media. Inc.

0)

is the hvperbolic speed of y at the in­ stant t (here 1C'c)') = V :\·2 + y 2 denotes the Euclidean norm ) . This is a model for the upper half-plane of Lobache\·sky­ Poincare, the classical example of a space v.·here the fifth axiom of Euclid does not hold . By a line in� we un­ derstand the image of a maximal geo­ desic, that is. a constant-speed curve in 'J-e defined on the whole real line which mmtmizes the (hyperbolic) length between any two of its points. They are \vel! kno\vn to be the curves obtained by intersecting� with verti­ cal (Euclide a n ) straight lines and with circles centered at points on the hori­ zontal axis. The importance of the hyperbolic plane stems from the fact, among many other things, that it is essentially the unique complete simply connected sur­ face with constant negative curvature.

A Metric on 9 Which Does Not Discriminate by Size Recall that we are looking for a notion of distance travelled hy a segment mov­ ing (and changing its size as wel l ) in the real line, and we want to treat the shorter intervals fairly. We identify a segment in .'J hy its midpoint a and its length e. that is, we parametrize :J via ¢ : :A = ((a,£) E IR:2,e > Ol-" .'f.

c/J(a,€) = [a - €!2.a + €/2].

(2)

Let y(t) = [at- €1/2, at + €1/2] be a curve in :J, and let a(t) = ¢- 1 (y(t)) = (at,ft) be the corresponding curve in :A. Now we see that the hyperbolic speed (1) of a (or equivalently of y) a t t h e instant t , that is,

is appropriate if we do not want to dis­ criminate by size. since it "relativizes" the Euclidean speed to the size of the interval at the instant t. In the following examples we ob­ serve that , at the infinitesimal level, each segment of .'J takes itself as the yard-

stick to measure the size of the neigh­ boring �egments and the distance from them. EXAMPLE 1: If an interval moves to the

right s times its own length without changing size , then its motion will be of s units if we consider the hyperbolic metric. In fact, for the curve y ( t) = [a- £/2,a + £/2] + t€, with 0 :::s t::::; s, the speed of the associated curve aU)= (a+ t£.£) in 'Je, with respect to the hyperbolic metric, is

II :t

(a+

te,nll

= IIC£,0)ll(a+ll't) =

:(f,O)j/t = 1 .

Hence the length of y is dt= s. More concretely. a segment of length 1 000 which moves 3000 units to the right tra\·els the same distance as a segment which measures .001 and moves .003 in the same direction. (By the travelled distance \Ve understand the length of the trajectory. which may be brger than the distance bet\veen the end points . )

J;�

EXAMPLE 2: Let y be the curve in j

defined

by

y(t) = [-e',e'],

and

let

a(t)= W.2e1) be the associated curve

in 'Je. Then the Euclidean speed at the instant t, 1a'( f) = 1(0,2e1), = 2e1, coin­ cides with the size of the interval at f. Since the segment perceives itself as be­ ing the yardstick at any time. \\'e are not surprised that y has constant (hyper­ bolic) speed equal to one: lla'(t)JJ = (0,2e' )'/2e1 = 1 .

Geodesics i n !P

Since the metric we are considering on j is copied through the function ¢ from that of the hyperbolic plane, for which the vertical straight lines are trajectories of geodesics, it turns out that the curve y of Example 2 is a geodesic in j. We also observe, considering the other geodesics of the hyperbolic plane, that in particular the trajectory of the

The segment of length L centered at h is in the trajectOJy of the best path joining the segments of length £ centered at a and c. respectively. best path in j joining two segments of the same length , say £, consists of seg­ ments of length greater that £. This can he explained by noticing that since each segment takes its own length as the standard by which to measure travelled distances or changes in its size, a longer segment will cover great distances more easily. Instead of simply moving to its final position keeping its size constant, as it would under the Euclidean metric on .1'. it does better to make an extra effort at the beginning and increase its size. so it \\·ill perceive distances as shorter, and finally make the effort of reducing its length at the end.

lsometries of !P

= z + h,

Higher Dimensions If one considers balls or spheres in IR", one obtains in the same way models for the hyperbolic space of dimension n + 1, by identifying the ball or sphere of center a E IR" and diameter d with the point (a,d) in the upper half-space of �n+I ACKNOWLEDGMENT

It is well known that the orientation­ preserving isometries of the hyperbolic plane '3£ are the Mobius transformations of the complex plane which preserve '3£. Some of them are

.ft,(z)

ment of equal length with initial point moYed hy b units. As we could have expected. after that transformation the segments do not notice any change. The same happens for Gc: the sizes of the segments and their relative positions all change in the same proportion c.

gc(z) cz, and h(z) = - 11z, =

with b, c E �. c > 0. The correspond­ ing isometries of j, induced by the iden­ tification ¢, which I denote with capital letters, are the following. The mapping Fh takes each segment of j to the seg-

I thank Pablo Roman for his help in draw­ ing the picture. REFERENCES

[1] Anderson , James W . , Hyperbolic geome­ try , Springer Undergraduate Mathematics Series. London (2005) . FaMAF-CIEM Ciudad Universitaria 5000 Cordoba Argentina e-mai l : salvai@mate. uncer.edu

© 2007 Springer Science+ Business Media, !nc

.. Volume 29. Number 2. 2007

7

•:Mti

A Sh o rt De r ivat i o n of Lord Bro u nc l<e r's C o nti n u ed F ract i o n for 7T PAUL LEVRIE

If we define Yn =

1

2n-

2n-3

1

--2n-'5

+

... + (-1)11-21+ (-1)11-1 + (- 1)11 '\

it fol lows immediately that

= 2n+ 1

Yn+1 + Yn

Yn+2 + Yn+I =

1 ,

1T

4'

1

2n+3

We can use these two equations to generate a second-order linear homogeneous recurrence relation by dividing the second one by the first Yn+2 + Yn+l

2n+ 1 2n+3

Yn+1 + Yn

and rewriting the result as .YH+2 + Yn+I =

2n + 1 + 2n+3 (Yn+I y,z).

(1)

Some rearranging leads to Yn+2 Yn+l

+1

=

2n+ 1 2n+3

( +�) 1

(2n+ 3).Y11+2 + 2

Yn+ 1

.Yn+l

1

=

-(2n+ 1)--Yn+l J'n

and finally we get

2n+ 1

Yn+I

Yn+Z.

2+ (2n+3)

Yn

Yn+1

This recurrence generates a continued fraction: 1

:...3-2+3· -__: 2+ 5. Yl

(2)

2 +3. -----"--3-2+ 5 . ---'-5 2+7·--'--72+

- ---

Y2

which converges to the value

.E: 4

Yo

Hence

i_ 1T

4

1 - .E:

.Yl

= 1

+

4

1=-z

___

2+

1.

1T

__ _

32

52

2+�,

and this is Lord Brouncker's famous continued fraction (see, for example, [1]).

SOME REMARKS. 1. That Yn =t- 0 for a l l n is a consequence of the standard proof of Leibniz's rule for alternating series applied to Leibniz's series for .E:. 2. To prove the convergence of the continued fraction, we note t hat z�,n .Yn and z�2J = ( -l)n are both solutions of the following recurrence relation:

=

Zn+2 + Zn+l =

8

THE MATHEMATICAL INTELL IGENCER © 2007 Springer Science+Business Media, Inc.

+ -2n -2n+3 1

(Zn+l + Zn)

¢:::>

z

n+ 2 _

_

2 z 1 + 2n+ 1 z n· 2n+3 n+ 2n+ 3

The continued fraction associated with this recurrence is 1

-

2 + :\

:\

-----­

-� + · . .

and its partial numerators A 11 and denominators B11 are those linear combinations of 1, A1 = 0, B0 = 0 , B1 = 1 ( see for instance [2]). It is easy and z;;' that satisfy A0 to verify that

z;/'

=

A11

=

(

)'11+ 1 -

.

;)

·

( -ll11

RII = rII -.!!. 4 ... < -u��

From

A11

are given hy:

=

B 11

+

+ (-I

=

< -n��-�

(-1)11

-­ !311

All

!311

=

1-

lim .:.:h

IJ�X

we

.:;

-

_!_ 3

+

1

_ _

')

-.

. . +

(- 1 )11 - I

1

2n -

1

)

.

)11 it follows that the approximants of the continued fraction ( 3)

Taking the limit for 11 � x \\·e get

where

. (1

B ,,

-----1

- J. + J.- . . . + ( -1)11-1 s :\

I

2n-l

1

1-11"'

4

have used Leihniz·s series for.!!._. Hence the continued fraction ( 3) converges

to 1 - . By constmction, the continued fraction (2) is equivalent to ( 3 ) , so (2) also converges. and by comparing ( 3) with (2), we see that convergence is to the value

-(1-

2). 7r

REFERENCES

[1 ] J . Arndt and Chr. Haenel, 71"-un/eashed, Springer-Verlag , Berlin, 2001 . [2] L. Lorentzen and H. Waadeland, Continued Fractions with Applications, Studies in Computational Math. , Vol. 3, North-Holland, New York, 1 992. Department of Computer Science K. U. Leuven

Celestijnenlaan 200A B-3001 Heverlee, Belgium Departement IWT Karel de Grote-Hogeschool Salesianenlaan 30 B-2660 Hoboken, Belgium e-mail: paul . [email protected]

tf) 2007 Springer Sc1ence+Bus1ness Media, I n c . , Volume 2 9 , Number 2 , 2007

9

•:Mti

Remark on Stirling's Formula and on Approximations for the Double Factorial F. L. BAUER

In his fascinating article 1 'e: The Master cifAll', Brian ]. McCartin refers to Stirling's formula of 1730 (1)

and to its history. Tabulating this formula shows that the relative error is asymptotically - 1 -

( -;" )" · \!,2- 1r·n -

n!

n

10

3628800

1 00

9.33262 1 544 . X 1 0 1 57

x

12n

8.295960443 . X 1 0-3

0.99551 5253273

9.324847625 . X 1 0 1 57

8.329834321 . . X 1 0-4

0.9995801 1 8588

4.023872600 .

4.023537292 .

1 0000

102567

2.846259680 .

2.846235962 .

1 00000

X 1035659

2.824229408 .

10456573

2.824227054 . X 1 04565 73

X

relative error

relative error

:

3.598695618 . X 1 06

1 000

X

1211

X

X

102567

8.332985843 . X 1 0-5

0.999958301 1 60

8.333298608 .

0.99999583301 2

10-7

0.999999583330

X 10-6

1035659

8.333329861 X

The formula is, apart from its beauty, known for its good approximation,2 and thus it was not necessary to make it widely known that even a tiny correction to Stirling's formula still improves the approximation . ;.,-----,------:::--:Let us first consider the equation n! = e-" n" V 27T ( n + o nJ which can be 11 solved for On: o11 = ( n' ( e/ n) ) 2!(27T) - n. A tabulation of o 11 for some values of n, ·

·

·

·

n

1 . 1 76004802 . . .

0.176004802 .

10

1 0. 1 68006975 .

0.1 68006975 . .

1 00

1 00.1 66805076 . .

0.1 66805076 .

1 000

1 000.1 66680550 .

0.1 66680550 ..

1 0000

1 0000.166668055 . .

0.166668055 . .

1 00000

1 00000.1 6666805 .

0.1 66666805 . . .

(n! · (e/n)")2/27T

On

not only checks Stirling's formula experimentally: it also gives reason to guess that limn->"' 011 = The conjectured modified Stirling formula reads

-2;.

(1*)

n!

�

e-11

•

n''

·

J

27T

·

( n + -2;)·

Tabulating this formula gives: n

n!

10

3628800

1 00

9.332621 544 . . X 1 0 1 57

1 000 1 0000 1 00000

102567

4.023872600 . . X

1035659

2.846259680 . . X

10456573

2.824229408 .. X

{;)" · � 21r {n + ·

i)

9.332615096 . . X 1 0 1 57

1035659

2.846259680 . . X

10456573

X

1Mathematica/ lntel ligencer 28(2006) , no. 2, 11}-2 1 .

2J. Arndt and Ch. Haenel, in their pretty book

0.0 1 % from n

10

2o

89.

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media, Inc.

6.908958656 . . X 1 0-7

0.994890046523

6.944089506 . X 1 0-11

0.999948888993

10-13

0.999994888889

X

1r : Algorithmen,

2007) , mention that the relative error is less than 1% from n

1 44n2

0.949 1 08 1 1 2341

0.999488899422

6.944408950 .

2.824229408 .

x

6.591 028557 . X 1 0-5

6.940895 1 34 . X 1 0-9

102567

4.023872572 . . . X

rei error

relative error

3.628560824 . . . X 1 06

r

Computer, Arithmetik (3 d ed . , Springer, Berlin

2o

9, less than 0.1 % from n

2o

28, and less than

A The approximation is now much better3 ; the relative error is asymptotically I_, l-�c4nstrict proof for the modified formula can be based on the well-known 1 asymptotic expansion _

= e-n .

n'

n". � . ( 1 +

+

1 __ 1211

For the double factorial (2n- 1 )!!

def =

_1_, 2R8n-

+ .. ·).

(2n - 1) · ( 2n - 3)· . . . ·3 · 1 =

( 2 n)! 2'' · n''

an approximation can be obtained from the Stirling formul a , eliminating (2)

( 2n

1 )!!

_

( 2n) ! 2" · n!

=

_

-, ,

( )

2 n " Vz. .

7r:

e

But perhaps we should use instead the improved Stirling formula, leading to

(----;-)

2n "

( 2n- 1 )11-

(2*)

(12n + 1 )

·

(6n + 1 )

·

The result of ( 2 * ) for n = 5, 9!! 945 . 1 7377 10, is distinctly better than with the un­ modified Stirling formula (2) , which gives 91! 952.8896030 (Table 1) 1 1 - for ( 2 ) , , for ( 2*): The relative error is asymptotically =

=

24 11

192 rr

--

-

(2) relative error

n

10

4.172926252 .

100

4.167510527 .

1000

4.166753230 .

10000

4.166675344 .

100000

4.166667534 .

. x

. x

rei error

w-3

x

10-4

1.000202526691

. x

10-·5

1.000020775286

w-6

1.000002082752

. x

10-7

1.000000208327

. x

(2*) relative error

24n

1 .00 1502300523

rei error

x

192n2

4.899537334 .

. . x

w-s

0.940711168172

5.205227728 .

x

10-9

0.999403723811

5.177287373 . . . X 10-7 5.208022763 .

. . X

10-11

5.208302276 . . . X 10-13

0.994039175688

0.999940370570 0.999994037039

Moreover, from the folklore5 of the growth of binomial coefficients:

( ) 2n n

d�f

-

( 2n)! n ! · n!

-

22"

\l7iYz

comes another approximation6 for (2n - 1 ) 11 which is free of ( 2n - 1 ) ! !-

(3a)

e:

2" · n ' �

Multiplying the left-hand side by (2n - 1 ) ! ! and the right-hand side by the equivalent (2n)! 2" · n ' (2n)! ) . one obtams ((2n - 1 ) 11)-- , � , i . e . , the slightly better variant 2" · n ! v 1r · n 2" · n! '

(2n - 1 )!'-

( 3b )

n 10

rei error

(3a) relative error 1.25731 9341.

. x

100

1.250776360 . . .

1000

1.250078076 . . .

10000

1.250007812.

100000

1.250000781 .

X

X

X

. x

V(2;1)! �

x 8n

(3b) relative error . . X

rei error

x 16n

10-2

1.005855472905

6.266959316 .

10-3

1.002713490580 1.000308395968

10-3

1.000621088744

6.251927474 . . . x 10-4

10-4

1 .000062460932

6.250195056 . . .

10-5

1.000006249609

6.250019528 .

10-6

1.000000624996

6.250001953 . .

x

. . X

.

x

10-5

1.000031208982

10-6

1.000003124590

10-7

1.0000003 j 2496

3The relative error is less than 0.1% from n 2: 3, less than 0.01% from n 2: 9, less than 0.001% from n "'= 27, and less

than 0.0001% from n 2: 84.

4See for example: George Marsaglia and John C. W. Marsaglia, A New Derivation of Stirling's Approximation to

n!.

American Mathematical Monthly, 87, (1990), 826-829. 5Max Koecher, K/assische elementare Analysis. Birkhauser, Basel 1987, p. 76 (Korollar 2); Reinhold Remmert et a/., Zah/en, Springer-Verlag, Berlin 1983,

p. 117.

6As soon as even small handheld computers like the Hewlett-Packard hp the double factorial, this formula became interesting.

11C

had a key for the factorial, but none for

© 2007 Springer Science+ Business Media, Inc., Volume 29. Number 2. 2007

11

Table 1 shows that (3b) is better than (3af (the relative error is asymptotically _l_ for H11 1 6 for (3b)) and that both are not very different from (2), but that the improved (3a) , 1

II

(2*) is dramatically better:

Table 1. Comparison of (2*), (2), (3a), (3b)

(2n)n. Vj(12n+1}

(2n- 1)!!

n

(6n + 1 )

e

(2ne) .

v;:-:n

v'2

��--------

--

-------===- ------

V(2rlj!

2n. n!

�

1.128379167

1.062251932- \11.1.128379167

2

3

3.002820329

3.062287889

3.191538243

3.094287435

=

V3

3

15

15.00701675

15.20846222

15.63528038

15.31434640

=

\115 . 15.63528038

4

105

105.0291997

106.0955133

108.3244000

106.6492475

=

V105

5

945

945.1737710

952.8896030

968.8828884

956.8669341

=

\1945 . 968.8828884

1.002670344

1.040520190

3.191538243

·

·

108.3244000

However, the folklore formulas may also be improved. Before doing so, we observe that the samples both for (3a) and (3b) give upper bounds. Lower bounds are obtained in the following way: The Wallis product reads as follows: .

.

. .

the elements of the sequence are 2 1 times the square of I l. -'- S2. n ·

thus asymptotically

y:;:

( n + _1__2 )

·

�

2 1

2 +1

·

2. 3

(2n- 1)11

(4a)

·

S2. �

.. .

·

·

\I

__liz_ and H 2n-1

·

:3

·

S

)

' ·

• ·

·

·

)

2 n-l ' 211

·

2" n' ·

�

J

( n + 1).

7T.

Multiplying, as above, the left-hand side by (2n- 1)!! and the right-hand side by the

· I ent eqmva

(2n)! . . a s 1·tgI1t1y I )etter vanant . , we o IJtam agam 2" n!

--·

(2n)!

(2n- 1)!!

(4b)

�

Table 2 shows that we have indeed lower bounds now. Table 2. Comparison of

n

(2n- 1)!!

(4a) I

2n

and (4b) ·

n!

(2n)!

I

�7T· (n + �)

v7T·(n+�)

0.921317732

0.959852974

=

\11.0.921317732

2

3

2.854598586

2.926396374

=

\13. 2.854598586

3

15

1 4.47545684

14.73539455

=

\115.14.47545684

4

105

102.1292238

103.5546643

=

\1105 . 102.1292238

5

945

923.7935873

934.3366310

=

\1945.923.7935873

The relative error is now again asymptotically _l_ for (4a), -1- for (4b ) : 16n

Hn

n

(4a) relative error 10-2

. . X

1 .249992968 . . . X 10-6

100

1.243002794 .

1000

1.249397216 .

. . X

100000

, \l(2rlj! J

v;-:-n

---

=

(2n- 1 )!!

8The folklore simplified the (n +

12

THE MATHEMATICAL INTELLIGENCER

x

8n

(4b) relative error

0.946379734734

5.932470444 . .

10-3

0.994402235964

6.216946495 .

10-4

0.999437773329

6.246681188 . .

0.999943752734

6.249667983 . .

0.999994375027

6.249966797

1.249929690 . . . X 10-5

10000

7Actually,

rei error

X

1.182974668 . . .

10

·

2n n!

·

v;-:n

--

.

�) in the denominator of (4a) and (4b) to n.

rei error

x

16n

. X

10-3

0.949195271180

10-4

0.994711439354

. X

10-5

0.999468990150

10-6

0.999946877402

10-7

0.999994687524

. X . X

. . . X

We could now use the arithmetic means of (3a) and ( 4a ) or of ( 3 b ) and (4b) . ob­ taining for n = 5 in the case ( 3a l / ( 4a ) 946.3382380, in the case C3b) / ( 4b ) 945.601782')­ quite respectable when compared vvith 945 . 1 7377 1 0 from ( 2*) . Note that ( 3a )/(4a) yield a n inclusion interval 946.3382380':::: 2 1 .2064 1 270, (3b)/(4b) yield an inclusion interval x- 945.6017825 1 :::: 1 1 . 26')1') 1 5 . However . \Ve are now going t o obtain even slightly better results b y improving the formulas ( :3a ); ( 4a J and ( :3b ) / ( qb J . looking for the best On· To this end . we observe that the ( a )-variant and the ( b )-\·ariant give the same oil. We consider for the (a)-\·ariant the equation:

lx-

(2 n - 1 )!1 and solve it for oil:

o ��-

(

( 2n n! )2 (2n-

YTT· (n + o;J

2 1l . n! (2 1 ) !!

n-

A tabulation of oil for some values of

n

2 1 l · n!

=

n.

)2

.,.

1)!!

. _l._ 7T

- n.

On

1.2732395447351

0.2732395447351

10

10.2530447201518.

0.2530447201518 ..

100

100.2503117163367 .

Q2503117163367 .. 0.2500312421850 . .

1000

1000.2500312421850 .

10000

10000.2500031249218 .

0.2500031249218 . .

100000

100000.2500003124992 .

0.2500003124992 . .

1000000

1000000.2500000312500 .

0.2500000312500 . .

gives reason to guess that lim11_,x 011 =_!_ ( more precisely: oil-_!_ + 1 J Thus � . � 3211 -

�

( 2n - 1 l1'-

(3a*)

'VITT·(n+±)'

.

and

( 2n)!

( 3b*)

(2n-ll!!-

/.

1

'J'TT (n + -:;-)

improves the accuracy considerably . as shown in Table :39 Table 3. Comparison of (3a*) and (3b*)

n (2n- 1)!!

2n n! (n + �) ·

\ rr .

rei error

rei error x

64n2

(2n)! (n + �)

\/ n-.

rei error rei error

x

128n2

1.009253009

0.009253009

0.592192576

1.004615851

0.004615851

0.590828928

2

3

3.009011112

0.003003704

0.768948224

3.004502178

0.001500726

0. 768371712

3

15

15.02189149

0.001459433

0.840633408

15.01094175

0.000729450

0.840326400

4

105

105.0901043

0.000858136

0.878731264

105.0450425

0.000428976

0.878542848

5

945

945.5328815

0.000563896

0.902233600

945.266403

0.000281998

0.902393600

Reversal of ( 3a *) leads to

(Bauer 1 994). 9F. L.

Bauer, Decrypted Secrets-Methods and Maxims o f Cryptology, 4th edition. Springer, Berlin 2007, p . 48. ( I n

cryptography, i t i s important t o have good and nevertheless simple approximations for (2n- 1)!!, the number o f prop­ erly self-reciprocal permutations, of 'genuine reflections' of 2n elements).

© 2007 Springer SCience- Bus1ness Media, Inc., Volume 29, Number 2, 2007

13

The relative error is asymptotically

64\2 1

-2 28 17 1

as can be seen here: (3a·) rei error

n

for (3a*),

rei error

x

for (3b*),

64n2

(3b') rei error

1 0-4

0.950476200605

1 0-6

0.99500399221 8

7.773465667 .

. . . X

1 0-8

0.999500039148

7.808594025

1.485 1 1 9063 .

1 00

1 .554693737 .

1 000

1.56 1 7 1 881 1

1 0000

1 .562421 875 .

1 00000

1 562492 1 87 .

. . X

. . X . . x

1 0-10

1o-· 12

rei error x 1 28n2

7.42531 9640 . . . X 1 0-s

. . X

10

. . X

. . . X

0.999950000391

7.8 1 21 09377 .

0.999995000004

7.81 2460937 .

. . X

. . X

0.9504409 1 3967

1 0-7

0.995003605487

10-9

0.999500035246

1 0-11

0.999950000352

1 0-11

0.999995000004

The different approximations are summarized in Table 4 . For n 1, n 10, a comparison of the numerical results (correctly rounded) is given. =

Table 4. Summary: Calculation of Double Factorial for 1!!

=

1, 5!!

=

945, 10!!

=

=

5, and n

=

654729075; with

asymptotic relative errors.

(4a)

( 4b)

(2•)

( 3b.)

(3a·)

(2)

(3b)

(3a)

0.921 31773

0.95985297

1 .00267034

1 .00461585

1 .00925301

1 .0405201 9

1.06225 1 93

1 . 1 28379 1 7

923.793588

934.336631

945. 1 73771

945.266403

945.532882

952.889603

956.866934

968.882888

646983796. 1 8n

65084491 4. 1 1 6n

654761 1 54. 1 1 92n2

654777691. 1 1 28n2

654826310. 1 64n2

6574612 1 1 . 1 24n

658832235. 1 1 6n

662961 1 1 0. 1 8n

Moreover, Table 4 shows upper bounds: a group of three with asymptotic relative er­ rors proportional to .l and a group of three with asymptotic relative errors proportional to

\; and lower bound� : a group of two with asymptotic

,r

relative errors propo1tional to .l.

ACKNOWLEDGMENTS

Thanks go to Christoph Haenel for computational support, using MATHEMATICA. Nordliche Villenstrasse 1 9 D-82288 Kottgeisering Germany

Opinion Survey The Best \lathematical Books of the Twentieth Century

(MathematicallutelliRencer.

volume 29. number 1, p;.�ge 2 1)

R ·aders ;.�re reminded rhatthe dosing date for contribmions to th<: sur"l:<:) is ...,e p­

tcmb ·r 1, 2007. Pleas· .'end \"Otes to Eric Gmnwald on cricgmm..""
14

THE MATHEMATICAL INTELLIGENCER

11

Otto Neugebauer

Raps My l
PHILIP J. DAVIS

��

ome years back I used to have lunch fairly regularly with the famous Austrian-American historian of math­ � ematics and science Otto Neugebauer. I was an "hon­ orary member" of the History of Mathematics Department at Brown University. We were joined at lunch, off and on, by other members of the Department that might include Abraham Sachs, David Pingree. Gerald Toomer, and ·'visit­ ing firemen" such as Bernard Goldstein and Noel Swerd­ low. I have written about these lunches in Otto Neugebauer: Reminiscences and Appreciation ( Ame rican Mathematical Monthly 1 0 1 0994), pp. 1 29-131). During these lunches Neugebauer talked freely about

personalities. methodologies, history, politics, etc. He \YOre his prejudices ''on his sleeve . . , Philosophy was a tabu sub­ ject. In the first place Neugebauer hated philosophy and philosophers, and secondly, insofar as he had formulated a philosophy of mathematics. it was in considerable op­ position to mine. But as regards the history of mathemat­ ics and science, I sat at his feet. One day, over lunch, I happened to suggest that Ptolemy 's model of the solar system with its cycles and epicycles was an early an d interesting exa mp le of curve fi ni ng by means of trigonometric series (i/11 = cos(8) + i sin( 8)) . This thought came to me through my work on approximation theory; but

Transcription of the Letter Ptolemy-but no Curve-Fitting I. Let u:-. a sume that .til planets mm e '' ith uniform .In­

gular 'L'locit} in drcle., .1bout the sun.

2. To te 1 thb a-....umption h} "olbef\ allon", we lllllst tr.ms­

flor helicx:entrk coordinate., to gee ·t:ntric. Our h) p<>thesis 1s then represented hy an epkyde Ill< del. TI1erl.· is noth·

mg gO<xl" or had" alx>ut an epkyde m<xld {"hieh. in­

cid ·malty. gh es a 'er� rea,onahle cstimat

·

for th

·

dis­

,.

t.mu:s of the plam:ts from the r�llios of the epic}de radii.)

3.

A qualitativ

recording of the planetaf} orbits I tim­

ing. �a}. (io ) ears for la�tronomical "} mho! for the pbnl:t •

aturn) and lasrronomical

"llpports on

trongl} tht•

e

}mhol for the plan ·t Jupnerl>

btence of epicycles, s

·n edge

nd moving along a slightl) indined defer ·nt c cf.Hg ·

J';') p 12'1-1 in Ill} "lhst. .\nc \I,Hh. Astron. ) .t. th •

of g.1l • I.

·

\ic''

1es in Kant-Laplace!

\-; a r '"lilt of \ Cf) sophisticated obscn·ation._ (p •r­ .

haps in part based on Bah} Ionian arithmctiral mcxlels!)

Hipparchus and Ptol 'Ill) found de\ iations from the um­

formity of motion. It turned out that tht• ...implc

de,·ke

of a">:-.uming ec<.cntric orbits {.Is in the solar theOI) ) did not suffKe. Ptol 'Ill) found

the ingenious \\ ay out b)

-1. Copernicth ' ith hi" dopn·1 of aJ unifonn b)

assumin� uniformit) of an­ gular motion about a point

motions
ric to the ob ...en·cr 0. But

u...ed -.ome

l.
no\\ here in hi-. theory is a

seumd epiq de ust•d.

16

"Ddc.:n.:nt = drdt• .tbout 1\1.

THE MATHEMATlCAL INTlELLIGENCER © 2007 Springer Science+ Business Media. Inc .

1

•

circle

I 1!>. lie eliminat d th A

de

irnll.u

ah ut t ht• !Jl!:.illl -.un, not tht• true sun!) for the - plan ·ts (l'tok·m} 2•·

=

C<JU.lllt

far as I kno\\ the -.!Of) al out the epiq lt:-epil �­

hecame ''lonunon knc)\\ ledg ,. through .1 r ·mark of

t'zt,..,,.., "'� l ,I, ••1 •J .., f"'� I) t' ,..,l.r '(.L+ ,/,.f f.:. ' .,./ .i, fr,., 1) ...,j ,, " 1., f .1.. l ,!..,}, !!.&...) ./of ') 1ft .1., /J ,¥,. 2-"

..,,j,,._, S:

-·

•

-··-

"

"

,._

•.

;f, f" ... I .f...., J.• ..t.) .I..J I.. 1)1,1. 'I'' 1., J.,., <-.., J:..,.,(,.{ ' J.- l • ,.._,J �r.l,·• .{ 1),(.,.

.. 11..1 "- f·l.lt· /n.t.

J

I l ••,

.... /.

"/"'�" ./ J

•.

I

4'

•• •••

••

••

K/• ,. ..J,.-4.,...{ ,_ ., ...� (./,. (� 1... ..,t.,,L.,J :

l

•<

I.. I- ,.IJ... ! IJ 1.. .... ,{ 'M J,.. {If.. ..;., ,J r17·;yl., -r� "'v'"·· u �..J - ..1 ;., /f.. 1 ,�, 1;... .£-1-a!J ••

.•

•

{" J.. '•"/1. 0 ... /.no) /I- f.. "'"l'J J,l I."'""'"'� •-•// ,(,..,. J' . ( Jff I. ./. /.( � 71-"'1 k ,.,_ ,j,.f j., ,,.._/- � � 2 •.( 4·-. _,. 1•1 1J 71.7 ,I• <•r ./ � ' -'r� �-/.-... ....>{ f·.' n !. { ,t, ""' · •

•.•

...••

-· •

__

·�·

...... ,J., , ,..,.h, /)�·E. _, 1... ....... • .•

"

1 N�l..... . 11.... ,(!..

" ..... •

I.� 1 I''" �� •....,,"

7t,

1.1.....

"" ..... f··,· •

_,/...,

••

,I.. ......,:.! •...,..

J

P,J. f2, iJ."J. • •• ,(, /,) 4n '7-,..,,;..,,' ,..Jt.,.. ·-/ . ·/• I. ·r;r·�:�J, I. ih!... ..� -.u ·�

•

rn:IC. �

-1...,..1.. ...J. ..... P.r·1f� .1-. H.,t

• '/'"�'/'' /.,

.,;,5

e

Cardinal lklarmine in thl· time of the (�alileo triab. I h.i\'e

no
"kt thl'lll
6. K1.·pler r ·introdtt<:l.'d th · c.:quant (.tlso for thl' solar th ·· ory-1.e. ft>r the.:

·arth's motion>. th<.:n he.: rc.:alizL·d that

the.: l"l'lllaining sm.tll ohs ·n alional

dflot"t-.

can IK· dimi­

nat<.:d h� r ·plating th1.· motion .tbout the

·quant h} the.:

1.'\\ ton

c.

plaincd thl' Kepler motion .ts th

qut:IKI.' of .t d�n:11nic hl\\ of gra\ ll:tlion.

·

um�t·­

P.S. The only ·cun t·-fittin!-( 111 this :-.to� art· Copernint.�' rather inept allempts to appro. imale

thl.'

Ptol!.'rnait'

mmk•l as closely as poss1ble h) a ... up<.:rimpo... ition ol cir· t:ular motions. Respectful!) �ubmittcd

<.:qu.tl-area tht·orem, and final!� h) repladng the.: drntlar by an dliptit' motion \\ ith tht· fo<:i rc.:placing the t\\ o s� mmctric l.'l'ntcrs 0 and E and tht· sun in one f
in point of fact I had heard it much earlier from another man whom I admired greatly: in the lectures of the physicist and philosopher of science, Philipp Frank. PHILIP DAVIS is so well known for his com­

mentaries

on

mathematics-in

his

recent

books, some in collaboration with Reuben Hersh, in

The lntelhgencer;

and in

SIAM

that some readers may think of him

News­

as

com­

mentator more than practitioner. To restore

just perspective. one should (for example) cast a glance at his Interpolation and Approximation ( 1963), or Circulant Matrices ( 1979). Division of Applied Mathematics Brown University

Providence, Rl 029 12

USA

e-mail: philip.davis@brown edu

Neugebauer bridled at my suggestion. "No' No' No! I will explain to you what Ptolemy was up to.·· And he got up and left the lunch group. A few clays later, I received a two-page hand-written letter from him explaining in detail .. ,vhat Ptolemy was up to. " I was touched that Neugebauer took the time to write to me and felt it was another indication of his total devotion to his historic craft, as well as his regard for me. The knuckle-rapping led to no diminution of our friendship. Our lunch meetings continued. and my wife and I visited him and his daughter several times in their summer home on Deer Isle in the Penobscot Bay, Maine area. Recently in clearing out one of my desk drawers. I found his letter to me, and I think the mathematical community might find both this st01y and the letter interesting aspects of a major mathematical personal ity. I give both a photocopy of the letter and a transcription. Notice that his signature is the backside of an elephant. its tail in the form of the letter "e . " In his circles, Ncugeha ucr was known as "The Elephant, . . ancl I never once heard his colleagues refer to him in his pres­ ence as "Otto" or "Prof. ;\eugehauer. .. Always: "The Elephant . ..

© 2007 Springer Science-+ Business Media. Inc .. Volume 29, Number 2. 2007

17

lj¥1(9-\H•i

David E .

Fe I i x I< I e i n, Ado l f H u rw itz, and the ''J ewish Quest i o n '' i n G e r ma n Acad e m ia DAVID E. ROWE

Send submissions to David E. Rowe,

Fachbereich 08-lnstitut fur Mathematik,

Johannes Gutenberg U n iversity, 055099 Mainz, Germany. rowe@Mathemati k . u n i-mainz.de

18

Rowe ,

Editor

athematicians love to tell sto­ ries about people they once knew or perhaps only heard about . If the story happens to sound be­ lievable, others are apt to repeat it, pos­ sibly embellishing the original tale. Such mathematical folklore occasionally finds its way into print, and once it does, readers are apt to take such stories at face value, lending them additional credibility. Occasionally, though, al­ leged facts come under scrutiny, and es­ tablished stories are exposed as fiction. Yet even when someone comes along with decisive evidence refuting an ear­ lier account, it can easily happen that the original story just refuses to die. The case I have in mind here con­ cerns a version of events in the career of Adolf Hurwitz, who taught as an as­ sociate professor in Konigsberg from 1 884 u ntil 1 892, when he received a full professorship at the ETH in Zurich, sue-

ceeding Georg Frobenius. A rendition of what supposedly occurred at that time can be found via the Internet cour­ tesy of MacTutor: Hurwitz remained at Zurich for the rest of his life, . . . [but] not because he had not been offered a chair in Germany. Schwarz, who was profes­ sor at Gottingen, succeeded Weier­ strass by accepting his professorship in Berlin in 1892. Gottingen ap­ proached Hurwitz and offered him the vacant chair only weeks after he had accepted the Zurich chair, but he turned down the offer. This must have been a remarkably hard decision for Hurwitz since at that time a chair at a leading German university such as Gottingen would have been much more prestigious to any German than a chair in Switzerland. However Hur­ witz was an extremely loyal person, and having given his word that he

This balcony photo from 1 9 1 2-1 3 , which graces the cover of George P6lya's Picture Album, shows a familiar scene at the Hurwitz home: Einstein and Lisbeth Hurwitz playing a violin duet while her father conducts. Courtesy of Gerald Alexanderson.

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Sc�ence+ Business Media, Inc.

The Main

Building of

there. Photo courtesy

year Hurwitz arrived

I would prefer to say ) . And it is tru e , a mathematician \\·ho is not something of a poet will never he a complete math­ ematician ··3 Such stereotypes were commonplace during this period and went hand-in-hand with na·ive attitudes or prejudices about race and ethnicity. For present purposes. I will overlook the complicated issues of Jewish iden­ tity as these affected German Jews who felt torn by the cont1icting pressures they faced. Jewish mathematicians came from a variety of backgrounds and held widely divergent views about Judaism and their identification with it. What they shared. however, was a sense of standing on the outside. of being per­ ceived as somehow alien. and these perceptions could make all the d iffer­ ence.

In writing about Jewish mathemati­ cians in Germany during the Second Empire, one faces the ditlicult issue of deciding what should count as "Jew­ ish. " 2 Many Jews during this period were no longer practicing their religion. and in such cases they were often inclined to convert to Christianity in order to better their professional prospects. Candidates of "Mosaic con­ fession" were at a distinct disadvantage nearly everywhere; at several universi­ ties in the heavily Catholic south they had no chance at all. At the same time, baptized Jews were not generally dealt with as Christians, though overt signs of discrimination were comparatively rare. Academic anti-Semitism took on subtle guises, and the various forms of de facto discrimination against those who were perceived as Jewish can in most cases only be discerned through private communications. Thus, Karl Weierstrass confided to Sofia Ko­ valevskaya that his nemesis, Leopold Kronecker, ''shares the shortcoming one finds in many intelligent people , espe­ cially those of Semitic stock: he does not possess sufficient fantasy ( intuition

Adolf Hunvitz ( 1 859- 1 9 1 9 ) was a Wun­ derkind. The youngest of three sons of a Hildesheim manufacturer, he began to make a name for himself as a teenager. Young Adolf's mathematical talent was discovered by his teacher at the Hildesheim Realgymnasium. This teacher's exotic name was Hermann Cisar Hannibal Schubert ( 1 848--1 9 1 1 ) . the inventor of the Schubert calculus in enumerative geometry. Although an internationally recognized mathemati­ cian, Schubert spent his entire career teaching in secondary schools, first in Hildesheim and later in Hamburg. Not that this was so rare: H. G. Grassmann was also a Gymnasium teacher. and even Weierstrass taught "school mathe­ matics" for over a decade before he made the leap to an institution of higher education, the Institute of Trade in Berlin. It took another eight years be­ fore Weierstrass gained a professional appointment at Berlin University in 1 864. In 1 874. at the age of 26, Schubert won the Gold Medal of the Royal Dan­ ish Academy of Sciences for his work extending the theory of characteristics to cubic space curves. ' Around this time

the Konigsberg University in 1 884 .

of ETH

the

Bibliothek. Zuri c h .

would accept the Zurich position he \Youlcl not renege on his promise. 1 An earlier \·ersion of this story was told by Hun\ itz's former student, Ernst Meissner, in a memorial speech deliv­ ered after his teacher"s death in 1919. The text of that speech was reprinted in volume one of Hurwitz's Werke. published in 1932. and thereafter it be­ came a standard source for biographi­ cal information ( Meissner 1 932) . About twenty years ago I wrote an article on various forms of anti-Semitism in Ger­ man mathematics, part of \vhich dealt with the case of Adolf Hunvitz ( Rowe 1986, 433-435 ) . As a point of record I noted then that Meissner's account of the events of 1 892 was olwiously untrue. Presumably he \Yas merelv elaborating on a story he hac! once heard, perhaps even from Hur\vitz himself. Unfortu­ nately, Meissner"s version of the events obscures a far more poignant stmy, one that deserves to be widely known. In retelling it here I hope not only to cor­ rect the record hut also to suggest some larger themes relating to the vulnerabil­ ity of German Jews who pursued aca­ demic careers in mathematics. 1 www -history. mcs. st-and. ac. uk/historyI

Klein's M ost Talented Student

2The position taken here on Jewish identity follows the one adopted for the recent exhibit "Judische Mathematiker i n der deutschsprachigen akademischen Kultur," held

in

Bonn in July 2006 in conJunction with the annual meeting of the Deutsche Mathematiker-Verein1gun g .

3We1erstrass to Kovalevskaya, 27 August

1 883;

Weierstrass pointed to other examples : Abel vs. Jacobi, and Riemann as opposed to Eisenstein and Rosenhain 1n this

letter, first made public by Gbsta Mittag-Leffler, "Une page de Ia vie de Weierstrass, " Compte Rendu du Oeuxieme Congres International des Mathernatic1ens. Paris: Gauthier-Villars,

1 902,

p.

1 49.

4The problem was posed by Chasles's former student, H. G . Zeuthen, an authority on enumerative geometry; see (Zeuthen).

© 2007 Springer Science+ Business Med1a, Inc., Volume 29. Number 2 , 2007

19

Solomon Hurwitz with his three sons, Max, Julius, and Adolf, age three. Photo from 1 August 1 862, the year the boys· mother died. Photo courtesy of ETH Bihliothek. Zurich. Hurwitz was spending Sundays at Schu­ bert's home, \vhere he was given pri­ vate lessons. It did not take long before the youngster had absorbed the main techniques of enumerative geometry, and soon he collaborated with Schubert in writing a paper on Michel Chasles's characteristic formula , an enumerative technique devised by the French geometer in 1 864 to count the number of curves satisfying certain algebraic conditions within a one-parameter fam­ ily of conics. Chasles was the grand old man of French projective geometry and a con­ noisseur of classical Greek mathemat­ ics . He thus appreciated the fact that the

20

THE MATHEMATICAL INTELLIGENCER

new field of enumerative geometry had classical roots that went back to the fa­ mous problem of Apollonius: to con­ struct a circle tangent to three gi\ en cir­ cles. ( \'Vith the aid of conic sections, the solution is simple . but the ancients de­ manded a solution hy means of straight edge and compass alone . ) In enumera­ tive geometry this traditional type of problem gets a new twist: the question is no longer how to construct a required figure hut rather how many different so­ lutions does the problem have. In the case of three circles in the plane, the answer ( no surprise ) is 8 ( at most ) . Schubert took up the analogous prob­ lem in 3-spacc: how many spheres arc

tangent to four given spheres? ( answer. again no surprise: at most 1 6 ) . Such questions are easy to pose. hut can be ,·ery hard to solve. Jakob Steiner claimed that the number of conics tan­ gent to five given conics in the plane was 7 .776: that's a lot. in fact way too many. Using his theory of characteris­ tics. Michel Chaslcs showed that the correct answer was a mere 3,264. Much later. Hurwitz would return to this field with fundamental vv ork on the most general algebraic correspondences on Riemann surfaces, generalizing Chasles·s correspondence principle. After graduating from the Realgym­ nasium, Hurwitz hoped to study math­ ematics at the university. but his father \vas less than enthusiastic about this plan . Solomon HunYitz was a practical man of limited means. and he surely knew that career opportunities in math­ ematics were few and far between. par­ ticularly for u nbaptized Jews. Schubert. however, persuaded him to reconsider. He also told him about a friend, a young geometer at the Munich Institute of Technology who was very eager to at­ tract young talent; his name was Chris­ tian Felix Klein. A U7zmderkind of another kind. Klein burst upon the scene in the late 1 860s as a pupil of Julius Pllicker, \\·ho taught in Bonn. Pllicker was a talented experimental physicist as well as a mathematician . hut his work received far more recognition in England than within his native Germany. After Pli.i cker's death in 1 868. Klein gravitated into the circle surrounding Giittingen's Alfred Clehsch, whom he greatly ad­ mired . Still, he was itching to see the mathematical world, which in those days centered in two cities: Berlin and Paris. He was visiting the French capi­ tal in July 1 870 when the Franco-Pruss­ ian War broke out, and he had to make a wild dash to the train station to es­ cape France. After a brief stint helping the Prussian troops, he rejoined Clebsch in Gottingen. His mentor then paved the way for his appointment as full pro­ fessor in the sleepy university town of Erlangen. For this occasion Klein pre­ pared his famous "Erlangen Program . " spelling out the role of transformation groups and their invariants for geomet­ rical investigations. The author was barely twenty-three years old, but he had the benefit of consulting with the

a small group of talented students at the p ol �· tec hni c school in .\lunich. Among these \\·ere a number of audi tors from the nearby uni\·ersity. including Walther von Dvck and :\lax Planck. These two belonged to an informal group of math­ ematics students that gathered periodi­ cally in Munich pubs. At Klein's urging. thev joined with six others in May 1 877 to form the M u nich Mathematics Club, a formal organization that survived for decades afterward ( Hashagen 2003, 63-68 >. Since the founding members

Yie'\\· of Hildesheim from !VIoritzberg. Photo counesv of ETH 13 i bl io t hek . Zurich.

brilliant i\onvegian. Sophus Lie. his older friend and collaborator. Soon thereafter. in November 1 872. Clebsch suddenly died. Most of his students in Gottingen decided to join his protege in Erbngen. and they soon found that Klein knew how to keep them busY. One of these Clehsch pupils. Ferc.li­ nand Lindemann. vva s given the task of editing the master's lectures. The first edition of the resulting volu me. \Yhich came to he known as Clebsch-Lincle­ mann. appeared in 1 H76: at least that \\·as the official date of publication. Ap­ parently the book came out earlier. as Lindemann sent a copy to the Gottin­ gen mathematician, !VI . A. Stern. \\·ho \\'rote back on 27 November 1 87 5 thanking h i m : "I have t i l l n o w on ly looked at the book very superficially. but I believe I can say that the presen­ tation is very illuminating. perhaps more than it would have been under Clebsch's own editorship. For despite all appreciation for the richness of his intellect. one must admit that his pres­ .. entations -;vere often quite obscure . , Moritz Abraham Stern. who had stud­ ied under Gauss, \Yas evidently still n?ry sharp when he wrote this at 68 years of age; indeed, he would not retire for 5M.

A

Stern to Ferdinand Lindemann,

6Felix Klein to M .

A

Stern,

26

July

another ten years. \X-'hen he did finally take his leave in 1 8H 5 . his chair \Yas as­ sumed by Felix Klein. As we shall see . Stern. Klein . and Lindemann were all destined to phl\· a major part in Hur­ \\·itz's career. Klein also mainta ined close ties \Yith t\vo older members from Clebsch's cir­ cle. Paul Gorda n and Max Noether. both of \\ hom-like Stern-were Jewish . I n lir4 Klein had t h e opportunity to fill a ne\v post in Erlangen as associate pro­ fessor. I !e consulted with Stern about this. and Gordan soon thereafter ac­ cepted (' Klein's second choice was l'
enjoyed the social status associ:! ted \Yith being enrolled at a university. they were keen to maintain their traditional class privileges in the newlv founded dub. Thus students at the poktechnic \vere allm:ved to join. but onlv as associate members. Hunvitz entered \\'ith this ju­ nior status soon after it was founded . At the M unich polytechnic Klein and his colle ague Alexander Brill. another for­ mer member of the Clebsch circle. held a seminar for students \vho wished to pursue mathematical research . A mod­ est young newcomer. Hurwitz no doubt made a strong impression on both of them. and he would later go on to be­ come Klein's star pupil. Yet. like Klein earlier. Hur"':itz clearly had greater ambitions. and he left Munich for Berlin after only one se­ mester. At that time Berlin held far more attraction for aspiring voung mathe­ maticians than any other university in Germany. Led by Ernst Eduard Kum­ mer. Karl \'X'eierstrass. and Leopold Kro­ necker. the Berliners made their influ­ ence felt in more \vays than one: insiders realized that they virtually mo­ nopolised uni\·ersity appointments throughout Prussia. They had thereby succeeded in keeping the " southern German"

mathematicians.

particularly

those associated with the Clebsch group and its journal , J1athematiscbe A n ­ nalen. o u t of t h e Prussian system. For H u rwitz. who was far too young to be concerned about these professional ri­ valries, Berlin offered the opportunity to broaden his kno\\·ledge of complex analysis . Already familiar \\'ith the geo­ metric approach based on Riemann sur­ faces . a staple of Klein's school. he now i m me rsed himself in rhe competing the­ ory de\·eloped by \\.eierstrass and his

2 7 November 1 875, Lindemann Teilnachlass, Universitat Wurzburg.

1 874.

Kle1n Nachlass. Niedersachsische Staats- u n d Universitatsbibliothek. Goltingen.

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students. which relied on constructive methods such as analytic extension of local power series representations. Karl Weierstrass never wrote a text­ book on function theory. although many students must have \vished that he had, for he was anything hut a bril­ liant lecturer. :'\Jevertheles.s. his courses attracted huge numbers of students who struggled to u nderstand the master's message. This was imparted in a cycle of five lecture courses delivered every four semesters. beginning with an in­ troduction to the theory of analytic func­ tions (6 hours) followed by another 6-hour course on elliptic functions. Dur­ ing the third semester. \XIeierstrass gen­ erally offered two 4-hour courses, one on applications of elliptic functions and another on the calculus of variations. He then topped off the cycle with a 6-hour course on Abelian functions. Al­ though this pattern was fairly consistent. he revised the content of the courses each time he taught them, so that stu­ dents never emerged with a truly canonical set of lecture notes. Hurwitz was a hit unlucky; when he arrived in Berlin for the winter semester of 1 H77-78 \XIeierstrass was lecturing on Abelian functions, the most difficult topic in the cycle. He decided to attend anyway. and dutifully wrote up his notes for the course. It is likely that he confided to his former teacher that he found the subject very ditlicult, as Schubert consoled him with the thought that probably no more than six mem­ bers of the class could follow Weier­ strass's presentations 7 Hurwitz felt at home in Berlin, where he befriended Carl Runge, who along with Planck had freshly arrived from M unich. The latter would soon profit from the physics courses taught by such luminaries as Hermann von Helmholtz and Gustav Kirchhoff, whereas Runge was primarily interested in Weierstrass's cycle of lectures. Because he planned to spend several semesters in Berlin, however, he did not plunge in imme­ diately as H u rwitz did but rather joined

him when the master started the new cycle during the summer semester of 1 878. During the following winter se­ mester they attended \XIeierstrass's course on elliptic functions. which left a profound impression on Hurwitz -" Runge became a leading expert in applying \'17eierstrassian techniques without imbibing the orthodox views of the Berlin school. In this connection. his friendship \Vith Huf\vitz may \vell have been a factor. Privately. Hurwitz con­ ceded to Runge that he favored the geo­ metric approach of Cauchy and Rie­ mann over the purely analytical style cultivated by the Weierstrassians 9 A few years later. he pointed out to Runge that the latter could simplify his proof of the Runge approximation theorem by using the Cauchy integral formula. Gottfried Richenhagen conjectures that this in­ sight may have contributed to Runge's grmving disillusionment with the methodological strictures demanded by Weierstrass and his followers ( Richen­ hagen 1 985. 60-66 J . Berlin offered another great attrac­ tion for Huf\vitz; complementing Weier­ strass's more methodical lectures were those of his brilliant younger colleague, Leopold Kronecker. In a letter to his friend Luigi Bianchi, written during his second stay i n Berlin. Huf\Vitz related that he had returned primarily to attend Kronecker's l ectures on number the­ ory. 10 In the same letter. he called the controversial Berlin algebraist "that great, but very vain mathematician"; clearly he found Kronecker's personal­ ity less appealing than his mathematics. After three initial semesters in Berlin, Hurwitz returned to Munich for the summer semester of 1 879 . At first he felt lonely and wrote Runge that he had a "great longing" to he back in Berlin. 1 1 He arrived in Munich just in time to make the acquaintance of a student from Pisa, Gregorio Ricci-Cubastro , who would later become famous as the in­ ventor of the Ricci calculus. Ricci re­ turned to Italy at the end of that sum­ mer. but the following semester Hurv\·itz

met another Italian. Luigi Bianchi, with whom he struck up a warm friendship ( Hashagen . H9-90). This was a critically important time for both young men. particularly for Bianchi, who \\'as able to complete his cli.ssertation u nder Klein's direction in August 1 HHO. His friend only learned about this by letter. however. as Hurwitz "" as hack home in Hildesheim recovering from a near physical collapse. This was a recurrent theme, as H ur­ witz suffered serious health problems throughout his life; with advancing age these became a matter of deep concern. His father also continued to have strong reservations about the \\'isdom of pur­ suing an academic career, given the rise of anti-Semitism in Germany. Klein saw matters otherwise. He reassured his star pupil about his future prospects, hut counselled him to take the semester off in order to regain his strength. I Ie also wrote to HunYitz's father with the same advice: Above ali i want to stress that among the totality of young people \Vith whom I have u p until now worked there was not one who in specifi­ cally mathematical talent could mea­ sure u p to your son. From now on your son will enjoy a brilliant scien­ tific career. which is all the more cer­ tain because his gifts are combined with endearing personality traits. The only dangerous point is his health. Your son probably already long ago weakened himself through overv·.:ork in his studies . . . . Let me close with the assurance that no one will be hap­ pier than I when your son's health . . . fully returns. I need his intensive collaboration for my latest mathemat­ ical investigation. 1 2 H u rwitz was. indeed. a charming. modest, and unusually warm individ­ ual. He \Vas also a talented pianist. who loved to gather with friends and fam­ ily for music-making. Years later i n Zurich , Albert Einstein was a regular participant in such festivities and he­ came a close friend of the Huf\vitz clan.

7H. C. H. Schubert to Adolf Hurwrtz, 8 December 1877, cited i n (Hashagen 2003, 1 06). 8Hurwitz wrote up lecture notes for all three of the Weierstrass courses h e attended. These Ausarbeitungen are numbers 112, 113, and 115 in h i s Nachlass a t the ETH. 9Adolf Hurwitz to Carl Runge, 1 4 May 1 879, cited rn (Richenhagen 1985, 62). 1 0Hurwitz to Bianchi, 20 March 1882, in Luigi Bianchi. Opere , vol. XI, Rome: Edizioni Cremonese, 1 959, p. 80. 1 1 Adolf Hurwitz to Carl Runge, 14 May 1879; cited in (Hashagen 2003, 90). 1 2Felix Klein to Solomon Hurwitz, 10 May 1880, Mathematiker-Archiv, Niedersachsische Staats- und Universitatsbibliothek, Gbttingen.

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THE MATHEMATICAL INTELLIGENCER

At the time this letter to the senior Hur­ \Yitz was \\Titten, Klein already knew that he would he offered a new chair in geometry at Leipzig University. He may even have confided this informa­ tion to Hurwitz so that he could pre­ pare to submit his doctoral dissertation in Leipzig rather than Munich. At any rate, the following semester Hurwitz was one of 89 students who attended Klein · s lecture course on geo­ metric function theory. the first of sev­ eral courses in which Klein promoted a Riemannian alternative to the theory taught by Weierstrass in Berlin. Hur­ \Vitz"s dissertation . published in 1 88 1 in Mathematische A nnalen, used Kleinian techniques to de\·elop a new founda­ tion for modular functions. He applied these results to a classical problem in number theory: the classification of class numbers of hina1y quadratic forms \\'ith negative discriminant. At age twenty-two he had already demon­ strated his mastery of several dbci­ plines within pure mathematics.

Hurwitz in Gottingen and Konigsberg As a graduate from a Realgymnasium, Hurwitz lacked a strong background in Latin and Greek the mainstays of the curriculum at the humanistic Gym­ nasien. This presented no difficulty for pursuing his doctorate in mathematics at Leipzig, but the next hurdle posed a real problem: to pursue an academic ca­ n:er meant applying to habilitate, and the Leipzig regulations required appli­ cants to he graduates of a humanistic Gymnasium. Klein must have realized that this stipulation, together with Hur­ ·witz"s Jewish confession, virtually ruled out any chance of his habilitating in Leipzig. In a nearly simultaneous case involving Walther von Dyck, who also attended a Realgymnasium, Klein faced strong opposition within the Leipzig faculty. In the end, he barely managed to push through Dyck"s candidacy (Hashagen 2003, 1 1 8-1 2 1 ), but in Hur­ witz"s case he did not even try. Luck­ ily, another option presented itself. Klein still had good connections in Gottingen, going back to the days when Clebsch served as his protege there. Un­ der the latter's watchful eye, Klein had

habilitated there in 1 87 1 , which gave him the opportunity to befriend M. A . Stern, a senior member o f the faculty. Stern had been a fixture in Gottingen for many decades. Klein recalled many years later how Stern told him that the young Bernhard Riemann ··sang like a cana�y" before his tragic illness. Perhaps Klein thought about Hurwitz in similar terms. Certainly he knew that Stern. the first unbaptized Jew to become a ful l professor a t a German university, would offer his talented pupil the kind of sup­ port and friendship he needed. So af­ ter a second brief stay in Berlin, Adolf Hurwitz became a Pri1 •atdozent in GC.it­ tingen in 1 8H2. Little has been written about Stern's career. which in many ways typifies the hardships faced by Je,vish academics. 1 5 He was born in 1 807 in Frankfurt am Main, where the ghetto had long housed the largest Je\vish community in Ger­ many. There he was educated at home by his grandfather and father who, hop­ ing he would become a rabbi, arranged special lessons in Latin, Greek. Chal­ daic, and Syriac. In 1826 he entered Hei­ delberg University to study mathemat­ ics. but j u st one year later he transferred to Gottingen, where he spent the rest of his long l ife. From beginning to end he was an active member of the local Jewish community, and during his stu­ dent clays he \vrote home in Hebrew. In 1 829 Stern was awarded a doctorate with distinction. and the following year he was appointed as a Privatdozent, an unsalaried position. To eke out a living he translated Poisson's textbook on mechanics and published two popular works on astronomy. Many years later, in 1 838 the Hanoverian Ministry agreed to pay him a modest annual salary of 1 50 taler, but noted his special status: "as a Jew, Stern cannot become a pro­ fessor. " Nevertheless, he was nominated by the faculty in 1 840 for a minor post. Five years later the ministry deigned to answer, "as a Jew it is completely out of the question. " In the wake of the po­ litical events of 1 848, however, Stern fi­ nally received an appointment as an as­ sociate professor, and throughout the 1850s he was given small salary in­ creases. Finally, in 1 859 he was made a full professor.

Stern ·s career was in one sense unique. Before his time , the only pos­ sibility open to Jews who wished to pur­ sue academic careers was converting to Christianity . as had C . G. J. Jacobi. Be­ tween 1 848 and the founding of the Sec­ ond Empire in 1 87 1 , practicing Jews were granted various legal rights within Germany: they vvere henceforth free to live in cities. move about . attend uni­ versities, and take part in civic and po­ litical affairs . Yet these gains came at a heavy price, namely the advent of mod­ ern anti-Semitism during the 1 870s. The wild speculation that took place after the founding of the Reich was followed by a severe financial crash t\VO years later, leading to a depression that lasted some twenty years. Predictably . Jewish banking interests were blamed for this and every other e\·il brought about by capitalism and swift modernization. At the universities, the historian Heinrich Treitschke published a widely read pamphlet titled ··A Word about our Jewry . " containing the infamous phrase ··The Jews are our misfortune. " This message helped mobilize anti-Semitic elements to form the new Association of German Students in 1 880. Raucous student groups like these may never have crossed Hurwitz's path during his brief stay in GC.ittingen, but he surely must have realized that the at­ titude toward Jews had become more openly hostile in some quarters. What­ ever notice he may have taken of this, though, he went about his business undistracted. During these two years ( 1882-1884) , Hurwitz published a series of interesting results in function theory. One of these papers confirmed a con­ jecture of Weierstrass, namely that a sin­ gle-valued function in n variables that can be locally represented as a rational function has a global representation as a rational function. Hurwitz gave an el­ egant proof of this claim by means of a newly found result of Georg Cantor. who showed that the continuum was an uncountable set. Fifteen years later H u rwitz would deliver a lecture on applications of Cantor's controversial 1l1enRenlehre to problems of analysis at the First International Congress of Mathematicians. held in Zurich. In the meantime . the Clebsch school

13The discussion of Stern ' s career below is based on (Kussner 1 982).

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finally gained a foothold in Prussia, when Ferdinand Lindemann was called to Konigsberg. This turn of fortune came after he won international acclaim in 1 882 for his proof that 7T is a tran­ scendental number, thereby finally es­ tablishing the impossibility of squaring the circle. One year later he was ap­ pointed to succeed Heinrich Weber. Lindemann's memoirs ( Lindemann 1 97 1 ) contain many interesting anecdotes about mathematical life in Kc)nigsberg during the ten years he taught there. When he arrived he found the facilities woefully inadequate. There were no special lecture halls for mathematics, not even a n adequate blackboard. The mathematics seminar library, which consisted of little more than twenty-odd volumes of the Mathematische A n ­ nalen, was housed in the university's detention quarters (the Karzer), along with some of the rowdier students. Af­ ter several years of futile negotiations, Lindemann managed to obtain a sepa­ rate room for the mathematics seminar, although the mathematicians were forced to share this with a professor of medicine who used it to store pharma­ ceutical preparations. Lindemann also managed to persuade the Minist1y of Education that a new associate profes­ sorship be established, a position that would eventually go to Hurwitz. Lindemann was particularly im­ pressed by some of the advanced stu­ dents he inherited from Weber and even admitted that a few of them had surpassed him in their knowledge of mathematics. He was no doubt refer­ ring to David Hilbert and Hermann Minkowski, names that would become famous together. Minkowski was born in Russia, the son of jewish parents, Lewin and Rabel Minkowski, who set­ tled i n Konigsberg as naturalized Pruss­ ian citizens when their son was eight years old. Unlike Hilbert, whose math­ ematical gifts took long to ripen, Minkowski was a child prodigy who completed the curriculum at the Alt­ statisches Gymnasium in Konigsberg by the age of fifteen. Two years later he submitted a paper to the Academic des Sciences in Paris in response to a prize announcement for proving Eisenstein's formula for the number of ways a pos­ itive integer can be expressed as the sum of five squares. Although it was written in German, contrary to the stip-

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THE MATHEMATICAL INTELLIGENCER

ulations of the Academy, Minkowski's paper was awarded the Grand Prix des Sciences Mathematiques, creating a minor sensation in the world of math­ ematics. One year later, in 1 880 Minkowski began his studies at the Al­ hertina alongside Hilbert. His principal teachers there were Welx.:r and the physicist, Waldemar Voigt. After five se­ mesters, however. he left Konigsberg to study in Berlin. where he attended the lectures o f Weierstrass, Kummer, Helmholtz, Kirchhoff, and, above all, Leopold Kronecker. He then returned to Konigsberg, passed his doctoral ex­ ams, and submitted a dissertation on the thecny of quadratic forms; it was pub­ lished soon thereafter in Mittag-Leffler's

A cta Mathematica. During this second stay in Ki'inigsberg Minkowski met Hurwitz, who had come to Kcinigsberg from Gottingen in the spring of 1884 to assume a newly cre­ ated associate professorship. Minkowski

was a sharp-witted, brash young man, and Hilbert clearly enjoyed his sarcas­ tic sense of humour. 13y contrast Hilbert's relationship with Hurwitz was far more formal, befitting that of pro­ fessor and student, despite their close­ ness in age. Nevertheless, it was the quiet H urwitz who exerted the more de­ cisive impact on Hilbert's mathematical sensibilities, in particular his longing for universal breadth of knowledge. Once a week, Lindemann held a col­ loquium in his home, which he orga­ nized with the assistance of Hurwitz. The two most active students at these meet­ ings were Hilben and Emil Wiechert, who later became an outstanding geo­ physicist and a colleague of Hilbert's in Gottingen, though Minkowski also took part in these colloquia. These meetings were sometimes attended by old-timers like johann Georg Rosenhain, a jew who studied under jacobi. Afterward the group woul d usually gather in a

Adolf Hurwitz ::1s a teenager. Photo courtesy of ETH Bihliothek, Zurich.

Berlin mathematics came to an end, lea ving behind a power vacuum of ma­ jor proportions. The issue of succession also brought about some interesting behind-the-scenes discussions when a committee of Berlin faculty members met on 22 Januaiy 1892 to propose can­ didates to fill these two vacancies. For although the committee members aired a variety of views on the subject, on one point they were unanimous: under no circumstances would they counte­ nance the candidacy of Felix Klein . The historian Kurt R . 1.3iermann brought the following excerpts from the committee's protocol to light: WuEKSTKASS: Klein dabbles more. A dazzler. [Klein nascht mehr. 13/endeli

Prof. Dr. Adolf Hurwitz during his later years in Zu rich . Photo courtesy of ETI I Bihl iothek. Zurich. restaurant for dinner. talking mathe­ matics the whole time. l n the spring of 1 887 Minkowski de­ parted for l3onn. where he began teach­ ing as a Pricatdozent. Thereafter. Hur­ witz became Hilbert's primary source of intellectual stimulation ( Caratheodoiy 1 943. 3S2 ) . During their nearly daily walks they wandered through nearly e\·et)· corner of mathematics. including the problems Hilbert was struggling 'v\·ith in his lecture courses. Konigsberg was j ust about the last place one would have expected to find mathematics stu­ dents during the late 1 880s and early 1 890s, and Hilbert's courses vvere often attended by no more than two or three :.�uditors, sometimes even fewer' He complained about these circu mstances occasionally but never seemed to be re­ ally bothered by them. His goal. no doubt. was clear enough: he wanted to become a truly universal mathemati­ cian. His lecture courses, supplemented by discussions with Hurwitz, served as his prim�uy vehicle for attaining that purpose. and they spanned practically e\·et)' area of higher mathematics of the day: from invariant theory. number the­ ory, and analytic, projective, algebraic, and differential geometry to Galois the­ OIY, potential theoi)', differential equa­ tions. and function the01y .

In KC.i nigsberg. HUI\Yitz was at the height of his powers, and he opened up whole new mathematical vistas to Hilbert. who looked up to him with ad­ miration mixed with a tinge of envy. He later said about him that " Minkowski and I were totally overwhelmed by his knowledge and we never thought we would ever come that far" ( Bl umenthal 1935. 390 l . It was through H u rwitz that Minkowski and Hilbert learned how to ., "talk mathematics, the subtle art o f con­ versing about the essential ideas and problems that characterise a particular mathematical theory. In the years ahead, these three brilliant upstarts would become the stars of their gener­ ation in Germany. Yet only two would attain full professorships in Germany, for reasons that take us into the com­ plex realm of power and authority in German academia over a century ago.

Playing the Game of "Mathematical Chairs" The year 1 892 marked a turning point for mathematics in l3erlin, where a se­ ries of events took place that reverber­ ated throughout the German u niversi­ ties and beyond. With Leopold Kronecker's sudden death o n 29 De­ cember 1891 and Karl Weierstrass's sub­ sequent retirement. the "golden age" of

HELMHOLTZ: Kronecker spoke very disparagingly of Klein . He regarded him as a windbag [Faiseu r] . Ft ·cHS: I have nothing against his personal qualities. only his perni­ cious manner when it comes to sci­ entific questions. ( B iermann 1 988. 20-)-206 ) As the mathematicians on the com­ mittee certainly knew, Lazarus Fuchs had been involved in a public dispute with Klein after the latter openly criti­ cized Henri Poincare for naming a cer­ tain class of automorphic functions af­ ter Fuchs. Weierstrass, now an a i l i ng old man. had been hoping for a long time to install Klein's Gottingen col­ league. Hermann Amanclus Schwarz. as his successor. Kronecker had opposed this plan. causing Weierstrass to post­ pone his retirement. Now that his nemesis was gone, Weierstrass merely had to overcome Fuchs's evident aver­ sion to Schwarz. Fuchs tried to finesse this issue by suggesting that he and Schwarz were too close scientifically, hut Weierstrass easily dismissed this claim. He then demanded that a good analyst he nominated \Vith the poten­ tial to draw students, and he asserted that only two candidates met this cri­ terion: Klein and Schwarz. Fuchs fee­ bly tried to nullify this argument, not­ ing that he and not Weierstrass was the one who would have to get along with Schwarz. But then he admitted that Sc hwarz was a much better choice t han Klein, at vvhich point the rest of the committee fell in line. In the faculty's official petition to the Ministry from 8 February, the Berlin

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consensus on Klein was stated eve n more sharply: Above all else. however, account must he taken that the appointed successor be suitable for continuing the generations-long tradition of our university to lead students in serious and self1ess probing of mathemati­ cal problems. For this reason. per­ sonalities like Professor Felix Klein ( horn 1 84 9 ) must be set aside, al­ though opinions among scholars about his scientific achievements are very divided. since the entire effect of his writings and teaching stand in opposition to the tradition of our university as characterized above (Biermann 1 988. 307-308) . These views ref1ect far more than just personal aversion to Klein and his style of mathematics. As the still young leader of a large group of mathemati­ cians formerly associated with Alfred Clebsch, Felix Klein represented a threat to Berlin ·s hegemony vvithin the German mathematical community, par­ ticularly within the Prussian u niversity system. G6ttingen. where Clebsch had taught until his unexpected death i n 1 87 2 , afterward fel l under Berlin's in­ f1uence through the appointment of Weierstrass's star pupil, Schwarz . How­ ever, as already noted, Klein still had considerable residual inf1uence in G6t­ tingen, and in 1 886, against strong op­ position from Schwarz, he managed to obtain a chair there. Thereafter, lead­ ing representatives of the Berlin school did their best to keep Klein boxed i n . H e , i n turn, hoped t o gain t h e support of Friedrich Althoff, the powerful min­ isterial official who oversaw appoint­ ments i n the Prussian educational sys­ tem. This conf1ict set the stage for the events that followed. Rather than looking toward the chal­ lenges of the future, the Berlin mathe­ maticians were mainly content to keep their own house in order. This conser­ vative outlook was apparent from the start. Thus , it came as n o surprise when the appointment committee decided to bring two Berlin products back into the

fold by nominating Frobenius to fill Kro­ necker's chair and Schwarz to succeed Weierstrass. These recommendations were quickly approved by the Prussian Ministry ( meaning Althoff) , at \vhich point the game of mathematical chairs could begin. One month later Klein wrote to Adolf H unYitz to com ey his plans for filling Sclnvarz's vacant post: Althoff was here for three days and has decided on the calls to Berlin . [Concerning Schwarz's replacement] you will probably have guessed that I want to recommend you and Hilbert as the only two who, together with me, are in a position to assure Ght­ tingen a place of scientific distinction. . Naturally, I will name you first and Hilbert behind you. 1 ' Klein had been "·arching Hilbert's de­ velopment closely. \Vhereas most ob­ servers regarded him as an expert on invariant theory, Klein realized that his interests went far beyond this field. In a report to Althoff. written in October 1 890. he described Hilbert as ''the ris­ ing man .. in mathematics. 1 " Still , as a mere Privatduzellt in Konigsberg he had no chance of leaping to the chair i n Gi)ttingen once occupied by Dirich­ let and Riemann. As for Hul\Vitz, Klein expressed three issues of concern: There are, however. a series of dif­ ficulties associated with your ap­ pointment . . . First, there is the problem of your health. . . . Sec­ ondly, there is the much subtler dif­ ficulty that you are, not only per­ sonally but also in your scientific style, much closer to me than is Hilbert. Your coming here could therefore perhaps give our Gottin­ gen mathematics a too one-sided character. There is thirdly-! must touch on it, as repugnant as the mat­ ter is to me , and knowing full well your j ustified sensitivity to this-the Jewish question. Not that your call as such would present difficulties; these I would be able to overcome. The problem is that we already have [Arthur] Schoentlies. for whom I

\Vot!ld like to create a firm position as salaried E':xtraurdinarius. And having you and Schoentlies together is something I will not get past ei­ ther the faculty or the Minister. J (, As this final remark implies. the Giittingen Philosophical Faculty hac! an un­ otficial policy aimed at restricting the number of Jewish Dozenten within a given discipline. Klein e\'idently raised this issue directly with Althoff. \\ ho ap­ parently informed him of the ministry's position on this matter. The Prussian government clearly sa\,. no reason to in­ tercede in such cases of de facto dis­ crimination against Jews. So Klein faced an impasse. Shortly before the faculty ccmvened to deal \Yitb Schwarz's suc­ cessor, he wrote to Althoff that. in view of the anti-Semitism within the faculty. .. he would be \Villing to .. sacrifice Sch(in­ tlies to bring Hunvitz to Gi1ttingen. 1 Two weeks later. on 1 7 March 1 892. Klein wrote to Hul\Vitz again. this time to inform him that he was no\\ the only serious contender for Schw·arz·s posi­ tion. It had been impossible even to get Hilbert's name on the list of candidates since he was still a Pri�·atduzenl. 1 " This assessment of Hurwirz·s chances, hmY­ ever, did not retlect the true state of af­ fairs. Klein's rivals, Schwarz and Ernst Schering, pushed for a list that contained the names of ( 1 ) Heinrich Weber. ( 2 ) Ferdinand Lindemann. and (3) Georg Hettner, whereas Klein favored ( 1 ) Hur­ witz, (2) Hilbert, and (3) Friedrich Schot­ tky. After long and intense debate. the G6ttingen Philosophical Faculty com­ promised on ( 1 ) Weber, ( 2 ) Hul\Vitz, and (3) Schottky. These negotiations took place during the week of 6-10 March, a full week hefure Klein wrote to Hurwitz about his chances. 1 '! Thus, Klein was ap­ parently banking on the autocratic Al­ thoff, knowing full well his reputation for ignoring faculties' wishes. Presum­ ably Klein had received some kind of as­ surances from him regarding this ap­ pointment; othen.vise Klein's optimistic letter to Hurwitz would he inexplicable. Clearly he expected Althoff to pass over Weber and choose Huf\vitz instead.

1 4Felix Klein to Adolf Hurwitz, 28 February, 1892, Mathematiker-Archiv, Niedersachsische Staats- und Universitatsbibliothek, Gbtt i n9en. 1 5Felix Klein to Friedrich Althoff, 23 October, 18 90, Rep. 92 Althoff B N o . 92, fols. 16Felix Klein to Adolf Hurwitz, 28 February, 1 892. 1 7Felix Klein to Friedrich Althoff,

7

76-77,

Geheimes Staatsarchiv, Berlin.

March, 1892, Re p . 92 Althoff AI No. 84, Bl. 21-2 2, Geheimes Staatsarchiv, Berlin.

1 8Felix Klein to Adolf Hurwitz, 17 March, 1892, Mathematiker-Archiv, Niedersachsische Staats- und Universitatsbibliothek, Gbttingen. 1 9Kiein Nachlass 22L Personalia,

26

s. 5 ,

THE MATHEMATICAL INTELLIGENCER

Niedersachsische Staats- und Universitatsbibliothek, Gbttingen.

-y--

P. .//(Jl I&/ /

Paul G o t d an s letter to Klein. 1 6 April 1 892. in reaction to the latter's dashed pl a ns for Hun,·itz (a partial transcription and trans­ lation appear on the next pttingen. ·

·

This r la n might well have been re::tl ized had not an unexpected circum­ stance diverted Klein from his original objective Frohenius, who had not yet accepted the offer from Berlin began to contemplate whether he might not pre­ fer Gottingen. Delighted by this turn of events, Klein assured Althoff that Frobe­ nius '\Youlcl be welcomed with open a rms by the Gottingen faculty. He e\·en admitted in a tastless letter to H wwitz that. had he known there was any po s­ sibility of winning Frobenius, he would have placed his name first on his list of nominees 2 0 Sensing that his long-term strategy· of "divide and conquer" was fi­ nally about to pay otl, Klein rejoiced at the prospect of stealing away from Berlin the a lgebraist generally regarded as the natural heir to Kronecker's chair n Not o nlv would this solve the second con­ cern he had confided to H urw itz since ­

.

.

-

Frobenius's style clearly complemented his ovvn-but it would also solve the problem posed hy the third issue. the 'jewish question," opening the way for Schoent1ies·s appointment. Klein invited Frobenius to visit him in Gc)ttingen on 20 March, but the meet­ ing evidently went badly. Klein recalled that Frobenius was exhausted from overwork during the stay. Not surpris­ ingly, he opted for B erl i n and imme­ diately afte rward Althoff offered the va­ cant post in Gottingen to H einrich Weber. Klein was furious when he learned that Althoff had chosen to honor the faculty's wishes rather than his own, although it is not implausible that his opportun i stic fence-jumping may ultimately have ruined Hurwitz's chances. Anothe r possibility is that Hur­ \\·itz \vas the victim of an "anti-Semitic ,

backlash" within the Prussian Ministry,

a suspicion Klein h imse lf raised . For this reason, he wrote Hurwitz, he consid­ ered the latter's "chances of succeeding Weber i n Marburg or obtaining a call anywhere else in Prussia as unfavor­ able. " 22 This assessment turned out to be prophetic although Hurwitz did re­ ceive an offer shortly thereafter from the ETH in Z urich to succeed Frobenius which he accepted. Little did he know that he would spend the remainder of his career there. With Hurwitz ' s depar­ ture from Konigsberg, Hilbert could novv move into his mentor's former po­ sition, thus ending his clays as an im­ poverished lecturer. Minkowski, still in Bonn, was also elevated to associate professor at this time. Whether or not Klein really believed that anti-Semitism had quashed Hur­ \\'itz·s candidacy for G()ttingen. he was clearly disappointed. However, his ,

,

20Felix Klein to Adolf Hurwitz, 23 March, 1 892, Mathematiker-Archiv, N iedersachsische Staats- und Universitatsbibliothek. Gbttingen . 2 1 Felix Klein to Friedrich Althoff, 21 March, 1 892, Rep. 92 Althoff AI No. 84,

81. 27-28,

Geheimes Staatsarchiv, Berlin.

22Felix Klein to Adolf Hurwitz, 7 April, 1 1 April 1 892, Mathematiker-Archiv, Niedersachsische Staats- und Universitatsbibliothek, G6ttingen.

© 2007 Spnnger Science ...:- Business Media. Inc . . Volume 29. Number 2. 2007

27

older frien d and colleague, Paul Gor­ dan, who presumably knew someth ing about anti-Semitism in the German uni­ versities from firsthand experience, had a vety different perspective of this mat­ ter, and wrote Klein accordingly. I am sorry to hear that you were not called t o Berlin, as your all-embrac­ ing spirit would have brought order to the mathematical relationships i n Germany . . . . I t was j u s t that you recommended Hurwitz for Gottin­ gen; Hurwitz deserves this distinc­ tion. That your recommendation did not go through, however, is a for­ tune for which you cannot thank God enough . What would you have had with H u rwitz in Gottingen? You would have taken o n the complete responsibility for this Jew; every real or apparent mistake by Hurwitz would have fallen o n your head, and all his u tterances in the Faculty and Senate would have been regarded as influenced by you . H u rwitz would have been considered nothing more than an appendage of Klein 25 Although disappointed, Klein was not one to take such a defeat lying clown. He fired off an angry letter to Al­ thoff, complaining about the loss of face he had sutiered in the Gottingen faculty; he had fought tenaciously for Hurwitz's cause, only to have Weber, the candi­ date proposed by his opponents, called instead. This turn of events, he asserted, . . . coul d only be somewhat reme­ died by having Schoent1ies named Extraordinarius. On the one hand, it is known that I have been work­ ing on his appointment for years; on the other, that my efforts have only met with resistance, so that I only dispensed from doing so as Hur­ witz's call stood in question. Should Schoent1ies now be passed over, this impression [i. e . , of Klein's impo­ tence in academic affairs] will be­ come a virtual certainty. I woul d then be forced t o advise young mathematicians not to turn to me if they hope to make further advance­ ments in Prussia. 2"' Shortly after this letter was written, Arthur Schonflies was appointed ausserordent-

. . na g

Paul Gordan to Felix Klein, 16 April, 1892

ie nkht nach B ·ri m g ·J..onHnL·n �md r h u t m i r leid. het l h r

f:.tssL"ndL"n Gl:'bt h:i t t L· n "'l O rdnu ng i n den Deutschland" gdmtt lrt

Giill mgen

\IPr fiir Sk· ist

\ orgesl·hlagen

l''o

''

lit:nt diL'SL'

hl.tg nkht durchgeg.mgen ist. das ist

h.il!en SiL' ' on l l u m itz in ( ;(>ttingen?

ie h:men die ganze \ erant\\ onung

e i n G l tick fli r sie, flir

\\ · ld t e"

Sil' Gott nicht genug danken kiinnen. \\ a ...

fur diesen J uden iilx•rnommen; jeder merklidu.� od<..•r sdwinhare 1-ehler ' on l l u m itz \\:ire a u f l i nt: Kapp • g 'kommen und aile ,\eul�eru ngL·n \ on l l u r­ " i t z in Fakultiit und \\

en.ll h:iiiL'll als ' on I h nen 1 1L·einfl us... t gt:goltl·n. ll u r­

ilz h:me n u r a l s e i n \pp ·ntli.

\ on Klein gq�olten

lieher Professor in Gottingen, where for the nt:xt seven years he taught de­ scriptive geometry and strengthened the u niversity's offerings in applied mathematics. As it turned out, Klein got along vety well with Heinrich Weber, a gifted and highly versatile mathematician whose re­ search interests complemented Klein's. Shortly after Weber's arrival, he and Klein co-founded the Giittingen Mathe­ matical Society, which thereafter played an instmmental role in forging the kind of tightly knit mathematical community that Klein had long hoped to lead. Just two months after Weber ac­ cepted the call to Gi.ittingen, Klein got a tremendous break, though not from Althoff nor even from within Prussia . I t came rather in t h e form o f a very at­ tractive otJer from the Bavarian Ministry of Culture, which hopt:d to induce Klein to accept the chair at Munich University vacated by Ludwig Seidel's retirement. Even before Klein had the otJicial offer in hand, he took up negotiations with AlthotJ, who matched the Bavarian con­ ditions right down the line, thereby en­ abling the rejuvenated Gi:ittingen mathe­ matician to decline graciously 2" In the process of doing so, he advised the Bavarian Ministry regarding other poten­ tial candidates, including Ferdinand Lin­ demann, who was ultimately appointed. In the meantime, Hilbert stood wait­ ing in the wings, making evety etlort to signal his loyalty to Klein. Less than a year after succeeding HUtwitz, he in­ formed Klein that Althoff was going to

24Felix Klein to Friedrich Althoff, 10 April, 1892, Rep . 92 Althoff AI No. 84, 25For details, see (Toepell 1996, 208-212, 370-378). 26David Hilbert to Felix Klein, 13 August 1893 (Frei 1 985, 96).

THE MATHEMATICAL INTELLIGENCER

\ LT

or-.l

81. 32-34,

�

recommend his appointment as Linde­ mann's successor in Konigsberg. 2c' At this point, Hilbert and Klein began trading notes on suitable candidates for the as­ sociatt: professorship that Hurwitz had occupied only a year earlier. Through­ out these deliberations, Hilbert indicated that he saw no chance of acquiring Minkowski, in part because his friend had excellent chances of moving up in Bonn. Time ticked on. Four months later, Althoff requested that Hilbert come to Berlin to discuss the matter with him per­ sonally, probably because of various other complications with personnel. When asked for advice, Klein merely re­ sponded that AlthofJ was totally unpre­ dictable, but he also thought that Minkowski's candidature was unrealistic. Six weeks later he congratulated Hilbert on pulling it off all the same ( Frei 1 985, 1 04-106). The issue of religious affilia­ tion never arose, perhaps because the position was not a full professorship ( O r­ dinariat). At any rate, the fact that Hilbett got to name Minkowski as his successor was a dear sign of things to come. Heinrich Weber was not the young dynamo that the empire-builder Klein hoped to find, and Klein now became more determined than ever to bring Hilbett to Gi:ittingen. When Weber ac­ cepted a chair in Strasbourg, he quickly seized the opportunity by notifying Hilbert in a letter of 6 December 1894: . . . Perhaps you do not yet know that Weber is going to Strassburg. This very evening the faculty will meet, and although I cannot know ahead

23Paul Gordan to Felix Klein, 16 April, 1892, Klein Nachlass, Niedersachsische Staats- und Universitatsbibliothek, Gbttingen .

28

um­

e i n ( ; ! ttl k. f );! I� .._k' l l u m iV in

h.tlx•n, ist recht ge'' L'SL'll: l l un\ll Z

,\uszekhnung. a her dal' di "'en \

·m

i� ensch;tftiKhen \ erh:l ltn isse

Geheimes Staatsarchiv, Berlin.

of time \Yhat the commission ,,·ill rec­ o mme nd . J st il l wish to inform you that I will make evety effort to see that no one other than you is called here. You are the man whom I need as 111\" scientific complement: due to the direction of your work. the power of your mathematical thinking. and the bet that you now stand in the middle of your productive career. I am counting on you to gi,·e a new inner strength to the mathematical school here, which has grown and, as it appears. ,,·ill continue to grO\Y a gre:tt deal further. and perhaps you \\·ill exert a rej ll\·enating influence on me as " ell. I cannot kno"· whether I \Viii preYail in the faculty, even Ies.� so whether the recommen­ cl�nion '' e make ,,·ill ultimately he fol­ lowed in Berlin. But this one thing you must promise me . e,·en today: that you \vii! not decline the call if it comes to vou1 ( Frei 1 9H5. 1 1 5 ) . It \Vas the chance of a lifetime. and I Iilhen knew it. Klein no longer showed

anv interest in pursuing Hurwitz's candi­ dacy. As !Jek.an of t h e Faculty, he \vrote up the list of recommendations himself and it contained just two names, rather than the usual three (or more) : Hilbett and MinkmYski.T Some of Klein's col­ leagues criticized his choice, presuming that he wanted to appoint an easy-going. amenable ( hequem) younger man. To this he replied: "Ich berufe mir den allerunbequemsten-l \vant the most dif­ ficult of air < Hlumenthal 193'), 399). Klein clearly knew what he wanted as well as \\·hat he ,,·as getting. and. for his part. Hilbert gladly joined forces with him.

The "Jewish Question" Reconsidered Many German intellectuals disdained. even more feared. the autocratic Friedrich Althoff, who could make or break an academic career at whim ( Kayser 1996. 1 69-177). Hi l bert , on the other hand. could only sing his praises. and for good reason: e,·erything 'i\'ent his ,�vay . Less than two weeks after he

received Klein's letter. he met with Al­ thoff again, this time ro seal the Gi>rrin­ gen deal. They quickly reached agree­ ment. whereupon Althoff asked Hilhen whom he would like to have as his suc­ cessor. practically inviting him to name Minkowski. When Hilbert obliged, Al­ thoff then asked whom he might want to fill Minkowski's associate professor­ ship ( Frei 1 9H5, 1 1 7- 1 1 8 ) . Whatever ib drawbacks. Althoff's style was surely a model of bureaucratic efficiency. Soon a fter this conversation, Hilbert wrote Lindemann. informing him that he had recommended Minkowski for the Kiinigsberg chair. He received the follm,·ing illuminating reply. sent on New Year's Day 1 H9 '5 : It also would appear to me most natural to have Minkm\ ski ap­ pointed as your successor. If Althoff is really of the same opinion. then it really does not matter who else you place on the list . Still it seems to me possible that he will collide against [Minkowski's] Judaism; at

Two Tributes to Adolf Hurwitz Max Born recalling Hurwitz as a teacher \t the end of th

'' Inter "eme ...ter

1 1902· 1 903) 1

ol m� 'otudent l ife. I ca refu lly worked out the \\ hole cour!-.e, a�ain de­

cided to sp ·nd th · summer htud) in�] at another unh er­ ·

sit� in ord ·r to \\ id ·n my ' ie\\ s on stience and lifl•. It

was only natural that I should consider Zuri<.h. \\ here, in addition to th

Cantonal l niver it)', there '' .ts the l!itle­

Rilti.· · iscbe Tec:hnische lloc:hsc:hu/e, an important science and

·

"d10ol of

·ngineerin�. M) friend lOtto] Toeplitz ap­

in<. luding the�e pri\ .He appcnd it·e . and my noteh<Xlk wa� u ...ed h} Coumnt \\ hen he, many years later and a fter H u r­ \\ itz' death. puhlbhed hts wel l-kncm n hook . . . the t�t lll'd Cour.mt-Hum ttz
.,o­

George Polya on Hurwitz as a colleague J I U f\\ II/ had great mathematit.tl breadth, .1� much a., was

prm ed my choice of Zuri<.h, as a mathematician of great

pm.!-.ihlc in hi� time, lie had learned algebr.t and number

renown . H u rwitz, lh ed there . . . . I ha,·e to confess that

theol) from Kummer and Krone<.ker, comple

I \\ ent on ly to two mathematical cour...es. one ( four hours a week) by Hurwitz on ell iptic functions, the other (t\\ o hour.,) by [Heinrich] Bu rkhardt on fourier anal} sis. H u l"\\ itz wa., a tiny man ,,·ith the emaciated face of an ascetic in '' hi h bu rned two unnaturall} la rge eye

. l ie

\·ari.thlcs

from Klcm and \\ eier tr.tss. It " as H u f\Yitz " ho .1rrangcd

for m · 111} fiN i.!ppoimment at the ETH (The S\\ i. s Fcd­ er..tl I nstitute of Technolo1-w>. From the t i me of m poinuncm there in 1 9 1 1 unttl his death in

con�t.tnt touch '' ith him

1 9 1 9.

·

ap­

I \\ as in

\\ e had a "Pecial \\ ay \\ e

was ail in� and \·ery frail. But his lectures '' ere brilliant,

\\ ork ·d. l "·ould ' is it him and we would .,it in his �rudy

perhaps the mo!-.t perfect I ha\ e e' er heard. The cour'>e

and talk mathemat ics-�eldom anything ebe-u nt i l he fin­

was the cominuation of another. on anal} tic fu nctions,

i-.h ·d hb dgar. Then we would go for

which I had not attended: I therefore had some diffi< ulty

in� t h · math ·matical discussion. H i. health '' as not too

in fol lowing and had to " ork hard. read in� man} hooks. Once when I mbsed a point in a lelture I '' ent to l l ur­ '' itz aftef\\ ;uds and a..,ked for a prh at

· e

planation. l ie

im ited me and another student from Br '!-.lau. K} na!-.t , . . .

to his house and ga\ e us a .,eries of pri\ ;He lett u re!-. on sonK· ch.tpter., of the th<.'Of) of fu nuion., of complc · ' a ri­

ahlcs, in particu lar on ,\ l itta�-I.eft1er"., theorem. \\ hit h I st ill consider as one of the most impr •ssh·

·

·

pcricncc.,

..1

\\ a l k , continu­

good, so wh ·n \\ C.: wal ked it h.td to be on le\·cl ground,

not al W:t} s ·asy in the hilly part ot Zi.i rkh, and if we \\ ent

uphill. \\ C wal ked \ ef)' slcl\\ 1} . I \\fOte a joint paper \\ ith ! f u m itz. In fac:t , it is a pap ·r of mine and a paper of his, l i n ked in a pocti

form ot corrc�ponc.lence . .\ ly connec­

t ion \\ ith Ihm\' il/ '' a., del·�r and 111} debt to him �rca ter ·

than to any oth · r coli •,tgue. I played a large role in edit­

ing hi� collec:ted \\ orks ( P6lya 1 9H... . 2'5).

27Kiein's list of 13 December, 1 894 can be found in the Personalakten Hilbert, Universitatsarchiv Gbttingen.

© 2007 Springer Science + Business Media, Inc., Volume 29, Number 2 , 2007

29

any rate, he said to me once ( in con­ nection with Hurwitz's appointment in Konigsberg ) , that there would be no misgivings about appointing a Jew to an associate professorship, hut that the matter would he other­ wise were the appointment to be a ful l professor's position. Of course , Althoff changes his vie\YS . too ' 2H Althoff's personal views \\'ere probably beside the point. It ,,·ould seem that Minkowski, unlike Hurwitz earlier, was in the right place at the right time. Thus the 'jewish question" never reared its head and Althoff simply followed Hilbert's recommendation. In the mean­ time. mathematics in Berlin languished: Schwarz largely gaYe up research. while Frobenius became increasingly frus­ trated as he saw how Klein managed to ingratiate himself with Althoff and the Prussian Ministry of Culture. Minkowski remained in Konigsberg only a little more than a year before joining Hurwitz in Zurich. In his letters to Hilbert (Minkowski 1 973, HG-1 54) he provides many details about the times he and Hurwitz spent together. A criti­ cal test then came in 1 902. when Hilbert was nominated to succeed Lazarus Fuchs in Berlin. No mathematician had ever before declined a formal offer from the leading German university. and Klein knew i t would not be easy for Hilbert to refuse either. He urgently ap­ pealed to Althoff for support. and the latter obliged by creating a new Ordi­ naria! in mathematics at Gottingen out of thin air, as it were. and appointed M inkowski to fill it. This unprecedented action was all the more daring consid­ ering that Minkowski's appointment also overturned the umvritten policy of the Gottingen Philosophical Faculty that restricted the proliferation of Jewish Dozenten within a given discipline. (Arnold Schoenflies \vas by now gone, but in the meantime Karl Schwarzschild had been appointed Professor of As­ tronomy in 1 90 1 . ) Seven years later came Minkowski's sudden death from appendicitis. and the chair had to be filled anew. The Gottin­ gen faculty deliberated over three possi­ ble candidates: Adolf Hurwitz. Otto Blu­ menthal, and Edmund Landau. All three

were Jewish and all three ended up shar­ ing first place on the faculty's Bem ­ jimgsliste, hut there the similarities encl. Huru·itz had longstanding friendships with Klein and Hilbert. and Blumenthal. at that time the managing editor of J:fath­ ematische Annalen. had been Hilbert's first doctoral student. Landau, on the other hand . \Yas not only an outsider. a Berliner, he was a proud, independent­ minded. rich. and self-confident man whose mathematics and teaching ex­ uded the n:ry purism that Klein had op­ posed for so long. Unlike the notoriously diflicult and conceited Landau, both Hur­ witz and Blumenthal were known for their Liehenswzh·digkeit, or kind disposi­ tions. In other \\ ords. they \\·ere the kind of self-effacing "good Jews" that liberal­ minded Germans tended to like. as op­ posed to the ·'others"-those "haughty Berlin Jews " and the 1101/l 'eatl riches who--so the thinking went-had the au­ dacity to flout established social con­ vention by tlying to hobnob \Vith Pruss­ ian high society. According to a story that Richard Courant l iked to tell, the faculty's deci­ sion to pass oYer Hurwitz and Blu­ menthal in favor of Landau was swayed by a statement made by Klein, who supposedly said something like this: "Landau is very disagreeable, very dif­ ficult to get along with. But we. being such a group as we are here, it is bet­ ter that we have a man who is not easy. '' 2 9 Whether true or not, this story not only fits Klein but also captures an important feature of the Gottingen community he wanted to build . Adolf Hurwitz lacked the ruthless arrogance that went with the territory. True, Klein regretted that Althoff saw no place for Hurwitz in the Prussian u niversity sys­ tem, and he even queried Hilbert about this i n 1 904, but those who knew Hur­ witz in Zurich (see the accompanying tributes to Hurwitz by Max Born and George P6lya ) found that his burning love for mathematics never died.

Blumenthal, Otto. 1 935. "Lebensgeschichte, " i n David H ilbert, Gesammelte Abhandlungen , Bd . 3, Berlin: Springer-Verlag, pp. 388-429. Born, Max. 1 978. My Life. Recollections of a Nobel Laureate. New York: Charles Scrib­

ner's Sons. Caratheodory,

Constantin.

1 943.

"David

Hilbert," Sitzungsberichte der Bayerischen Akademie der Wissenschaften zu MOnchen ,

Math. -nat. Abt 1 943, 350-354 . Frei, Gunther, Hrsg. 1 985. Der Bnefwechsel David Hilbert- Felix Kle1n (1 886- 1 9 1 8). Ar­

beiten aus der Niedersachsischen Staats­ und Universitatsbibliothek Gbttingen, Bd . 1 9 , Gbttingen: Vandenhoeck & Ruprecht. Hashagen ,

Ulf.

2003.

Walther von

Dyck

(1856-1934). Stuttgart: Franz Steiner Verlag. Hurwitz,

Adolf.

1 932-33.

Mathematische

Werke, 2 vols . , Basel: Birkhauser.

Kayser,

Heinrich.

1 996. Erinnerungen aus

meinem Leben. Munchen: lnstitut fUr Ges­

chichte der Naturwissenschaften Munchen. Kussner, Martha. 1 982. "Carl Wolfgang Ben­ jamin Goldschmidt und Moritz Abraham Stern , zwei GauBschuler j udischer Herkunft , " Mitteilungen der GauB-Gesellschaft 1 9:37-

62. Lindemann ,

Ferdinand.

1 971 .

Lebenserin­

nerungen . Munich.

Meissner, Ernst 1 932. "Gedachtnisrede auf Adolf Hurwitz," in (Hurwitz 1 932, xxi-xxiv). Minkowski, Hermann . 1 973. Briefe an David Hilbert,

ed .

Lily

Rudenberg

and

Hans

Zassenhaus, New York: Springer-Verlag. P61ya, George. 1 987. The P61ya Picture Album: Encounters of a Mathematician , ed. G . L.

Alexanderson . Boston: Birkhauser.

Reid, Constance. 1 970. Hilbert. New York: Springer-Verlag. Richenhagen,

Gottfried.

1 985. Carl Runge

(1 856-1 927): von der reinen Mathematik zur Numerik.

Gbttingen:

Vandenhoek

&

Ruprecht Rowe, David E. 1 986. " 'Jewish Mathematics' at Gottingen in the Era of Felix Klein , " Isis 77:422-449. Toepell, Michael.

1 996. Mathematiker und

Mathematik an der Universitat MOnchen. 500 Jahre Lehre und Forschung, Algorismus,

Heft 1 9 . Munchen: l nstitut fUr Geschichte der BIBLIOGRAPHY

Biermann, Kurt-R. 1 988. Die Mathematik und

Naturwissenschaften. Zeuthen, H. G. 1 91 4. Lehrbuch der abzahlen­

ihre Dozenten an der Berliner Universitat,

den

1 8 1 0- 1933. Berlin: Akademie-Verlag.

Teubner.

Methoden

der Geometrie.

Leipzig :

2BFerdinand Lindemann to David Hilbert, 1 January 1 895, Hilbert Nachlass, Niedersachsische Staats- und Universitatsbibliothek, Gbttingen. 29(Reid 1 970, 1 1 8) Unfortunately, the surroundin g i nformation in Reid's account about those mathematicians who were considered to succeed Minkowski conflicts with the documentary evidence, which makes it difficu lt to place great confidence in the anecdote cited.

30

THE MATHEMATICAL INTELLIGENCER

•:• Mi• i§l.f hl¥111'1 =1§.11

The proof is in the pudding.

C o l i n Adam s , E d itor

�

Wo rst Case Sce na r i o Su rviva l Hand bool< : M athe mat i cs COLIN ADAMS

Opening a copy of The Mathematical

Intelligencer you may ask yourself wzeasiZJ', "What is this anyzl'ay-a

mathematical journal, or what?" Or you may ask, "Where am /?" Or ez •en "W"bo am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularzv. It's mathematical. it's a humor column, and it may even be harmless.

y its \·erY nature . mathematics is a . L_.'') ' .) risky endeavor. You may have al-

.k.

ready experienced injury-physi­ cal or psychological-from attempts to learn mathematics, or perhaps just from lidng a mathematical lifestyle. To help \Yard off further peril . \\'e have com­ piled the following scenarios with the help of our experts. But keep i n mind that the safest course is always to con­ sult a Ph.D. mathematician. UNDERTAKE THE ACTI\ ' ITIES IN THIS ARTICLE 01\i YOL'R 0\'\'.'\ 0:\'LY WHEN NO P h . D . IS A\'AILABLE. The publisher, author. and experts d isclaim liability for any harm caused by the use of the infor­ mation contained in this article. Good luck1 ·

How to Give a Job Talk

Column editor's address: Colin Adams,

l'nless you are Gauss. at some point you will have to giYe a job talk. This is perhaps the most important talk you \\'ill ever give . so unless you want to teach Remedial Fractions for the rest of your life . consider following these in­ structions for the various sections of your talk:

Department of Mathematics, Bronfman

Science Center, W i ll i ams Col lege, W i l l i amstown , MA 01267 USA e-mai l : Col i n . C.Adams@wil l iams.e du

First quarter: Start off simply, defining ,·our terms carefully and drawing lots of pictures . l u l ling the audience into the

impression they might understand the talk. Then continue to build one defi­ nition on top of another. each depen­ dent on the pre\·ious one. Lots of Greek letters. arro\\·s. and supersubsuperscripts giYe an impression of intellectual depth. Second quarter: Include numerous the­ orems. corollaries and lemmas. Number them . making it easier for the a udience to keep track of the running total. I n passing. mention the names o f as many members of the department as possible. If someone·s work is related only he­ cause the two areas start with the same letter. that's good enough reason to in­ clude them . Third quarter: This is where vou make the talk incomprehensible to eYeryone. including yourself. If you can under­ stand it. someone else might. in \\'hich case it cannot be serious work . Of course. making the talk incomprehen­ sible is not difficult. A selection of para­ graphs from three or four randomly chosen papers out of an old issue of 2\!latematicheskie Issledovaniya of the Akademiya Nauk Republiki Moldova usually does the trick. Last quarter: Here is \Yhere you make it sound like vou are on the verge of solving the greatest t\\'O or three prob­ lems in mathematics. Just a year or two more. and \\·hate\·er institution has had the foresight to hire you will be bask­ ing in your reflected glory. And then. after making this point, \\·hate\·er you do . end on time. This is the most sacred of all rules. Ending on time is the human equivalent of rolling over on your back and exposing your neck . If you do not end on time. you are implying that you believe you are more important than your audience is. Your chances of being hired then drop to epsilon. which in this case is an ex­ ceedingly small number.

© 2007 Spnnger SCience - Business Med1a. Inc . Volume 29. Number 2 . 2007

31

How to Escape from a Sinking Department Your department is floundering. It has lost its VIGl{E grant. The state legisla­ ture has cut the budgets. Salaries have been frozen, and the university is shrink­ ing the size of the department while en­ rollment remains high. You need to make the leap to a different school. Here are the simple steps to follmv: 1 . Choose the school to which you

wish to jump. Spend a sabbatical there, so they have a chance to get to know you, unless that would be to your dis­ advantage. 2. Prepare your cu rriculum z ·itae be­

forehand. Bulk it up with papers, notes, and technical reports. Often a single pa­ per can make more than one appear­ ance in a variety of guises. Even an er­ ratum to a paper can add to the length of the vitae if carefully titled ( e . g .. Notes on the Work of . . . ) . 3 . Wait until both departments are al­ most at the same level. Do NOT WAIT to jump until your department is already i n the dumpster. Then it is too late. As your department deteriorates, and the department to which you wish to j u mp improves, there will he a moment when the t\VO are comparable. It is the instant before then that you must jump. 4 . Typically, the landing will be bumpy. Expect resentment on the part of mem­ bers of your new department who do not appreciate the addition of a senior member in an area unrelated to their own interests. A disgruntled chair may give you unpleasant teaching assign­ ments for the first few years. But very

32

THE MATHEMATICAL INTELLIGENCER

quickly, you vYill he absorbed into the fabric of the department. 5. If you a re the Glli.�e of departmen­ tal decline, then keep in mind that you \Viii have to jump again \-eiT soon.

How to Survive a Counterexample from the Audience

You are delivering a keynote address at the National Mathematics Meetings when an audience member interrupts to point out an obvious counterexam­ ple to your main theorem.

1 . Remain calm. Do not break down in tears or run from the room. 2. Make it clear that you understand this material to a depth this interloper could never hope to achieve. For instance. say, "'Of course, your example does not ap­ ply. Your space is not Hasselhof. .. If the audience member asks the definition of Hasselhof. act incredulous. Do not admit that it is meaningless he­ yond being the name of the actor David Hasselhof of Baywatch fame. Simply say, " Ho\v can I be expected to have a fruitful conversation with someone who is not familiar with Hasselhoficity . " Then storm out of the room.

Analysis: What matters is not the cor­ rectness or incorrectness of your theo­ rem. What matters is the permanent im­ pression formed in the minds of the audience. Better to shoulder through, and present the appearance of being fully in control of the situation. Later, appropriate "additional hypotheses" can be added to make the theorem true, vacuously if necessary.

How to Survive a Lecture to Your Daughter's Second Grade Class There are several simple rules for speaking on mathematics to an unusu­ ally young audience. Rule 1 : Bring candy. Offer to hand it out only when they are quiet . Rule 2 : Wear colorful clothing. Consider a red rubber ball on the end of your nose. Rule 3: Talk down to their level. Know your audience. If you \Yish to speak about Haar Measure, speak about Lebesgue measure instead. If you want to explain exotic n-spheres, stick to di­ mension 7. Rule 4 : Know \vhen to stop. Never run into recess. or you might have a riot on your hands.

How to Survive Running Over a College Administrator 1 . Do not panic. You have tenure.

2. Stop the car. get out, and call fran­ tically for help. Act vet-y concerned.

3. Ride with the administrator in the ambulance. Take this opportunity to ex­ plain to him or her the transforming ef­ fect you have had on the field of sub­ liminal jet bundles. 4 . Once at the hospital, offer to donate a kidney. It is u nlikely a kidney will be necessary, but the gesture will make you look good. If the offer is accepted, excuse yourself to use the bathroom, sneak out the window, and change pro­ fessions. It's not worth a kidney.

The H exagonal Parquet T i l i ng 1<-lsohedral M onoti l es with Arbitrari ly Large I< JOSHUA E. 5 . SOCOLAR

he interplay between local constraints and global struc­ ture of mathematical and physical systems is both sub­ tle and important. The macroscopic physical properties of a system depend heavily on its global symmetries , b u t these are often difficult t o predict given only informa­ tion about local interactions between the components. A rich history of work on tilings of the Euclidean plane and higher dimensional or non-Euclidean spaces bas brought to light numerous examples of finite sets of tiles with rules governing local configurations that lead to surprising global structures. Perhaps the most famous now is the set of two tiles discovered by Penrose that can be used to cover the plane with no overlap but only in a pattern whose sym­ metries are incompatible with any crystallographic space group [ 1 , 2] . The Penrose tiles ''improved" on previous ex­ amples due to Berger [3] and others (reviewed by Grun­ baum and Shephard [4]) showing that larger sets of square tiles with colored edges (or several types of bumps and complementary nicks) could force the construction of a non­ periodic pattern. This naturally led to serious rumination about the possible existence of a single tile, or monotile, that forces non-periodic global structure. Though I will con­ fine my attention here to Euclidean space, the problem of non-periodic tilings in hyperbolic space has also been suc­ cessfully addressed [5] . The non-periodic monotile problem may be thought of as a limiting case of a more general problem. Any tiling can be classified according to its isohedral number k, de-

fined as the size of the largest set of tiles for which no two can be brought into coincidence by a global symmetry (any reflection, rotation, translation, or any combination of these) that leaves the entire tiling invariant. A set of tiles and matching rules for which the smallest isohedral number of an allowed tiling is k is called a k-isohedral set. If the set consists of a single tile, the tile is called a k-isohedral monotile. The challenge is to find a k-isohedral monotile for arbitrarily large k. To gain some intuition about the isohedral number, con­ sider the two tilings shown i n Figure 1 . The tiling o n the left has k = 1; any tile can be mapped to any other by a translation that leaves the entire tiling invariant. The red tile can be mapped into the yellow one by a 1 80° rotation about the midpoint of their common edge; into the blue one by a counterclockwise 90° rotation about the lower left corner of the red tile; and into the gray one by a clock­ wise 90° rotation about the upper left corner of the red tile. Combining these rotations with the square lattice of trans­ lations generated by the vectors shown allows any tile to be mapped into any other. The tiling on the right has k = 2. It has the same symmetries as the one on the left, but there is no symmetry that maps the green tile into the orange one; there are two "inequivalent types" of tiles in this tiling. The answer to the question " Is there a k-isohedral monotile?" for arbitrarily large k depends crucially on how the question is posed. As I will show below, there are many

© 2007 Spnnger Sc1ence+ Bustness Media, Inc , Volume 29, Number 2 , 2007

33

Figure I . A 1 -isohedral tiling and a 2-isohedral tiling.

subtly different versions of this and similar questions, and versions that may at first glance appear equivalent turn out not to be. Forcing nontrivial global structure of a certain precisely defined type can be accomplished in a variety of ways depending on what types of local matching rules are deemed permissible. Shall we require that the monotile be completely defined by its shape alone? O r shall we allow coloring of the edges and specification of which colors are a llowed to coincide? Shall we insist that the monotile be a simply connected shape? Shall we insist that the tiling cover the entire plane, or j ust that it have the highest possible density? Recent exhaustive searches of polyomino monotiles con­ sisting of square, triangular, or hexagonal units have pro­ duced k-isohedral examples with k as large as 1 0 (so far! ) [6] , but there appears to be no systematic way to construct such examples analytically. In these examples, the match­ ing rules are enforced by shape alone and the entire space must be covered. Below I present several variations of a class of monotiles and matching rules that can force tilings with arbitrarily l a rge k. The tilings formed all have the basic structure of the hexagonal parquet shown in Figure 2. Each rhombus in the figure is composed of L = 5 monotiles. Generaliza­ tion to arbitrarily large L is clearly possible. Exercise: Find the isohedral number of the hexagonal parquet in terms of L (Figure 2). I n each of the following four sections, I present a monotile and matching rule that forces a tiling with the symmetry of the hexagonal parquet. The difference between the tilings lies in the way in which the rule is expressed. Defining a tile to be the closed set of points bounded by the tile edges (and faces in higher dimensions), we have the fol lowing four cases. In a l l cases, we a llow tiles to over­ lap only along edges.

a: � --

0 %

...

:::�
.... •

"

- .

.

.

; � A • �

l �.

•

- . ..;

"'"

constraining which colors can coincide, and the tiles cover the entire space. 2. The edges are not colored. The rule is that the tiles must cover the entire space, but the tile is not a simply con­ nected shape. 3. The monotile is simply connected, and the rule is that the tiling must maximize the density of tiles (without necessarily covering the entire space). 4. The (3D) monotile is a simply connected shape, and the only rule is that the tiles must fill the space. In this work we allow only rotations and translations of the monotile, not reflections. All of the results can be easily ex­ tended to the case where reflections are allowed by re­ placing the disks, bumps, and nicks with chiral shapes. Perhaps just as important as the discovery of monotiles that force any desired isohedral number, these examples show that subtle differences in the rules of the game may generate dramatically different results. Note that I have not exhibited a simply connected, uncolored, two­ dimensional monotile that forces the hexagonal parquet structure. In fact, I show below that this is impossible. If you want to restrict the problem to these terms, the record in two dimensions is still Myers's polyomino consisting of a simply connected cluster of 1 6 hexagons. [6]

A 20 Monotile with Color-Matching Rules The tile shown in Figure 3a is endowed with a matching rule requiring that no two red edges may touch. As the as­ pect ratio of the tile is increased in integer steps, the min­ imal isohedral number of a space-filling tiling formed with this tile increases without bound. THEOREM 1 Let T be a parallelogram tile with angles of60°

and 1 20° and side lengths

JOSHUA SOCOLAR received his PhD in Physics from the University

of Pennsylvania in 1987, with a thesis on quasilattices and quasicrystals. He has been on the facutty at Duke since 1 992, where he is attached both to the Center for Nonlinear and Complex Systems and to the Center for Systems Biology. H is hobbies include music and (his son's in­ fluence) bird-watching. Physics Department and Centers for Nonlinear and Complex Systems Duke U niversity Durham, NC 27708 USA e-mail: [email protected]

34

1 . The edges of the monotile are colored, there are rules

THE MATHEMATICAL INTELLIGENCER

1

and L >

1,

where L is an integer.

(a)

:... .'-, ,.. ________,� /

Figure 2. The L = 5 hexagonal parquet tiling of the plane. Each "board" is a copy of the same tile.

Color the short edges of T red (not including the vertices) and the long edges (and vertices) black. The minimal isohedral number qf a tiling in which no points are covered twice with red is l (l + 1)/2 J .

Proof Consider the tile Ti shown in Figure 3a. The match­ ing rule and requirement of space filling immediately im­ ply that 12. must be present. The only way to continue the tiling is then to place T3 and T4 as shown with dashed out­ lines. The process of adding forced tiles stops only when the ends of 7i are reached, as shown in Figure 3b. At this point, 75 is forced and the process repeats until the hexa­ gon of Figure 3c is produced. The existence of this hexagon in the tiling ensures that the isohedral number of the tiling is at least l (l + 1)/2 J . Each tile can be characterized by the distance of its center from the center of the rhombus containing it, with tiles on opposite sides of the center possibly related by rotation of 180° about the center. By inspection it is clear that the hexagons can tile the plane while respecting the matching rules, forming a stan­ dard honeycomb lattice i n which the isohedral number is exactly l (L + 1)/2j . D THEOREM 2 For L > 2, the color-matching rule for the monotile T cannot he enforced by alterations of the tile shape

alone; i.e., by placing bumps and nicks on the tile edges. Proof Let the shape of one red edge be designated R1 and the shape of the other red edge by R2. Further, let R1 and

Figure 3. The L

=

(c)

5 hexagonal parquet monotile : (a) the tile;

(b) forced tiles; (c) the forced hexagon.

R2 be the complementary shapes that fit onto R1 and R2, respectively. The color-matching rule implies that neither R1 nor R2 is congruent to R1 or R2. We will now show that the number of instances of Rx on the tile must be greater than the number of instances of R�-, which immediately im­ plies that T cannot tile the plane. Consider any tile i n the interior of a rhombus, which has both black edges matching black edges of its neighbors. Let the shapes of the two black edges be B1 and B2 . There are two possibilities: ( 1 ) B1 = B 2, which implies B2 = B1 ; or (2) rB1 = B1 and rB2 = B2, where rX indicates rotation of X by rr. I n either case, any instance of R.� fou n d on B1 must be matched by an instance of Rx either on B2 or on rB1 . Thus the number of instances of R� on black edges cannot exceed the number of instances of Rx. This means that the single Rx on the red edge makes the number of R.x's larger. The conclusion is that in order to tile the plane, the red edge matching rule has to be relaxed, but this in turn permits simple periodic tilings with isohedral numbers or one or two. D

Forcing the Hexagonal Parquet with a M ultiply Connected Monotile The color-matching rule for the hexagonal parquet monotile can be enforced by shape alone if one does not insist on T being simply connected. The proof is by construction, as displayed in Figure 4. The seven black regions at the left of the figure form the monotile. By inspection, it is clear that there is no way to have two short edges of the basic

' ' ' ' ... ••• ••

1

• •

\

-

'

• •

Figure 4. An L 5 multiply connected monotile (gray) that forces the hexago­ nal parquet tiling. Colors are guides to the eye to help identify individual tiles. =

© 2007 Springer Science+ Business Media, Inc . , Volume 29, Number 2 , 2007

35

Figure 5. A simply connected monotilc that forces hexagonal parquet layers that can be stacked to fill space. Top, bottom. and tiling \"iews.

Figure 6. A monotile that forces a double-layered hexagonal parquet and a unit cell of space-filling tiling. Top and bottom views.

parallelogram coincide. (The nearby disks or protruding rods get in the way.) Thu s the rules for how the parallel­ ograms can be placed are at least as restrictive as the color­ matching rules discussed above. The figure clearly shows, however, that the hexagonal parquet tiling can still be formed.

Forcing the Hexagonal Parquet with a Simply Connected 3D Monotile The color-matching rule required for the hexagonal parquet tile can also be implemented with a simply connected monotile in three dimensions. The simplest way to do it is to promote the multiply connected 2D monotile on the right in Figure 4 to a 3D parallelepiped with shallow protruding rods and grooves, as shown in Figure 5 . The complete tiling is a stacking of identical hexagonal parquet layers. The low­ est permitted isohedral number for the space-filling 3D tiling is the one in which the layers are in perfect registry. This can be forced, if desired, by placing bumps on the rods at the positions corresponding to the disk centers in the

monotile of the left panel of Figure 4 and corresponding dents in the bottom of the parallelepiped. Note that the pat­ tern of disks in Figure 4 is not a triangular lattice, so the registry is indeed forced. The multiply connected tiling on the left in Figure 4 sug­ gests a different strategy for constructing a 3D monotile. The tiling now consists of stacks of double layers, each double layer being a hexagonal parquet with flat top and bottom surfaces. The enforcement of the matching rule for the top of the double layer is provided by the pieces of tile on the bottom of the double layer. The protrusions and indentations on the top surface of the bottom-layer pieces do not fit properly into those in the top-layer piece when one attempts to match the top pieces end to end. Thus one is forced to form a hexagonal parquet in a manner quite similar to the multiply connected 2D tiling above, with the bottom-layer pieces playing exactly the same role as the isolated disks in the 2D monotile. One realization of this 3D monotile and one unit cell of the double layer are shown in Figure 6 , each being shown

Figure 7. A simply connected monotile that forces a double-layered hexagonal parquet.

36

THE MATHEMATICAL INTELLIGENCER

Figure 8. An L 5 monotile (top) for which the hexagonal parquet is the maximum density tiling. =

from viewpoints above and below the plane of the double layer. The monotile of Figure 6 would not he simply connected if we took the tile to be the open set not containing edges. The construction can be modified, however, so as to make even this open set simply connected. The L = 4 version of the modified tile is shown in Figure 7 . The protruding '' legs" from the top-layer piece will fit into the grooves in the bot­ tom-layer piece, with two legs (one from each of two neigh­ boring tiles) filling each hole formed by neighboring tiles on the bottom layer. The legs protruding from the end of the top-layer piece and fitting into half of the groove on the end of each bottom-layer piece form a connection that makes the whole tile simply connected.

Forcing the Hexagonal Parquet with a Maximum Density R ule The hexagonal parquet can be enforced by a simply con­ nected shape in 2D if one replaces the space-filling constraint with the demand that the tiling have the maximum possible tile density. The shape in Figure 8 can form a hexagonal par­ quet tiling as shown. The color-matching rule is enforced by the bumps on the ends of the tile. The parquet tiling is then the maximum density tiling that can be achieved with this tile. l3ecause the smallest excluded area around a tile occurs when its ends are fitted into notches, every tile in the par­ quet tiling excludes the smallest area possible.

Conclusion I have exhibited several types of monotiles with matching rules that force the construction of a hexagonal parquet. The isohedral number of the resulting tiling can be made a s large as desired by increasing the aspect ratio of the monotile. Aside from illustrating some elegant peculiarities of the hexagonal parquet tiling, the constructions demon­ strate three points. 1 . Monotiles with arbitrarily large isohedral number do ex­ ist; 2. The additional topological possibilities afforded in 3D al­ low construction of a simply connected monotile with a rule enforced by shape only, which is impossible for the hexagona l parquet in 2 D ; 3 . The precise statement of the tiling problem matters­ whether color matching rules are allowed; whether mul-

tiply connected shapes are a llowed; whether space­ filling is required as opposed to just maximum density. So what about the quest for the k = oo monotile? Schmitt, Danzer and Conway have exhibited a 3D monotile that forces a non-periodic tiling [ 7 , 8, 9] . The tiling is a stack­ ing of identical layers and each layer is a periodic packing of the monotile. The non-periodicity arises because the pla­ nar lattice directions in successive layers are rotated by an angle incommensurate with 2 7T. This tiling has an unusual feature: the number of local configurations around a monotile is infinite. That is, no two tiles in a given layer are covered in exactly the same way by the tiles in the lay­ ers above and below it. Consequently, the layers can slide over each other to form an infinite number of tilings that are not related by any global symmetry. Any attempt to en­ force a finite set of local environments for this monotile will require a commensurate rotation angle and render the isohedral number finite, though it could be arbitrarily large. Another example of a k = x monotile is the decagonal tile together with matching rules allowing certain types of overlap first presented by Gummelt [ 1 0] . Steinhardt and Jeong proved that the overlap rules and the requirement that the tile density be maximized force a structure with the same symmetries as the Penrose tiling [ 1 1 ] . At present there is no general theory distinguishing pat­ terns that can be enforced by color-matching rules from those that can be enforced by shape alone or by maximum density constraints. The maximum density criterion is of particular interest in physics-and is particularly vexing be­ cause of the difficulty of linking this global criterion to lo­ cal constraints that can be exhaustively checked. I n some cases , such as the one above, it can be proven that satis­ fying certain local constraints will guarantee maximum den­ sity. The recent proof that the FCC packing of spheres i n three dimensions h a s maximum density is another exam­ ple [ 1 2] . On the other hand, there is some evidence that the maximum-density sphere packing in many dimensions is actually a random packing [ 1 3] , which would have an in­ finite isohedral number and a n infinite number of local con­ figurations around a single sphere (monotile). As the examples described above suggest, there are sur­ prisingly simple links between local rules and global struc­ ture yet to be discovered or placed within a well-defined theoretical framework. Tiling enthusiasts around the world are looking for new ideas and examples, enjoying the recre­ ational nature of the puzzles that crop up, and appreciat­ ing the visual and logical structures that emerge along the way. ACKNOWLEDGMENTS

I thank C. Goodman-Strauss and M. Senechal for their gener­ ous mathematical and editorial advice. REFERENCES

[ 1 ] R. Penrose, "Pentaplexity," Math. lntelligencer 2 (1 979), 32-37. [2] M . Gardner, "Extraordinary nonperiodic tiling that enriches the the­ ory of tiles," Sci. Am. 236 (1 977), 1 1 0-1 2 1 . [3] R. Berger, "The undecidability of the Domino problem , " Mem. A mer. Math. Soc. 66 (1 966), 1 -72.

© 2007 Springer Science+ Business Media, I n c . , Volume 29, Number 2, 2007

37

[4] B. Grunbaum and G. C. Shephard, Ti!ings and Patterns, Freeman,

[9] C. Radin, "Aperiodic tilings in higher dimensions," Proc. American

[5] C. Goodman-Strauss, "A strongly aperiodic set of tiles in the

1 0 . P. Gummelt, "Penrose tilings a s coverings o f congruent decagons, "

Mathematical Soc. 1 23 (1 995), 3543-3548.

New York (1 987).

hyperbolic plane," lnventiones Mathernaticae 1 59 (2005), 1 1 91 32 .

11.

[6] J . S. Myers, " Polyomino tiling , " http://www.srcf.ucam.org/jsm28/

431 -433 (1 996). 1 2 . T. C. Hales, "A proof of the Kepler conjecture, " Ann. of Math. 162

or quasiperiodic tiling , " in Aperiodic '94 , edited by G. Chapuis World Scientific, Singapore (1 995).

P. J . Steinhardt and H . - C . Jeong, "A simpler approach t o Penrose tiling with implications for quasicrystal formation," Nature 382,

tiling/ (2005).

[7] L. Danzer, "A family of 3D-spacefil lers not permitting any periodic

Geometriae Dedicate 62 (1 996), 1 -1 7 .

(2005)' 1 065-1 1 85 . 1 3 . S . Torquato and F. H . Stillinger, "New conjectural lower bounds on

[8] M. Baake and D. Frettloh , "SCD patterns have singular d iffractio n , "

the optimal density of sphere packi ngs," to appear, Experimental

J. Math. Phys. 46 0335 1 0 (2005).

Mathematics 1 5, Issue 3 (2006).

� Springer

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38

THE MATHEMATlCAL INTELLIGENCER

I\'Jfflj.i§ .. fih¥11§.B§4fii,j11i§.id

A C urious C u b ic I d e ntity and Se l f­ Si m i l a r Su ms of Sq u ares ALF VAN DER PooRTEN ,

KURT THOMSEN , AND MARK WIEBE

M ichael Kleber and R avi Vaki l , Ed itors

To Richard K Guy on his 90th birthday "There are just four numbers (af­ ter 1) which are the sums of the cubes of their digits, viz. 153 = 13 + 53 + 33, 3 70 = 33 + 73 + ()3, 3 71 = 33 + 73 + J3, and 407 = 43 + ()' + 73 This is an odd fact, very suit­ able for puzzle columns and likely to amuse amateurs, but there is nothing in it which appeals much to a mathematician. The proof is neither difficult nor interesting­ merely a little tiresome. The theo­ rem is not serious; and it is plain that one reason (though perhaps not the most important) is the ex­ treme specialty of both the enunci­ ation and the proof, which is not capable of any significant general­ ization. " -G. H . Hardy, A Mathematician 's

Apology This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on.

H

=

5882 + 2353 2

=

5882353 = (lOR + 1 )/17,

Please send a l l submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 125, USA e-mail: vakil@math. stanford .edu

1 63 + 503 + 333 = 3 66 + 5003 + 333 3 = 1 5 666 + 50003 + 33333 1 =

1 6 5033 1 66500333 1 666 50003333

This is so surprising that it must be triv­ ial. Indeed, both sides of

a3 + b3 + c3 = l02k a + 1 0 k h + c k = 1 , 2, . . are 3 6 a 3 + 66 a 2 + 42a + 9, whenever 1 0 k = 1 + a + h + c, with h = 3a + 2 , and c = 2a + 1. W e are indebted to Gery Myerson for several helpful remarks, particularly for reminding us that we should immedi­ ately have recognised the number 1 53.

I ntroduction

endrik Lenstra's cute observation that 12 2 + 33 3 = 1 233 is readily generalised. If a 2 + b2 = lQ k a + h then 1 02k + 1 = (2a - 1 0 "')2 + (2h 1 )2 , so to discover the example it suf­ fices to decompose 1 04 + 1 as the sum of two squares. Noting that 1 0 001 137 73, one readily finds the trivial 1 04 + 1 = 1002 + 1 2 , and 1 04 + 1 = 762 + 332 . The latter yields the opening remark, and also 882 + 33 2 = 8833. Quite as readily, see [3], one finds de­ compositions of all lengths 2 k, helped by the fact that if n is a divisor of l0 2k + 1 , then 1 0 k = - 1 (mod n) . A striking special case is the sequence of integers ooR<4u+l) + 1)/ 1 7 , u = 0, 1 , . . . . Thus ·

Contributions are most welcome.

commences a not quite so well-be­ haved sequence for k = 1 2 , 28, . . . . Corresponding decompositions in cubes are far less natural, so one ex­ pects the example 1 3 + 5 3 + 3 3 = 1 53 to be no more than an isolated curios­ ity. Surprisingly, however, also

5 882 352 941 176 470 5882 + 23 529 4 1 1 764 705 882 3532 = 588 235 294 1 17 647 058 823 529 4 1 1 7 64 705 882 353 = (1024 + 1)/17, and so on. By the way, 5882352941 2 2 + 2352941 1 76482 = 5882352941 2235294 1 1 7 648

I report here on work done some years ago with Kurt Thomsen and Mark Wiebe, at the time undergraduate students at the University of Manitoba. I had promised that I would convert that work into possibly publishable form, but the priority I gave my commitment turned out to be too low. Happily, the topic promised a suitable talk for a cel­ ebration of Richard Guy's 90th birthday at the Summer Meeting of the Canadian Mathematical Society, 2006 in Calgary. I narrate that talk below. Aif vdP. The visit by Kurt and Mark to the ceNTRe for Number Theory Research, then at Macquarie University, Sydney, was in part supported by a n Australian Research Council international research exchange grant.

A Very Nice Observation In his fine text Getaltheorie voor he­ ginners, Frits Beukers quotes Hendrik Lenstra, at §10.4 Kunstjes met decimalen, as making the 'heel fraaie observatie'

© 2007 Springer Science+Business Media, Inc., Volume 29, Number 2 , 2007

39

and remarks that such examples may be generated systematically, citing

1 8261 4781 22 + 38635038882 182614781 23863503888. =

as 'een indrukwekkend voorbeekl " . I had received the book direct from the author as ·one of the few foreign­ ers who speak Dutch and may he able to appreciate some elementary number theory' and was indeed appreciative, both of Hendriks ve1y nice observation and the impressive example. Frits points out in the text that of course

a2 + fi2 = 10.(' a + h

( a2 + tl)(cz + dz) = (ac + hd)2 + (ad :±: be?

details composition of the quadratic form X 2 + Y 2 with itself. In very brief. one uses: if - 1 is a square modulo }; say m2 "" - 1 mod .f then the Euclidean algorithm applied to m and f etliciently writes f as a sum of two squares. The key notions are Cor­ nacchia's algorithm [2] ( in this special case, Hermite-Serret) , and reduction of a definite quadratic form. If f divides 1 02k + 1, then m = lOk will do. For ex­ ample

882 353 = 588 . 10 000 + 2 353 10 000 4 . 2353 + 588,

5

=

entails

(102k - 4 10k a + 4 a2) + ( 4 h2 - 4 b + 1) = (lOk - 2a) 2 + (2 b - 1 )2 = 102k + 1 . ·

S o finding examples i s a matter of rep­ resenting 1 02k + 1 as a sum of two other squares, with the even other square providing a and the odd one h. We all know-in any case I knew­ that finding such representations is a matter of factorising 1 02k + 1 , repre­ senting the factors as a sum of two squares . and hoping that the subtlety of consequent representations of 102k + 1 is not spoilt by trailing zeros in a or b, let alone leading zeros in h (a and b must both be k digits i n length). Specifically, for k = 2 it is easy to see that

and we have already obtained the first two remainders less than \15882353. It turned out not to be entirely evi­ dent that seemingly interesting sums of squares yield amusing decompositions. Indeed. the example 1010 + 1 = 101 3541 27961 has an encouraging three prime factors but the 2:l - l - 1 = 3 possibly amusing decompositions are ·

·

2584043776 = 258402 + 437762, 1 765038125 = 1 76502 + 381252, which are pretty interesting, but also the unacceptable 99009901 = 9902 + 099012 In all, it is not immediately completely compelling that there are infinitely many delightful decompositions. However, I did feel a frisson of ex­ citement on noticing that 108 + 1 = 17 5882353 and, without any fuss, ·

1 052 - 210 = 1 1 025 - 1024 = 1 04 + 1 = 73 . 137, readily providing

1 04

+

1 = ( 1 02 - 2 a)2 + (2 b - 1)2 = 762 + 652

82 + 32 and 137 = 42 + by w a y o f 73 1 1 2. Thus we know that a = 1 2 and b = 33 must give 1 2 2 + 332 = 1 233 (all this, without our having to be able to com­ pute 332 = 2500 - 1 700 + 1 72 = 1089). I was provoked [3] to compute other interesting examples, not so much con­ centrating on the issue of factorising 102k + 1 , but rather on that of repre­ senting its factors as sums of two squares and 'composing' such sums; here I recalled that =

5882353 = 5882 + 23532 The numbers (108(2u+ l l + 1)/17 all have analogous a utomatic decompositions. An embarrassment

The preceding remarks were the sort of thing that I showed Kurt Thomsen and Mark Wiebe back in 200 1 . They imme­ diately embarrassed me by remarking that. of course, also 882 + 332 = 8833. I had failed to see actively that, if

a + a' = 10k, a2 + h2 = 10k a + b immediately entails a'2 + fi2 = 10k a' + b. Notwithstanding my protests, Kurt and Mark then proceeded to play with cu-

bic identities , silencing me by making the u seful remark already quoted in my opening summary. We then drifted on to quite different topics, and I only be­ came aware that Kurt and Mark had po­ litely returned to the subtle sums of squares when they reported on their work at the end of their visit. Kurt and Mark avoid representing in­ tegers as sums of squares and the like. They favour a viewpoint closer to their cubic diversion. By the way, further re­ sults for cubes seems to call for merely tiresome activity.

Sequences of Self-Similar Sums of Squares As a first example. consider integers so that

a = 3(3 t + 1)

b = 3(8t + 3)

a' = 8(8t + 3) 10k = a + a' = 73t + 27. The choices k = 2, 10. 18, . . . all are admissible and after 1 22 + 332 = 1 233 yield 1 232 876 7 1 22 + 3 287 671 2332 =

1 2 328 767 1 23 287 671 233 . . . . . What we have here is a not all that curious property of 73 = 32 + 82 In­ deed, 32 137 = (3 4)2 + (3 1 1)2 The curious matter is just how much more curious such an identity seems to be when 3 is replaced by 1 . Compare the examples •

a= t a= t

·

·

h = 4t + 1 h = 4t

and

a' = 4(4 t + 1 ) a' = 4(t - 1 )

10k = 17t + 4 10.(' = 17t - 4,

which a r e admissible, respectively, for k = 4 , 20, . . . , and k = 1 2 , 28, . . . , and provide sequences associated with 17, as mentioned above. To see what's going on here, sup­ pose that

a' = u.] t + u2

h

= bi t + hz

lQk = ( UJ + ul)t + Cu2 + u'z ) = n1 t + n2 ,

and a2 + h2 = 10k a + b; here the co­ efficients all are integers. Viewing that last equation as equat­ ing the coefficients of polynomials in t yields by = u1 u1 and, oBdA,* both u1

'When lecturing, many of us use the abbreviation 'wlog ' , confusing our students by suddenly introducing a logarithm, and not recognising that 'with loss of generality' would be similarly denoted. The German version plainly is an abbreviation of something, and it unambiguously alleges no Beschrankung of the Allgemeinheit.

40

THE MATHEMATICAL INTELLIGENCER

and u] are squares, say u1 = a 1 , u ] = 2 a] . \Yith b1 a1 a] . It follows that a1 ' u2 and, j ust so. that a ] ' u2 . A re-examination o f the equations next shows that b2 must be either a1a2 or a ] a2 ; and a second's thought shows such an ambiguity is necessary. The cases entail a1a2 - a ) a2 = 1 or a ] a2 a1a2 = 1, respectively. One readily confirms-a critical 2 step-that n1 = af + a{ divides 102 k + 1 , for example by noticing that n§. + 2 1 = Ca1 a2 + a{a2 ) + ( al a2 - a ) a2) 2 = n1(a� + a'/). To sum up, =

a = a1( a1 t + a2) d

=

a] ( a ) t + a2)

b

lOk

a1 ( a j t + a!z)

=

=

( ai + ajl) t + C a1 a2 + a ] a!z),

2 and. mind you . a� + a j divides 1 Q 2 k + 1 ; equivalently. a1 a2 - a; a2 = 1 . Those with negative inclinations can readily ring changes on this summary, say by both changing the sign of t and of a, a' , and 10 k, and, say a1 .

By the way. complaints of decima l­ ism are baseless. The number 10 ap­ pearing here may be replaced by any other base. Computers may prefer 1 6 . some of u s will like 64: 9 0 might be considered peculiarly suitable to the occasion at which this talk was pre­ sented. The upshot of all this is that one can construct all examples of sequences of self-similar sums of squares at will. ( 1) Select a1 and ai so that a1 + a]2 divides one and therefore lots of 1 Q2 k + 1 . (2) Compute a2 and a2 so that a1az - aia2 = 1 . (3) Write the four equations and compute pairs ( k. t). Conversely. given a self-similar sum of squares. one easily identifies sequences to which it belongs. In all, the boring completeness of this solution removes all subtlety. To make up for that, I 've left several cute asides for readers to discover for themselves.

REFERENCES

[ 1 ] Frits Beukers, Getaltheorie voor Beginners. Epsilon Uitgaven, Utrecht, 1 999. [2] Abderrahmane Nitaj, ' L'algorithme de Cor­ nacchia' , Exposition. Math. 13 (1 995), 358365. [3] Alfred J . van der Poorten, 'The Hermite-Ser­ ret algorithm and 1 22

+

332 ' , Proc. Work­

shop on Cryptography and Computational

Number Theory (CCNT'99), (National Uni­ versity of Singapore, 22-26 November, 1 999), K.-Y. Lam, I. E. Shparlinski, H. Wang and C. Xing eds . , Birkhauser 2001 , 1 29-1 36 . ceNTRe for Number Theory Research, Sydney 1 Bimbil Place Killara NSW 2071 Australia e-mail: [email protected] (Aif van der Poorten) e-mail: [email protected] (Kurt Thomsen) Frantic Films, Research and Development, http :1/software. franticfilms. com/

e-mail address: [email protected] (Mark Wiebe)

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THE MATHEMATICAL INTELLIGENCER P lease send your old address ( o r label) and new address to:

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© 2007 Springer Science+Bus1ness Media, Inc., Volume 29, Number 2, 2007

41

C yc l o i ds i n Lo u i s I . l
ou is I. Kahn ( 1901-1974) , one of the most outstanding modern American architects, is notecl for the use of simple geometric forms and

L

spatial compositions in massive con-

crete huildings . Kahn's work shows his great reverence for making accessihle the value of clear structural systems, ma­ terials and light. Pushing the structural syste ms and materials to t h e i r bound-

J I N-HO PARK, YONGSUN Joo, AND JAE-GUEN YANG

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

42

Figure I . Exterior views on the Kimbell Art Museum.

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media, Inc

aries is one of the essential components of his architectural approach . Kahn also sought to optimize natural light for the occupants of the building. Born on the island of Saaremaa, Es­ tonia, in 1 90 1 , Kahn emigrated to the United States and practiced there until his death in 1 974. Among his best­ known buildings is the Kimbell Art Mu­ seum in Fort Worth, Texas 0 966-1 972 ). one of the most-admired architectural pieces of the time. Other masterpieces are the Salk Institute in California (USA ) . the Indian Institute o f Management in Ahmedabad (India), and the National Assembly at Dacca ( Bangladesh) . The Kimbel l museum sits o n the gentle slope of the Amon Carter Square Park, about two miles from downtown Fort Worth ( Figure 1 ) . The structure . covering a gross area of 1 20 ,000 square feet, lies in a trapezoid-shaped area of 9 . 5 acres, bounded by Camp Bowie Blvd . , Arch Adams St. , and West Lan­ caster Ave. There are a number of cul­ tural institutions in the vicinity: the Amon Carter Museum of Western Art, the Fort Worth Museum of Science and History, and, directly opposite the Kim­ bell, along Arch Adams St. , the new Modern Art Museum of Fort Worth ( Fig­ ures 2 and 3 ) . The Kimbell Art Museum i s home to one of the world's finest art collections, with acquisitions ranging from antiquity to the twentieth century [Loud, 1987). Among the interesting architectural fea­ tures of the building are the unique vault roofs, which exhibit a rare exam­ ple of a strategic use of the mathemat­ ical properties of the cycloid [ Gast, 1 998; Benedikt, 1 99 1 ] . An art museum with a mathematical property: could a mathematical tourist ask for more?

plan. with cycloid vaults, and that was the design that was executed. The imposing structure opened four years later, in October 4, 1972. The mu­ seutn incorporates the slope of the site and has entrances at both the lower and upper levels. Public access for pedes­ trians is offered on the west side, at the upper level, whereas the parking lots are located on the east side. at the lower level ( accessed from Arch Adams Street). There is a central axis passing through the front and the rear en­ trances, the lobby and the stairs; two water pools outside the porticos are symmetric too, along that axis. This ax­ ial approach is typical of Kahn's de­ signs. On June 2 5 , 1 969, he wrote, in a letter to Mrs. Kimbell (Johnson 1 975] : Two open porticos flank the en­ trance court of terrace. In front of each portico is a reflecting pool which drops its water in a continu­ ous sheet about 70 feet long in a basin nvo feet below. The sound would be gentle. The stepped en-

trance court passes between the por­ ticos and their pools with a fountain around which one sits, on an axis designed to be the source of the por­ tico pools. The west lawn gives the building perspective . Kahn continues: The south garden is at a level 10 feet below the garden entrance ap­ proached by gradual stepped lawns sloped to be a place to sit to watch the performance of a play, music or dance, the building with its arched silhouette acting as the backdrop of a stage. When not so i n use it will seem only as a garden where sculp­ ture acquired from time to time would be. The north garden , though mostly utilitarian, is designed with ample trees to shield and balance the south and north sides of the building. The series of 16 structural units with their vaulted roof are supported inde­ pendently and clustered with preserva­ tion of their independent structural

W 7th St

W 7th St

T r i n ity Park

A Man, a Plan: Kahn Kahn started the design for the Kimbell in 1 967, arriving at his final scheme around September 1 968. August Komendant, Kahn's structural engineer, describes this period as "very difficult and agonizing. " The design process seemed to have been an ordeal for Kahn: the first scheme was a square plan, with a circulatory street, an arcade around the complex, and polygonal ceilings. The second plan was an H­ shape, with two connected structures covered by a walkway and curved roofs . The final layout was a C-type

Ci5 c::­ <1>

E

0

.9

c 0 �

Figure 2. Partial map of Fort Worth, Texas, showing the location of the Kimbell Art

Museum.

© 2007 Springer Science+ Business Media. Inc . . Volume 29, Number 2 , 2007

43

The Cycloid Choice

N

E9

Figure 3. Site plan of the Kimbell Art Museum.

character. At the same time, each vault is an inseparable element of the whole, making the order in the composition clear: the Kahn approach. These are structural elements grouped on a tripartite scheme, with northern, central, and southern parts. Whereas each of the north and the south parts has six vaults, the central part between them has fou r vaults . The permanent collection, an auditorium, a cafe and the galleries occupy the north­ ern part. The central part, with only four vaults, provides an entrance court in which even the trees are arranged in an orderly fashion. I t includes the main en­ trance hall, the lobby, a bookstore, the museum shop, and the library. The southern part holds galleries for tem­ porary exhibitions, the reception hall, and a kitchen. All are arranged around a large open block dedicated to staff fa­ cilities including offices, shops, storage, conservation and photography studios,

44

THE MATHEMATICAL INTELLIGENCER

and workplaces for shipping and re­ ceiving (Figure 4). The three courts are all open to the sky and allow natural light into the museum. The largest square court on the northern side bor­ ders the cafe, while two smaller square courts are located on the south (Figure 5 ) : one is two stories high so that the light penetrates to the floor below, to the conservator's studio . Kahn provides an account of his intent for having sep­ arate gardens and naming each court separately [Latour 1 99 1 ] : Added t o the sky light from the slit over the exhibit rooms, I cut across the vaults, at a right angle, a coun­ terpoint of courts, open to the sky, of calculated dimensions and char­ acter, marking them Green Court, Yellow Court, Blue Court, named for the kind of light that I anticipate their proportions, their foliation, or their sky reflections on surfaces, or on water will give.

The most distinctive design element of the museum is the series of s ixteen parallel cycloid vau lts. Kahn did not arrive at the cycloid roof at the begin­ ning of the design process; it was determined after a series of trials in collaboration with Komendant and Marshall D . Meyers , project manager for the museum: Kahn's presentation injune had pro­ posed a semicircular ceiling, which Brown had found ostentatious and thus, dictated by its geometry, too high . Kahn's response in September had been a flattened arch. \vhich met Brown's requirements but was not particularly graceful. Meyers had explored the possibilities of a flat roof with quarter-round edges, el­ lipses, and segmented circles, none of which proved up to the task." [Leslie, 2005] Rather than using the equations of a cy­ cloid , the design was made using a sim­ ple drawing of the curve generated by a point on a rolling circle, as shown in Fig­ ure 6 . The rhythmic sequential cycloid structure is easily recognized from a dis­ tance. Each vault consists of 100 X 23 foot shells poured in place by post-ten­ sioning cables and reinforced concrete. and sheathed by a calcium-lead layer. The roofs are about six feet apart, sepa­ rated by flat-roofed channels for air con­ ditioning and electrical distribution ducts . The height of a vault from the upper level is 20 feet and the height from the lower level is 40 feet. Each concrete cycloid vault is supported by four 2 X 2 foot corner concrete piers, thus providing column-free spaces (Figures 7 and 8).

Seeing Cycloid Light

Each vault has a longitudinal 2 1 /z-foot narrow slit at the roof, over the entire length of the structure, covered by aoylic Plexiglas. Perforated metal re­ flectors are installed u nder the light slit of the cycloid vault. Kahn's series of sketches show an evolution, in which he experimented with various relation­ ships between roof shapes and light re­ flectors (see Figure 9). The cycloid vaults and the lighting reflectors were carefully designed to control the qual­ ity and intensity of daylight in the mu­ seum. Kahn writes Uohnson, 1975]: The scheme of enclosure of the mu­ seum is a succession of cycloid

10

�

6

ll

12

2

····

m

i 1lc81i li����1111 :tt

m

m

m. 8

9

I

the light, sufficiently so that the in­ j urious effects of the light are con­ trolled to whatever degree of con­ trol is now possible. Uohnson 1 975]. m

•

I,

----········-----····················· . ··-

-

-

··:::::.::::::::::::::::::::::::::

J

Figure 4. Gallery level (top) and lower level floor plan (bottom) of the museum (re­

drawn by the authors). [Key: 1 . portico, 2 . entrance hall, 3 . gallery, 4. kitchen, 5. con­ servator's court, 6 . north court, 7. fountain court. 8. gallery, 9. two-storied conserva­ tor's studio . 1 0 . snack bar, 1 1 . bookshop. 1 2 . shop. 1 3 . offices and laboratories, 1 4 . east entrance. 1 5 . mechanical room. 1 6 . shipping and receiving]

vaults each of a single span [ 1 00] feet long and [23] feet wide, each form­ ing the rooms with a narrow slit to the sky, with a mirrored shape to spread natural light on the side of the vault. This light will give a glow of silver to the room without touch­ ing the objects directly, yet give the comforting feeling of knowing the time of day. The natural light was a n integral part of the program for the museum design from the beginning [Loud 1 987]. The Kimbell's first director, Richard Brown. was especially concerned that " natural light should play a vital part in illumi­ nation. " It may be conventional know!-

edge that natural light distracts the visual experience of viewers in a mu­ seum. but Kahn succeeded in using natural light as the main source for lighting the museum galleries. Natural light penetrates through the slit in the vault and reflects off the mirrored shape of the convex perforated metal reflec­ tors back to the side of the interior vault and then diffuses to the exhibition room, as shown in Figure 1 0 . Kahn de­ scribes 'the luminosity of solver' of the perforated metal reflectors with cycloid vaults as follows: . . rather a new way of calling something; it is rather a new word entirely. It is actually a modifier of

Unusual Computational Complications The key to the museum design is the structural novelty of the cycloid roof, with a span of 1 00 ft. supported by four corner columns. The monolithic struc­ ture is only four inches thick, so thin it looked ·'fragile,'' even to Kahn himself. The roof, with its longitudinal 2 1 I2 foot narrow slit at the apex of each vault, is braced by cross concrete struts every 1 0 feet, transferring the load along the vault, longitudinally and vertically . to the lower corner of the vault and to the four corner columns. As shown in Fig­ ure 1 1 , the forces of the longitudinal load are absorbed i n two ways by three post-tensioning cables. a technique com­ monly employed to improve the load­ bearing properties of concrete by in­ corporating high-tensile steel stretched cables into the structure. The di­ aphragms at the end absorb the forces of the vertical load. Traditionally, a barrel roof can he supported along the longitudinal edges or on the curved ends. When a barrel is supported along the longitudinal edges, it behaves like a row of arches, one next to the other, developing hori­ zontal forces pushing outward and ab­ sorbed by buttress walls. When a barrel vault is supported on its curved ends, it behaves like a beam conveying the load to the walls at the end and to the ground ( Salvadori, 1 980). However, in the Kim­ bell Art Museum design, the structure of the roof is pushed beyond the con­ ventional knowledge of typical vault structures. Kahn's design employs an unconventional mixture of techniques using those complicated post-tension­ ing cables and diaphragms. It seems Komendant's contribution to the cy­ cloid structure for the museum was es­ sential (Komendant, 1 975; Leslie, 2005). Mathematicians have long admired the distinctive properties of the cycloid, and more generally of the cycloid fam­ ily. including epicycloids, hypocycloids, epitrochoids, hypotrochoids. or car­ dioids. The length of one arch of a cy­ cloid is 7T times the diameter of the rolling circle, while the area under the arch is three times the area of the circle.

© 2007 Spnnger SCJence + Buslness Med1a, Inc., Volume 29, Number 2 , 2007

45

Figure 5. Fountain court with garden water fountain (see Figure

4).

Figure 6. The generation of the cycloid (after Louis I . Kahn). 46

THE MATHEMATICAL INTELLIGENCER

Figure 7. Section and elevation of the museum (left) and a joint where roof and corner column meet (right).

Figure 8. Detailed photo showing the roof structure of the museum (left), and

porticos with a side access path and reflecting pool (right ) .

v Figure 9. A series of Kahn's preliminary sketches for the roof. made in 1 967, and how the light should infiltrate (Redrawn

by the author).

Figure I 0. A gallery, showing the transparent ceiling reflector (left), and a cafe and

gallery areas, showing the north garden court with Maillot's "L'Air" (right).

© 2007 Springer Science ...... Business Media, Inc., Volume 29, Number 2 , 2007

47

3 Post-tensioning cables

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ACKNOWLEDGMENT

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coid roof design and the natural light fixture of the museum in only one other building, the Wolfson Center for Me­ chanical and Transportation Engineer­ ing 0968-1977) on the university cam­ pus of Tel Aviv .

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·

.

.

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REFERENCES

The Kimbell Art Museum website: http://www. kimbellart.org/ Benedikt, Michael. 1 99 1 . Oeconstructing The

:

�. .

.

.

·

We thank Kazi Ashiraf for allowing the use of his Kimbell Museum photographs in this article. This work was supported by an Inha University research grant .

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Kimbell. New York: Sites Books. Brownlee, David B . , and De Long, David G. 1 99 1 . Louis I. Kahn: In the Realm of Archi­ tecture. New York: Rizzoli International Pub­ lications. Buttiker, Urs. 1 994. Louis I. Kahn: Light and Space. New York: Whitney Library of Design. Gast, Klaus-Peter. 1 998. Louis I. Kahn: The Ideal Order. Basel, Switzerland: Birkhauser. Johnson, Nell E. 1 975. Light is the Theme:

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Louis I. Kahn and The Kimbell Art Museum.

·.·• · ·

Cables .

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...

Fort Worth: Kimbell Art Museum. Komendant, August E. 1 975. 18 Years with Ar­ chitect Louis I. Kahn. Englewood, NJ: Aloray Publisher.

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Latour, Alessandra. Intra and ed. , 1 991 . Louis

Figure I I . Structural details of the cycloid shell (after August Komendant), (a) Par­

Leslie, Thomas. 2005. Louis I. Kahn: Building

c

A - ·A'

B - B'

..'

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-

·

tial side elevation of the cycloid roof, with post-tensioning cables; (b) Partial plan,

where post-tensioning cables are installed; (c) End of a roof with a glass separa­ tion and walls (A-A ' ) , and a cross-section (B-B ' ) .

I. Kahn: Writings, Lectures, Interviews. New York: Rizzoli International Publications. Art, Building Science. New York: George Braziller, Inc. Loud, Patricia Cummings. 1 987. In Pursuit of Quality; The Kimbell Art Museum: An Illus­ trated History of the Art and Architecture.

The cycloid shape of the Kimbell Art Museum is perhaps the most singular example ever applied in building design . The visitor feels the full effect of a quintessentially dignified cycloid

space form embodying a mathematical curve into a physical building . Kahn used different types of arcs such as cir­ cle, semi-circle, and circle sectors in other buildings, but he repeated his cyl-

Kimbell Art Museum, Fort Worth, TX.

Ronner, Heinz, and J haverti, Sharad. 1 987 Basel, Switzerland: Birkhauser.

Salvadori, Mario. 1 980. Why Buildings Stand Up. New York: W.W. Norton & Company.

Jin-Ho Park

Yongsun Joo

lnha U niversity

Kyodo Architects & Associates

lnha University

Department of Architecture

9-9 N I K Bldg. , Sanei-cho, Shinjuku-ku

Department of Architecture

and Architectural Engineering 253 Yonghyun-dong, Nam-gu Deoul 1 36-701 Korea e-mail: [email protected]

48

THE MATHEMATICAL INTELLIGENCER

Tokyo 1 60-0008 Japan e-mail: [email protected]

_

Louis I. Kahn: Complete Work 1 935- 1974 .

Jae-Guen Yang

and Architectural Engineering 253 Yonghyun-dong, Nam-gu lncheon 402-751 Korea e-mail: [email protected]

Mathematical Table-tu rni ng Revisited BILL BARITOMPA, RAINER LOWEN , BURKARD POLSTER, AND MARTY Ross

ou sit down at a table and notice that it is wobbling, because it is standing on a surface that is not quite even. What to do? Curse, yes, of course. Apart from that, it seems that the only quick fix to this problem is to wedge something under one of the feet of the table to sta­ bilise it. However, there is another simple approach to solv­ ing this annoying problem. Just turn the table on the spot! More often than not, you will find a position in which all four legs of the table are touching the ground. This may seem somewhat counterintuitive. So, why and under what conditions does this trick work?

Balancing Mathematical Tables­ A Matter of Existence In the mathematical analysis of the problem, we will first as­ sume that the ground is the graph of a function g : �2 � � , and that a mathematical table consists o f the four vertices of a rectangle of diameter 2 whose center is on the z-axis. What we are then interested in is determining for which choices of the function g can a mathematical table be balanced locally that is, when can a table be moved so that its center remains on the z-axis, and all its vertices end up on the ground. We first observe that it is not always possible to balance a mathematical table locally. Consider, for example, the re­ flectively symmetric function of the angle e about the z­ axis with

,,OJ� {;

THEOREM 1 (Balanc ing Mathematical Tables) A mathematical table can always be balanced locally, as long

as the ground.fu nction g is continuous.

This result is a seemingly undocumented corollary of a theorem by Livesay [ 1 5] , which can be phrased as follows:

For any continuous function .f dEfined on the unit sphere, we can position a given mathematical table with all its vertices on the sphere such that .f takes on the same value at all .four vertices. Note that since our mathematical table has diagonals of length 2, its four vertices will be on the unit sphere iff the centers of the table and the sphere coincide. Choose as the continuous function the vertical distance from the ground, .f: § 2 � � : (x,y,z)

f--c'>

z - g(x,y).

lf 0 ::=; e < - or w ::=; e < . 2 2 ' otherwise. .

w

3w

So, the ground consists of four quadrants, two at height 1 and two at height 2; see Figure 1 . It is not hard to see that a square mathematical table cannot be balanced locally on such a clifflike piece of ground. On the other hand, w e can prove the following theorem:

Figure I . On this disc ontinuous ground a square mathemat­ ical table cannot be balanced locally.

© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 2, 2007

49

Figure 2. The initial position of the table, on the left, and its end position, on the right. In both positions, two opposite legs are on the ground and the other two are at equal vertical distance from the ground (above the ground in the initial position and below the ground in the end position). Also, in both positions the center of the associated mathematical table is on the z-axis. Also check out the Quicktime movie at www.maths. monash.edu.au/-bpolster/table.mov to see what happens when we rotate around the z-axis.

So, one of our highly idealized tables can be balanced lo­ cally on any continuous ground. However, being an exis­ tence result, Theorem 1 is less applicable to our real-life balancing act than it appears at first glance. Here are two problems that seem worth pondering:

1 . Mathematical us Real Tables. A real table consists of four legs and a table top; our theorem only tells us that we can balance the four endpoints of the legs of this real table. However, balancing the whole real table in this position may be physically impossible, as the table top or other parts of the legs may run into the ground. To deal with this complication, we define a real table to consist of a solid rectangle with diameters of length 2 as top, and four line segments of equal length as legs. These legs are attached to the top at right angles, as shown in Figure 2 . The end points of the legs of a real table form its associated mathematical table. We say that a real table is balanced locally if its associated mathematical table is balanced locally, and if no point of the real table is below the ground. 2. Balancing by Turning. A second problem with our analy­ sis so far is that Theorem 1 , while guaranteeing a balanc-

BILL BARITOMPA Bill Baritompa is a PhD in in­

RAINER LOWEN Rainer Lowen studied mathe­

finite-dimensional topology from Louisiana State

matics at Tubingen, Gottingen, and Warwick; his

Note that here and in everything that follows the vertical dis­ tance of a point in space from the ground is really a signed vertical distance; depending on whether the point is above, on, or below the ground, its vertical distance is positive, zero, or negative, respectively. Now, we are guaranteed a position of our rectangle with center at the origin such that all its ver­ tices are the same vertical distance from the ground. This means that we can balance our mathematical table locally by translating it this distance in the vertical direction. Easy!

Balancing Real Tables . . . By Turning the Tables

1/) a: 0 % 1:I "'

�� .. ..

. A�

I

University. He is now retired from academic

PhD is from Tubingen. He works on geometry,

dnudgery, but still lectures part-time at the Univer­

topology, and the interplay between the two. He

sity of Canterbury. He is a mathematical clown,

has been at TU Braunschweig since 1 987. Watch

banjo player, and dance caller, and is a past direc­

for his (co-authored) book The Classical Fields. His

tor of the Dance

funded by the

non-mathematical activities include singing and

Royal Society (see 'M'\J\!V.danceofinathematics.com).

photography; there have been several exhibitions

of Mathematics,

lnstitut fUr Analysis und Algebra

Christchurch

Technische Universitat

New Zealand e-mail : B. Baritompa@ math . canterbury. ac. nz

50

of his photographic work

Department of Mathematics and Statistics University of Canterb ury

THE MATHEMATICAL INTELLIGENCER

381 06 Braunschweig Germany

e-mail: r. [email protected]

Balancing a Square Table by Turning­ The Intermediate-Value Theorem In Action on�i<J<:r a wobhlmg square tabk-. \X'e \\ Obhlc the t<.1hlc until two oppw.itc vertic ·s of the associated mathematical table a rc on the ground. and the other two vcrtkc� a rc the same vert kal tlhtancc above the

ground; sec the lett diagram in figure .2 . Let's call tlw. position of the table its

i11ilit1l position.

By pushing

dm\ n on the table. we can make t he h m c n ng vcr­ ticc-. touch th

•

ground and, in doing so, w e have

�hovcd the "touching" \ Crt i<:cs that -.am • vcrti<;al dis­ tanl·e into the grou nd. We call t h i-. nC\\ po-.ition of the table ih

end posiliml;

-.cc t he right diagram i n Fig­

ure .2. Starting i n the i n itial po-.ition. we now rotate

the table around the z-axis; in doing so, we ensure that at a l l times t he center of t he mathematical table

i'> on t he z-axis, t hm the same pair of vertices as in the initial position arc touching the ground, a nti that the other two \ ertices arc a n equal \ crtical distance from the ground. E\·entua lly. \\ e will arrive at the end position. So, w

started out with two vertices hover­

ing above t.he ground, and we finished with the same vertices shm·ed below the ground. furt hennore. the vertical diswncc of the hm ering vertices depends n>n­ tinuou-.ly on t h · rotation angle. Hen<:e. h} the Inter­ mediate- aluc 'Il1corem. somewhere during the rot:l­ tion these vertices

�trc also tou hing the ground: that

is. t he table has been balan

·d lcx·al ly.

ing posJtJon, provides no practical method for finding it. After all, although we restrict the center of the table to the z-axis, there are still four degrees of freedom to play with when we are actually trying to find a balancing position. The accompanying rough argument (see box) indicates

how, by turning a table on the spot in a certain way, we should be able to locate a balancing position, as long as we are dealing with a square table and a ground that is not "too crazy. " Unlike some other material applications o f the I nterme­ diate-Value Theorem, it seems that this neat argument is not as well-known as it deserves to be. We have not been able to pinpoint its origin, but from personal experience we know that the argument has been around for at least thirty-five years and that people keep rediscovering it. In terms of proper references in which variations of the ar­ gument explicitly appear, we are only aware of [7], [8] , [9] (Chapter 6 , Problem 6 ) , [ 1 3 ] , [ 1 1 ] , [20], [22], [23], and [ 1 6] ; the earliest reference i n this list, [7] , i s Martin Gardner's Mathematical Games column in the May 1 973 issue of Sci­ entific A merican. 1 Note that an essential ingredient of the argument is the simple fact that a quarter-turn around its centre takes a square into itself-to move the table from the initial position to the end position takes roughly a quar­ ter-turn around the z-axis. Closely related well-documented quarter-turn arguments date back a lmost a century; see, for example, Emch's proof that any oval contains the vertices of a square in [3] or [ 1 7] . Section 4. At any rate, we defi­ nitely do not claim to have invented this argument. At first glance, the argument appears reasonable and, if true, would provide a foolproof method for balancing a square table locally by turning. However, for arbitrary con­ tinuous ground functions, it appears just about impossible to turn this intuitive argument into a rigorous proof. In par­ ticular, it seems very difficult to model the rotating action in such a way that the vertical coordinate of the hovering vertices depends continuously upon the rotation a ngle, and that we can always be sure to finish in the end position. As a second problem, it is easy to construct continuous grounds on which real tables cannot be balanced locally. For example, consider a real square table with short legs, together with a wedge-shaped ground made up of two steep half-planes meeting in a ridge along the x-axis. Then it is

1 Martin Gardner credits Miodrag Novakovic of Belgrade and Ken Austin of Chesham in England for drawing his attention to the table-turning problem. We were able to get in touch with Ken Austin. According to him the table-turning problem and rts intuitive solution are due to Miodrag Novakovic, who discovered it in the late 1 950s.

BURKARD POLSTER Burkard Polster received

MARTY ROSS Marty Ross got his PhD in mini­

his PhD from the University of Erlangen-Numberg

mal surfaces at Stanford. He is now an ftinerant

in 1 993. Readers of The lntelligencer

are

already

mathematician. Together with Burkard Polster, he

tuned in to his brand of "fun" mathematics. by his

is currently working on a book on mathematics in

articles in these pages and his book The Mathe­

the movies.

matics ofjuggling, reviewed in our vol.

28,

no.

2.

He says that when not doing fun mathematics he

studies perfect mathematical universes-a

promise of a future lntelligencer article?

P . O . Box 83

Fairfield, Victoria 3078

Australia

e-mail: martinirossi@gmai l . com School of Mathematical Sciences Monash University, Victoria 3800

Australia

e-mai l : Burkard. Polster@sci. monash. edu .au

© 2007 Springer Science+ Business Media. Inc.. Volume 29. Number 2, 2007

51

clear that the solid table top hittin g the ground \Vill pre­ vent the table from being balanced locally on this ridge. By restricting ourselves to grounds that are not too \Yilcl, we can prove that balancing local�v by turning really works. THEOREM 2 (Balancing Real Tables) Suppose the ground is described by a Lipschitz continuous functi01z2 with Lipschitz constant les�� than or equal to � - 7ben a real table V2 with ratio

length sh011 side . length long side can he balanced locally on this ground by turning !l its legs have length greater than or equal to V l 1+ r-, . r=

Since

0<

r :S 1 , the maximum of

. 1 , V J + r-

in this range is 1 ,

while all our tables have diagonals of length 2. Thus we conclude that any real table whose legs are at least half as long as its diagonal can be balanced locally by turning on any "good" ground. If we are dealing with a square table, then this table can definitely be balanced locally by turn­ ing if its legs are at least half as long as its sides. Because of the half-turn symmetry of rectangles, we can be sure to reach a balancing position of a rectangular table whilst turning it 180 degrees on the spot. As we indicated earlier, to balance a square table, we never have to turn it much more than 90 degrees. For an outline of the follovdng proof for the special case of square tables, aimed at a very general a udience, see [22]. Furthermore, it has just come to our attention that Andre Martin has also recently published a proof of this result in the special case of square tables and Lipschitz continuous ground functions with Lipschitz constant less than 2 - \13. In terms of angles, Martin's Lipschitz constant corresponds to 1 5 degrees and ours, which is optimal for local turning . to approximately 3 5 . 26 degrees. PROOF. We start by considering a mathematical table ABCD with diameters of length 2 and center 0. Our approach is to bound the wobblyness of our ground by a suitable Lipschitz condition such that putting the two opposite vertices A and C of this mathematical table on the ground, and wobbling it about A C until B and D are at equal vertical distance from the ground, are unique operations . This ensures that everything in sight moves continuously, as we tum the mathematical table on the spot. Following this, it is easy to conclude that we can balance this table locally by turning it.

O u r intuition tells us that to successfully place the four corners, we need fou r degrees of freedom, fou r separate motions of the table. Putting our intuition into effect, we approach our balancing act as a succession of four I n ter­ mediate-Value Theorem ( IVT) arguments, taking one "di­ mension" at a time. 2Recall that for the ground function g round is at most k :

jg(P) - g(Q) i

g to

be Lipschitz continuous means that there

:s: k,P-

Q

for any P,

Q

3We emphasize that the slope of a line in space is always nonnegative.

THE MATHEMATICAL INTELLIGENCER

Start out with the table hovering horizontally above the ground so that OA lies above the positive x-axis, and lower the table until A touches the ground. Second vertex: C

We now show that since the ground function g is contin­ uous, A can he slid along the ground. tmvards the z-axis ( with 0 sliding up or down the z-axis ) . so that C also touches the ground. This is intuitively clear. For example, if C starts out below the ground. we slide 0 up the z-axis. Since C will definitely be above the ground when A has reached the z-axis , it will end up on the ground somewhere along the way. More formally, consider the function

D( t)

= 1(t,O, g( t,O)) - ( - t,O,g ( - t,0))24t2 =

+

( g ( t,O) - g ( - t,0))2.

Since our table has diagonals of length 2 , we want a value of t :S 1.' = 1 such that D( t) = 2 2 = 4 . Because D is continu­ ous,- D(O) 0, and D( l ) 2: 4, this follows trivially from IVT. =

Uniqueness of C

Assuming that g is Lipschitz with lip(g) :S 1 , we show that the above positioning of C on the ground is u nique. This follo-ws from the fact that the function D is strictly mono­ tonic. This in turn can be seen by differentiating D, noting that ! g ( t,O) - g(- t,O)I :S 2 t with equality only if the function g(t,O) is linear with slope ::±:: 1 in the interval under consid­ eration. Note that Lipschitzness is sufficient to apply this dif­ ferentiation argument: see [2 1 , chap. 5 , §4]. Alternatively, the strict monotonicity of D follows easily from a direct algebraic argument: we note that gCO, 0) = 0, and twice apply the dif­ ference of two squares.

f) E

Equal-hovering position

We now rotate the table through an angle [ - "IT., "IT.] about 2 2 the diagonal AC. We choose the direction so that rotating the table through the angle brings the table into a ver2 tical position, with B lying above A C. We want to prove the existence of a f) for which the points B and D are at an equal vertical distance from the ground: we call such a position an equal-hovering position. To show that there is such a special position, we first choose The table 2 is now vertical, with B above AC and D below AC. Since the segments AB and BC are orthogonal, one of the slopes3 of these segments will be greater than or equal to 1 . Hence, since Lip(g) :S 1 and since both A and C are on the ground, we conclude that B is above or on the ground; similarly, we conclude that D is below or on the ground . If we now rotate the table about AC until fJ = then B is below or on the ground and D is above or on the ground. No\v, a straightforward application of IVT guarantees a value of f) for which B and D are an equal vertical distance from the ground.

-"IT.

f) = -"IT..

f,

exi sts

E IR2 . The Lipschitz constant of

Lipschitz continuous function is automatically continuous.

52

First vertex: A

a k such that the slope of the line segment connecting any two points on the

g

is then defined to be the optimal (smallest) choice of k. Also, recall that any

B'

Uniqueness of the equal-hovering position

�'e now fix k s 1 and take the ground to have Lipschitz constant at most k. We show there exists a choice of k which guarantees the uniqueness of the hovering position. Take A and C to he touching the ground as above . with AC then inclined at an angle ¢. In the following. we some­ times need to express the various objects as functions of 8, the rotation angle about AC ( when assuming the incli­ nation angle ¢ to be fixed , which is the case when we are referring to a particular ground ) ; then, for example . AB would he expressed as AB( 8). At other times, we need to express the objects as functions of ¢ and 8 (when we are not referring to a panicular ground) ; AB would then be expressed as AB(¢, 8). Here ¢ E [-�,_1':] and 8 E [ - _1':, _1':] . 2 2 q 4 \Ve first note that for any equal hm·ering position the slopes of both AB and BC must be at most k in magnitude. To see this, suppose AB has slope greater than k. Then, clearly, B is either abm·e or helmY the ground. Since CD is parallel to AB. it has the same slope as AB: further. if B is higher than A, then D is lower than C and Yice versa. There­ fore. if B is above the ground. then D is belmv the ground. and vice versa . I t follows that equal hovering is impossible . Second, let tangentB( 8) and tangentD(8) be the tangent vectors to the semi-circles swept out by the points B( fJ) and D( 8) , and let vertB( 8) and vertD( 8), be, respectively, the vertical distances of B and D to the ground. l\iote that we have an equal-hovering position itl vel1B( 8) - uertD( 8) = 0. I t is easy to see that in the 8-interval where the slope of tangentf3(8) is greater than or equal to k, then vertB(8) Also, since tangentB( 8) = is strictly decreasing . - tangentD( 8) , ue11D ( 8) is strictly increasing in this inter­ val . Thus L'eJ1B( 8) - certD( 8) is strictly decreasing. :.Jow, let's choose

{

k = min max slopeAB( ¢, 8), e slope BC( ¢, 8) , slopetangentBC ¢, 8) .

}

Of course. slopeAB( ¢, 8) . slopeBC( ¢, 8) . and slopetan ­ gentB( ¢, 8) denote t h e slopes o f AB. BC and the tangent vector at B. respectively, and the minimum is taken over , _1':] and 8 E [ - _1': , _1':] . Also, because of all choices of ¢ E [ - _1': ·-! - 1 2 2

compactness, the minimum above is actually achieved. Given this choice of k, we shall show that in the interval where equal hovering is possible the slope of the tangent is at least k. So, in this interval, um1B(8) - ue11D( 8) is strictly decreas­ ing, and thus the equal-hovering position must be uniqu e . · Note that the vectors AB( ¢, 8) , W e first show that k =

,� v2

BC( ¢, 8) , a n d ta ngentBC ¢, 8) are mutually orthogonal. I f we then write (0,0, 1 ) in terms of this orthogonal frame and take norms, it immediately follows that 1

=

sin 2 {3 1 + sin 2 {32 + sin 2 {33,

where {3 1 , {32 • and {33 are the angles the three vectors make with the �v-plane . Therefore. at least one of the sin 2 {3i is at least _l. and thus the vertical slope ( = I tan f3/l of the cor­ It follows that k 2: responc.l ng vector must be at least 1 1 T o cI en1onstrate t l 1at t l1e n1mm1un1 v . -:= 1s ac h 1' eve d , . . k -;=. �� show that any table can be oriented in tfi'J critical posi­ tion, with all three slopes equal to and with associated tilt angle ¢ between _1': and E TZ/do this, consider the 4 4 . tripod formed from three edges ot a cube ti l ted to have

f

�.

=

+,,

_

.

p

Figure 3 Three mutally orthogonal segments of length 1 bal­ anced on the .\JLplane . Then (gray) rectangles of any shape can be fitted as indicated.

vertical diagonal shown in the left diagram in Figure 3. These edges are mutually orthogonal, a n d o n e easily cal­ culates that the slopes of all three edges are . Notice that ' 2 v 1 es. s l1own every table is similar to one ot the gray rectang in the right diagram. created by moving the point A' from P to B'. Furthermore, it is clear that the slope of the diag­ onal A' C ' is less than the slope of B' C' . which is equal to �. guaranteeing that the tilt angle ¢ is i n the desired

+

'v 2

range. By scaling and translating the rectangle suitably, and relabelling the vertices A', B' , and C ' as A, B, and C, re­ spectively. we arrive at the desired orientation of our table. I t remains to show that k = � implies that the slope of y ) the tangent is at least k i n the interval where equal hovering is possible. Note that i n this interval , slopeAB and slopeBC are at most Then the equation 1 = sin 2 {3 1 + sin 2 {32 + sin 2 {33 implies that slopetangentB is at least

�-

Continuity of A, B, C, and D

�·

All of the above calculations were performed with OA pro­ jecting to the positive x-axis . We now consider rotating the table about the z-axis (while of course being willing to tilt the table as we rotate ) . So let y be the angle the projec­ tion of OA makes with the positive x-axis. B y our Lipschitz hypothesis, for any y there is a u nique equal-hovering po­ sition (with the projection of OA making the angle y, and tilting the table around AC a n angle between _1': and _1': in 2 2 a fixed direction) . We need to show that the positions of the four vertices A, B, C, and D, of the equally hovering table are continuous functions of y. To do this, consider a sequence l'n � y, with corresponding corner positions A n, En, C , and D1 1 and A, B, C, and D. We want to show that " A n � A, Bn � B, C11 � C� and Dn � D. By compactness, we can take a subsequence so that the corners converge to something: A 11 � A* . B11 � B*, etc. But, by continuity of everything in sight, A* and C* are touching the ground, B * a n d D * are a n equal vertical distance from the ground, and all four points are corners of the kind of table we are con­ sidering, with the projection of OA* making a n angle y with the positive x-axis. By uniqueness . we must have A* = A, B* = B, C* = C, and D* = D, as desired. _

Balancing position

With the uniqueness of the equal-hovering position for a given y, and with the continuous dependence of this po© 2007 Springer Science +Business Media. I n c. . Volume 29. Number 2 . 2007

53

sition of the table upon y, we conclude that the distance that

B and D are hovering above the ground is also a con­

tinuous function of y. If we are dealing with a square table,

we can now finish the proof using IVT one more time, as described in the intuitive table-turning argument presented at the beginning of this section. For a general rectangle, let the

initial position

be an

equal-hovering position for which the z-coordinate of the center of the table is a minimum, and let the

end position

be an equal hovering position for which the z-coordinate of the center of the table is a maximum. Note that the hov­ ering vertices in the initial position must be on or below the ground: if not, we could create a lower equal-hovering position, contradicting minimality, by pushing vertically

Figure 4.

The four conic sections and the table top in the

case of a s q uare table with legs of length

horizontally.

� that is

balanced

down on the table until the hovering vertices touch the ground. Similarly, in the end position the hovering vertices

tains the whole table top. It is clear that the four conic sec­

are on or above the ground. Now, IVT can be applied to

tions are congruent and that any two of them can be brought

guarantee that among all the equal-hovering positions there

into coincidence via a translation. Furthermore, given any

is at least one balancing position.

point of one of these conic sections, this point and the re­ spective points in the other three conic sections form a

Balancing real tables

To balance a real table of side lengths ratio r, we determine a balancing position of the associated mathematical table,

a s described above. We now show that legs of length at least

� guarantee that, balanced in this position, none

of the

pcii�ts of the real table is below the ground. We give

square that is congruent to our table top. Finally, since the legs have slope of at least

�, the end point of a leg of

our table on the plane is cont iined in the conic section as­

sociated with the other end point of this leg.

�· �. This

To show that we may need legs of length at least we concoct a special ground with Lipschitz constant

�ircle,

the complete argument for a square table, and then de­

ground coincides with the xy-plane outside the unit

scribe how things have to be modified to give the result

and above the unit circle it is the surface of the cone with

for arbitrary rectangular tables. I n the following, we will re­

vertex (0,0,

fer to the four vertices of the table top as corresponding to the vertices

A ' , B' , C' , D' ,

A, B, C, D. respectively, of

the mathematical table.

�) and base the unit circle . As the diagonals

of our table a fe of length 2 , the mathematical table will bal­ ance locally on this ground iff its vertices are on the unit

circle. This means that the legs have to be at least as long

We first convince ourselves that no matter how long the

as the cone is high if we want to ensure that no point

legs of our table are, no part of a leg of the balanced table

of the table top is below the ground; it follows that we

will be below the ground. Let's consider the orthogonal tri­

have to choose the length of our legs to be at least

pod consisting of

length of the legs is equal to

AB, AD, and the leg at A. Since the Lip­

schitz constant of our ground is at most

�, t h e slopes of

AB and AD are less than or equal to this

talue;

thus, ar­

�· If the � and the table is bilanced

on this ground, then the four co'h ic sections are circles that

intersect in the center of the table top as shown in Figure

guing as above, we see that the leg must have slope at

4.

least

union of these four circles. If we make the legs longer, the

.Jz· This implies that no l e g of o u r balanced table will

dip bel�w the ground.4

It remains to choose the length of the legs such that no

point of the table top of our balanced table will ever be below the ground. First, fix the length of the legs and con­ sider the inverted

souo

circular cone, whose vertex is one

of the vertices of our mathematical table, whose symmetry axis is vertical, and whose slope is

J:z.

Intersecting this

As you can see, the table top is indeed contained in the

circles will overlap more. If we make the legs shorter, the circles will no longer overlap in the middle. Now consider any ground, and take the legs to be of length

�; clearly, if we can show that this table does not

dip bela� the ground, then the same is true for any table

with longer legs. When we tilt the table away from the hor­ izontal position, the intersection pattern of the conic sections

cone with the plane in which the tabfe top lies gives a

gets more complicated. The critical observation is that tilting

conic section which is an ellipse, a parabola, or a hyper­

the table results in the conic sections getting larger; it can

bola . 5 Note that since we intersect the plane with a solid

be shown that each conic section contains a copy of one of

cone this conic section will be "filled in". We can be sure

the circles in Figure

that a point in this plane is not below the ground if it is

these conic sections, it and the corresponding points in the

46

Since, given any point of one of

contained in the conic section. Therefore, what we want to

other three conic sections form a square that is congruent to

show is that the union of the four conic sections associ­

our table top, the union of these conic sections will contain

ated with the four vertices of our mathematical table con-

a (possibly translated) image of the union of the circles, that

4For certain grounds with Lipschitz constant

0· it is possible that a leg of a balanced table may lie along the ground.

5Parabolas and hyperbolas can occur because the plane that the table top is contained in can have maximum slope greater than

0·

6To see this, note that what we are looking at are the possible intersections of a given cone with planes that are a fixed distance from the vertex of the cone.

54

THE MATHEMATICAL INTELLIGENCER

Figure 5.

If the table is not horizontal, the conic sections are larger than the circular (gray)

sections in the horizontal case (left). Their union is simply connected (right) .

A'

A

B'

B'

CAJ

B

Figure 6.

A

The points of intersection of the two l ines of slope

tionings of the rectangle AA' BE ' are on a circle .

A'

B'

A

B

~

---k for the different possible vertical posi­ 'v 2

we encountered before ; see Figure 5. Our previous picture

everything that we have done so far to end up with another

has, so to speak, just grown a little bit and been translated.

circle segment. However, the apex of this circle segment will

(Note, however, that it is not immediate that the conic sec­

be closer to

tions together cover the table top rather than some transla­

fact, the more we tilt, the closer we will get; see Figure

tion of the table top). Using this fact and the simple possi­

A' B' than the one we encountered before. In 7.

Since t h e apex of o n e of these circles is t h e point clos­

ble convex shapes of the conic sections that we are dealing

est to

with, we can conclude that no matter how we tilt the table,

horizontal, we now calculate just how close this apex gets

A' B ' , and since the apex corresponds to AB being

the union of these conic sections will always be a simply

when we tilt around

connected domain. This means that we can be sure that the

This maximal possible angle is attained if

table top is contained in this union if we can show that the

thogonal to

boundary of the table top is contained in it. We proceed to show that for all possible positions of

AB though a maximal possible angle. AD (which is or­

�

AB) has slope · It i s a routine exercise to 2 check that in this position the slope of the line connecting

A with the midpoint of A' B' i s

� · This means that in this

our table in space the sides of the table top never dip be­

position the apex will be contained i n

low the ground. Clearly it suffices to show this for one of

that if we choose the legs of our square table to be at

the sides of the table top, say

A' B ' . For this we consider

the possible positions of the rectangle with vertices A, A ' , B, and B' i n space. W e start with the rectangle vertical and AB horizontal. Draw lines of slope ending in A and B;

�

see Figure 6 (left) . Since the point of int� rsection of these two

lines is not above

least

A' B' . We conclude

� long, then the boundary of the table top, and hence

also th � table top itself, will not dip below the ground.

For tables that are not square, the same arguments ap­

ply up to the point where we start tilting the rectangle

A' B ' , no point of this segment can be

B'

below the ground when the rectangle is positioned in such a way. Now rotate the rectangle around its center, keeping it in a vertical plane, and keeping the slope of AB less than or equal to

�· Again, draw lines of slope � ending in

A

and B; see FigJre 6 (middle) . Again, the position of the point

at which these two lines intersect tells you whether

A' B '

can possibly touch the ground with the rectangle in this position. Since the two lines always intersect in the same angle, we know that the points of intersection are on a cir­ cle segment through

A and B; see Figure 6 (right) .

Now tilt the original rectangle around AB. We repeat

Figure 7.

The more we tilt, the closer the apex of the circle

segment gets to A 'B ' . As long as the apex is below or on A'B ' , we can be sure that A'B' does not dip below the ground.

© 2007 Springer Science + Business Media, Inc., Volume 29, Number 2, 2007

55

AA' B' B around AB. We now have to worry about two dif­

a sphere that is large enough to ensure that all legs of our

ferent rectangles corresponding to the longer and shorter

table end up on this part of the sphere whenever the table

r is the ratio of the lengths of the

is locally balanced on this ground. Then intersecting this

sides of the table, then it is easy to see that the critical

sphere with the plane that the leg points are contained in

sides of the table top . If

length of the legs that we need to avoid running into the

gives a circle that all leg points are contained in. Hence the

ground is the length that makes the longer of the two rec­

leg points of the table are concircular. Now, let's consider a

tangles similar to the rectangle that we considered in the

ground that includes part of an ellipsoid which does not con­

square case. This critical length is

tain a copy of the circumcircle of the leg points of the table;

�.

V l + r2

Other Balancing Acts

D

moreover. we choose the ellipsoid large enough so that all leg points of our table end u p on the ellipsoid whenever the table is locally balanced on this ground. Then intersecting

Horizontal balancing

the ellipsoid with the plane containing the leg points gives

When we balance a table locally, the table will usually not

an ellipse that is different from the circumcircle of the leg

end up horizontal, and a beer mug placed on the table may

points. However, this is impossible if the table contains more

still be in danger of sliding off. It would be great if we could

than four leg points because five points of an ellipse deter­

arrange it so that the table is not only balanced but also hor­

mine this ellipse uniquely. We conclude that an always lo­

z­

cally balancing table must have three or four leg points and

axis and balancing it somewhere else on the ground. Just

that these points are concircular. Note that requiring concir­

imagine the ground to be a tilted plane, and you can see that

cularity in the case of three points is not superfluous, for we

this will not be possible in general. However, Fenn [4] proved

need to exclude the case of three collinear points.

izontal, maybe by moving the center of the table ofi the

the following result:

lf a continuous ground coincides with the xy-plane outside a compact convex disk, and if the ground never dips below the xy-plane inside the disk, then a given square table can be balanced horizontally with the center of the table lying above the disk. Let's call the special kind of ground described here a Pen n ground and the part of this ground inside the distinguished compact disk its hill. The problem of horizontally balancing tables consisting of plane shapes other than squares on Fenn grounds has

Livesay's theorem, which made the proof of Theorem 1 so easy, has a counterpart for triangles, due to Floyd [6]. It is a straightforward exercise to apply this result to prove the following theorem:

THEOREM 3 (Balancing Triangular Tables) rr

the ground function is continuous, a triangular table whose three leg points are contained in a !>phere around its center can he balanced local£y.

also been considered. Here "horizontal balancing on a Fenn ground" means that in the balancing position some interior points of the shape are situated above the hill. It has been

Of course, one should be able to prove a l ot more when it comes to balancing triangular tables'

shown by Zaks [29] that a triangular table can be balanced

In the case that the center and the (three or four) leg

on any Fenn ground. In fact, he showed that if we start

points of an always locally balancing table are coplanar,

out with a horizontal triangle somewhere in space and mark

we can say a little hit more about the location of the cen­

a point inside the triangle, then we can balance this trian­

ter point with respect to the leg points. Begin by balanc­

gle on any Fenn ground, with the marked point above the

ing the table in the xy-plane and drawing the circles around

hill, by just translating the triangle. Fenn also showed that

the center that contain leg points; if one of the leg points

tables with four legs that are not concircular and those form­

coincides with the center, then also consider this point as

ing regular polygons with more than four legs cannot al­

one of the circles. Now it is easy to see that there cannot

ways be balanced horizontally on Fenn grounds. Zaks men­

be more than two such circles. Otherwise a ground that

tions an unpublished proof by L. M. Sonneborn that any

coincides with the xy-plane inside the third smallest circle

polygonal table with more than four legs cannot always be

and that lies above the plane outside this circle would

balanced horizontally on Fenn grounds. It is not known

clearly thwart all local balancing efforts. Therefore, if we

whether any concircular quadrilateral tables other than

want to check whether our favorite set of three or four con­

squares can always be balanced horizontally on Fenn

circular points is the set of leg points of a locally balanc­

grounds. See [ 1 4] , [ 17] , [ 1 8] , and [ 1 9] for further results re­

ing table, there are usually very few positions of the cen­

lating to this line of research.

ter relative to the leg points which need to be considered.

Local balancing of exotic mathematical tables

ter of the circle that the leg points are contained in. As a

Perhaps the most natural choice for the center is the cen­ Taking things to a different mathematical extreme, we can

corollary to the above theorem for triangles, we conclude

n 2: 3 leg points in 3-space together with a n additional center point. We then ask

that a triangular table with this natural choice of center is always locally balancing. In the case of four concircular

whether, given any continuous ground, it is always possi­ ble to balance this table locally, that is, move this config­

whether any tables apart from the rectangular ones are al­

uration of

ways locally balancing. However, a result worth mention­

consider a table consisting of

n + 1 points into a position in which the n leg

points are on the ground, and the center is on the z-axis. The example of a plane ground shows that the leg points of an always locally balanceable tabl e have to be coplanar. Let's consider the example of a ground that contains part of

56

THE MATHEMATICAL INTELLIGENCER

points with this natural choice of center we do not know

ing in this context is Theorem 3 in Meyerson's paper [ 1 7] (see also the concluding remarks in Martin's paper [ 1 6]). It can be phrased as follows:

Given a continuous ground and one of these special fou r-legged tables in the xy-plane, the

tahle can he rotated in the xv-plane around its center to a posit ion ll 'here the jciltr points 0 1 1 the f!,f"C){ { nd ahol'e the leg points are coplanar. The quadrilateral formed by the copla­ nar points on the ground will he congruent to the table if and only if the plane containing it is horizontal . in \Vh ich case we have actually found a balancing position for our table. In all other cases, the quadrilateral on the ground is a deformed version of the table. Still , if the ground is not too wild . the quadrilaterals will he very similar. and lifting the table up onto the ground should result in the table not wobbling too much. Livesay's theorem is a generalization of a theorem by Dyson [ 2] , which only deals with the square case. A higher­ dimensional counterpart of Dyson's theorem arises as a spe­ cial case of results of Yang [2')]. Theorem 3 and Joshi [ 1 2 1 , Theorem 2 : Gicen a contimwus rea!-t •cilued jimction de­ fined on the n -.,phere, there are 11 mutual!)' 011hof!,Oilal di­ ameters of this sphere such that the ji m ctio n takes on the same t •aluc at al/ 2 n endpoints of" these dimneten;. Note that the endpoints of 1 1 mutually orthogonal diameters of the //­ sphere are the \·ertices of an //-dimensional orthoplex. one of the regular solids in 1 1 dimensions. ( For example, a 1 -dimensional orthoplex is just a line segment and a 3-dimensional orthoplex is an octahedron . ) esing the same simple argument as in the case of Livesay·s theorem . we can prove the follcnving theorem: THEOREM 4 (Balancing Orthoplex-Shaped Tables) An ( 1 1 - 1 ) -dimensional m1hoplex-shaped table in IR '1 can be balanced /ocal{l' on any growzd git •en h1· a con­

rectangul a r tables the ends of whose legs do not form a perfect rectangle are not uncommon and , as our simple ex­ ample shows. those uneven legs may conspire to make our anti-wobble tactics fail. Considering our examples of a discontinuous ground at the beginning of this article, it should be clear that a wob­ bling table on a tiled t1oor may a lso defy our table-turning efforts. How to turn tables in practice

In practice, it does not seem to matter how exactly you turn your table on the spot, as long as you turn roughly around the center of the table. Notice that you needn't ac­ tually establish the equal hovering: as you rotate towards the correct balancing position. there will be less and less wobble-room until, at the correct rotation, the balancing position is forced. \X'ith a square table . you can even go for a little hit of a journey , sl iding the table around i n your ( continuou s ) backyard . As long as you aim to get hack to your starting position, incorporating a quarter-turn in vour overall mm·ement. you can expect to find a balanc­ ing position. REFERENCES

[ 1 ] de Mira Fernandes, A. Funzioni continue sopra una superficie sfer­ ica. Portugaliae Math 4 (1 943), 69-72.

[2] Dyson , F. J. Continuous functions defined on spheres. Ann. of Math. 54 (1 95 1 ) , 534-536.

[3] Emch, A. Some properties of closed convex curves in a plane,

tinuousjitnctimz: [Ril- l � R

[41 Fenn, Roger. The table theorem. Bull. London Math. Soc. 2 ( 1 970),

For other closely related results see [ l l . [24] . [ 26] . [27 ] . [ 281. [ ') 1 , and [ 1 0] .

[51 Fenn, Roger. Some applications of the width and breadth of a

Balance everywhere

Imagine a square table \Vith diameter of length 2 suspended horizonta lly high above some ground, vvith its center on the z-axis. Rotate it a certain angle about the z-axis . release it, and let it drop to the ground. It is easy to identify con­ tinuous grounds such that all four legs of the table \\ ill hit the ground simultaneously. no matter what release angle you choose. Of course, any horizontal plane \\·ill do, and so will any ground that contains a \·ertical translate of the unit circle . We l eave it as a n exercise for the reader to con­ struct a ground that is not of this type but admits horizon­ tal balancing for any angle. Also. readers may wish to con­ \'ince themselves that the fol l o\\·ing is true: Consider a ground as in Theorem 2. If the center of the table has the same z-coordinate in all its equal-hovering positions ( posi­ tions in which A and C touch the ground and B and D are at equal vertical distance from the ground) , then in fact the table is balanced in all these positions.

Amer. J. Math. XXXI/ (1 91 3), 407-4 1 2 . 73-76.

closed curve to the two-dimensional sphere. J. London Math. Soc.

(2) 10 (1 975). 2 1 9-222 .

[61 Floyd , E. E. Real-valued mappings of spheres. Proc. Amer. Math. Soc. 6 ( 1 955), 957-959.

[71 Gardner, Martin. Mathematical Games column in Scientific Amer­ ican (May 1 973), 1 04 . [8] Gardner, Martin. Mathematical Games column in Scientific Amer­ ican (June 1 973), 1 09-1 1 0.

[9] Gardner, Martin. Knotted Doughnuts and Other Mathematical En­ tertainments. W . H . Freeman and Company, New York, 1 986. [1 OJ Hadwiger, H . Ein Satz uber stetige Funktionen auf der Kugelflache. Arch. Math. 1 1 ( 1 960), 65-68.

[ 1 1 1 Hunziker, Markus. The Wobbly Table Problem . In Summation Vol. 7 (April 2005), 5-7 (Newsletter of the Department of Mathematics, Baylor University). [1 21 Joshi, Kapil D. A non-symmetric generalization of the Borsuk-Uiam theorem. Fund. Math. 80 (1 973), 1 3-33.

[ 1 31 Kraft, Hanspeter. The wobbly garden table. J. Bioi. Phys. Chem. 1 (200 1 ) , 95-96.

Some Practical Advice

[ 1 4] Kronheimer, E. H. and Kronheimer, P. B. The tripos problem. J.

Short legs and tiled floors

[ 1 5] Livesay, George R . O n a theorem of F. J. Dyson . Ann. o f Math.

Note that if you shorten one of the legs of a real-life square table, this table will wobble \vhen set clown on the plane, and no turning or t ilting will fix this problem. In real life.

[ 1 6] Martin, Andre. On the stability of four feet tables. http://= 20

London Math. Soc. 24 ( 1 98 1 ) , 1 82-1 92 . 59 ( 1 954), 227-229. arxiv.org/abs/math-ph/051 0065

© 2007 Springer Sc1ence- Bus1ness Med1a. Inc

Volunle 29 Number 2 . 2007

57

[ 1 7] Meyerson, Mark D. Balancing acts. The Proceedings of the 1 98 1

bles: feasting from a rnathsnack. Vinculum 42(4) , 4 November

Topology Conference (Blacksburg, Va. , 1 98 1 ) , Topology Proc. 6

2005, 6�9 (also available at www.rnav.vic.edu .au/curres/math­

snacks/mathsnacks. html).

( 1 98 1 ), 59�75.

[ 1 8] Meyerson, Mark D. Convexity and the table theorem. Pacific J.

[24] Yamabe, Hidehiko and Yujob6, Zuiman . On the continuous func­

[ 1 9] Meyerson , Mark D . Remarks on Fenn's "the table theorem" and

[25] Yang, Chung-Tao. O n theorems of Borsuk-Uiam, Kakutani-Yam­

[20] Polster, Burkard ; Ross, Marty and OED (the cat). Table Turning

[26] Yang, Chung-Tao. On theorems of Borsuk-Uiam, Kakutani-Yam­

tion defined on a sphere. Osaka Math. J. 2 (1 950), 1 9�22 .

Math. 97 ( 1 98 1 ) , 1 67�1 69.

Zaks' "the chair theorem" . Pacific J. Math. 1 10 ( 1 984), 1 67�1 69.

abe-Yujobo and Dyson I. Ann. of Math. 60 ( 1 954), 262�282 . abe-Yujob6 and Dyson. I I . Ann. of Math. 62 ( 1 955), 2 7 1 �283.

Mathsnack in Vinculum 42(2) , June 2005. (Vinculum is the quar­ terly rnagaz1ne for secondary school teachers published by the

[27] Yang, Chung-Tao. Continuous functions from spheres to euclid­

ean spaces. Ann. of Math. 62 ( 1 955), 284�292.

Mathematical Association of Victoria, Australia. Also available at www. rnav. vic .edu. au/curres/mathsnacks/mathsnacks. html).

[28] Yang, Chung-Tao. On maps from spheres to euclidean spaces.

[2 1 ] Royden, H. L. , Real Analysis . Prentice-Hal l , 1 988. [22] Vinculum, Editorial Board. Mathematical inquiry- from a snack to a

Amer. J. Math 79 ( 1 957), 725�732.

[29] Zaks, Joseph. The chair theorem. Proceedings of the Second

meal. Vinculum, 42(3), September 2005, 1 1 1 2 (also available at

Louisiana Conference on Combinatorics, Graph Theory and Com­

www.mav.vic.edu.au/curres/mathsnacks/mathsnacks.html).

puting (Louisiana State Univ . , Baton Rouge, La. , 1 97 1 ) , pp.

�

[23] Polster, Burkard; Ross, Marty and QED (the cat). Turning the Ta-

CAMB RIDGE

557�562. Lousiana State Univ., Baton Rouge, 1 97 1 .

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58

T H E MATHEMATICAL INTELLIGENCER

M athem a tic a l

E n tertain ments

S ud o l
The problems appeared on pp. 4 1 -44 of the winter issue (vol. 29, no. 1 )

� ust as with the problems, the solu­ , . � tions are gi\·en here in their origi­ J nal form . as published in the French

...,.

·

press of that time . without the slightest change in the text ( \Yhen there was tex t ) o r i n the grid . For example, when the 3 X 3 sub-squares are set off by heav­ ier lines. it means they really \\"ere giYen so at the time . After the original statements. com­

mentaries in italics haue been supple zen tmy deta ils.

gil'ing

added

1

17

24

62

69

73

47

36

7

14

33 37 11

27

72

76

40

50 30 79

59

53 4

56

43

66

31

38

2

18

63

67

61

68

75

22

32

39

52

57

10

74

48

34

1 2 25

5

41

70

77

51

28

19

8

80 60

44

64

15

3

78

55

42

46

35

26

6

81

58

49

29 45

9

65 13

of contagious mathematics that trauel from person to person in the community, because they are so elegant. suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Comment. 77.Jis square is interesting the poillf r!f l 'iezr qfSudoku in that the magic su m 369 also u·orks for the nine numbers ofeach 3 X 3 sub-square. Amcmg the readers zDho found the solutimz at the time u·ere A . Huber (au­ thor t!{ Proh. () helou·J and B . .Hepziel (author of Proh. 9).

jivm

Solution to Problem 2, Le Siecle, 11 July 1891. 62

43

27

69

33

14

76

37 56

21

60 34

1

38

Please send a l l submissions to the Mathematical Entertai n ments Editor,

Ravi Vakil, Stanford U n iversity,

Department of Mathematics, Bldg. 380,

Stanford, CA 94305-2125, USA

e-m a i l : [email protected] rd . edu

66 19

40

57

4

17

36

54

70

18

80

48

22

75

29

10

61

26

71

32

78

67

23 52

51

58 9

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50

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47

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77

31

25

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28

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24 59

2

7

15

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44

6

68

49

55

39

20

42

4356 1 600 289 1 296 6241 2209 576 3481

1 089 1 96 5776 1 369 31 36 441 5184 2809 49

1444 361 3249 2916 4900 64 5929 961 225 324 6400 2304 484 1 681 3600 1 1 56

4489 2601

4 4096

25 5476 144 784 625 3969 1 936

5625 841 1 00 3721 676 2025 36 4624 2401 9

2 1 1 6 4225 1764 256 6561

5041 1 024 6084 1 69 3025 1521 400

1 6 23

71

20

1

2704 81

54

1

3844 1 849 729 4761 1 6 2500 1 21 5329 900

529 3364 1 225

n

Solution to Problem 1, Le Sitkle, 30 June 1888.

21

This column is a place for those bits

f\/1_!ch <3_�el Kleber and R avi Vaki l , E di tors

16

81

The two diagonals and the two lines crossing at the center each give 1 ,225,449. that is, 1/9 the sum of the cubes of the first 81 integers. Comment. This note is misstated. The

jimr lines passing through the center­ the tu•o diagonals. the central rou', and the central column-are not on�}' himagic (as can he seen hy adding their squares. shcnl'/z in the second grid) hut also trimagic. Name!}'. when their ele­ ments are cubed these fou r lines giue the same sum. 1,225.449. Amon?, the readers who found the solution at the time Zl'ere Luet (author of Proh. 1 ahoue), A . Huber (author qf Proh. 6 ) , and B. Meyn iel (author qf Proh.

9 J.

Solution to Problem 3, Revue des Jeux, 4 September 1891. By subtraction from 1 23, the given numbers allow one to fill in squares 1 and 3 . B y subtraction from 8 2 , square 1 yields square 9. and square 3 yields square 7. By superposing them and sub­ tracting from 1 23 . squares 1 and 3 g ive square 2 . squares 3 a n d 9 g ive square 6 , s q u a res 7 ancl 9 give square 8 . and squares 2 and 8 or 4 and 6 give square 5 . Comment. 1be solu t io n should also say that squares 1 and 7 f!, il 'e square 4 .

© 2007 Spnnger Scrence +Busr ness Media, Inc.,

Volume 29,

Number 2 , 2007

59

2 1 4 46

63

55

18

50

67 2 1

35

79

6

38

74

7

54 59

4

7

30 26 2

42

71

34 69

10

65

25

22

77

1

24 29 9

41

66

37 81

43 78

49 57

31

8

40

70 44

76

3

73 47

61

20 75

33

4

39 80

45

16 51

56

5

11

52

60

1 2 53

58 32 64

62 1 3

48

17

8

Solution to Problem 4, Le Siecle, 3 December 1892.

72 19

15

27

23 28 36 68

1 7 20

3

70 73

45 48 19

6

9

4

5

55 67

79 51

36 39

33

2

26 76

61

58 30 42 54 23 16

41

47 32

44

69

59

71

74

27

31

43

46

75 60 6

72

18 21

14

53 81

1 3 25

52

3

11

64

29 63 66 78 57 1

7

1 0 22

35

38 50

65

77 62

1 5 37 49

34

24

9

28 40

80 56

8

68

12

One can change the order of rows within each band, for example putting horizontal rows in the order 2, 3, 1 , or 3, 1, 2 ; in the second band 5 , 6, 4 , or 6, 4 , 5 ; in the third band H, 9, 7, or 9, 7, 8, etc . , etc. Also one can shift three columns together from left to right or from right to left; likewise one can shift three rows together from top to bottom or from bottom to top.

Comment. Among the readers who .found the solution at the time U'as B . Portier (author of Proh. 7 below). Here are the two Sudokus associated tl'ith this solution: 2

8

6

6

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7

8

9

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7

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2

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1

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7 1

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2

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1

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THE MATHEMATICAL INTELLIGENCER

9

5 3

60

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5

8 9

Solution to Problem 5, Le Siecle, 1.4 January 1.893. 9

7

2

2

8

6

7 8

2 6

6

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1

4 3 5

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7·2 5·1 3·3 1 ·7 8·9 6·8 4·6 2·5 9-4

1

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7

1

3

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9

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4·7 2· 9 9·8 7·6 5·5 3-4 1 ·2 8·1 6·3 3·1 7·3 5·2 6·9 1 ·8 8·7 9·5 4-4 2·6

8

2

6

8 -4 6·6 1 ·5 2·3 9·2 4 · 1 5·8 3·7 7·9

2

7

8

2·8 9·7 4·9 5-4 3·6 7·5 8·3 6 ·2 1 · 1

9

9

7

6·5 1 -4 8·6 9 · 1 4·3 2·2 3·9 7·8 5·7

9·9 4·8 2·7 3·5 7-4 5·6 6 · 1 1 ·3 8·2

6

5·3 3·2 7·1 8·8 6·7 1 ·9 2-4 9·6 4·5 1 ·6 8·5 6 -4 4·2 2·1 9·3 7·7 5·9 3·8

1

Com /1/ent. The solution to the prohlem turns out to he sillljile. in tho! eoch rou · is the scune cts the ruu · {{/Jut·e it shijied to the h:fi. ?11 is is not ct Sudoktt. ji;r the .:) X .:) suh­ squares do not contain all the 1 / T t m heJ:,·. hut a simple latin square like !he oue qjLuler \ P, il 'e/1 as Fi12,. 1 ill the 1 H9.) ar­ t 'eJy

ticle fro/. 29. 11/J. 1 . p. 57 J. One u/ the diagonals consL\ts 1m!)'


=

1 +2+3+

magic stun

+9

=

like the otheJ:.;:

i 'i .

.

/tmm(� the reade1:' u 'ho jinmcl the solution o t the time ll 'Cls H. ,He�'lliel I author o/ l 'ro hlem 9 J . EHRAT\ '\1: /11 tbe ji;� ure j(n·problem :; published in the on�r.;­ inal a rticle ( !\1athematical Intelligencer. 1'01. 29. n o . 1 . p. -+2 J. the 3 X 3 borders U'ere errcmeous!J • drctu•n ll 'itb heal 'ier lines. The orif!,inal problem. as published in Le Siecle. did n ut hif.!,b­ light the 3 X :3 su h-squares heca use that u·ould hal 'e heen m em z ingless in tbis case.- The r:ditors.

Solution to Problem 6, Les Tablettes du Chercheur, 1. May 1.894.

The square is a solution of the problem of Hl officers. I n ea ch entry. the first nu mber indicates t h e rank a n d t h e secom! the regiment. or \'icc \'l'lXl ( J. Cum meut. One sees clea r/1' the tu • o Sudukus i111 ·oll 'ed. The .fi'J:o;t Sudoku is made up nj' the p,:,'f ji;r.;u res jimn ea c h eutr) ·, . the other of' the second jlgu res 'J(; get the associated binw,r.; ic square. s i mjiiJ ' subtract 1 j iwn the second J l / t m her o ne/ re­ mol 'e the S�i!, ll (!/ multiplicatio n . '! bus 7 · 2 hecomes 7 L 'i · 1 heco i / J es SO, etc.. and 3 · H lwcol/les 57. •

Solution to Problem 7, La France, 2 July 1.894. 16

5

2 1 5 5 80 69 4 9 3 8 36

42 2 8 53

9

22 1 1 7 5 70 5 9

6 5 63 76 32 48 43 26 1 5 35 5 1 3 7 20 1 8

58 74 72 52 4 1 3

4

1

68 57 79

30 1 0

8

24

25 1 4 78 64 62 45 3 1 4 7 6

19 17

27 6 1

77 66

8 1 67 56 39 34 50

23 1 2

7

7 1 60 7 3 2 9 5 4 4 0

46 44 3 3 1 3

2

C.'O IIIIIIent. Here is its deco nzpos itiO J z inlu lll 'li Sudokus:

2

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7 1

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3

5

© 2007 Springer Sc ience -! Business Med1a. Inc . Volume 29, Number 2. 2007

61

7

5

3

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9

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5 1

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6

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0

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6

1

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6

We received a large number of answers, but only seven of them satisfied the stated conditions. ( . . . ) The most pleas­ ing figure, given hy K . D . RousseL is reproduced here:

0

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The remaining "macaroons'' do make u p a 3 X 3 magic square. 1\'amely, they amount to the following: 3 2 7

8 4 0

1 6

5

giving for every row . column, or diagonal the same sum 12. Notice above that i n each compartment the remaining macaroons are arranged like the points on dice or domi­ noes, only with 4 and 5 subjected to a quarter-turn. Comment. On 1 September 1894, les Tahlettes du

Chercheur gave another solution, the first of the two grids that follow. The array in the second grid corresponds to the same problem hut with 5 (instead of 4) as the number qf macaroons to remain in each horizontal, vertical, or diag­ onal line. These two new solutio ns are complementary in the sense that superposing them produces the complete ar­ ray with a macaroon at every place.

62

THE MATHEMATICAL INTELLIGENCER

v

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Solution to Problem 8, L 'Echo de Paris, 24 July 1894.

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cs1 p.ll''"""

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There is another soluticm to this as a Sudoku:

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\\'e receiYecl no correct solutions.

Com ment. \\" hereas all the preceding pmhlenzs had read­ eJ�\· sull 'illi� them o/ the lime. Olllllsingl)' the one c!uses/ to u11r Suduk11s had none-' J7Jis one had all the features u/ o S11dok11. except that its 3 X 3 s11 h-sqzwres Zl 'ere nut nwrked: yet the)' did. it tllrned Oi l/. each contain all the nu nzben· I thro11gh 9 .

6 7

9 1

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9

9

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But B . lfe)'l z ie! did rip,hl JWI to p11b!ish it. .filr it doesn ·t .. .fiilji'!l the conditions he stales. not being "diabolic .. the bro­ ken diagonals do not oil add to 4 5: the solution published is the cml)' one that is magic pandiagonal. .

53 rue de Mora 95880 Enghien les Bains

France e-ma1l: [email protected]

Mere Mathematician?

J'rom [uga r Allen Poe's The Pttrloilled l.eller. as printed in t he l leritage Edit ion

of Poe':-. :-.tories. at paj.�e 2Ho.� :

'' But i s t his rea l l) t h e poet!" I askl·d . "Ther

both h:n e atta ined reputat ion in lcw.:rs. Th

·

ar

t \H l brothers. I kmm : and

m i n ister I belie\ e ha'> \Hi tten ·

learned ly on the D i llerentia l Calcu lu�. He is a rnathematkia n . and no poet . " •

You are mis ta ken , I knm' h i m \\ ell: he is bot h . A� poet and mathemati­

cian. he \\'ould r ·a son '' e l l , as m ere: mathematinan. he coulu not ha\ e rea­ soned at a l l . and t h u s '' ould h;n c.: h ·�.:n at the merq of the Prcfe t t . " Th

·

s ·cond s p a k ' r i.s Auguste Dupin. Poc'.s

c

·lchratl'd dch:cth e, '' hose rather

muted praise: of mathcmat iu.l n'> in the quotl'd pass.tgl' turns negati\ e in the p.t ra­

gra ph s that follm\ , as a courtl's} w l\1. Dupin I a m not including them hcrl'.

Ralph A. Rrum

Un1vers1ty of Rochester Rochester, NY 1 4627

USA

e-ma1l: [email protected]

© 2007 Springer Science -Business M edia. Inc .. Volume 29. Number 2. 2007

63

la§l)l§l.'fj

Osmo P ekonen , Editor

i

Ramsey M ethods i n Analysis Spiros A . A lg)'ros and Stel'o Todorccl'ic BASEL, BIRKHAUSER, 2005, PP 257, €38, ISBN 3-7643·7264·8

REVIEWED BY HANS-PETER A. KUNZI

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

C o l u m n Editor: Osmo Pekonen, Agora Centre, 40014 U n iversity of Jyvaskyl a , Fi n l a n d e-m a i l : [email protected]

64

I

'""'"'C"'"l he book introduces the reader to sophisticated Ramsey-theoretic methods that have recently been used in the theory of Banach spaces. Before describing its contents in some detail. I discuss the combinatorial and analytic background of the presented results. In a first course of comhinatorics one is taught that Ramsey theory [ 1 1] deals with the insight that many large structures-no matter how disor­ dered-necessarily contain highly or­ dered substructures. Its simplest in­ stance, the pigeonhole principle, asserts that n + 1 pigeons cannot sit in n holes so that every pigeon is alone in its hole. This principle is non-con­ structive in the sense that it states the existence of a pigeonhole with more than one pigeon in it, hut it says noth­ ing about how to find it. One usually meets the first nontriv­ ial applications of the theory when studying graphs. Unavoidably, one learns, in any collection of six people either three of them mutually know each other or three of them mutually do not know each other. Theorems of similar type hold for infinite graphs. If G is a graph with infinitely many ver­ tices. then G or its complement contains a complete suhgraph on infinitely many vertices. The exact formulation of the usual Ramsey theorem [13] for infinite sets re­ quires some terminology. Given a set X, let [XJA' stand for the set of all sub1A

sets of X of size k. If r is a positive integer. an ,�colouring of a set D means a function 7r from D to the set l l , . . . , r } . If a set D bas an r-colour­ ing 7T and C <;;; D, then C is said to be monochromatic if 7T restricted to C is constant. We can nmv formulate the usual ( infinite) Ramsey theorem as fol­ lows: Let k and r he pusitiz •e integers.

Then for ez •e1y r-coluuring of' the set lNJ" there exists an il?('inite subset X rij' N such that [XJ " is monochromatic.

Generalizations to higher cardinali­ ties are possible hut require some care: there are graphs with u ncountahly many vertices that neither have an un­ countable complete suhgraph nor an uncountable independent set of ver­ tices. In connection with the present hook, however, a further variant of the described colouring problem is more important: If we let f::J ixl denote the set of infinite subsets of N, then given an r-colouring of f::J ixl there need not be a n infinite subset X <;;; N such that x l x l is monochromatic. To see this ( com­ pare [9] ) define infinite subsets A. B <;;; N to be equivalent if their symmetric difference is finite. and from each equivalence class choose a representa­ tive. Then colour A green or yellow ac­ cording to whether the symmetric dif­ ference of A and the representative of its equivalence class is of odd or even size. Since deleting an element of a set always changes its colour, an infinite X such that xlxJ is monochromatic can­ not be found. Note that this "had" colouring was constructed with the help of the axiom of choice. Fortunately, positive results can still he proved for colourings that are de­ scribed in a more constructive way. In topological Ramsey themy one equips f::j locl with its (metric) subspace product topology, after having identified subsets of N with their characteristic functions. The Galvin-Prikry theorem [4] states that if Cf6 is any Borel suhset 1 of f::J i xl , then there exists an infinite subset JYJ of N such that either M i xJ <;;; Cf6 or M lxl n Cf6 = 0.

set is a Borel set if it belongs to the smallest u-algebra containing the topology.

THE MATHEMATICAL INTELLIGENCER © 2007 Springer Sc1ence+Bus1ness Media, Inc.

Another topology finer than the metric topology on N i"' l , called the Ellentuck topology, allows one to describe in topological terms those colourings for which a Ramsey theorem holds. Important applications of topological Ramsey theoJY can he found in the the­ ory of well-quasi-ordered sets [ 1 2) . A quasi-ordered set is well-quasi-ordered if for each of its infinite sequences ( X1 1 ) there exist indices 111 and II, with m < 11 such that .1,·1 11 s X1 1 • I ndeed. well­ quasi-orders are characterized as those quasi-orders in \vhich every strictly de­ scending sequence is finite and e\·ery set of painYise incomparable elements is finite. Kruskal ' s theorem states that if Q is �veil-quasi-ordered. then so is the class of all finite Q-trees. for non-tri\·­ ial generalizations to higher cardinali­ ties. :"-Jash-Williams's theory of better­ quasi-ordering was developed. One of its applications yields La\·er's theorem that the class of countable linear or­ derings is well-quasi-ordered under em­ hedclahility. Some readers will certainly he a \Yare of the recent progress in the theory of separable infinite-dimensional Banach spaces. Tsirelson·s example [ 1 4] of a re­ flexi\T Banach space \\'ith an uncondi­ tional basis showed that such a space need contain neither the space c0 of null sequences nor the space fp ( for any 1 s p < :x: ) . The definition of the norm of his example \vas inductive and mo­ tivated by Cohen's forcing method. Tsirelson·s Banach space is now con­ sidered the first truly non-classical space and stands at the beginning of the re­ cent developments. In 1 97 3 En flo showed that a separa­ ble Banach space need not possess a Schauder basis. Gowers and Maurey [ 1 0] discovered hereditarily indecomposable ( HI for short) separable Banach spaces in 1 99 1 . independently exhibiting a space that does not contain an uncon­ ditional basic sequence. Their tech­ niques allowed Govvers to answer neg­ atively Banach's hyperspace problem . since a HI space cannot he isomorphic to any of its proper suhspaces; in par­ ticul ar, it is not isomorphic to its hy­ perplanes. We recall that a Banach space is HI ( compare [2] ) if no infinite­ dimensional closed subspace is the topological direct sum of two further infinite-dimensional closed subspaces of it. Equivalently, for any two infinite-

dimensional subspaces the distance be­ tween their unit spheres is zero. The combinatorial approach to ana­ lytic problems from l3anach space the­ ory started in the 1 970s. However, the importance of the so-called combinato­ rial forcing method became clear only during the last decade , after some im­ portant open questions in Banach space theory had been answered with its help. It started with the concept of a spread­ ing model of a basic sequence on a Banach space clue to Brune! and Su­ cheston, which helped to exhibit con­ nections between the local theory of general Banach spaces and the local theory of classical Banach spaces. We cite another simple, but beautiful result of Ramsev tvpe about sequences in normed spaces . due to Elton and Odell [3]: Every infinite-dimensional normed space contains an infinite normalized sequence ( xu ) of \·ectors such that for some E > 0 and all rn * II. ll x"1 - xull > 1 + E. Probably the best-knmvn result of this kind is Rosenthal's theorem that a Banach space X does not contain an isomorphic copy of f 1 if and only if every bounded sequence in X has a \Yeakly Cauchy subsequence. The use of infinite-dimensional Ramsey theory in Banach spaces was developed fur­ ther by Gowers during investigations which were part of the work for �·hich he �·as awarded the Fields Medal in 1 998. As an example, we mention his positive solution to the homogeneous space problem of Banach. Gowers es­ tablished that if a Banach space is iso­ morphic to all of its infinite-dimensional closed suhspaces. then it is isomorphic to a Hilbert space. The contribution of Ramsey-theoretic ideas to this theorem is covered by his dichotomy result [7,Rl. which asserts that every Banach space contains a subspace vvhich either has an unconditional basis or is hereditarily in­ decomposable. This dichotomy. com­ bined with some analytical results due to Komorowski and Tomczak-]aegem1ann. implies a positive solution to the homo­ geneous space problem. It is natural to attempt to use con­ cepts from Ramsey theory to obtain di­ chotomy results in mathematics ( sec [9] ) . The basic idea is to colour appro­ priate objects according to whether they support some good property or not . However. first. suitable objects have to he identified and a reasonable conjec-

ture has to he formulated. I n Banach space theory one useful a pproach turned out to he the stat<.:ment: if a space fails to contain a subspace pos­ sessing some good property, then it contains a subspace in which the good property does not hold in some very strong sense. One should remark that, i n the con­ tinuous setting, Ramsey methods in general determine sets that are only ap­ proximately monochromatic. Indeed. often classical Ibmsey theory cannot be directly applied to continuous prob­ lems. However. the known ideas and methods from combinatorial Ramsey theory inspire new techniques and prin­ ciples which are applicable to Banach spaces. We mention an instructive ex­ ample. The reader will note that the function f below can be interpreted as a kind of continuous colouring. Given a Banach space X. let S(X) de­ note its u nit sphere. A function f : 5( X) --> IR is called oscillation stable on X if for all infinite-dimensional closed subs paces Y of X and E > 0 there exists a closed infinite-dimensional sub­ space Z of }' such that sup{l /(x) ­ f(y) : x.y E S( Z)) < E. A combination of difficult results of several mathemati­ cians yields the following remarkable conclusion ( compare, e . g . , [5] ) : F'or an iz?fi'n it!!-dimensional Banach spac!! X !!1'!!1)' Lipschitzjiozction f : S( X) --> IR is

oscillation stahl!! if and on�v tf every closed inj!zzite-dimensional suh:,pace Y q/X contains an isomo1ph of c0.

The hook under review contains m·o sets of notes that were originally pre­ pared for an Advanced Course on Ram­ sey J\lethods in Analysis gi\·en at the Cen­ tre de Recerca Matematica. Barcelona. in January 2004. Part A is titled "Saturated and Con­ .. ditional Structures in Banach Spaces ( with t\vo Appendices) and is due to Spiros A. Argyros. It describes a general method of building norms with desired properties and presents in particular the theory of H I Banach spaces. The approach to HI extensions of a ground norm shares many ideas with the extension of models in set theory. The ground norm can be considered the initial model and its HI extension a new Banach space which is H I and at the same time presen·es properties of the initial space. Part A also presents a non­ sepa rable ret1exive Banach space con-

© 2007 Spnnger Setence -r- Bus1ness Med 1a . Inc , Volume 29. Number 2 . 2007

65

tammg no unconditional basic se­ quence (compare [ 1 ]) . Part B is titled "High-Dimensional Ramsey Theory and Banach Space Geometry·· and is due to Stevo Todor­ cevic. It explains in fou r sections Ramsey-theoretic methods relevant to modern Banach space theory: Finite­ dimensional Ramsey theory. Ramsey theory of finite and infinite sequences. Ramsey Theory of finite and infinite block sequences, and approximate and strategic Ramsey theory of Banach spaces. In particular, Nash-Williams's methods are used in the proof of Rosen­ thal's theorem stating that every weakly null sequence ( x,z) in a Banach space contains either a subsequence ( x,,) all of whose subsequences are Cesaro sum­ mahle, or a subsequence ( x,,,) whose spreading model is isomorphic to e j . Furthermore Part B gives a detailed exposition of the block-Ramsey theory developed hy Gowers. If F I N denotes the collection of all finite nonempty subsets of N. a finite or infinite se­ quence (xz) of elements of F IN is called a block sequence if xi < :c1 whenever i < j. ( For x. y E F I N. x < y denotes the fact that max(x) < min(_y) . ) Todorcevic argues that while the space N [x J has numerous interesting ap­ plications to Banach space theory, the block spaces such as F I N 1"'l of all in­ finite block sequences of finite sets seem to he more relevant to the deeper problems of that theory. The basic Ram­ sey-type result about block sequences is a pigeonhole principle for F I N due to Hindman which says that if F I N is coloured with finitely many colours, then there exists an infinite block se­ quence (arz) such that all nonempty unions of finitely many of the sets a 11 have the same colour. Among other things, Todorcevic dis­ cusses how F I N [xJ , endowed with some appropriate topology, satisfies analogues of results that hold for N lxJ equipped with the Ellentuck topology. In the last section of the book, he then shows that Gowers's dichotomy theorem suggests a corresponding Ramsey theory of finite and infinite block sequences in Banach spaces with Schauder bases. He notes that in this setup an unexpected new phenomenon occurs: the classes of ap­ proximately and strategically Ramsey sets are, in general, no longer closed un­ der the operation of complementation.

66

THE MATHEMATICAL INTELLIGENCER

The lecture notes should serve their purpose to give a first condensed in­ troduction to some of the most recent advanced investigations in its area fairly \Vel!. To this end. it might he advisable to reverse the alphabetic order and read the second part of the book first. After having mastered the basic techniques in the four discussed version.s of abstract Ramsey theory . the reader will then be able to appreciate fully the sometimes necessarily highly technical and delicate methods in the first part of the volume. which often are related to the recent re­ search of the authors. In the light of all the rather compli­ cated constructions outlined above. one is likely to hegin to wonder-like Gow­ ers at the end of his address [6]­ whether there might he a theory of .. easily described .. Banach spaces, ,-ery different from the general theory. that would eliminate many of the peculari­ ties discussed in this review. At present, however. it seems unclear hem· such a theory can be built. The approach taken by the authors of the presented hook is remarkably dif­ ferent. They propagate the message that apparently unpleasant spaces form an integral and interesting part of classical Banach space theory and are something we have to get used to.

nach spaces, Geom. Funct. Anal. 6 (1 996), 1 083-1 093. [8] W. T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies, Ann . of Math. (2) 156 (2002), 797-833.

[9] W. T. Gowers, Ramsey methods in Ba­

nach spaces, Handbook of the Geometry of Banach Spaces, vol. 2, North-Holland, Amsterdam, 2003, pp 1 07 1 - 1 097.

[ 1 0] W T. Gowers and B. Maurey, The un­ conditional basic sequence problem , J .

Amer. Math. Soc. 6 (1 993), 851 -874. [1 1 ] R. L. Graham, B. L. Rothschild , and J. H. Spencer, Ramsey Theory, (2nd ed.), Wiley lnterscience, New York, 1 990. [ 1 2] J. B. Kruskal, The theory of well-quasi­ ordering: a frequently discovered concept, J. Combinatorial Theory Ser. A 13 (1 972), 297-305. [1 3] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1 930), 264-286. [1 4] B. S. Tsirelson, Not every Banach space contains an imbedding of fp or c0, Funct

Anal. Appl. 8 (1 974), 1 38-1 4 1 . Department of Mathematics and Applied Mathematics University of Cape Town Rondebosch 7701 South Africa e-mail: kunzi@maths. uct.ac.za

REFERENCES

(1 ] S. A. Argyros, J. Lopez-Abad, and S. Todorcevic, A class of Banach spaces with few non-strictly singular operators, J . Funct. Anal. 222 (2005), 306-384. [2] S. A. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Mem. Amer. Math. Soc. 1 70 (2004), no. 806. [3] J. Elton and E. Odell, The unit ball of every infinite-dimensional normed linear space contains a (1 + E)-separated sequence,

Gode l's Theorem: An I ncomplete G u ide to Its U se and Abuse h.v Torkel Franzen

Colloq. Math. 44 (1 98 1 ) , 1 05-1 09.

A. K. PETERS, WELLESLEY, MASSACHUSETIS,

( 1 973), 1 93-1 98.

t the Godel Centenary Confer­ ence, ·'Horizons of Tr�nh, " held I at the University of Vienna in April 2006. Solomon Feferman paid tribute to the work of the late Torkel Franzen. Feferman's comments, printed on the back cover of Code! :> Theorem:

[4] F. Galvin and K. Prikry, Borel sets and Ramsey's theorem, J. Symbolic Logic 38

[5] W. T. Gowers, Lipschitz functions on clas­ sical spaces, European J. Combin. 13 (1 992), 1 4 1 -1 5 1 . [6] W . T . Gowers, Recent results in the the­ ory of infinite-dimensional Banach spaces, Proceedings of the International Congress

2005, 172 pp, ISBN 1-56881-238-8, $24 . 9 5

REVIEWED BY GARY MAR

L

Birkhauser, Basel, 1 995, pp. 933-942 .

A n Incomplete Guide to Its U1·e and A huse, succinctly pinpoint Franzen's

(7] W. T. Gowers, A new dichotomy for Ba-

distinctive achievement: "This unique

of Mathematicians, Vol. 1 ,2 (Zurich, 1 994),

exposition of Kurt Geidel's stunning in­ completeness theorems for a general audience manages to do what none other has accomplished: explain dearly and thoroughly just what the theorems really say and imply and correct their di,erse misapplications to philosophy, psychologv. physics . theology. post­ modernist criticism and what have you . " Franzen's hook will h e o f interest to three audiences: ( 1 ) beginning logic stu­ dents who want a concise and self­ contained explanation of what Gi'Jdd's theorems do say: ( 2 ) non-mathematically trained scholars and educated layper­ sons who want a logically correct ex­ planation of what Geidel's theorems do not say: and (:3 J professional logicians \Vho \\·ant a comprehensive, and criti­ cal. sun ey of the philosophical per­ spectin�s opened up by Giidel"s \York. 0 0 0

Logic students now have access to manv popular accounts of Geidel's life and \vork . among them '\fagel and New­ man's classic exposition Coder, Proq/ ( 19"i9) and Douglas Hofstadter's Puli­ tizer-Prize-winning G'6de/. Escher. Bach 0979 ) . and, more recently, John Casti and Werner DePauli"s G6del: A L!f"e ()/ Logic ( 2000 ) , based on an Austrian na­ tional television documentary, as wdl as Rebecca Goldstein"s novelistic biog­ raphy. Incompleteness: The Pro()/ and Paradox c;j"Kzt 11 Gc!del ( 2005 ) ( revie\ved in The "tfathematical IntelliRencer. \ ol 2H. no. 4. 2006 ) . Hmve\·er. these books tend to sacrifice technical correctness for public comprehensibility: none of them comment in detail on the many misstatements and missapplications of Geidel's theorem. and some commit the very errors Franzen exposes. Steering the beginning student clear of some common confusions. Franzen explains technical terms and poses instructive questions: Godel published the completeness theorem 0930) for his doctoral dis­ sertation and then in the following year published his celebrated in­ completeness theorem 0 93 1 ) . The latter is not the negation of the for­ mer. What are the t\\·o quite distinct meanings of completeness in these two landmark theorems hy Gi:idel­ the former concerning first-order logic and the latter concerning Pea no Arithmetic? •

Although it is common to speak of the incompleteness theorem, there are ac­ tually tzro incompleteness theorems, known as Geidel"s First and Second Incompleteness Theorems. Contem­ porary formulations of both theorems talk about formal systems that "con­ tain a certain amount of arithmetic. " What two different requirements are meant by this single phrase? One important simplification of Gi'Jdel's first incompleteness theorem was discovered by J. Barkley Rosser 0 936 ) . What is the difference be­ tween Rosser's notion of simple con­ sistency and Gi.'l del"s original formu­ lation of his first completeness theorem in terms of w-consistency ? Goldbach"s famous unproven con­ jecture states that every e\·en num­ ber greater than 2 is the sum of t\vo primes. How is Rosser's simplifica­ tion related to the fact that Gold­ bach-like statements ( i .e . . statements with the same logical form a s Gold­ bach's conjecture. known as TI-0-1 statements) that are undecidable must be true? GiJdel 's incompleteness theorem, con­ trary to some misstatements, does not imply that euerv consistent fom1al sys­ tem is incomplete. The Theory of Real Numbers. for example, is complete. Hmv is this possible since the Real Numbers include the Natural Num­ bers of arithmetic' Moreover, certain subtheories of Peano Arithmetic such as Presberger Arithmetic 0928 ) , are decidable. Four years after the publication of Godel's incompleteness results. Ger­ hard Gentzen ( 1935 ) published a proof of the consistency of elementary arithmetic making use of a generalized version of mathematical induction, known as transfinite induction. Why doesn't Gentzen·s result conflict with Gi'ldel's Second Incompleteness The­ orem, which concerns the unprov­ ability of consistency for a wide spec­ trum of fonnal systems? Chapter 2, "The Incompleteness Theo­ rem: An Overview, " introduces the reader to the First Incompleteness The­ orem, its relation to Hilbert's Non Ig­ nora himus view of mathematics, and its irrelevance with regard to explaining the "Postmodern condition.'' Chapter 3 . "Computability, Formal Systems. and In­ completeness," explains the conceptual •

•

•

•

connections among the logical notions of computability , formal systems, and incompleteness. These initial chapters of Franzen's hook, then, give the he­ ginning logic student a correct and con­ cise account of what the Giidel incom­ pleteness theorems actually do say. 0 0 0

Readers who are not mathematically in­ clined hut are intrigued by the many claims about the implications of Godel"s work will find Franzen a sober and re­ liable guide in explaining what Geidel's theorems do not say. For example, does Godel"s theorem show that a Theory of Everything ( TOE) in theoretical physics is impossible? Do Gi'ldel"s theorems re­ fute the strong Artificial Intelligence ( A I ) thesis that the human mind c a n be mod­ eled by a computer? ·'No mathematical theorem," Franzen notes. "has aroused so much interest among nonmathemati­ cians as Gi'Jdel's incompleteness theo­ rem. '' Indeed, Franzen's book grew out of taking on the exhausting task of com­ menting on the seemingly inexhaustible erroneous references on the Internet to Godel"s incompleteness theorems. Franzen discusses misuses of the in­ completeness theorems in theoretical physics and theology (Chapter 4), in skeptical arguments about mathematical knowledge (Chapter 5 ) , and i n the Lu­ cas-Penrose arguments about the limita­ tions of Artificial Intelligence (Chapter 6 ) . He dispatches his task with great clarity and a little self-ref1ective humor. After ac­ knowledging his colleagues in the pref­ ace. Franzen drolly comments: "For any remaining instances of incompleteness or inconsistency in the book, I consider myself entirely blameless, since after all, Godel proved that any book o n the in­ completeness theorem must be incom­ plete or inconsistent. Well, maybe not." ··Godel's theorem is an inexhaustible source of intellectual abuses, " note Alan Sokal and Jean Bricmont i n Fashionable

:Von sense: Postmodenz Intellectuals ' A buse ()/ Science 0 997) , a continuation of the discussion raised by the famous hoax in which Sokal's parody of a postmodern article was accepted for publication in a literary journal . Had Franzen limited his sites to debunking postmodern, political, or poetic invoca­ tions of Godel's theorem that were "ob­ viously nonsensical, · · this book could easily have settled into a smugness that

© 2007 Springer Sc 1ence- Bus1ness Med1a. Inc . . Volume 29. Number 2 . 2007

67

comes from dispatchi ng strawm an ar­ guments. Franzen aims higher. In Chapter 4 Franzen discuss es the claim that, because of Gi.idd 's theorem. the physicist's dream of a Theory of Everything is not only unattained, hut theoretically unattainable. In his essay. "" The World on the String"" in the 1\cu • York Reuiew q/ Books ( 2004 ) , Freeman Dyson argued: ""Another reason Yvhy I believe science to he inexhaustible is Giidel"s theorem . . . . His theorem im­ plies that pure mathematics is inex­ haustible. No matter ho\\" many prob­ lems we soh·e. there \Vill always he other problems that cannot be soh·ed within the existing rules. :'\ow I claim that because of GC">del"s theorem. . physics is inexhaustible too. . In his talk. "GC">del and the End of Physics. " Stephen Hawking has argued similarly: ""In the standard positivist approach to the philosophy of science. physical the­ ories live rent-free in a Platonic heaven of ideal mathematical models . . . . But we are not angels who view the uni­ verse from the outside. Instead, we and our models are both part of the universe we a rc describing. Thus, a physical the­ ory is self-referencing. like in Giidel"s theorem. One might therefore expect it .. to be either inconsistent or incomplete . Do GC">del"s theorems ha,·e such uni­ versal implications' Drawing on Feter­ man's reply to Dyson in the Neu• York Reuieu• of Books (www.nybooks.com/ articles/1 7249 ) . Franzen explains: "The basic equations of physics. whatever they may he. cannot indeed decide every arithmetical statement, but whether or not they are complete considered as a description of the physical world, and what completeness might mean in such a case, is not something that the in­ completeness theorem tells us anything abou t . " In other words. the incomplete­ ness of the arithmetic component of a physical theory need not imply any in­ completeness in the description of the physical world. In Chapter 5 Franzen critically dis­ cusses the claims advanced by J R. Lu­ cas ( 1 961 ), and updated more recently by Roger Penrose in his Emperors New Mind ( 1 989) and Shadows c�l the Mind 0 994 ) . Lucas argued that no matter how complicated a machine we constru ct, it will correspond to a formal system, which, in turn, will be subject to a Godelian construction for finding a for-

68

THE MATHEMATICAL INTELLIGENCER

mula unprovable in that system. De­ fending Lucas's conclusion. Penrose up­ dates the argument in an attempt to show that the aspirations of strong Ar­ t ificial Intell igence ( AI ) are doomed to failure . going on to conjecture that a non-computational extension of quan­ tum mechanics will someday provide a theory of consciousness. Geidel's own remarks on the subject ( in his unpublished 1 9 5 1 Josiah Willard Gibbs Lecture at Brown Cni,·ersity, see vol. III, Collected Works r:l Kl/11 Gr)del. edited by Feferman et a/. ) are more cau­ tious and nuanced: The human mind is incapable of formulating ( or mechanizing) all its mathematical intuitions. I . e . : If it has succeeded in formulating some of them, this very fact yields new in­ tuitive knowledge. e.g . . the consis­ tency of this formalism. This fact may be called the "incompletahility" of mathematics. On the other hand. on the basis of what has been proved so far, it remains possible that there may exist ( and even he empirically discoverable ) a theorem­ proving machine \vhich in fact is equivalent to mathematical intuition, hut cannot he fJn!l 'ed to he so . nor even proved to yield only correct theorems of finitary number theory. The second result is the follow­ ing disjunction: Either the human mind surpasses all machines ( to he more precise: it can decide more number-theoretic questions than any machine) or else there exist number-theoretic questions unde­ cidable for the human mind. Criticizing Lucas and Penrose, Franzen argues that ""we have no basis for claim­ ing that we ( ' the human mind') can out­ prove a consistent formal system " be­ cause Gi.i del"s theorem only implies the equiL•alence of the consistency of the formal system and the Goclel statement asserting its own unprovahility. In gen­ eral, however, we have no guarantee that the formal system in question is consistent. an assumption required for u s to draw the conclusion there is a truth u nprovable in the formal system. And what about the weaker claim that there could not be any formal sys­ tem that exactly represents the human mind as far as its ability to prove arith­ metical theorems is concerned? Franzen criticizes Hofstadter·s reflections to this

eflect from Gudel. E-;cher, Bach. noting Hofstadter's informal remarks have at least '"the virtue of making it explicit that the role of the incompleteness theorem is a matter of inspiration rather than implication ·· : The other metaphorical analogue to GC">del"s Theorem which I find provocative suggests that ultimately. v.:e cannot understand our own minds/brains. . . . All the limitative theorems of mathematics and the themy of computation suggest that once your ability to represent your own structure has reached a certain critical point. that is the kiss of death: it guarantees that you can never represent yourself totally. In such metaphorical statements. Franzen notes. the inability of a formal system to prove its own consistency is interpreted as the inability of the sys­ tem to "analyze or justify itself. or as a . kind of blind spot . . The problem with such a view is that ""the metaphor un­ derstates the difficulty for a system to prove its O\vn consistency. . [T]he unprovability of consistency is really the unassertihility of consistency. A sys­ tem cannot truly postulate its own con­ sistency. quite a part from questions of analysis and j ustification . although other systems can truly postulate the .. consistency of the system . 0 0 0

As noted above, Franzen's first two goals were to explain accurately what Geidel's theorems do say to the begin­ ning logic student and to curb the en­ thusiasm of the nonmathematically in­ clined who have heard exaggerated claims about the philosophical and mathematical implications of Gi'l del"s theorem by pointing out what they do not say. Franzen's book will also he of interest to logicians who \vant a model of sober clarity for explaining the philo­ sophical perspectives opened up by Gbdel's work. Goclel's theorems are stunning and significant enough "with­ out any exaggerated claims for the[ir] revolutionary impact." In Chapter 7 Franzen discusses the conceptual connections among GC">del's Completeness Theorem, non-standard models of arithmetic, and the Incom­ pleteness Theorems. Chapter 8 covers misleading fonnulations of incomplete­ ness in terms of Kolmogorov-Chaitin

complexity. Gregory Chaitin is known for his information-thecJretic interpretation of Giidel's theorem ( 196S ) and for his dis­ covery of the Halting Probability !l ( also known as Chaitin"s number). As Chaitin touts his results in 1be l/nknowable 0999 ) : "'In a nutshell. Giidel discovered incompleteness. Turing discovered un­ computahility. and I discovered random­ ness-that"s the amazing bet that some mathematical statements are true for no reason, they're true by accident. " How­ ever, Chaitin's informal explanation that ·· . . . if one has ten pounds of axioms and a twenty-pound theorem. then the theorem cannot he derived from those axioms"' is misleading. I n a recent book "V!cta/1/ath ( 200') ), Chaitin expands upon this informal account: we·re really going to get irreducible mathematical bets. mathematical facts that ·are true for no reason. · and \Vhich simulate in pure math. as much as is possible. indepen­ dent tosses of a fair coin . . . . " The prob­ lem with Chaitin's informal explanation. as Franzen points out, is that Chaitin " s version of the Gi'ldel theorems does not deal with the complexity of the theorems themselves but instead with theorems that are statements about complexity. There is. moreover. an intriguing con­ nection between Giidel's incompleteness theorem and axioms of infinity: postu­ lating the existence of various infinite sets has formal consequences for ele­ mentary number theory that cannot he proved by elementary means. Most of the mathematics done today can he for­ malized within Zermelo-fraenkel set the­ ory with the axiom of choice ( ZFC ) . ZFC minus the axiom of infinity. ZFC- '", is equivalent in its arithmetic part to Peano Arithmetic, and so the Geidel incom­ pleteness theorems apply. Therefore, zrc - w is incomplete and does not im­ ply its own consistency. It turns out that ZFC (which includes the axiom of infin­ ity) can prove the consistency of zrc··w. So here we have a n example in which adding an axiom of infinity to a theory ( in p�llticular, ZFC -w) yields ne\V arith­ metical theorems ( the consistency of ZFC- "i) not provable within that original theory. Stronger axioms of infinity ex­ tending versions of ZFC also yield ne\Y arithmetical theorems not provable in the theories they extend. Franzen remarks: ·· rrom a philosophical point of view, it is highly significant that extensions of set theory by axioms asserting the existence

· NmY

of very large infinite sets have logical consequences in the realm of arithmetic that are not provable in the theory that they extend . " As yet. no arithmetical problem of tra­ ditional mathematical interest is known to be among the new arithmetical theo­ rems of extensions of ZFC by axioms of infinity. However, a step in this direction was taken with the Paris-Harrington The­ orem ( 1977 ) . The Paris-Harrington The­ orem is related to Ramsey's theorem that. for each pair of positive integers k and l greater than 2, there exists an integer R( k, / ) ( known as the Ramsev number) such that any graph with R( k. / ) nodes whose edges are colored reel or green will ei­ ther have a completely green subgraph of order k or a completely red suhgraph of order !. for example. at any party with at least six people. there are either three people who are all mutual acquaintances or mutual strangers. The Paris-Harrington Theorem. a combinatorial strengthening of Ramsey's theorem, was the first "nat­ ural" statement found to be true hut un­ provable in Peano Arithmetic. At the 1930 " Epistemology of the Ex­ act Sciences" Conference in Konigsberg, Gi'lclel quietly announced his First In­ completeness Theorem. Among confer­ ence participants were such eminent logicians as Rudolf Carnap and Arend Heyting. but only John von Neumann ap­ preciated the profound significance of Goclel's I ncompleteness Theorem. Not long afterward, von Neumann realized that the Second Incompleteness Theo­ rem could be obtained by formalizing the argument for the first. Communicating his discovery to Gi'ldel in a letter. von Neumann graciously declined to publish \vhen Godel informed him that this stun­ ning theorem was already discussed in his forthcoming "On Formally Undecid­ able Propositions in Principia Mathe­ matica and Related Systems I " 093l l. which would become a celebrated achievement of twentieth centmy logic. What was Gi'ldel's Second Incom­ pleteness Theorem and what effect did it have on Hilbert's program? In addition to constructing the Godel statement G for the formal system S, the argument es­ tablishing the implication "if S is consis­ tent. then G is not provable in S" could be carried out within S itself. Moreover. the property of being a Godel number of a proof in S is a computable one. and is consistent' is a Goldbach-like so

·s

statement. a statement which if blse, can he shown to be false by a computation. Thus. Giidel's Second I ncompleteness Theorem follows: if S proves the state­ ment Con( S"') expressing · S is consistent' in the language of S, then S proves G, and hence S is in t�ICt inconsistent. Hilbert's metamathematical program call­ ing for consistency proofs for formal sys­ tems such as arithmetic in which all fini­ tistic arguments can be formalized \\ as etlectively clashed by the Second In­ completeness Theorem. Franzen carefully points out three common misconceptions about the Sec­ ond Incompleteness Theorem. "first. it is often said that Geidel's proof shows G to be true. or to be ' in some sense · true. But the proof does not sh v G to be true. What we learn from the proof is that G is true if and only if S is consis­ tent. In this observation, there is no rea­ son to use any such formulation as 'in some sense true' . . . " Second. Godel's theorem does not rule out consistency proof'i using methods not formalizable within Peano Arithmetic. Third. ·'[a]nother aspect of the second incompleteness the­ orem that needs to be emphasized is that it does not show that S can only be proven consistent in a system that is stronger than S" For example. Gentzen proved the consistency of Peano Arith­ metic ( PA ) in 1 936 by application of an arithmetically expressible instance of transfinite induction up to Cantor's ordi­ nal e0 ( the least fixed point under orcli­ nal exponentiation to the base w), while otherwise using arguments that can be formalized in a Yery weak subsystem of PA. So the consistency of P A is proved in a system that overlaps in part with P A hut is not an extension of it. On the other hand, it has been ar­ gued that if a system S like PA has been accepted as intuitively true, then one ought to accept the consistency state­ ment Con( S ) for S. That will give rise to a new formal system S' obtained by ad­ joining Con( S ) to S Now S' is also in­ tuitively true, so the process can he it­ erated. in t�Kt . through the constructive transfinite. Alan Turing ( 1939) showed that one could obtain completeness for Goldhach-like statements for ordinal logics obtained by iterating consistency statements into the constructive transfi­ nite starting \Vith PA. Later, Fefennan 0 96-4 ) showed that one could obtain a progression that is complete for all arith-

m

© 2007 Spnnger SCience t-Bus1ness Media, Inc., Volume 29, Number 2, 2007

69

metical statements by iterating certain reflection principles . Franzen's other book, Inexhaustihilitv: A Non-exhaus­ tive Treatment (ASL Lecture Notes in Logic "'16, 2004) contains an excellent exposition of the incompleteness theo­ rems. and the reader is k:d step-by-step through the technical details needed to establish a significant part of Feferman·s completeness results for iterated ref1ec­ tion principles for ordinal logics. Torkel Franzen's untimely death on April 1 9 . 2006 came shortly before he "\v as to attend. as an invited lecturer. the Godel Centenary Conference, "Hori­ . zons of Truth, . held at the L'niversity of Vienna later that month. This. and his invitations to speak at other conferences featuring a tribute to G()del. testifies to the growing international recognition that he deserved for these \\·orks . ACKNOWLEDGMENTS

I thank Solomon Feferman for substan­ tive and insightful correspondence dur­ ing the preparation of this revie\\·. and Robert Crease. Patrick Grim, Robert Shrock, Lorenzo Simpson, and Theresa Spork-Greemvood for their intellectual and material support for my participa­ tion in the Godel Centenary ·'Horizons of Truth'' Conference in Vienna.

Department of Philosophy Stony Brook University Stony Brook, New York 1 1 794-3750 USA e-mail: [email protected]

The Art of Conjecturing

together with Letter

to a Friend on Sets in Court Tennis by jacob Bemoulli

translated with an introduction and notes hy Edith Dudley .�vlla BALTIMORE, THE JOHNS HOPKINS U N IVERS ITY PRESS, 2006, xx + 430 PP. £46.50 ISBN 0-80188235-4.

REVIEWED BY A . W. F. EDWARDS

70

THE MATHEMATICAL INTELLIGENCER

l

n 1 9 1 5 the young stat1st1c1an R. A. Fisher. then 2 5 . and his former stu­ dent friend C. S. Stock wrote an ar­ ticle [ 1 1 bewailing the contemporary ne­ glect of The Origin q/ Species: So melancholy a neglect of Darwin's work suggests reflections upon the use of those rare and precious pos­ sessions of man-great books. It \vas. \\ e believe. the custom of the late Pro­ fessor Freeman [2] to warn his stu­ dents that mastery of one great hook \\·as worth any amount of kno\\·ledge of many lesser ones. The tendency of modern scientific teaching is to ne­ glect the great hooks, to lay far too much stress upon relatively unimpor­ tant modern work. and to present masses of detail of doubtful truth and questionable weight in such a way as to obscure principles . . . . Hmv many biological students of today have read The Origin? The majority know it only from extracts, a singularly ineflecti\·e means. for a work of genius does not easily lend itself to the scissors: its unity is too marked. Nothing can re­ ally take the place of a first-hand study of the work itself. With her translation ofJacob Bernoulli's A r:s- Conjectandi in its entirety Edith Sylla now makes available to English­ speakers without benefit of Latin another great book hitherto known mostly from extracts. As she rightly observes, only thus can we at last see the full context of Bernoulli s theorem. the famous and fundamental limit theorem in Part IV that confirms our intuition that the propor­ tions of successes and failures in a sta­ ble sequence of trials really do converge to their postulated probabilities in a strict mathematical sense, and therefore may be used to estimate those probabilities. However, I must resist the tempta­ tion to review A rs Cm�jectandi itself and stick to Sylla's contribution. She thinks that it ·deserues to he considered the

founding document qj' mathematical probability ', hut I am not so sure. That honour belongs to Bernoulli's prede­ cessors Pascal and Huygens, who math­ ematized expectation half a century ear­ lier: Bernoulli's own main contribution was 'The Use and Application of the Preceding Doctrine in Civil, Moral, and Economic Matters· (the title of Part IV) and the associated theorem. It \vould he more true to say that Ar:s- Conjectandi is the founding document of mathemati-

cal statistics. for if Bernoulli's theorem were not true. that enterprise would be a house of cards. ( The title of a recent hook by Andres Hale! says it all: A His­

tmy q/ Parametn'c Statistical h?/erence from Bemoulli to Fisher. 1 713-- 193 5 [3] . ) When I first became interested in Bernoulli's hook I was very fortunately placed. There v;as an original edition in the college libra1y ( Gonville and Caius College, Cambridge) and amongst the other Fellows of the college was Pro­ fessor Charles Brink. the University's Kennedy Professor of Latin. Though I have school Latin I was soon out of my depth, and so I consulted Professor Brink about passages that particularly interested me. Charles \vould fill his pipe. settle into his deep wing-chair and read silently for a while. Then, as like as not, his open­ ing remark would be ·Ah, yes, I remem­ ber Fisher asking me about this passage'. Fisher too had been a Fellow of Caius. I'\ow, at last, future generations can set aside the partial. and often amateur. trans­ lations of An; Conjecta11di and enjoy the whole of the great work professionally translated, annotated. and introduced by Edith Sylla. in a magisterial edition beau­ tifully produced and presented. She has left nu stone unturned, no correspon­ dence unread, no secondaty literature un­ examined. The result is a work of true scholarship that \viii leave every serious reader weak with admiration. Nothing said in criticism in this review should be construed as negating that. The translation itself occupies just half of the long book 2 1 3 pages. An­ other 146 pages are devoted to a pref­ ace and introduction. and 22 to a 'trans­ lator's commentary'. Next come 4 1 pages with a translation of Bernoulli's French Letter to a Friend 011 Sets in Cow1 Tennis which was published with A rs Conjectandi and which contains much that is relevant to the main work; a translator's cmnmenta1y is again ap­ pended. Finally. there is a full bibliog­ raphy and an index. In her preface Sylla sets the scene and includes a good survey of the secondary literature (Ivo Schneider's chapter on Ar:> Conjectandi in Landmark Writings in Western Mathematics 1640- 1940 [4] ap­ peared just too late for inclusion). Her introduction 'has four main sections. In the first, I review briefly some of the main facts of Jacob Bernoulli's life and its so­ cial context. . . . In the second, I discuss

Bernoulli's other vvritings insofar as they are relevant . . . . In the third, I describe the conceptual backgrounds . . . . Finally . in the fourth. I explain the policies I fol­ lowed in translating the work . · The first and second pans are extremely detailed scholarly accounts \Yhich will be stan­ dard sources for many years to come. The third, despite its title ' Historical and Conceptual Background to Bernoulli's Approaches in A1:s- Cmzjectandi · . turns into quite an extensive commentary in its own right. Its strength is indeed in the discussion of the background, and in par­ ticular the placing of the Llmous 'prob­ lem of points' in the context of early busi­ ness mathematics, but as commentary it is uneven. Perhaps as a consequence of the fact that the book has taken manv years to perfect, the distribution of material be­ tween preface, introduction. and trans­ lator's commentary is sometimes hard to understand. with some repetition. Thus one might have expected com­ ments on the technical problems of translation to be included under 'trans­ lator's commentary', but most-not all-of it is to be found in the intro­ duction. The distribution of commen­ taJy between these two parts is con­ fusing. but even taking them together there are many lacunae. The reason for this is related to Sylla 's remark at the end of the intro­ duction that 'Anders Halcl, A. W. F. Eel­ wards, and others, in their analyses of A rs Conjectandi, consistently rewrite what is at issue in modern notation . . I have not used any of this modern no­ tation because I believe it obscures Bernoulli's actual line of thought. · I and others have simply been more interested in Bernoulli's mathematical innovations than in the historical milieu, whose elu­ cidation is in any case best left to those. like Sylla. better qualified to undertake it. Just as she provides a wealth of in­ formation about the latter, she often passes quickly over the former. Thus C pp . 73, 345 ) she has no detailed comment on Bernoulli's table (pp. 152-153) enumerating the frequencies with which the different totals occur on throwing n dice. yet this is a brilliant tab­ ular algorithm for convoluting a discrete distribution, applicable to any such dis­ tribution. In 1 865 Todhunter [5] ·espe­ cially remark[ eel]' of this table that it was equivalent to finding the coefficient of x'"

in the development of (x + x2 + .x .3 + x ' + x� + .xr, ) 11, where n is the number of dice and m the total in question. Again C pp . 74-75. 345 ) . she has nothing to say about Bernoulli's derivation of the bino­ mial distribution ( pp . 1 65-167), which statisticians rightly hail as its original ap­ pearance. Of course, she might argue that as Bernoulli's expressions refer to ex­ pectations it is technically not a proha­ hili(V distribution. but that would be to split hairs. Statisticians rightly refer to 'Bernoulli trials' as generating it, and might have expected a reference. Turning to Huygens · s Vth problem C pp. 76. 345 ) , she does not mention that it is the now-famous 'Gambler's ruin' problem posed by Pascal to Fermat, nor that Bernoulli seems to he t1oundering in his attempt a t a general solution ( p . 1 9 2 ) . And she barely comments ( p . 80) on Bernoulli's polynomials for the sums of the powers of the integers, although I and others have found great interest in them and their earlier derivation by Faulhaber in 1 6 3 1 , including the 'Bernoulli numbers'. Indeed. it was the mention of Faulhaber in A rs Con­ jectandi that led me to the discovery of this fact ( see m y PascaL'> A rithmetical Triangle and references therein l6L Sylla does give some relevant references in the translator's commentary, p . 347). I make these remarks not so much in criticism as to emphasize that A rs Conjectandi merits deep study from more than one point of view. Sylla is probably the only person to have read Part III right through since Isaac Todhunter and the translator of the German edition in the nineteenth century. One wonders how many of the solutions to its XXIV problems contain errors. arithmetical or otherwise. On p. 265 Sylla corrects a number wrongly transcribed, but the error does not af­ fect the result. Though one should not make too much of a sample of one, my eye lit upon Problem XVII (pp. 275-8 1 ) , a sort o f roulette with four balls and 3 2 pockets, four each for the numbers 1 to 8. Reading Sylla's commentary (p. 83) I saw that symmetry made finding the ex­ pectation trivial, for she says that the prize is 'equal to the sum of the num­ bers on the compartments into which [the] four balls fall' (multiple occupancy is evidently excluded). Yet Bernoulli's calculations cover four of his pages and an extensive pull-out table.

It took me some time to realize that Sylla's description is incorrect, for the sum of the numbers is not the prize it­ self, but an indicator of the prize, ac­ cording to a table in which the prizes corresponding to the sums are given in two columns headed nummi. Sylla rea­ sonably translates this as ·coins', though ·prize money' is the intended meaning. This misunderstanding surmounted, and with the aid of a calculator. I ploughed through Bernoulli's arithmetic only to disagree with his answer. He finds the expectation to be 4 349/3596 but I find 4 1 53/1 7980 ( 4 . 097 1 and 4 .0085) . Bernoulli remarks that since the cost of each throw is set at 4 ' the player's lot is greater than that of the peddler' hut according to my calcula­ tion. only by one part in about 500. I should be glad to hear from any reader \Vl10 disagrees with my result. Sylla has translated Circulator as 'peddler' ( ' ped­ lar' in British spelling) but 'traveler' might better convey the sense, espe­ cially as Bernoul l i uses the capital A n d s o w e are l e d to t h e question of the translation itself. How good is it? I cannot tell in general, though I have some specific comments. The quality of the English is. however. excellent. and there is ample evidence of the care and scholarly attention to detail with which the translation has been made. I may remark on one or two passages. First, one translated in m y Pascal 's A nlhmetical Triangle thus: This Table [Pascal's Triangle] has truly exceptional and admirable properties; for besides concealing within itself the mysteries of Com­ binations, as we have seen, it is known by those expert i n the higher parts of Mathematics also to hold the foremost secrets of the whole of the rest of the subject. Sylla has (p. 206): This Table has clearly admirable and extraordinary properties, for beyond what I have already shown of the mystery of combinations hiding within it, it is known to those skilled in the more hidden parts of geom­ etiy that the most important secrets of all the rest of mathematics lie con­ cealed within it. Latin scholars will have to consult the original to make a judgment, but, set­ tling down with a grammar and a dic­ tionary 25 years after my original trans-

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lation ( with which Professor Brink will have helped) , I think mine better and closer to the Latin. I might now change 'truly' to '\vholly' and prefer ·mystery' in the singular ( like Sylla l , as in the Latin. as well as simply · higher mathematics'. But her ·geometry' for G'eometria is surely misleading. for in both eighteenth­ century Latin and French the word en­ compassed the whole of mathematics. Second. there is an ambiguity in Sylla 's translation ( p . 1 9 1 l of Bernoulli's claim to originality in connection \Vith a ' propertv of figurate numbers'. Is he claiming the property or only the demon­ stration? The latter, according to note 20 of chapter 1 0 of Pascal�' A rithmetical

have come to light: p. xvi, lines 1 and 2. De Moivre has lost his space; p. 73 . line 1 4 . Huygens has lost his ·g'; p. 1 ')2, the table headings are awkwardly placed and do not ret1ect the original in which thev clearly label the initial columns of Roman numerals; p. 297 n, omit diario ; and in the Bibliography . p. 40H. the reference in Italian .� houlcl ha\·e 'Accademid. and Bayes's paper \Yas published in 1764; p. 4 1 '), Kendall not Kendell; and, as a Parthian shot from this admiring re\·ie\ver. on p . 4 1 4 the ti­ tle of my hook Pascal 's Arithmetical Triangle should not be made to suffer the Americanism 'Arithmetic ·.

Triangle.

Gonville and Caius College

Third. consider Sylla's translation ( p . 329) of Bernoulli's comment on his great theorem in part IV: This. therefore . is the problem that I have proposed to publish in this place. after I have already concealed it for twenty years. Both its nm·elty and its great utility combined \Vith its equally great difficulty can add to the weight and value of all the other chapters of this theory. Did Bernoulli actively ccmceal it' In col­ loquial English I think he j ust sat on it for twenty years ( pressi ' l ; De Moivre [7] writes 'kept it by me· . And does it add weight and value. or add to the weight and value? De Moivre thought the former ( actually · high value and dignity' ) . This is also one of the pas­ sages on which I consulted Professor Brink . His rendering was: This then is the theorem which I have decided to publish here. after considering it for twenty years . Both its novelty and its great usefulness in connexion with any similar diffi­ culty can add weight and value to all other branches of the subject. I n one instance Sylla unwittingly pro­ vides two translations of the same Latin. this time Leibniz's ( p . 48n and p. 92 ) . The one has 'likelihood' and the other 'verisimilitude' for ·uerisimilitudo � And j ust one point from the French of the ' Letter to a Friend' ( p. 564 ): surely 'that it will finally be as probable as any given probability', not ·all'. Finally, in vie\v of the fact that this irreplaceable hook is sure to remain the standard translation and comme ntary for many years to come, it may be help­ ful to note the very few misprint s that

72

T H E MATHEMATICAL INTELLIGENCER

Cambridge, CB2 1 TA UK e-mail: [email protected] REFERENCES AND NOTES

[ 1 ] R. A. Fisher, C. S. Stock, "Cuenot on preadaptation. A criticism, " Eugenics Re­

view 7 ( 1 9 1 5) , 46-61 .

[2] Professor E . A. Freeman was Regius Pro­ fessor

of

Modern

History

at

Oxford,

1 884-92. [3] A. Hald, A History of Parametric Statistical Inference from Bernoulli to Fisher, 1 7 13-

1 935, Springer, New York, 2006. [4] I. Grattan-Guinness (ed.), Landmark Writ­ ings in Western Mathematics 1 640- 1 940,

Elsevier, Amsterdam, 2005.

[5] I. Todhunter, A History of the Mathematical Theory of Probability, Cambridge, Macmil­

lan, 1 865. [6] A. W. F. Edwards, Pascal's Arithmetical Triangle , second edition, Baltimore, Johns

Hopkins University Press, 2002. (7] A. De Moivre, The Doctrine of Chances, third edition, London , Millar, 1 756.

James Joseph Sylvester: Jewish M athematician in a Victorian World by Karen Hzmga Parshall ------- ·-----

T H E JOHNS HOPKI N S U NIVERSITY PRESS, BALTIMORE. 2006. xiii + 461 PP. $69.95, I S B N : 0-8018-8291-5.

REVIEWED BY TONY CRILLY

l

ames Joseph Sylvester OH14-1897l 1s well known to mathematicians. Was he not the scatter-brained ec­ centric who wrote a poem of four hun­ dred lines. each rhyming with Rosalind' And , lecturing on it . spent the hour nav­ igating through his extensive collection of footnotes, leaving little time for the poem itself:' Another story told by E . T. Bell is of Sylvester's poem of regret titled "A missing member of a family of terms in an algebraical formul a . " Such scraps inevitably ewJke a smile today. hut is his oddity all there is-stories and tales to spice a mathematical life? I3ell·s essays in A1en of /V!athematics have been int1uential for generations of math­ ematicians. hut his snapshots could not claim to be rounded biographies in any sense. This. then, is a review of the first full-length biography of the extraordi­ nary mathematician J. J. Sylvester. How can we judge a mathematical biography' On the face of it, writing the life of a mathematician is straightfor­ ward: birth. mathematics, death. Thus t1ows the writing formula: describe the mathematics, and top and tail with the brief biographical facts and stories. A possible variant is the hriet1y written life . followed by the mathematical her­ itage. There are many approaches, hut these are consistent \Vith William Faulkner's estimate of literary biogra­ phy, "he wrote the novels and he died . " According t o this hard-line view, biog­ raphy should not even exist. Yet Sylvester deserYes to be rescued from Bell's thumbnail sketch of the "Invariant Twins" in which he lumped Cayley and Sylvester together in the same chapter. Perhaps only in the genre of math­ ematical biography do possible subjects outnumber potential authors. A stimu­ lating article on writing the life of a mathematician. and an invitation to con­ tribute, has recently been published by John W. Dawson, the biographer of Kurt Godel. 1 Writing about another per­ son's life is a voyage of discovery about one's own life, and surely the biogra­ pher is different at the end of such a project. Writing about a period of his­ tory different from one's own also in­ volves some exotic time travelling. A central problem for writers whose subjects' lives were bounded by tech­ nical material is to integrate technical developments with the stories of those lives. This is almost obvious, hut it is

worth reconsidering in the l ight of Par­ shall ' s S)'!t •estcr. The mathematician turning to biography faces the challenge of not giving an exposition of the cur­ rent state of the subject's mathematical field in isolation from the milieu in \\·hich it was created. At the other end of the spectrum. a writer concentrating on the l ife may give no more than a side\\·ays glance at the mathematical de­ velopment and thereby descend into a refined form of gossip mongering. E. C. Bentley's catchy clerihe\v does not solve this central problem of integration: The art of Biography Is different from Geography. Geography is about maps. But Biographv is about chaps. Nor does Lance Armstrong offer a so­ lution in the succinct title of his ( auto lhiographv: " It's not about the . hike . . Greater weight cou ld he placed on both these dictums if the exclusiv­ ity \\ ere removecl. Biography is about chaps and maps. and the title of Lance Armstrong's biography should he " It's . not only about the hike . . alt hough ad­ . mittedly to lesser effect. Parshall' � title of her book about the mathematician Sylvester places him in the Victorian world. revealing her intention to make the l ife an undivided whole. Professor Parshall has spent the last twentv years in Sylvester's company and knows him wel l . She has brought her­ self to the point where she understands the breadth of his mathematics and its detail. itself no mean feat. More than this, she has come to u n derstand the historical background to Sylvester's mathematical problems and most im­ portantly his background to these prob­ lems. While she cannot know every­ thing about him, she knows a great deal. even aspects of his l ife he did not know himself. All this takes time-hut one should not be put off by the apho­ rism "that it takes a l ife to know a l ife . .. Mathematical biography can prof­ itably be compared with literary biog­ raphy because mathematicians and lit­ erary people are members of groups concerned with producing original work. James Boswell described the craft of [literary! biography as not only relat­ ing all the important events of a l ife in their order. hut interleaving what the subject privately wrote, and said. and

thought. In Dr Johnson, Boswell had a life coterminous w ith his own. and he \\·as familiar with the day-to-day cir­ cumstances which shaped the life of his subject. Sylvester's biographer lives in a different century from her subject and has had to face the additional challenge of becoming familiar with events and customs from a "foreign country" so nec­ ess;uy to faithfully immerse her subject in his own time and space. And while Boswell was in daily contact with John­ son, Parshall only ever met Sylvester in her imagination. In common with all bi­ ographers of the long-since dead. all she had were the surviving sources. among them Sylvester's own writings ( poetical and mathematical ) . ne\vspa­ pers. diaries. notebooks. and ahm·e all. the extant Sylvester letters-thousands of them scattered in archives all over Europe and America. Handling these documents. manv touched hy Svh-ester himself. brought him alive to the biog­ rapher and thence to the readers. Ac­ cording to .\lichael Holroyd. biography allows readers to read bet�·een the lines of a person's work: and in this case. mathematicians may he encouraged to read Sylvester's papers in their original \·ersions. In the end a completed biographv is what the writer makes of it-there are no general rules of what it should con­ sist of. apart from being constrained by the sources. The sources suggest a tale. hut it is the artist who selects and shapes the material. pumps the blood back into the subject. and ultimate!\� tells the storv . In some branches of literary bi­ ography it seems permissible to invent dialogue. hut this trend has not yet reached mathematical biography. For Parshall. the sources cannot he trans­ gressed, and time and place are to be respected. Thus we have a birth-to­ death biography of the traditional kind. There is still the question of tone, how­ ever; while Parshall is sympathetic to Sylvester's plight she is no hagiogra­ pher-her subject is certainly not a great man placed on a pedestal whose plinth is strewn with mathematical results. ;\lor is she an E . T. Bell. whose essays \Yere drawn from only a few sources-chiet1y the principal obituaries. I nstead she takes us through Sylvester's life, alert­ ing us to its pivotal points and to his mathematical contributions as he made them. Parshall lays before us Sylvester's

anxieties and predicaments and oh­ seiYes how he deals \\ ith them. I\ot all the way stations would have been cho­ sen hy Sylvester-indeed an autobiog­ raphy by Sylvester would have been \·et)' different. The structure of the hook is strictly chronological . Sylvester was the youngest son of a large \Veil-to-do Jewish family li\'ing in England. The family moved around the country hut settled in Lon­ don's East End by the time the hoy reached school age. Adopting life's . "slippet)' path . as a leitmotif. Parshall hegins the l ife of the young Jewish hoy living in a class-ridden land ruled by Anglicans for Anglicans. Quite early on, teachers recognised his mathematical t1air which combined \\'ith his Jewish­ ness to separate him from classmates and expose him to exclusion. A recur­ ring theme i n this portrait is Syh·ester the outsider yet the hov. the man. and the old man always ready to he in­ volved in battles and stand up to any kind of injustice. After a private education at primary schools in London. at the age of four­ teen Sylvester enrolled at University College London ( l'CU. \\·here Augustus De 'Vlorgan \Yas so int1uential . There oc­ curred the first of Sylvester's stumbles on the slippet)' path . in \vhich he de­ fended himself against school bullies. and his school education was com­ pleted at Liverpool's Royal Institution School. To his teachers, his mathemat­ ical talent was obvious, and hi.s family \\·as advised that their son shou ld at­ tend university. As mathematics was to he the subject of study, this meant Cam­ . bridge . the " holy city of mathematics . . Practising Jews coul d not graduate from Cambridge University. since they could not in conscience affirm the thirty-nine articles of the Anglican religion. but they were allowed to follmv the u niversity courses as any undergraduate would . Cambridge d id not close its doors o n talent, a n d Sylvester w a s free to enjoy a ful l university life except the right to put B.A. after his name. In 1 H37. he achieved the position of being second in the order of merit. a feature of the Cambridge system which ranked the hundred or so students according to their performance in the famed !'vlathe­ matica l Tripos examination. Each year the students were listed from the "Se­ nior Wrangler" at the top to the

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''Wooden Spooner" at the bottom, the rankings being grouped into three classes: Wranglers, Senior Optimes , and Junior Optimes. ( Here Parshall's list of the leading students in 1 837 is slightly amiss. A. J Ellis graduated sixth wran­ gler in 1 837 to Sylvester's second wran­ gler, whereas R. L. Ellis ( no relation) was the Senior Wrangler in 1 H40. ) His undergraduate career behind him, Sylvester applied for the vacant professorship of natural philosophy at UCL, \\·here he had been a pupil. Par­ shall reveals him as an astute young man, gathering an impressive list of sponsors willing to write him testimo­ nials. He was successful in gaining the appointment and set about fitting him­ self into a natural philosophy mould and learning the necessary lecturing skills-the latter, a considerable chal­ lenge. As De Morgan noted of Sylvester's performance in this period, ·'[w)hen he was with us he was an entire failure: whether in lecture room or in private exposition, he could not keep his team of ideas in hand. " Ever restless, Sylvester became embroiled \Vith the university authorities in London, \\·hich prompted his move to America , where it was re­ ported that a ·'little, bluff, beef-fed En­ . glish cockney . appeared in Virginia as the new professor of mathematics. Here Parshall is on home territory, herself a professor at the University of Virginia, the site of Thomas Jefferson's inspired educational experiment at Charlottesville. In America, Sylvester's path became more slippery and after only a short pe­ riod he was hack in London, this time without a job, his previous tenured po­ sition at UCL had been filled by another, and he was thrown back on his own devices. How could Sylvester earn a living in London during the tumultuous 1 840s? Unsettled in his personal life and in the doldrums mathematically, he needed a respectable position which paid a salary. As Parshall informs us, he found a niche in the Equitable Law Life Com­ pany, and a career in the embryonic in­ surance industry. Installed as a man of business in the City of London, he also found time and money to study for the legal Bar. A barrister had gentlema n sta­ tus in the stratified Victorian world of subtle class distinctions. At this time , he met the great prop in his life, Arthur Cayley, and so began their famous as-

74

THE MATHEMATICAL INTELLIGENCER

sociation . The idea that this collabora­ tion was uninhibited teamwork, as might he inferred from Bell, is dispelled by Parshall. From Sylvester's perspec­ tive, it \Yas an alliance built on both co­ operation and competition-the coop­ eration enriched by mutual support and daily discourse on mathematics, the competition ingrained by the education at Cambridge, \Vhich thrived on intensi\·e rivalry. Sylvester wrote only one joint mathematics paper (with James Ham­ mond ), towards the end of his life, hut Cayley never did, much to his regret. The 1 8')0s was an exciting time for Sylvester. Now qualified as a barrister and established as a well-known figure in the actuarial world. he felt that some­ thing was missing. He had done little mathematics in the previous decade; no\\· he aimed to set this right. In con­ cert with Cayley, he made new discm·­ eries in invariant theory, the modern al­ gebra of the day. This is the scene for the fairly well-known image of Sylvester ''sitting , with a decanter of port wine to sustain nature's flagging energies, in a . hack office in Lincoln's Inn Fields, . while for Parshall he ''was catching fire mathematically with Cayley fanning the . flames . . As he made inroads in invari­ ant theory \vith a train of hastily writ­ ten papers ( and driving his editors to despair) , he turned his attention to seek­ ing a career in the academic world. The painful reality was that there were few openings, especially for Jews. He tried for a position at the Woolwich Military Academy in London hut failed. A year later the death of the recently appointed professor at Woolwich gave him an un­ expected opportunity. Sylvester mar­ shalled his testimonial writers once again and was again successful. Many would regard Sylvester as nmY settled­ he had a springboard from which to re­ launch his mathematical career while treading water with regard to the teach­ ing of army cadets. But we are talking about] ames Joseph Sylvester. It was not long before he was at loggerheads with the Woolwich military authorities. These quarrels took their tolL and on the slippery path, Sylvester veered be­ tween the ecstasy of mathematical dis­ covery and the gloom of acute depres­ sion . Life at Woolwich for Sylvester was never easy. At the end of the 1 860s, a governmental enquiry into the educa­ tional operation at the Academy re-

suited in the amalgamation of profes­ sorships, and Sylvester, on the wrong side of the new retirement age, found himself without a job. He could do lit­ tle about it, but he fought the authori­ ties for a fair pension: no less a figure than William Ewart Gladstone came to the rescue in correcting a " lamentable departmental error'' and setting his pen­ sion to rights. Nevertheless, Sylvester was still in his fifties and needed to fill his days with useful activity. He set to work on his poetry, took singing lessons, made contributions to Penny Readings ( high-flown educational amusements for the working classes), and took an in­ terest in the broader issues of secondary school education . In between. occa­ sional enthusiasms for current mathe­ matical topics took hold of him, and he promulgated these with fervour. While all these activities buoyed him, they were not enough, and he knew it. A stroke of good fortune, after five years of intellectual tourism, gave Sylvester a chance to return to the aca­ demic mainstream. The new Johns Hop­ kins University was founded and its guiding light, Daniel Coit Gilman, was looking for European talent to establish a research reputation at Baltimore. Thus in 1 876 Sylvester was gainfully em­ ployed once again as a professor of mathematics, in this new graduate in­ stitution (with a salary he insisted should be paid in gold). As Parshall points out, Sylvester now found himself with a new set of problems. He had not taught at graduate level-ever-yet was expected to inspire students in research­ level mathematics. He had to give reg­ ular lectures, albeit to small audiences, hut students would now look to him for research guidance; the stuff of Penny Readings and the latest theories on the writing of poetry would cut no ice in this environment. What should Sylvester do? His research had always been sub­ ject to whim; now he needed a bulk of organised lecture material on which to draw. Sylvester fell hack on his great­ est success: invariant theory. In that sub­ ject in the 1 850s he had made gains, was recognised by the mathematical community as an authority, and could see that there were outstanding prob­ lems to solve. From the beginning of her book, Par­ shall reveals her enthusiasm for her un­ dertaking, and what better time to he-

gin the story than on a bright day: "In Baltimore, 22 February 1877 was a day of celebration. Bright. cloudless, un­ characteristically springlike, the birth­ day of America's first president was feted in fine style. Bunting hung from the bal­ conies. . . " His inaugural address. in Baltimore, suggests that Sylvester relished the chance of starting again in a coun­ try without the prejudices that he had endured in England. He addressed his newly found audience in 1 877: "Happy the young men gathered under our wing, who, u nfettered and untram­ melled by any other test than that of diligence and attainments. have here af­ forded to them an opportunity of filling . up a complete scheme of education . . This \Yas a pivotal point in Sylvester's life. and Parshall uses it as the opening scene of the book very effectively to in­ troduce a flashback. Without enthusi­ asm. the biographer cannot succeed. and here and throughout the book, her enthusiasm for writing the l ife matches Sylvester's own for mathematics. As with any of Sylvester's current re­ searches there was now, in the year 1 877, no field more worthy of study than invariant theory. It had everything-it was topical , there were routine calcula­ tions for students. and there were also challenging problems for himself to con­ sider. One desideratum was plugging the gap in ''Cayley's theorem" which purported to count the number of lin­ early independent invariants and co­ variants of a binary form. Both Cayley and Sylvester believed it true, and in­ deed, their calculations were based on its truth. But Cayley had made a crucial assumption, and while this caused no flurry in the 1 8 50s . the heightened im­ portance of "mathematical proof" in the 1 870s suggested it was a result which definitely needed proof. It was indeed a proud moment in Baltimore when Sylvester proved it. and. in announcing his success to the mathematical world. he blew his trumpet with all the Sylves­ trian puff he could muster. Less suc­ cessful were his attempts to reprove Gordan's theorem in which the German mathematician Paul Gordan had demon­ strated that the number of irreducible in­ variants and covariants was finite in the case of an algebraic binary form. A proof using only plain algebra would vindi­ cate the English methods of algebra and put the German semi-abstract calculus

in the shade. The quest took on a na­ tionalist dimension because Sylvester also believed that the so-called symbolic methods of Alfred Clebsch and Gordan had been appropriated from one of Cay­ ley's early discoveries in the theory of hyper-determinants. These objectives were theoretical, but of equal value in invariant-theory circles of the nineteenth century were the find­ ing and recording of the actual algebraic expressions for the invariants and co­ variants. For this Sylvester started up an ambitious calculatory program for the bi­ nary forms of the first ten orders (in mod­ ern language . for polynomials up to de­ gree ten ) . For the first four. the task was simple and had been recognised since the 1840s: the case of the binary quintic was thought to have been recently com­ pleted-though details in that case still awaited attention. Sylvester and his band of students took up the calculatory chal­ lenge with alacrity, and if a "truth" in the theory had not yet been confirmed by mathematical proof, they plunged on in the hasty heat of calculation. This was the inductive approach par excellence, and it is a measure of how Sylvester dif­ fered from mathematicians today ( at least he was able to publish work based on pure surmise') Sylvester did much to create a re­ search atmosphere at Johns Hopkins. The students were invigorated by hav­ ing a mathematician with a sound Eu­ ropean reputation among them, and if his teaching methods were idiosyncratic, stil l , he was a stimulating presence. Sylvester was instrumental in bringing the American journal of Mathematics into being, the first successful research journal in the country. Nevertheless, af­ ter seven successful years in Baltimore, Sylvester increasingly missed his former l ife in England, panicularly his London social circle based at the Athenaeum Club and at the Royal Society. The last act in Sylvester's dramatic career began with his appointment to the Savilian Chair of geometry at Ox­ ford in 1883. Even at the age of seventy he was not prepared to retire to the sidelines and rest on his laurels. He staned a mathematical society in the university and set out further grand the­ ories. One was a theory of reciprocants, a theory akin to invariant theory (he was to discover it had been substan­ tially developed elsewhere) . He became

entranced with matrix algebra, which he attacked by considering a plethora of low-dimensional cases-for example, his system of nonions involving 3 X 3 matrices. It was again the inductive ap­ proach which permeated all his mathe­ matical endeavours. Sylvester was fortunate that, like Cay­ ley's, his attainments were recognised in his lifetime. To some extent, his huge ego was soothed. As the Oxford years ran on. his health deteriorated, his eyes gave him trouble, and he retired to May­ fair, London's fashionable West End. His last years were spent writing poetry and delving into problems in number theory. In Professor Parshall, Sylvester has found a worthy chronicler of his life. Through the exercise of a great deal of care and diligence, Parshall has over­ come the practical difficulties of a scat­ tered archive of thousands of notes and letters . some of them almost indeci­ pherable. Her earlier work, james

Joseph Sylvester: L{le and Work in Let­ ters ( Oxford University Press, 1 998), provided some of the raw materials, but writing a biography is a different propo­ sition. In this quest, now completed, she has succeeded admirably. Sylvester is once more before u s , and rejuvenated he excites our enthusiasm for the great mathematical enterprise. Tony Crilly [email protected]. uk

A3 & H is Algebra : H ow a Boy from

C hicago's West Side Became a Force i n American M athematics by Nancy E. Albert i U N IVERSE I N C . , LINCOLN, NEBRASKA, 2005, 352 PP., ISBN-13:978-0-595-32817-8 U S $23.95.

REVIEWED BY JOSEPH A. WOLF

L

I

.

drian Albert was one of the most important American algebraists of the twentieth century. He was

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a remarkable person on many acco unts. Most obviously. he was a first-rate math­ ematician and bore much responsibility for bringing modern algebra into the current mathematics curriculum . He \\·as especially famous for his work on nonassociative algebras. on divisio n al­ gebras. and on Riemann matrices. Per­ haps less known. he: bore major re­ sponsibility for persuading the U . S . government to support basic research in mathematics. Even before the na­ tional importance of science and math­ ematics was underlined by Russia ' s launch o f the first satellite. "sputnik," i n 1957, Adrian Albert played a key role in establishing and increasing the amount of ONR and NSF research sup­ port for active mathematicians at all lev­ els of seniority. Some of this int1uence certainly must have been based on his defense work during and after World War I I , in particular his effective use of algebraic methods in cryptology. Perhaps known mostly to those who had direct personal contact with him. Adrian Albert put a lot of time. thought. and effort into encouraging and smoothing the \Yay for students and young researchers. On many occasions. some of which are mentioned in this biography, he arranged financial sup­ port for students to enable full-time study. He encouraged women in the study of mathematics, and to the extent then possible. he facilitated their career paths. The biography contains several instances of this . and there were in fact quite a few others. Adrian Albert usu­ ally did this sort of thing in a very warm­ hearted way: he would arrange the ben­ efit and then gleefully surprise the recipient with the good news. (Abraham) Adrian Albert, often nick­ named "A Cubed , " was born in 1 905 and grew up in Chicago, received his B . S . in mathematics from the University of Chicago in 1926. and earned his P h . D . from Leonard Dickson at the Uni­ versity of Chicago in 1 928. After two years as Instmctor at Columbia Univer­ sity he returned to the University of Chicago as Assistant Professor in 1 93 1 , Full Professor i n 1 94 1 . Math Department Chair 1 958- 1 962, and Dean of Physical Sciences 1962-1 97 1 . He passed away in 1 972. Adrian Albert married Frieda Davis in 1927. They had two sons, Alan and Roy, and a daughter, Nancy, who is the

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author of the hook under review. Roy died shortly after earning a B .A . in an­ thropology. Alan earned a B.S. in physics, worked as an engineer. and has since passed away. Nancy earned a J.D. and \vorks as a Ia \vyer. I am reviewing the June 2006 revised edition of this biography. \X!ith modern publishing methods it is l'asy to correct typographical errors and to take into ac­ count newly available source materials . Nancy Albert has done j ust that. In ad­ dition . it is remarkable j ust how Nancy Albert was able to transpose her legal abilities and produce such a precise and accuratl' description of Adrian Albert's mathematics . of the honors he received, and of his interactions with the federal government and the other members of the scientific establishment of his clay. Her description of the history of the University of Chicago. especially its De­ partment of Mathematics, is accurate and to the point. She is precise and in­ cisive when she discusses the social set­ ting of both her father and her mother. his connections with the University of Chicago and the other members of the Department of !l,lathematics there, and ( rather delicately) the way he managed despite anti-Semitic attitudes that lasted into the late l 9'50s. She was well ac­ quainted \\·ith the many mathematicians of her father's age and a bit younger. and that certainly contributed to the quality of the book. Also, recently she had access to some relevant AMS archives. This biography has already been re­ vie\ved by Lance Small in the Al1S So­ tices ( December 200'5) and by Philip Davis in the SIAJf Xeu·s (June 2006) . Lance Small is a n algebraist who started his graduate studies at the University of Chicago toward the end of Adrian Al­ bert's career. I defer to his description of Adrian ·s mathematical results be­ cause I work in areas (geometry and analysis) more or less orthogonal to Adrian's research. There is one non­ trivial point of historical disagreement between Lance Small's review and Nancy Albert's book. It concerns Adrian Albert's efforts to generate an offer from the University of Chicago to Nathan Jacobson and the possibility that these efforts were initially impeded by anti­ Semitic attitudes. Nancy Albert has clar­ ified this situation, with better docu­ mentation and a better arrangement of

the text, in the June 2006 revised edi­ tion of her hook. Adrian Albert and my father were close friends since their undergraduate days in the 1 920s. When Adrian Albert died he left a list of young mathemati­ cians to be invited to each take some books from his library. and I was on the list. Somehow that last gesture was typical of his generosity toward his stu­ dents and younger colleagues. As is clear at this point, this review is \vritten from the viewpoint of a friend. student, and colleague, rather than from the viewpoint of a historian of mathe­ matics or a mathematical critic. With that caveat, I definitely recommend this biography to all mathematicians inter­ ested in the interplay between mathe­ matics and public policy, and especially those in pure or applied algebra and those \\·ho had contact with the De­ partment of Mathematics at the Univer­ sity of Chicago any time from the 1920s through the 1 960s. Department of Mathematics University of California Berkeley, CA, 94720-3840 USA e-mail: jawolf@math. berkeley.edu

Saunders Mac Lane. A M athematical Autobiography hy Saunders Mac Lane WELLESLEY, MASSAC H USETIS, A. K. PETERS, 2004, 358 P., $39.00, HARDCOVER. ISBN 1-56881-150-0

REVIEWED BY HENRY E. HEATHERLY

�aunders Mac Lane ( 1909-2005) was one of the epoch-making � ;: mathematicians of the 20th cen­ tury. He knew and interacted with many of the outstanding figures of 20th cen­ tury mathematics. Add to this the well­ known lucidity of his expositions, his profound insight into the nature of mathematics, and his experience with scientific organizations and mathemati­ cal centers of excellence, and one comes to the pages of his a utobiogra­ phy with high expectations. These ex-

� W

pectations are-for the most part-\vell met. I never met :\lac Lane, but having read all or parts of severa I of his hooks and many of his expository papers and letters to the Notices. I feel I am ac­ quainted with Mac Lane the mathe­ matician and Mac Lane the commenta­ tor on the state of the mathematics profession. The hook consists of fifteen parts ( 64 chapters ) , ranging over Mac Lane's long and active life. It hegins with family background, early years, and his formal education. After undergraduate years at Yale, Mac Lane began graduate studies in the mathematics department at the University of Chicago. where he had a fellowship. Overall he found his year there to he disappointing. especially he­ cause he saw no possibility of doing a dissertation on logic at Chicago . He did write a master's thesis. learned how to play bridge. and met Dorothy Jones ( a graduate student i n economics whom he later married ) in that year. He won an Institute for International Education fellowship and decided to continue his studies in Giittingen, where he could write a thesis on logic. which had be­ come his main area of interest. Gottin­ gen in 1 9 3 1 was the premier mathe­ matical center in the world . There Mac Lane began a thesis under the direction of Paul Bernays. Hilbert's chief assistant. and became active in the mathematical life of late Weimar Germany. Regulations issued by the new Nazi regime in 1 933 forced Bernays to leave Gottingen and Mac Lane officially fin­ ished his doctoral thesis under Her­ mann Weyl. In July 1 933 Mac Lane took his doctoral exams and two days later got married-then back to the U.S.A. with a postdoctoral Sterling Fellowship at Yale. Professor Oystein Ore super­ vised all postdoctoral fellows in math­ ematics at Yale then, and Ore worked in algebra. So Mac Lane did research in valuation theory under Ore's direction. . "with considerable enthusiasm . . as Mac Lane recalls. Over the next several years Mac Lane held faculty positions at Har­ vard (where he was a Benjamin Peirce instructor) . Cornell. the University of Chicago, and then back to Harvard as an assistant professor. in 1 938. There he collaborated with Garrett Birkhoff to produce their influential book, A Sur­ uev qfModern Algebra ( 1 94 1 ) . I t was the first American undergraduate textbook

that presented the then new abstract ideas of Emmy Noether and B. L. van der \Vaerden. At Han·ard. Mac Lane began to di­ rect Ph.D. dissertations. His first student was Irving Kaplansky. who completed a dissertation. " Maximal Fields With Val­ u�ttions. ·• in 194 1 . Over the next sixty years Mac Lane directed Ph.D. disserta­ tion research for at least forty students. most of them at the University of Chicago. The dissertations were in many different areas: algebra, logic, al­ gebraic topology, category theory, topos theory, and theoretical computer science. He also co-directed two Ph.D. dissertations on the history of mathe­ matics. Among his many students was a future Fields Medal \Yinner. John Thompson. Another of his students. David Eisenhud . provided the preface for this hook. A list of forty of his doc­ toral students is given in the Mathe­ matics Genealogy Project website. with Steven Awodey listed as Mac Lane's last student. in 1 997. A photograph of Awodey and Mac Lane is given on page 3 1 4 of the autobiography. It is clear that Mac Lane greatly enjoyed his experi­ ence in the guidance of graduate stu­ .. dents. calling it "a splendid activity . While on a one-semester research leave in 1 94 1 Mac Lane started a col­ laboration with Samuel Eilenberg. This began with work on group extensions. homology. and cohomology that even­ tually led to their seminal work on cat­ egory theory. The latter is probably the mathematical work for which Mac Lane is best known . Mac Lane's collaboration with Eilenberg resulted in twenty-four joint papers in the period 1 94 1-19'5'5 . These papers ranged over several ma­ jor mathematical ideas: homotopy struc­ ture of spaces; homological algebra ( e. g . , the cohomology of groups) ; cate­ gory theory; and simplicial sets. The basic notions of category, functor, and natural transformation appeared in de­ finiti\·e form in their 1 945 paper. In the spring of 1 943 Mac Lane be­ came invoh'ed in mathematical work re­ lated to the war effort. He joined the Applied Mathematics Group at Colum­ bia . There he worked on problems that came from the Air Force, e . g .. gunnery guidance and pursuit cutves. The Co­ lumbia group had many outstanding mathematicians besides Mac Lane, in­ cluding Kaplansky, Eilenherg, Hassler

\Vhitney, Marston Morse, and Magnus Hestenes. At night Mac Lane and Eilen­ herg would \York on their own mathe­ matical problems in Eilenberg's apart­ ment. Category theory took shape during this period. A Guggenheim Fellmvship allc)'wed Mac Lane to spend the academic year 1 947-4H in Europe. In Paris he attended Leray's lectures on algebraic topology and got to know Armand Borel and Serre. Next Mac Lane went to ZOrich. where he stayed for a longer period of time to work with Heinz Hopf at the Swiss Federal Technical Institute. Mac Lane says it had been Hopfs ideas that originally started him and Eilenberg working on the cohomology of groups. After ZOrich. Mac Lane ,·isited several other mathematical centers in Germany . the Netherlands. Belgium. and Great Britain . In Oxford he began collabora­ tion \Vith J H. C. Whitehead on alge­ braic topology . the results of which were published in 1 9'50. Mac Lane and Whitehead had tentative plans to do fur­ ther joint research. hut to Mac Lane's regret this " never came to fruition . " Returning from Europe h e did not go back to Harvard, but instead accepted a professorship at the University of Chicago. This was the Chicago of what . has been called "The Stone Age . . he­ cause Marshall Stone had recently he­ come chairman there and was building the mathematics department to what ar­ guably became the premier one in the U . S.A. and perhaps the \Yorld . The se­ nior faculty were Albert . Chern. Stone . Wei!. Zygmund. and Mac Lane. Junior members included Kaplansky. Halmos . Segal . and Spanier. In 1 9 '5 2 Stone stepped down as chairman and Mac Lane took his place, serving from 19'52 to 1 958. He states that "the chairman's job was a troubled one, with its major policy issues and assorted bureaucratic .. regulations . He was frustrated by ad­ ministrative troubles ranging from keep­ ing an efficient department secretary to losing Andre Wei! from the faculty he­ cause the administration gave Wei! nothing to counter an offer from the I n­ stitute for Advanced Study. During the period 1 9'5 2-1 9'58. no new tenured a p­ pointments were made to the depart­ ment. although some excellent non­ tenured and tenure track appointments were made. Mac Lane considered his term as chairman "a prolongation of the

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Stone Age that continued , but did not expand, that tradition. " In 1 949 M a c Lane was elected to the National Academy of Sciences (NAS), leading to several decades of service with that organization. In 1 95H he was elected a member of the Council of the NAS and in 1959 he was named chair­ man of the editorial board for the Pro­ ceedings of the Academy, a position he held for eight years. He served two terms, 1 973-198 1 , as vice president of the NAS. His main activity during that term in office was in managing the Re­ port Review Committee. This commit­ tee reviewed reports on matters of high­ level science policy for various government agencies. He was elected president of the Mathematical Associa­ tion of America (MAA) for 1 9 52-1 953. and of the American Mathematical So­ ciety CAMS) in 1 972 for a two-year term. He also was an editor of the Bulletin and the Transactions of the AMS . In 1 973-1983 he served as a member of the National Science Board. Mac Lane had a long and distinguished career in the making of science policy at the highest level. This activity and his math­ ematical research continued until late in his long life. Mac Lane collaborated on research with many mathematicians over the span of his career. Besides the ones already mentioned, some of his co­ authors were Otto Schilling, Alfred Clif­ ford, and V. W. Adkisson. He co­ authored seven papers with Schilling in the period 1 939-1943, on algebras and number fields. With Adkisson he worked on geometric topology, and with Clifford, on group theory. From early in his career Mac Lane·s interest in mathematical logic led him to concerns about the philosophy of mathematics. He found "most of the subject misdirected and felt that one should be able to describe more accu­ rately what really is there in mathemat­ ics . " In 1 983 he gave a series of lectures at the University of Minnesota, which he later organized into a book, Mathe­ matics, Form, and Function ( 1 986) . He also wrote several papers on the gen­ eral subject of the philosophy of math­ ematics and what mathematics is-or should be-about. Throughout his autobiography Mac Lane weaves in personal items and details about his personal life. He gives

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lively descriptions of his wide-ranging travels, of people met, and of sights seen. There are many photos of math­ ematicians and of Mac Lane's family. Mac Lane gives us several examples of his well known humorous verses. Here is a sample from one he read at the banquet for the International Confer­ ence on Category Theory in Coimbra , Portugal, in 1 999. Sam Eilenberg said just one paper will do To introduce categorical notions so new We'll write it so \veil these ideas for to sell And publish it promptly the story to tell . He ends the book with a poignant thought: "All told, mathematics was a great career choice for me. , , There are subjects on which I wish Mac Lane had said more, e.g., his per­ sonal relationships with Eilenberg and Stone and the period while he was chairman at Chicago. While Mac Lane makes brief mention of his brother Ger­ ald, he does not mention that Gerald was also a mathematician, an analyst who was a professor at Rice University and Purdue. There is no mention of any mathematical interplay between the two brothers. The general reader will find a few chapters of the autobiography to be tough going mathematically. This is es­ pecially true for chapters twelve through fourteen, where some mathe­ matical sophistication is needed to di­ gest the concepts introduced and dis­ cussed-e. g . , crossed product algebras, covariant functors, Hom, and Ext. This caveat aside, a general scientific audi­ ence should find Mac Lane·s book both informative and delightful. On a tech­ nical note, for those wishing to use this autobiography as a source of informa­ tion , the format of the index will be dis­ appointing. Only the names of people are listed in the index. So, looking up items such as category theory, the Na­ tional Science Foundation, or Harvard, is made more difficult. For these rea­ sons, and to gain an external viewpoint, one might hope that a talented and mathematically knowledgeable biogra­ pher will now come along and give us another book-length view of this extra­ ordinary man, Saunders Mac Lane. As a supplement to the autobiogra-

phy under review, the references below should prove of interest to the reader. Note that the two AMS volumes contain five articles by Mac Lane , each well worth reading. REFERENCES

P. Duren (ed.), A Century of Mathematics in America, Parts II and Ill, Providence, R . I . ,

Amer. Math. Soc. , 1 989.

D. Eisen bud, Encountering Saunders Mac Lane, Focus 25 (2005), 5-7. Kaplansky (ed.), Saunders Mac Lane Se­ lected Papers , New York, Springer-Verlag,

1 979. J . MacDonald, Saunders Mac Lane, 1 9092005, Focus 25 (2005), 4. S. Mac Lane, Mathematics at G6ttingen Under the Nazis , Notices Amer. Math. Soc. 42(1 995),

1 1 34-1 1 38 . Mathematics Department University of Louisiana at Lafayette Lafayette, LA 70504- 1 0 1 0

U.SA

e-mail: [email protected] [email protected]

Alfred Tarski : Life and Logic by A nita Burdman Fefennan and Solomon Feferman CAMBRIDGE U N IVERSITY PRESS (OCTOBER 2004) ISBN: 0521802407, HARDCOVER, VI + 425 PP, $34.99

REVIEWED BY KRZYSZTOF R, APT

l·.

f you ask a well-educated person for the names of the three most promi­ . nent logicians in the twentieth cen­ tury, he will undoubtedly come up with Godel, Tarski, and Turing. While well­ known biographies of the first and last have been published-and their tragic deaths attracted peoples' sympathetic attention-no account of Tarski's l ife has appeared until recently. The book under review fills this gap excellently by providing a marvelously readable, informative, and gossipy account of his life and work. The book is far from be­ ing a dry account of Tarski's achieve­ ments: on the contrary. Tarski was a bon-vivant par excellence, and by delv·

ing into this part of his personality the authors transcend the genre of a cus­ tomary scientific biography. Before I proceed further, let me clar­ ify my admittedly very feeble connec­ tion with both Tarski and the second author of the book. This may explain my position of an interested yet impar­ tial bystander. In 1974 I defended a the­ sis in Mathematical Logic in Warsaw as the last PhD student of Andrzej Mostowski, who in turn \Vas the first PhD student of Alti·ed Tarski . A year earlier the renowned Banach Center in \varsaw organized a Logic Semester which brought to Warsaw several lu­ minaries in Mathematical Logic. One of them was Solomon Feferman. He prob­ ably does not remember the student who guided him on his first day from the Center to the neighbouring Mathe­ matical Institute of the Polish Academy of Sciences. I never met him afterwards and never met his wife. the first author. When I began reading this book I never expected to find so much sur­ prising material in it. It is a fascinating history of a huge fragment of mathe­ matical logic in the twentieth century. The reason for this is the enormous in­ fluence that Tarski wielded on the sub­ ject. After the Second World War he built in Berkeley, almost single-hand­ edly, an extremely successful school of logic that attracted and educated many of the best and smartest logicians in the world. In contrast to Godel and Turing. Tarski had many PhD students . 24 to he exact. Also, he influenced a large number of logicians throughout his ca­ reer. One can honestly say that the strength of mathematical logic in the United States is, to a very large extent, due to his efforts. The authors trace in detail Tarski"s life from his birth in 1901 in Warsaw, then in the part of Poland ruled by Rus­ sia, to his death in 1 983 in Berkeley. Tarski's original name was Tajtelbaum . I n 1 924 he changed it to Tarski and of­ ficially converted to Catholicism. even though he remained an atheist. In this way he hoped to circumvent the diffi­ culties facing scientists of Jewish origin in the newly reestablished Poland. Tarski was quickly recognized as a brilliant scientist. At the age of 23 he received a PhD degree from Warsaw University. Soon after that he and Ba­ nach proved the famous result on the

sphere decomposition, now called the Banach-Tarski paradox. In 1 929 he married Maria Witkowska. with whom he had t\YO children. Their marriage suf\·ived a six-year separation during the war ( the hook includes a re­ markable reproduction of a short note about the family reunion from the Oak­ land Tribune from 6 January 1 946 ) and several crises caused by his numerous love affairs with other women. As a PhD student in Warsaw, I heard that Wanda Szmielew, a renowned expert in the logical analysis of geometry, was emo­ tionally involved with Tarski. Little did I know that this was just the tip of an iceberg about which the authors make no secret. Indeed, they go to great lengths in describing Tarski "s numerous romances with various co-authors. PhD students, and secretaries . and the book features photos of many women who fell under his apparently irresistible charm. Wanda Szmielew even l ived for a year in the Tarski's family home in Berkeley. while having a relationship with him. Of course, all this did not contribute to a successful family life. Tarski"s wife emerges from the book as an almost angelically patient person who sacrificed her life to help her hus­ band's career. Eventually she moved out of their house to another place in Berkeley where she rented rooms to . . . logicians visiting her husband. Tarski's l ife-style also did not help much in fos­ tering warm contacts with his two chil­ dren. The authors discuss a striking scene in the lobby of the imposing Eu­ ropejski Hotel in Warsaw in which Tarski was to meet his son Jan during a brief visit to Poland in 1964. The meet­ ing was spoiled by the unexpected ap­ pearance of Wanda Szmielew. In spite of his name change and excellent mathematical record, Tarski never succeeded in getting a professor­ ship in Poland. He left Poland for a short visit in the United States on 1 1 August 1939. on the same ship as the brothers Adam and Stanislaw Ulam. Adam be­ came a brilliant historian of Russia and communism, while Stanis-law became one of the most famous American math­ ematicians involved in an essential way in the Manhattan Project. Because of the outbreak of the Second World War on September 1 Tarski ( and the Ulams) re­ mained in the United States. In 1 942, after holding a number of

temporary positions, Tarski joined the Department of :\lathematics of the Uni­ \·ersity of California . Berkeley which re­ mained his home institution until his re­ tirement in 1 968. Retirement did not prevent him from remaining scientifi­ cally active . including supervision of several PhD students. till his death in 1 983 . His fame steadily grew. In 1 96 5 , b e was elected a member of t h e Na­ tional Academy of Sciences. and other high honours followed. The book abounds in delightful anecdotes revealing Tarski"s magnetic personality. He was a famously brilliant teacher and fast thinker, as well as an exceptionally demanding and persistent supervisor. It suffices to say that t\YO of his students took nearly t\venty years to finish their PhD thes es. One of the stories concerns Dana Scott . a most renowned logician and re­ cipient of the Turing Award in computer science. who, as a PhD student, was fired by Tarski for procrastinating with the corrections of an amateurish trans­ lation of a compendium of Tarski"s early Polish texts. Scott subsequently finished his PhD thesis with Alonzo Church at Princeton. Eventually Tarski and Scott mended fences. and once Scott"s repu­ tation grew Tarski even suggested that he. Tarski, v-;ould call Scott his student. Throughout his life Tarski was a workaholic who slept little and worked till the wee hours, often relying on ben­ zedrine (a variant of amphetamine) to stay awake. He was also a heavy smoker and occasionally indulged in marijuana. While in Berkeley he regularly threw lav­ ish parties at which alcohol ( notably his homemade variant of slivovitz) would t1ow freely and higos, a Polish cabbage­ based stew. woul d be served . The au­ thors convey the jolly atmosphere of these parties by mentioning that John Myhill, a logician, once sang the statis­ tics on male homosexuality from the just published The Kinsey Repm1. Tarski"s entourage was regularly ex­ posed to his dictatorial behaviour. Chen-Chung Chang, who suffered from asthma. recalls in the pages of this book v,:hat his work as a PhD student of Tarski was typically like. They would start working about 9 p . m . , continuing till 4:30 a . m . , sitting in clouds of ciga­ rette smoke in an unventilated room at Tarski"s house . Around 2 a . m . Tarski would inquire whether Chang would

© 2007 Springer Sc1ence � Business Media Inc. Volume 29. Number 2. 2007

79

like a coffee; following his positive re­ ply he would yell his wife's name to wake her to make it for them. Tarski left a huge scientific legacy covering several areas of mathem atical logic. After his death the Journal of' Svmholic Logic published. in 1 9H6 and 1 9HH. more than ten surveys discussing his life and research in \ arious areas of logic. It is generally agreed that his most fundamental contributions are his for­ mal definition of truth, his theorem on the decidahility of the first-order theory of reals, and his work on relational and cylindric algebras. The hook allows the reader to learn in reasonable detail about Tarski's contributions by means of six 'Interludes· interspersed through­ out the text. These are highly readable accounts of the relevant background in mathematical logic and Tarski's contri­ butions. Tarski's work also had a notable im­ pact on computer science. especially his research on decision procedures and his approach to semantics. Interested read­ ers are referred to a very recent article by Solomon Feferman; see l l l . His \York on the definition of truth. meta-mathe­ matics. and generalizations of first-order logic was influential in philosophy and linguistics. Also. the renmYned Tarski's fixed-point theorem ( a ctually its weaker version. known as the Knaster-Tarski theorem) became a stanc!arc! tool in Mathematical Economics: see. e . g .. the classic lvficroeconomics Themy of Mas­ Collel. Whinston, and Green ( where Tarski is misspelled as Tarsky ) and ivfathematical Economics of M. Carter ( where he is misspelled as Tarksi). The authors took the trouble to check the spelling of the often confus­ ing ( for ·westerners· ) Polish names. I was impressed to fine! that all Polish words, except two. \\·ere spelled cor­ rectly. including those that contain non­ ASCII characters. The hook even con­ tains a Polish Pronunciation Guide. In summary, this is an excellent hook from which one can learn a lot about the history of mathematical logic in the twentieth century. the remarkab le in­ fluence of Tarski on this discipline . and, especially, about Tarski himself. REFERENCE

[1 ] Solomon Feferman. Tarski's influe nce on computer science.

Manuscript available

from http://arXiv.org/abs/cs/06080 62.

80

T H E MATHEMATICAL INTELLIGENCER

Krzysztof R. Apt CWI , Amsterdam The Netherlands e-mail: K.R.Apt@cwi .nl

Toward a Ph i losophy of Real M athematics h)' Dal 'id Co �fleld CAMBRIDGE U N I V ERSITY PRESS, 2003, 300 PP. $US 70.00, ISBN 0521817226

REVIEWED BY ANDREW ARANA

��

l:' h ilosophy of mathematics" L

} brings ,

to mind endless dehate about what numbers are and whether they exist. Since plenty of mathematical progress continues to he made without taking a stance on ei­ ther of these questions, mathematicians feel confident they can work without much regard for philosophical ref1ec­ tions. In t his sharp-toned . sprawling hook. Da\·id Corfield acknowledges the irrele\·ance of much contemporary phi­ losophy of mathematics to current mathematical practice. and proposes re­ forming the subject accordingly. Reading the introduction. it is hard not to he swept up by Corfielcl's revo­ lutionary fervor. Most contemporary philosophical writing on mathematics focuses on elementary arithmetic or logic, hut that is not a representative sample of mathematical practice today or at any time since Euclid. Corfiekl's push to \Viden the investigati\ e reach of philosophers of mathematics will he \Velcomed hy readers whose lo\·e of mathematics extends broadly. But is this \Videning important merely because it may he more attractive to lovers of mathematics? Or are there important philosophical questions about mathe­ matics that cannot he answered well without the widening? Corfield suggests that there are such questions. For instance. we can ask why some concepts ( such as groups and Hilbert spaces ) have received a great deal of attention while others have not. It could he that our interest in those concepts is just a fad . like wearing blue jeans; or is tied to matters of current

l

cultural and scientific interests ( or prac­ tices) that may change dramatically in time. Or it could be that there are some parts of mathematics that we can't help hut run into when our inquiry gets se­ rious enough. We can put the question this way: are some mathematical con­ cepts inel'itah!e? This question is far from a new one: defending a "yes" answer to it was one of Plato's chief goals, not just for math­ ematics but for every knowledge-seek­ ing activity. Furthermore, the view de­ ril'ed from Plato's that we now call "Platonism"-that mathematical objects exist. independently of human minds­ continues to attract followers. Notice how the question that "Platonism·· an­ swers differs from Corfield's. Corfield's question asks us to account for the in­ evitability that, mathematical practice suggests, groups and Hilbert spaces possess; "Platonism " addresses the question of whether mathematical ob­ jects exist ohjectiYely or are human con­ . structs. "Platonism . could he true yet not tell us why certain of these allegedly .. "objective features of reality. such as groups and Hilhe 11 spaces. turn out to he important. Corfield wants to turn the philoso­ phy of mathematics toward what is im­ portant for mathematics. I will consider one example in some detail. both to show what Corfield does and what he does not do. Chapter Four is a study of analogies in mathematics. cases where two apparently distinct domains seem to he related. His chief example. which I will survey in a moment, is the anal­ ogy between algebraic numbers and al­ gebraic functions that emerged from Dedekind and Weber's work in the 1 8HOs . Their work engendered an alge­ braic approach to the theory of alge­ braic curves that. as developed in the twentieth century by Chevalley and then Grothendieck. would rival Rie­ mann's geometric and Weierstrass's function-theoretic approach. Corfield motivates this discussion by quoting several famous mathematicians on the importance of this analogy, remarking that analogies might indicate a "deeper structural similarity " between the do­ mains that might in some sense he in­ evitable. Though Declekind and Riemann had a relatively dose personal relationship, stemming from their time together in

Giittingen, their approaches to mathe­ matics \Yere quite different. This is es­ pecially dear in their attitudes toward what we today call the Riemann-Rocl1 theorem. In lectures in l 8"i "i- 18"i6 that Dedekind attended. Riemann presented work on meromorphic functions over Riemann surfaces that he would pub­ lish in his 1 8"i 7 paper Tbeurie der A bel schell Fzmctiunen Among other things . Riemann considered the ques­ tion: given m points on f5, a Riemann surface of genus p, how many linearly independent meromorphic functions are there that have at worst simple poles at the m specified points? Riemann showed that there are 11 such functions, where 11 � m - p + 1 . ( His student Roch later identified the error term . in­ corporating Riemann's inequality into a more general equality. ) To prove it, Rie­ mann used topological considerations, in particular what \Ve no\\· call the "Dirichlet principle" . which states the existence of a function minimizing a particular integral involving that func­ tion. Dirichlet's principle was contro­ versial in the years following Riemann's work: it was unproved, and w·eierstrass sho\\·ecl in 1 870 that it failed in certain cases. Ho\vever . the Riemann-Roch the­ orem \\·as recognized as fundamentally important, and people tried to eliminate the use of Dirichlet's principle in its proof. Dedekind \\·as one of them: he hoped to avoid not only the Dirichlet principle hut also any " transcendental" , topological considerations whatsoever ( in practice. this meant avoiding conti­ nuity) . In 1882 , Dedekind and his colleague Heinrich \veber expressed the Rie­ mann-Roch theorem in algebraic terms involving fields of algebraic functions defined on a Riemann surface. What is striking about the Dedekind/Weber pa­ per (and of importance for Corfield's project) is the analogy Dedekind and Weber located between fields of alge­ braic numbers and fields of algebraic functions. In the 1 870s Dedekind had made great progress in algebraic num­ ber theory, developing his the01y of ideals in his numerous " supplements" to Dirichlet's lectures on number the­ ory. In rings of algebraic integers, ideals enjoy unique factorization into prime ideals and thus , said Dedekind. the al­ gebraic integers obey the same "laws of divisibility'' as the ordinary integers.

This seems to have confirmed his view, presented in his acclaimed 1 888 essay on the foundations of arithmetic:, "Was sind und was sollen die Zahlen?'', that "every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers-a declaration I have heard repeatedly from the lips of Dirichlet. ·· This vie\v was further confirmed in Dedekind's 1 882 work with Weber. What was needed. and what they found, was an analogue of ideals of al­ gebraic integers in fields of algebraic functions. Consider. for simplicity. al­ gebraic functions defined on the Rie­ mann sphere-the complex plane to­ gether with a point at infinity-which is a surface of genus zero. But. follow­ ing Dedekind and Weber, let's start with the field of algebraic functions IC< (l =

{'[; .

l : fg E ICl(l . g

=I= o .

}

Dedekind and \veher took the ring of algebraic integral functions IC[(l . to he the analogue of /L and the algebraic in­ tegers in this setting. and took the field 1[:( (! to he the analogue of ()) and the algebraic numbers. They used this anal­ ogy because ideals in the ring IC[(] en­ joy unique prime bctorization. like the integers and algebraic integers: and as noted, it was this "law of divisibility" that Dedekind had identified as critical . \'\'e can see the prime factorization in this setting by noting that IC[(] ( a nd hence each ideal of IC[(] ) consists of el­ + c1( + ements C1 1 ( 1 1 + C1 1 - 1 ( 1 1 1 + c0. \\ ith each c1 E. IC. and that . by the fundamental theorem of algebra, each such expression factors into products of linear terms ( ( - z1) for z1 E. C Ac­ cordingly . the prime ideals of this ring \\'ill be those generated by these linear factors, ( ( - z,), in addition to the zero ideal, which will turn out to be very im­ portant in the later development of scheme theory in algebraic geometry. All of the non-zero prime ideals are maximal, and these maximal ideals yield all points z1 E. IC, and thus all points of the Riemann sphere, except for the point at infinity, which cor­ responds to the maximal ideal ( g - 0 ) in the polynomial ring IC[g] under the ·

·

·

-;;

1 . identification g = · The maximal ideals ((

- z)

(with

z =I=

0 ) in IC[(] and the

( �)

maximal ideal g -

in ICl(J a re iden­

tified. As a result . we ha\·e that the maximal ideals of ICl(J and

1Cl1J

are in

one-to-one correspondence with all points on the Riemann sphere. This ob­ servation allowed Dedekind and Weber to shift talk of the Riemann surface to talk of the correlated ideals, u ltimately giving a proof of the Riemann-Roch the­ orem ( and others ) using purely alge­ braic considerations. ( Detlef Laugwitz's text Bernhard Rieman n 1826- 1 866, l3irkhauser . 1 999. especially p . 1 "i9. is quite helpful in understanding these parts of the Dedekind;Weher paper. ) This long detour into the details of the Dedekind/Weher paper shows how interesting Dedekind and Weber's work was, as they defended their vievv that fields of algebraic integers and of alge­ braic functions could and should be treated as obeying some of the same laws. On the significance of this, Cor­ field quotes Dieudonne: [T]his article by Dedekind and Weber drew attention for the first time to a striking relationship be­ tween t\\·o mathematical domains up until then considered very re­ mote from each other. the first mani­ festation of what was to become a " leitmotif ' of later \vork: the search for common structures hidden u n­ der at times extremely disparate ap­ pearances. ( p . 96 ) This detour also illustrates one of the problems with this book: Corfielc.l's ac­ count of this material only skims the surface of this deep topic. Corfield gives his own sketch of the DedekincV Weber analogy. discusses its connec­ tion with the modern-day notion of ramification, and then turns to another approach to Riemann's work, the " val­ uation-theoretic" approach developed first by Kronecker, later by Hensel, and extended in later work o n p-adic num­ bers. All this in six breathtakingly con­ cise pages' What Corfield intends to do with this sprawling case study is to d is­ cuss what is important about analogies in mathem atics . hut his purpose would have been better served if he had pro­ vided a more in-depth analysis of even j ust one aspect of this analogy. What we get instead is a series of Ho u rha­ kiste quotes, from Dieudonne . \veil, and Lang, extolling the virtues of analo-

© 2007 Springer SCience r Bus1 ness Media, Inc

Volume 2 9 , Number 2 . 2007

81

gies in revealing the ·structures· u nder­ lying the mathematics v,:e ordinarily ex­ perience. The quotes are interesting , but I am left wondering \vhat they (and hence, Corfield, who mostly lets them speak for themselves ) mean hy 'struc­ tures'. Corfield's indicated aim was to discuss how an analogy between two domains might indicate a '·deeper struc­ tural similarity" that could he said to be inevitable. However, we are not given any guide to what "structural similarity"" might be. aside from being shown an admittedly impressive analogy and a series of quotations from famous math­ ematicians commending this work. In­ deed . Corfield spends fewer than three pages analyzing the case study ( with four significantly sized quotations left largely unanalyzed ) , fewer than half the pages dedicated to the details of the case study itself. Mathematicians read­ ing Corfielc.l's book may get the wrong impression that the allegedly ""revolu­ tionary"" philosophy of real mathemat­ ics is primarily the narration of existing mathematics, and thus is by no means revolutionary. I say this is the " \vrong"" impression because I think Corfield has helped clear space for a variety of projects go­ ing far beyond both narration of exist­ ing mathematics and the types of ques­ tions ordinarily dealt with in the philosophy of mathematics-though he himself seems unwilling to occupy that space . I'll be more specific, hy turning back to the Riemann/Dedekind case al­ ready discussed. Corfield's heavy re­ liance on Bourhakiste quotations is no coincidence. As he makes clear else­ where in the book, he is sympathetic to the categmy-theoretic development of mathematics that followed Bourbaki. That gives one answer to the question of what are the structures revealed by analogies: they are categories. This plays right into the ongoing controversy between advocates of category theory and of set theory as to the "proper foun­ dation"" for mathematics. This is surely an interesting controversy, but one with opposing sides as entrenched as the sides of the Cold War. It is hard to see how taking sides in this debate is go­ ing to help Corfield's promotion of a new philosophy of mathematics. when he comes across looking like a mere partisan in an old battle. But worse , he brushes this debate u nder the rug, pro-

82

THE MATHEMATICAL INTELLIGENCER

viding little defense of the category­ theoretic view, and in fact marking such " " fou ndational" debates as having usurped the attention of philosophers for too long. What is needed here are new ideas. I would like to suggest two. one even hinted at by a quotation of \X7eil"s in Corfield's text. First. instead of talking of analogies as revealing "structures"". Dedekind talked instead of "" laws"" be­ ing obeyed in different mathematical settings. It's not a far leap from this ,·ie,,· to Hilbert's axiomatic view of mathe­ matics. Corfield describes Hilbertian ax­ iomatics as merely one step in an ""increasingly sophisticated"" series. in which Noether's algebra and Eilenberg and Mac Lane's category theory are fur­ ther developments ( p . R3) . I think Cor­ field is underappreciating the ,-ie\Y that by focusing on "laws" rather than on "objects" such as categories. Dedekind and Hilbert were able to focus their at­ tention on the statements-on the "laws" themselves-thus opening up metamathematical avenues of progress . In addition, it allows one ( though nei­ ther Dedekind nor Hilbert did this con­ sistently) to avoid talking of mathemat­ ical objects a ltogether, talking instead only of the statements we want to as­ sert. In practical terms, this makes little difference, since we ordinarily write mathematics in statements ( though this could change with the development of new notation or media, as Corfield dis­ cusses in Chapter Ten) . In philosophi­ cal terms. though, it means we no longer have to discuss vvhether certain mathematical objects ·'exist" , since we are no longer talking about objects. Since this is an outcome Corfield gives glimmers of favoring at times, I think it deserves our consideration. The other idea for thinking about analogies that I would like to suggest follows Weil's idea that in working out analogies, we are trying to decipher statements in the language of one do­ main into the language of other do­ mains. In a 1 940 letter to his sister Si­ mone ( pu blished in Sotices of the AJ1S 5 2 : 3 ( March 200 5 ) . pp. 334-341 ) . he de­ scribes himself as having worked in the ""Riemannian·· tongue for some time, hut wishing for the "translation·· of all the ideas of that work into the language of function theory (as developed by Weierstrass and his followers) and the

language of number fields ( in either the Dedekind/Weber ideal-theoretic dialect or the Kronecker-Hensel valuation-the­ oretic dialect). He saw work with <malo­ gies as attempts to fill in a "translation table" between the three languages , constructing a "Rosetta Stone ·· for math­ ematics. In explaining analogies this \Yay. Wei! made no appeal to "struc­ tures . " Instead, he emphasized learning to move t1uidly back and forth between these different ways of presenting math­ ematical things. Tme. each language of­ fers distinctive benefits: for instance . working in the "Riemannian" language may free us to think more visually or physically. At least as important, though, is the benefit of being able to switch languages freely which, in addi­ tion to granting us the advantages of each language whenever \Ye choose. lets us work simultaneously in several languages at once. Doing so lets us an­ ticipate results in one domain that we have not yet discovered but that we should expect, given the otherwise suc­ cessful translation (Wei! mentions cases of this in his letter). Weil's case of the Riemannian language is by no means the only example of this translation project in mathematics: a great deal of work since Descartes has been spent in geometry translating between analytic and synthetic languages. and in im­ proving the translation. I admit that my defense of the advantages of this ap­ proach is still quite tentative. My pur­ pose i n offering these two ideas is to show how interesting a problem Cor­ field has framed for us-and how much more remains to be said. For all the revolutionary talk given in the introduction, Corfield's views end up quite continuous with the usual top­ ics of the philosophy of mathematics. I have already highlighted his interest i n t h e quite traditional topic o f conceptual inevitability in mathematics, and of his advocacy of the category-theoretic side in the ongoing struggle over the "foun­ dations" of mathematics. When he con­ tinues discussion of the Riemann/ Declekind analogy in Chapter Eight ( otherwise dedicated to Lakatos's work), he turns our attention to Kronecker's role in its development, and highlights how later mathematicians such as Weyl took sides in the choice of an Dedekin­ dian ideal-theoretic or a Kroneckerian valuation-theoretic approach. Corfield

emphasizes that they based their deci­ sions in part on how "constructive" they perceived each approach to he. The value of constructive reasoning is yet another traditional topic in the philos­ ophy of mathematics. Here again. Cor­ field's project is not nearlv as radical as he would have us think. Stil l , for all my frustration with the hooks limitations . I think Corfield does a nice job of showing how the philos­ ophy of mathematics can begin to en­ gage areas of mathematics besides arith­ metic and logic-a shift I strongly favor. There are interesting chapters on auto­ mated reasoning. Bayesian reasoning. and Lakatos's work for those \vho are interested in these topics. I ha\·e tried to focus on the parts of the text that I think are the most daring. and proba­ bly of the widest interest among read­ ers of this magazine. Corfield deserves to be supported for his daring. and for his hope that the philosophy of math­ ematics will he revolutionized. even if his book is not the revolution \\"e might have hoped for. I thank my colleagues Zongzhu Lin and Scott Tanona for their helpful comments on earlier drafts. Department of Philosophy Kansas State University Manhattan, KS 66506-2602 USA e-mail: [email protected]

The 15 P uzzle:

H ow It Drove the

World C razy hv jerry

Slocu m an d Die Son nel 'eld

THE SLOCUM PUZZLE FOUN DATIO N , 2006. HARDCOVER. 144 PP, $30.00. ISBN-1890980153

REVIEWED BY AARON ARCHER

S"""''\ o me

�

years ago. I wrote an article about the 15 puzzle that began, -"'-·•·. · " In the 1 H70's the impish puzzlemaker Sam Lc>Vcl caused quite a stir in the United States. Britain, and Europe with his now-famous 1 5-puzzle·· [ 1 ] . I have always been pleased with myself for managing to slip the word ' 'impish" into a published mathematics article. As

it turns out, this devilish descriptor was the most accurate part of that sentence, as Slocum and Sonneveld document in their nev-.' hook 'lhc 1 S Puzzle: Hou.• it Drove the World Crazy. The 1 5 puzzle did once cause an intense craze that spread like \Yildfire across America and overseas. and it is indeed famous to this day. However, the initial fad did not oc­ cur until l imO. and Sam Loyd had noth­ ing to do with it until eleven years later, when he started to claim in print that he hac\ invented the puzzle. Through meticulous research using primary sources. Slocum and SonneYelcl not only expose Sam Loyd's fraudulent claims but also argue convincingly that the actual inventor was Noyes Chapman. the post­ master of Canastota. New York. The 15 puzzle is a sliding block puz­ zle consisting of 1 5 numbered square blocks placed inside a frame large enough to accommodate 16 blocks in a 4-by-4 grid. The empty space allows the solver to slide any of the adjacent blocks into the open space. Given a starting configuration, the puzzle is to reach some specified target configuration via a sequence of such moves. The in­ structions written on the cover of the original puzzle read . ·'Place the l3locks in the Box irregularly. then move u ntil in regular order" ( p . 8 ) . It did not spec­ ify exactly \\·hat is meant by "' regular or­ der."' but it did contain a picture of the blocks arranged as shown here. Later posers of the puzzle were more careful to explicitly state this as the tar­ get state. For the purposes of our dis­ cussion. let us call this the canonical state. Early fans of the puzzle discov­ ered that they could often reach a con­ figuration that differed from the canon­ ical state only in that the 14 and 1 5 were swapped, but tty as they might, they could not complete the puzzle. Thus. the problem of solving the puzzle from this starting state became the standard challenge, which we will call the 14-15

puzzle.

Beginning i n 1 880, numerous cash prizes of up to $ 1 000 American ( a princely sum i n those clays) were of­ fered for the solution of the 1 4- 1 5 puz­ zle. There have been numerous pub­ lished accounts of people spending endless hours engrossed in the puzzle, but nobody ever successfully claimed these prizes. An intriguing mathemati­ cal fact about the 15 puzzle is that for

exactly half of the 16! possible initial configurations. the puzzle is impossible to solve. It should come as n o surprise that the 1 4- 1 5 puzzle starts from one of these impossible configurations.

i�- T-lfE · aiM_ ·P u � �i i.�

: .. � I �

••

•

>

.:.

1 · 2

3

�:l

..

4 a� ., -

�

A

-. ...� �Ill 5 6 7 8 � § � = : ; � ; � :i 9 10 ll 12 J � , t-

� 4

. �

.,

�.

�

13 14 15

PI•<"• the Blocks in ·-. mllYe until

_}4i-.

p

> �

: � ·'

'" ' Bo• orrt_l'ularly. lh.en ,,, rro�_IJ!ar orti.:r. . .::-s. . .

I

The set of solvable configurations can be easily described using the the­ ory of even and odd permutations. First, it should be clear that the solvable con­ figurations are precisely those that can be reached starting from the canonical state. since each move is reversible. Sec­ ond, let us imagine the blank space to contain a block that we \Vi l l call the . "blank block. , Then each move consists of swapping the blank with one of the blocks that is adjacent to it horizontally or vertically. Third. let us focus on the set of attainable configurations where the blank lies in the lower right corner. Then the question is which of the 1 5! permutations of 1 6 blocks that fix the blank can be attained. starting from the canonical configuration. Let us consider an arbitrary sequence of moves that returns the blank to its initial position. Since each moYe swaps the blank with another block, we may write the resulting permutation as a product of transpostttons, one per move. Now color the spaces of the board white and black in a checker­ board pattern, with the lower right cor­ ner colored black. Since the color of the square occupied by the blank switches with each move and it starts and ends on black, it must make a n even num­ ber of moves, so the resulting permu­ tation may be written as a product of an even number of transpositions. Now recall the well-known fact that every permutation can he written as a prod­ uct of either an eYen or odd number of transpositions, but not both. Then the

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argument we just made shows that no odd permutation that fixes the bla nk is attainable. It is also the case that eve1y even permutation that fixes the blank is attainable, although that result is far less obvious. To complete the pictu re. a configuration of the puzzle with the blank in an arbitrary position is obtain­ able from the canonical configuration if and only if either the blank is on a black square and the permutation is even, or the blank is on a white square and the permutation is odd . It is against this mathematical backdrop that the re­ markable history of the 1 ') puzzle plays itself out. Slocum and Sonneveld pick up their tale with the first commercial produc­ tion of the puzzle, beginning in De­ cember 1 879, and proceed to trace its explosive spread o\·er the first half of 1 880. In December 1 879. a crude ver­ sion of the 1 ') puzzle had made its way into the hands of Matthias Rice, a Boston \\·ooc.b;orker who began man­ ufacturing the puzzles in his shop . Af­ ter some effort, he got one of the lead­ ing toy dealers in Boston to carry them. They became an immediate commercial success, attracting many other manu­ facturers to jump into the fray before the beginning of March. The craze spread from Boston to New York, all along the East Coast and across America, and to Canada, Europe , and beyond. It cut across all sections of American life . and references to the 1 ') p uzzle appeared in many venues of American pop culture. Stage produc­ tions. music. and poems featured the puzzle . and numerous newspaper and magazine articles appeared . discussing the hold the puzzle had taken on the nation's collective imagination. Most of these focused on people's obsession with the 1 4- 1 5 puzzle, and there was a great deal of public argument over whether it could be solved. Many of the articles contained ( likely exaggerated) reports of the puzzle's role in filling the insane asylums, and many sounded tongue-in-cheek alarms about the sup­ posedly deleterious effect the puzzle was having on the fabric of society. O n February 1 7 , 1 880, the Rochester De­ mocrat and Chronicle published an ar­ ticle titled, '· ' 1 5' : The Diabolical Inven­ tion of Some Enemy of Mankin d" ( p . 22). The New York Times followed u p with a whimsical story o n March 22 that

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T H E MATHEMATICAL INTELLIGENCER

opined. "No pestilence has ever visited this or any other country which has spread with the awful celerity of what is popularly called the 'Fifteen Puzzle . · . . . it no\Y threatens o u r free institu­ tions. inasmuch as from every town and hamlet there is coming up a cry for a · strong man· \vho will stamp out this terrible puzzle at any cost of Constitu­ tion or freedom" ( pp 'i3-'i 4 ). Much of Slocum and Sonneveld's narrative is driven by these numerous ne\\·spaper accounts that deal with the popularity and addictiveness of the puzzle. and some of which dig into the murky his­ tory behind its invention. I was struck by the authors' artful use of these articles to construct a se­ ries of tables ( pp. 'i8-60) providing an illustrated time line of the progress of the craze across the country and the \\·oriel. For each of a representative sample of 24 cities spread geographically from Boston to Los Angeles, a bar connects the dates of the first and last articles about the puzzle in the newspapers of that city, along with a special marker if the article listed the craze as having just started or just ended. It seems that the craze hit its peak in most American cities in February or March of 1880. and started to abate by April. The authors conclude that in most parts of the United States, the craze had ended by May. A similar chart shows the puzzle spreading through Europe in April and May. and even as far as Australia and New Zealand. Throughout the detailed documenta­ tion of the 1 ') puzzle craze of 1 880, there are many moments of comic relief. A dentist from Worcester. Massachusetts. offered a $2) set of false teeth for a so­ lution to the 1 4- 1 5 puzzle, which he later augmented with an additional 5 1 00 cash prize ( p. 1 6 ) . Shortly thereafter, another citizen of Worcester countered with an ad that read, "I have a cat that possesses a fine set of teeth. and I will give to any one the Entire Set and $ 1 00.00 (One Hundred Dollars), who will transpose two of her teeth and then put them hack into their original positions and not hurt the cat. Operators to take all the chances of getting bitten. Dentists are not al­ lowed to compete" ( p. 1 1 1 ). A large portion of the hook is de­ voted to documenting the initial 15 puz­ zle craze. This is interesting not only for the entertainment value of the craze-

inspired witticisms. hut also for the in­ sight it lends into the anatomy of a large-scale fast-moving fad. It also es­ tablishes definitively that the craze oc­ curred in 1 880. a key piece of evidence in support of the book's main thesis­ that the universally accepted belief that Sam Loyd im·ented the 15 puzzle is ac­ tually a myth . Slocum and Sonneveld second Mar­ tin Gardner's description of Sam Loyd as "America ·s G reatest Puzzlist, .. based on the work he actually did ( p. 7 'i ) . However, they also reveal a man who had a habit of trying to augment his le­ gitimate fame by claiming credit for the work of others. using, for example, in­ ventions of the English puzzlist Henry Dudeney without credit. He also claimed spuriously that he invented the game Parcheesi and a popular dexterity puz­ zle called Pigs in Clover. However. his most successful intellectual theft was the 15 puzzle. From 1891 until his death in 1 9 1 1 . Sam Loyd mounted an enduring disin­ formation campaign to take credit for the invention of this puzzle. His stories were not all consistent. and indeed it seems that part of his strategy may have been to muddy the waters. For instance, in various writings and interviews he claimed to ha\'e invented the puzzle at different times from 1 872 to 1 878. He was so successful in this undettaking that Slocum and Sonneveld found not a single article that questioned Loyd's claims. and every obituary that they found for him stated that he had in­ vented the 1 5 puzzle. For 1 1 'i years, Loyd's phony claims were accepted as established fact. until these authors painstakingly sorted through mountains of primary source material and made Swiss cheese of these claims. Having debunked the myth that Loyd invented the 1 5 puzzle, the authors then set out to discover the actual inventor. Through a tour de force of forensic so­ cial archaeology, Slocum and Sonneveld found that the puzzle was invented by Noyes Chapman, and they traced the puzzle's path from Mr. Chapman in up­ state New York to Matthias Rice in Boston. No step along this path is proven beyond a doubt, and it seems unlikely that it could be proven now after more than 1 2 5 years, but the evidence that the authors dug up strikes me as very com­ pelling. They even located a U.S. patent

application filed hy Mr. Chapman on February 2 l , J HHO, but found no expla­ nation of why the patent was rejected. Perhaps. they conjecture, Chapman"s in­ vention was judged too similar to a patent granted to Ernest Kinsey in 1 H7H for a different sliding block puzzle . Iron­ ically, it is precisely the slight difference in mechanical de.�ign that leads to the mathematical impossibility that fueled the incredible success of the 1 5 puzzle. The hook catalogs some variants of the 15 puzzle. such as the challenge of sliding the blocks to form a magic square , and to do so in the minimum number of mm·es. It also explains many solutions that involve cheating in some fashion. such as rotating the box 90 ck­ grees . or leaving the blank in the up­ per left corner, both of \Vhich exploit the amhigu it\· in the original instructions of what constitutes · ·regular order.·· The hook also gives a fair amount of attention to the mathematical litera­ ture on the puzzle. Here too, historical curiosities arise. A pair of articles by 'W"m. \\:oolsey j ohnson and William E . Story i n t h e American ]oumal C!f"Math­ ematics ( December 1H79 J have been \\ idely cited in the literature as the first to characterize exactly which configu­ rations of the 1 5 puzzle can he reached starting from the canonical configura­ tion--Johnson proved that all odd per­ mutations are impossible [2]. while Story demonstrated that all even ones are at­ tainable [5] . Slocum and Sonnevcld he­ came suspicious about the supposed publication elate because an attached note by the editors refers to the ongo­ ing 1 5 puzzle craze, \Yhich \Vould ha\·e been inconsistent with the preponder­ ance of documentary e\·idence, and Johnson's article refers to a newspaper account that turns out to have appeared on March 5 , 1HHO . The authors discov­ ered that the publication of this issue of the journal had been delayed until at least late March 1 8HO, and the issue was not received by the Library of Congress until April 1 7 ( pp. 66-67). The false date, together with Sam Loyd's machinations, caused later writers, myself included. to falsely conclude that the 1 5 puzzle craze had occurred in the 1 H70s. Meanwhile. in a newspaper article in the Hamhurp,iscber Corre,,pcmdellf of

April 6, 1 HHO, the German mathemati­ cian Hermann Schubert published a proof in German of the impossibility of obtaining odd permutations [ :)]. At this time it is unknown whether Johnson's or Schubert"s proof was published first , let alone \\ho submitted first. Schuhert"s proof went overlooked by later schol­ ars, possibly because it did not appear in a peer-reviewed journal. Mathemati­ cians are not used to looking for orig­ inal research in newspapers' Johnson"s and Schuhert"s proofs are explained in a short section guest­ authored by Dick Hess ( pp . 1 1 7-1 1 9 ) . He does a \ ery good job of digesting the arcane language used in the cJrigi­ nal proofs and presenting them in a fashion that the layman should he able to understand. In doing so, he does a sen·ice to our discipline bv demon­ strating to a broad audience how care­ ful mathematical reasoning can accom­ plish what experimentation cannot, in this case resolving the dispute over the solvability of the 1 5 puzzle that had raged in numerous newspaper editori­ als as part of a society-wide debate. He also gives another glimpse of the power of mathematical abstraction by explain­ ing Richard Wilson · s generalization of the 1 5 puzzle to a game of s"·apping labels between adjacent ,-ertices in ar­ bitrary graphs, and his remarkable the­ orem characterizing \vhich configura­ tions are solvable [6]. Unfortunately, Hess also makes sev­ eral small mathematical missteps. In his exposition of Johnson·s proof. he de­ fines a permutation of n objects as be­ ing even or odd according to the par­ ity of the number of cycles in its cycle representation. Luckily, this turns out to coincide with the usual definition when rz is even . as it is for the case under consideration, namely the 16 spaces in the 15 puzzle. But when n is odd, this definition conflicts \Yith the usual one. In explaining Wilson's result, he omits an important hypothesis of the theorem; namely, that the graph under consider­ ation contain no cut vertex. He also dis­ misses papers by Edward Spitznagel [4] and myself [ 1 ] a s '"essentially following Johnson·s approach, "" even though both of these papers actually give alternative proofs of Story·s more difficult result

that all even permutations are attain­ able. Much of the mathematical litera­ ture on the 1 5 puzzle obscures the dis­ tinction between proving that no odd permutation is attainable and proving that a ll even permutations are attain­ able, so I was sad to see that this hook did not do more to clarify that point. In addition to the superb historical research conveyed in the text, the hook contains dozens of beautiful illustra­ tions, including photos of many differ­ ent versions of the 1 5 puzzle. These, and an abundance of humor, make it a pleasure to t1ip through for the casual reader, \vhile the compelling historical account rewards those \vho read it con_'!" to co\·er. For those interested in harnessing the power of fads . it pro­ vides a valuable detailed case study. Rut most important is the uncloaking of Sam Loyd's wildly and enduringly successful hoax. Today"s citizenry would he wise to take this episode as a warning against the · ·he said, she said"" reporting that so often poses as journalism. It was a ba­ sic lack of fact-checking by journalists starting in 1 H 9 1 that allowed Loyd"s de­ ception to get otl the ground, followed by an echo chamber effect that elevated it to the status of accepted history. Ku­ dos to Slocum and Sonne\·elcl for set­ ting the record straight. REFERENCES

[1 ] A F. Archer, "A modern treatment of the

1 5 puzzle, " Amer. Math. Monthly 1 06

(1 999) 793-799. [2] W. W. Johnson, "Notes on the ' 1 5' puzzle 1 , " Amer. J Math. 2 (1 897) 397-399.

[3] H. Schubert, "The Boss puzzle," Hambur­ gischer Correspondent, 6 April 1 880.

[4] E . L. Spitznagel, Jr., "A new look at the fif­ teen puzzle," Math. Mag. 40 (1 967) 1 71 -1 74.

[5] W. E. Story, "Notes on the ' 1 5' puzzle I I , " Amer. J . Math. 2 (1 879) 399-404.

[6] R. M. Wilson, "Graph puzzles, homotopy,

and the alternating group," J Combin. The­ ory (Series B) 1 6 ( 1 974) 86-96.

Aaron Archer AT&T Shannon Research Laboratory 1 80 Park Avenue Florham Park, NJ 07932 USA e-mail: [email protected]

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© 2007 Spnnger SCience-+ Bus1ness Media Inc . Volume 29. Number 2 , 2007

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k1fl,l.iij.J4,i§i

Robi n W i l son

I nternationa l Cong resses of M athematicians

I

n recent years. several International Congresses ha \ e been commemo­ rated on stamps. The first of these was the ivlosc
Moscow (1 966)

Warsaw (1 98213)

SUOM I · FINLAND Helsinki (1 978)

Kyoto (1 990)

1,00

However, the u ncertain political situa­ tion forced the postponement of this congress to 1 983. The first International Congress to he held outside Europe or North America took place in Kyoto in 1 990; the stamp design shows an origami polyhedron. In 1994 the congress was held in Ziirich for the third time; the commemorative stamp featured Jakob Bernoulli and his law of large numbers. For the 1 998 con­ gress in Berlin. the stamp design in­ cluded a solution of the ·squaring-the­ rectangle' problem, dividing a rectangle with integer sides ( in this case, 1 76 a nd 1 7 7 ) into unequal squares with integer sides. The background consisted of spi­ rals made from the decimal digits of 71". For the 2006 International Congress in Madrid. the stamp design included a geometrical sunflower pattern and the earliest known written appearance (976 AD ) of the Hindu-Arabic numerals, found in the Codex Vigilanus in the l3ih­ lioteca del Matematicos of the Escorial monastery.

Zurich (1994)

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11 0 Please send a l l submissions to the Stamp Corner Editor,

-

Robin Wilson, Faculty of Mathematics, The Open U niversity, M i lton Keynes, M K7 6AA, England e-m a i l : r.j .wilson@open . a c . u k

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·-

Berlin (1998)

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Madrid (2006)