«·)·"I"·' I I
Paraconsistent Thoughts About Consistency Philip J. Davis
"You will notice, Pnin said, that when on a Sunday evening in May, 1876 Anna Karenina throws herself under that freight train, she has existed more than jour years since the begin ning of the novel. But in the life of the Lyovins, hardly three years have elapsed. It is the best example of rel ativity in literature that is known to me. "-Vladimir Nabokov, Pnin, ch. 5, sec. 5 (paraphrased). Most philosophers make consistency the chief desideratum, but in mathe matics it's a secondary issue. Usually we can patch things up to be consis tent. -Reuben Hersh, What is Mathe matics?, Really, p. 237.
mathematicians the opportunity to write about any issue of interest to
I am an occasional writer of fiction, en gaging in it as an amusement and a re laxation. Compared to professionals, I
the international mathematical
would say that my "fictive imagination"
community. Disagreement and
is pretty weak. This doesn't bother me
controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion
much, because I'm usually able to come up with something resembling a decent plot. I work on a word processor. The processor has a spelling check and a grammar check. The spelling check is useful but occasionally annoying. For example, it does not recognize proper
should be submitted to the editor-in
names
chief, Chandler Davis.
serted. One time it replaced President
unless
they've
bothered me after I finished was this:
Having produced a fairly long manu script, I was never quite sure whether it was consistent. I don't mean the consis
tency of a pancake batter or whatever the equivalent of that would be in prose style. I mean the everyday sort of con sistency of time, place, person, etc., to which I would add logical consistency. I'm not sure how to give a general and precise definition of consistency. What do I mean by it? I'll give some ex amples of what I think might be seen as inconsistencies. On page 8, of my first draft, I wrote that Dorothy's eyes were blue. On page
I
The Opinion column of f ers
I recently wrote a long short story ti tled "Fred and Dorothy." What really
been
pre-in
Lincoln's War Secretary Seward with Secretary Seaweed. You know the say ing: if something is worth doing, it's
worth doing poorly. In these gray days
my spelling check is the source of many laughs. Therefore it would be a pity if it were "improved." The grammar check in my word processor is only occasionally useful and mostly annoying. It occasionally
throws down a flag when I write a de tached sentence. For example: "Lon don, Cambridge, so why not Manches ter?" It often wants to change a passive formulation into an active one, and its
65, they're brown. On page 24, Fred grad�ated from high school in 1967. On page 47, he dropped out of high school in his ju nior year. On page 73 and thereafter, Dorothy, somehow, became Dorlinda. On page 82, there is an implication that World War II occurred before World War I. There are spelling inconsistencies: on page 34 I wrote "center" while on page 43, I wrote "centre." There are inconsistencies in the point of view.
Fred and Dorothy
is a
story told by a narrator; an "I." There is often an "I problem" in fiction, and here's how Peter Gay in The
Naked Heart: . The Bourgeois Experience from Victoria to Freud describes it: Often enough, the narrator [of the first person novelf--or, rather, his creator cheats a little, recording not only what he saw and heard, or was told, but also what went on in the minds of charac ters who had no opportunity to reveal their workings to him. Most readers, facing these flagrant violations of the narrator's tacit contract with them, suspend their disbelief . . .
suggestion for doing so often ends in a terrible muck. It doesn't catch non sense.
As
a test I wrote, "The man re
We learn to deal with inconsisten cies in books, and we do it in different
boiled the cadences through the mon
ways. Suspension of disbelief is only
key
and my spelling and
one way. Now, whatl would find really
grammar check simply changed "re
useful, if such a thing could exist,
boiled" to "rebelled."
would be a program that checks for
wrench,"
© 2002 SPRINGER·VERLAG NEW YORK, VOLUME 24, NUMBER 4, 2002
3
II
consistency as well as my copy editor
sides by 4, I get 0
=
4. Now is that an in
Louisa does. She's smart. She's careful.
I've now said enough about literary
She has read widely. She knows my
texts and I'm ready to get to logic and
mind. She's worth gold.
mathematics; to Boole and Frege and
that 0
Russell and Godel and Wittgenstein
cause Peano said so. Or did he? Well, if
and all those fellows. Whereas life does
he didn't, I would hope it can be deduced
Imagine now that I have bid Louisa goodbye and replaced her with a con sistency checker that I paid good money
not have a precise definition of con
for. Call the software package CONNIE.
sistency, mathematics has a clear-cut
I run Fred and Dorothy through CONNIE. It immediately comes back
definition. A mathematical system is consistent if you can't derive a con
consistency? It is in 7L, but not in 7L4!
Come to think of it, how do we know =
1 is a contradiction in 7L? Be
from his axioms about the integers. So, depending on where you're com ing from, a set of mathematical symbols may or may not be an inconsistency.
tradiction within it. A contradiction
Just as in fiction. In fact, a set of truly
Dorothy's eyes were blue and on page
would be something like 0
naked mathematical symbols is not
65 they're brown. What's the deal?" Did
sistency is good and inconsistency is
interpretable (or is arbitrarily inter
I have to spell out in my text that in the
bad. Why is it bad? Because if you can
pretable). By "naked" I mean that you
late afternoon October mist, Dorothy's
prove one contradiction, you can prove
have no indication, formal or informal,
eyes seemed brown to me?
anything. In logical symbols,
of where the writer is coming from.
with a message: "On page 8 you said
How did CONNIE handle metaphor? I wrote: "He saw the depths of the sea
(1) For all A and B,
=
1. Con
(A & �A)� B.
Now bring in G6del's famous and notorious Second Incompleteness The
in her eyes." Now CONNIE (a very
And so, if you allow in one measly in
smart package) knew her (its?) Homer
consistency, it would make the whole
erary texts. To state it in a popular way,
and recalled that
program of logical deduction ridicu
the GIT says that you cannot prove the
lous.
consistency of a mathematical system
Gray-eyed Athena sent them a favor able breeze, afresh west wind, singing over the wine-dark sea,
Aristotle knew about equation (1)
orem (the GIT). I want to apply it to lit
by means of itself.
and had inconsistent views about it. In
If mathematics is part of the universe
the literature of logic it's called the
of natural language, and I think it is, then
wine-dark isn't blue. For heaven's sake,
ECQ principle (ex contradictione quodlibet). But I call it the Wellington
get the GIT to imply that it is impossible
please make up your mind about
principle. (The Duke of Wellington
to build a universal consistency checker.
Dorothy's eyes."
1769-1852, victor at the Battle of Wa
Or, for that matter and much more im
and it blew the whistle on me: "Hey,
I believe that with a little thought, I could
Why did "Dorothy" morph into "Dor
terloo.) The Duke was walking down
portant these days, that it is impossible
linda"? That's part of my story: it's the
the street one day when a man ap
name the movie producers decided to
proached him.
to build a universal virus checker. If a
give her after she'd passed her screen test. CONNIE picked up a sentence in ar chaic English and screamed bloody mur
consistency checker can't be produced
The Man: Mr. Smith, I believe? The Duke: If you believe that, you can believe anything. Inconsistency is (or was) the primal
der. The sentence in question was part
for mathematics with its sophisticated and conventualized textual practices and with its limited semantic field, then I have serious theoretical doubts about literary texts.
of a movie script (within my story),
sin of logic. In 1941, in my junior year at
CONNIE might catch Dorothy's eyes
whose action was placed in the 17th cen
Harvard, I took a course in mathemati
being simultaneously blue, brown, and
tury. It rapidly becomes clear that the no
cal logic with Willard Van Orman Quine,
wine-dark, but there will be some in
who in the opinion of some became the
consistencies that CONNIE misses.
tion of consistency is not context-free. And so on and on. A writer of fiction
most famous American philosopher of
Inconsistency is how things appear
can explain away post hoc what appear
his generation. Quine hadjust published
in the world. We spend part of our life
to be inconsistencies. In technical lingo,
his
Mathematical Logic and it was our
cleaning up the confusions, trying to
often employed in mathematical physics,
textbook The course startedjust before,
impose some semblance of order. To
explanations that clear up inconsisten
or shortly after the shattering news
some extent we are successful, but
cies are called interpretations. I suppose that someone, somewhere,
came that J. Barkley Rosser had found
only in a limited sense and for a lim
an inconsistency in the axiom system
ited time. Heraclitus assured us that
has drawn up a taxonomy of textual in
Quine had set up. Well, Quine spent the
nothing is ever the same twice, and
consistencies. It must be extremely long.
whole semester having the class patch
when things begin to get fuzzy we
Mavens who analyze language often
up the booboo in our books; crossing
think, that's not the way we had per
split language into three systems with
out this axiom and replacing it with that;
ceived matters. So I'm afraid we all
replacing this formula with that-while
have to live with and deal with incon
sonal, ideational (i.e., ideas about the
we logical greenhorns were anxious for
sistencies. We learn to do it. Walt Whit
world in terms of experience and logi
man, the poet, knew this. He said,
cal meaning), and textual (ways of com posing the message). I worry mostly
him to get on with it and get to the punch
Q.E.D. as regards primal sins.
about the first two, and I'd limit my con
But back to business. If 0
Do I contradict myself? Very well then I contradict myself (I am large, I contain multitudes. )
different sorts of meaning: interper
sistency checker to work on them. 4
THE MATHEMATICAL INTELLIGENCER
line of logic, whatever that might be. =
1 is an
inconsistency, then by multiplying both
Mathematics is one way we try to
maticians are often smart enough to
"folk theorem" that bad software can
impose order, and we may do it incon
spirit away a contradiction-just as
often be useful.
sistently. Consider the arithmetic sys
Hersh says in the epigraph. Mathemat
Acknowledgments
tem that is embodied in the popular
ical inconsistencies are often exor
and useful scientific computer package
cised by the method of context-exten
I thank Ruth A. Davis and Kay O'Hal
known as MATLAB. Now MATLAB
sion. It is done on a case-by-case basis,
loran for providing me with some im
yields the following statements from which a contraction may be drawn: "le - 50
=
"2 +
0 is false" (i.e., w-50
(le - 50)
=
2 is
=
0),
true."
and it is worth doing only after the con
portant words and ideas.
tradiction has borne good fruit. So the notion
of
mathematical
consistency
may be time- (and coterie-) dependent just as in literature.
REFERENCES
George S. Boolos, John P. Burgess and Richard C. Jeffrey, Computability and Logic,
Well, we all recognize roundoff and
Logicians, who go for the guts of the
4th Ed. , Cambridge Univ. Press, 2002.
know its problems. And we know, to a
generic, and who are over-eager to for
Chandler Davis, Criticisms of the usual ratio
considerable extent, but not totally, how
malize everything, have come up with a
nale for validity in mathematics, in Physicalism
to deal with it; how to prevent it from
concept called
in Mathematics (A.D. Irvine, ed.), Kluwer Aca
getting us into some sort of trouble.
has even been a World Congress to dis
demic, Dordrecht, 1990, 343-356.
cuss the topic. Ordinary logic, as I have
Peter Gay, The Naked Heart, Norton, 1995.
noted, has the Duke of Wellington prop
Reuben Hersh, What is Mathematics, Really ?,
cally but can't exist numerically? At
erty that if you can prove A and not A,
Oxford Univ. Press., 1 997.
one point in history it was a highly ir
then you can prove everything. Para
Karl Menger, Reminiscences of the Vienna Cir
rational conclusion and one worthy of
consistent logic is a way of not having
Is it a contradiction that the diago nal of the unit square exits geometri
slaughtering oxen. Was it a contradiction that there ex ists a function on
[ -oo,
+ oo) that is zero
everywhere except at x
=
0, and whose
area is 1? It wasn't among the physicists
who cooked it up and used the idea pro
paraconsistency.
There
an inconsistency destroy everything.
cle and the M§Jthematc i al Colloquium, Kluwer ' AcademiC, Dordrecht, 1 994.
Contradictions can be true. Perhaps
Chris Mortensen, Inconsistent Mathematics,
such a system might be good for certain
Kluwer Academic, Dordrecht, 1 995.
applications to the real world where conflicting facts are common.
ductively. It was among the mathemati
consistent London, a man approached
cians until Laurent Schwartz came along
the Duke of Wellington.
in the 1940s and showed how to embed functions within generalized functions. More recently, in connection with Hilbert's Fifth Problem, Chandler Davis has written
I cannot see why we would want a lo cally Euclidean group without differ entiability, and yet I think that if some day we come to want it badly in which case we will have some notions of the properties it should have -we should go ahead! After jive or ten years of working with it, if it turns out to be what we were wishing for, we will know a good deal about it; we may even know in what respect it differs from that which Gleason, Montgomery, and Zippin proved im possible. Then again, we may not. . . .
A UTHO R
Walking down the street in para
The Man: Mr. Smith, I believe? The Duke: My dear Sir, don't let your belief bother you. When all is said and done, and para consistency aside, I don't think I can defme consistency with any sort of consistency. But I'm in good company. Paralleling St. Augustine's discussion of the nature of time, though I can't de fine a contradiction, I know one when I see one. In a very important paper written in the mid-1950s, the logician Y. Bar-Hil
PHILIP
J. DAVIS
Division of Applied Mathematics Brown University Providence,
Rl 02912
USA
e-mail:
[email protected]
lel demonstrated that language trans lation was impossible. This demon
Philip J. Davis, a native of Massachu
stration dampened translation efforts
setts and a Harvard Ph.D., has been
for a few years. illtimately it did not
in Applied Mathematics at Brown
deter software factories from produc
since 1 963. He is known for applied
ing language translators that have a
numerical analysis, and his tools are
Inconsistencies can be a pain in the
certain utility and that also produce ab
typically functional analysis and classi
neck, a joy for nit-pickers, and a source
surdities. I'm sure that the software
of tremendous creativity.
factories will soon produce a literary
cal analysis: some might say, he ap
Karl Menger, in his Reminiscences of the Vienna Circle and the Mathe matical Colloquium, tells the story
plies pure analysis to applied. But he
consistency checker called CONNIE. I
is known to many more as a com
will run to buy it. It might be just good
mentator on mathematics. Among his
enough for me. And if I've paid good
many nontechnical publications are
that in Wittgenstein's opinion, mathe
money for it, then, as the saying goes,
maticians have an irrational fear of
it must be worth it. The absurdities it
Hersh, The Mathematical Experience
contradiction. I've often thought as
produces will lift my spirits on gray
and Descartes' Dream.
much, but I also realize that mathe-
days and serve to remind me of the
the widely read books with Reuben
VOLUME 2 4 , NUMBER 4, 2002
5
HANSKLAUS RUMMLER
On the Distribution of Rotation Angles How great is the mean rotation angle of a random rotation?
�
• f you choose a random rotation in 3 dimensions,
its angle is jar from being uni-
formly distributed. And the [n/2] angles of a rotation in n dimensions are strongly correlated. I shall study these phenomena, making some concrete calculations in volving the Haar measure of the rotation groups.
The Angle of a Random Rotation in
3
Dimensions
IR3 has a well-defmed rotation angle a E [0, 1r], and, in the case 0 < a < 1r, also a well-defined axis, which may be represented by a unit vector g E S2• For the identity, only the angle a = 0 is well-defined, whereas any g E S2 can be considered as axis; if a = 1r, there are two axis vectors ± g. By a random rotation we understand a random variable in 80(3), which
Any rotation of the oriented euclidean 3-space
is uniformly distributed with respect to Haar measure. It is clear that the axis of such a random rotation must be uni formly distributed on the sphere S2 with respect to the nat
The Haar measure of 80(3) Proposition 1: If one describes 80(3) by the parame
trization p: [0, 1r] X S2 � 80(3), p(a, g):= rotation by the angle a about g, the Haar measure of 80(3) satisfies p*dJ1-s0(3)(a, 0
gles should give more weight to large angles than to small ones. In order to calculate the distribution of the rotation angle, I first express the Haar measure of 80(3) in appro priate coordinates.
6
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
�
2
= 47T2
It is certainly not uniformly distributed: The rotations by
a, let's say with 0::::; a < 1°, form a small neighbourhood U of the identity ll E 80(3), whereas the ro tations with 179° < a ::::; 180° constitute a neighbourhood V of the set of all rotations by 180°, which make up a surface (a projective plane) in 80(3). It is plausible that V has a greater volume than U, i.e., the distribution of rotation an
2
1
ural area measure, but what about the rotation angle? a small angle
=
sin2
( �) da dA(g)
(1 - cos a)da dA(g),
where dA is the area element of the unit sphere 82. Proof To begin with, observe that the restriction of p to ]0, 1r[ X S2 is a diffeomorphism onto an open set U in 80(3), and that the null set {0, 1r} X S2 is mapped by p onto 80(3) \ U, which is a null set with respect to Haar measure. We can therefore use a and g to describe the Haar measure of 80(3), even if they are not coordinates in the strong sense. The mapping
p is related to the adjoint representation is easy to calcu-
of the group Q of unit quatemions, and it
late the Haar measure of Q. Decomposing a quaternion into its real and imaginary parts, we may describe this group as follows:
with multiplication
(t, g) · (s, TJ)
=
(ts
-
(g, TJ),
l1J
+ sg + g X TJ).
The natural riemannian metric on Q = S3 c IR4 is invariant, and therefore the Haar measure of Q is just a multiple of the riemannian volume element. Using the parametrization cp :
[0, 1r]
x
S2� Q,
c.p(y, g):= (cos y, g sin y)
and taking into account the total volume of SS, we get for the Haar measure of Q
where dA denotes the area element of the unit sphere S2. To get from this the Haar measure of S0(3), we use the adjoint representation r = Ad: Q� S0(3), defmed by rq(O = qrii for q E Q and� E !R3. This is a twofold cover ing and r*df..Lso(3)
In the parametrization ljJ: = have therefore
=
To
2df..LQ·
c.p
: [0, 1r]
X
S2 � S0(3) we
The horizontal projection of the unit sphere S2 onto the tangent cylinder along the equator is an area-preserving map; thus we may choose a point on the cylinder and take the corresponding point on the sphere as axis. This means choosing a random point (A, h) in the rectangle [ -17, 1r] X [ -1, 1] and taking the rotation axis g = (v'f=h2 cos A , VI=h2 sin A, h). For the rotation angle a, we choose a random number a E [0, 1] and take a : = F-1 (a), where
F(a)
=
1
La f(t)dt = -(a - sin a) 1T
0
is the distribution function. Linear algebra tells us how to calculate from g and a the matrix g E S0(3). To test this generator of random rotations, I fixed x E SZ together with a tangent vector g E TxS2 and calculated with Mathematica the tangent vectors dg(x; g) for 600 ran dom rotations g E S0(3). The mapping g � (g(x), dg(x; g)) is a diffeomorphism from S0(3) onto the unit tangent bun dle of S2 and thus makes the rotations g visible by the "flags" (g(x), dg(x; g )) (Fig. 1). For the sake of curiositr,I calculated the mean rotation angle for 5,000 random rotations: The result E5,ooo(a) = 126°13'55" matches the theory, because an easy calculation gives the answer to the question of the subtitle as a con sequence of proposition 2: Corollary: The expectation of the rotation angle of a ran
dom rotation is To finish the proof, we observe that 1/J(y, g) is just the ro tat'ion by 2y about the axis g, i.e., p(a, g)= 1/1(�. n 0 See also [1], pp. 327-329, and [6].
+�
1T
=
1T
2
Random Rotations in
126° 28' 32".
Dimensions
4
The Haar measure of 50(4) The distribution of the rotation angle
The parametrization p is well adapted to our problem, be cause the subset of rotations by a fixed angle a is just the image of the sphere { a} X S2. If we integrate our expres sion for the Haar measure of S0(3) over these spheres, we obtain the following result: Proposition 2: The angle a E [0, 1r] of a random rotation is distributed with density f(a) = l. (1 - cos a):
� :il
7T
0.5
1
1.5
2
2.5
If we identify the euclidean IR4 with the skew field of quater nions IHI, the group Q S3 of unit quaternions acts on IR4 by left and right multiplication with q E Q, Lq : 11-0 � IHI and Rq : IHI � IHI, which are linear isometries, i.e., elements of S0(4). These special rotations generate the whole group S0(4): =
: Q
Q � S0(4),
(p, q) := Lp R-q o
is a group epimorphism with kernel {(1, 1), (- 1, -1)}. (See also [1], pp. 329-330.)
... ... ..
-:.
..
..
":f.:·; :_::: -:_�--�<.·::\.::. ,:. '..· '· . ..:. . .. . . . ·::. -· . -.. . '. : . . . .. ... . . : .·.: . .. · . . ·.· · .
3
./ ·:::
See also [7], pp. 89-93.
,.-:
;
· .
:
.
.
. ,.
: ,
.:
\� :>
•
i#Mii;IIM
.
.
. :.
'•'-t
.·· ..
·.
·.�;� .... . .·
.
.
·.
•
. ... ..... .. ·
Generating random rotations
Integrating our expression of the Haar measure of S0(3) over the segment [0, 1r] X {g) for any g E S2 confmns that the axis g of a random rotation is uniformly distributed with respect to the natural area measure dA on S2• Using this fact and knowing the distribution of the rotation angle, we can generate random rotations by choosing axis and angle as follows:
X
.
.:
..
•
.
.
••
.')�
. ..t.'
..·.
.
·: ·
",y .
..
.
�
. , , �..
:' •: •
}
. .. . .' .. .. ·�· .,..,·.:·��-· :,.. ,
.
·
VOLUME 24, NUMBER 4. 2002
7
Using the parametrization 'P for either factor of the product Q x Q, we obtain a parametrization of S0(4): '¥: [0, 7T1 X (0, 7T1 X S2 X S2 � S0(4), 'l"(s, t, g, TJ) : = ('P(s, g), 'P(t, TJ)). If we admit s, t E [0, 27T1 and calculate modulo 27T, '¥ be comes a fourfold covering'¥: T2 X S2 X S 2 � S0(4) with branching locus ({(0, 0)) U (1r, 1r ) X S 2 x S 2:
))
'l"(s, t, g, TJ) = '¥(1r- s, 7T- t, -g, -TJ) = '¥(7T + s, 7T + t, g, TJ) '¥(27T- s, 27T- t, -g, -TJ) =
for 0 ::s: s, t ::s: 7T, and even'¥(0, 0, g, TJ) '¥(1r, 1r, g, TJ)= 1 for all g, TJ E 82. The Haar measure of S0(4) therefore satisfies =
'l"*d�-tsoc4) =
c sin2 s sin2 t ds dt dA(D dA(TJ),
with a constant c. Pairs of rotation angles
(
)
(?
Proof We must find a u E Q with Tu(g) := ugu= i. But as
S0(3) acts transitively on 82, there exists a rotation which sends g to i, and as the adjoint representation T : Q � S0(3) is onto, there exists u E Q such that this rotation is Tu, i.e. Tu(D = i. 0 These lemmas allow us to show:
Proposition 3: Let p = cos s + g sin s, q = cos t + TJ sin t, where g and TJ are purely imaginary unit quaternions. Then the rotation (p, q) E SO(4) has the pair of rotation angles [s- t, s + t].
Proof By the two lemmas, the rotation (p, q) is conjugate to (cos s + i sin s, cos t + i sin t) and has therefore the same pair of rotation angles. Let us calculate the matrix of the latter rotation with respect to the canonical base (1, i, j, k) of !R4= !HI:
Leos s+i sin Rcos t-i sin
)·
Any rotation g E SO(4) is cm\iugate to a standard rotation Ro41
0 Ro42
0
with Ro4 :
c s 1'7 sm 1'7
=
- sin 1'7 cos 1'7
Choosing the rotation angles 1'71 , 1'1 2 in the interval [0, 27T1, the following pairs are equivalent, i.e., the corresponding rotations are conjugate:
(1'71> 1'7 2)- (1'7 2, 1'71) - (27T- 1'71 , 27T- 1'7 2) - (27T- 1'7 2 , 27T - 1'71).
will
s=
t=
Rots O R L O
0
t
Rots
0
) )
Rott
(
whence � "'
( (O
,
,
Rots-t . . . . (cos s + 1 sm s, cos t + 1 sm t) =
0
which fmishes the proof. 0
�Ots+t }
The class of these equivalent pairs be called the pair of rotation angles [1'71 , 1'7 21· This is an element of T21-, where the equivalence relation- is considered on the torus y 2= S1 x S1. Two rotations in S0(4) are cof\iugate if and only if they have the same pair of rotation angles [1'71> 1'7 21. The following lemmas are needed to determine the pair of rotation angles for an element (p, q) E S0(4).
Corollary (Fig. 2): The pair of rotation angles is dis
Lemma 1: For p, q, p', q' E Q, the rotations (p, q) and (p', q') in S0(4) are conjugate if and only if p is con
Herefis considered as a function on [0, 27T1 x [0, 27T1, i.e., it is normalized so that integrating it over (0, 27T1 x [0, 27T1 gives 1.
jugate to ±p' and q is conjugate to ±q' in Q, with the same sign in either case. Proof (p, q) is conjugate to (p', q') if and only if there exists aTE S0(4) with (p, q) = To (p', q') y-1 . As T = (u, v) for some u, v E Q, we have: a
(p, q) is conjugate to (p', q') u, v E Q with Lp
a
Rfi
o
o
a
o
R
a
= Lu R:v Lp' = Lu Lp' o L:u = Lup'u cvq'v o
if
tributed with density fiWt, 1'7z])
=
T
=
L:u
a
a
(up' , vq' v ).
The kernel of contains only the two elements (1, 1) and (- 1, -1); therefore we have shown that (p, q) is conjugate to (p', q') if and only if there exist u, v E Q such that p= ±up'u and q= ±vq'v , with the same sign in either case. 0 Lemma 2: Let g E S 2 be a purely imaginary quaternion with norm 1. the quaternion p= cos t + g sin t is conjugate to p' = cos t i sin t.
Then
8
+
THE MATHEMATICAL INTEWGENCER
s
1 (cos 1'11 - cos 1'1z) Z. 47T 2
'¥: (0, 7T1 X (0, 7T1 X S2 X S2 � S0(4)
Rfi· Rv R;u Rfi· Rvu o
:2 sin 2 ( 1'71 ; 1'7 2 ) in 2 ( 1'71 ; 1'1z )
Proof Starting with the parametrization
and only if there exist
o
=
+ildii;IIM
and using the relation [it1, it2]=[s- t, s new parametrization:
+ t], we obtain a
With respect to these parameters the Haar measure satisfies I/J*dJLso(4) =
( ; it2 ) sin2 ( it1 ; it2 ) dit1 dit2 dil.(g) dil.(
it1 C sin2
TJ).
Integrating over {(it1, it2)} XS2XS2 for fixed it1, it2 gives us the density
=
ma(gitg-1)=[it] for
The constant C' = 1hr 2 is obtained by integrating this func tion over [0, 27T] X [0, 27T]. 0 n �4
The results obtained in dimensions 3 and 4 can be gener alized to dimension n 2:: 4 using Hermann Weyl's method of integration of central functions on a compact Lie group. A central function is one which is constant on cof\iugacy classes.In the case of S0(3) this is simply a function of the rotation angle, and in the case of SO(4) of the pair of ro tation angles. In dimension n > 4 we can introduce the no tion of a multiangle characterizing the cof\jugacy classes.
The Haar measure of a compact Lie group
Let G be a compact and connected Lie group and T C G a maximal torus. There exists a natural mapping 1/J : GIT X T � G such that the diagram GXT�G
commutes, where cp(g, it):= rg(it)=gitg-1 and the verti cal arrow is the natural projection. The Lie algebra g is endowed with an Ad-invariant scalar product, and if t C g is the Lie algebra of the maximal torus T, its orthogonal complement ±-'- is stable under the map pings Ad g : g � g for g E .G. The restriction of �d g to±-' is denoted by Ad-'- g. With these notations, the Haar measure of a_ can be ex pressed in terms of that of T together with the invariant measure of G/T: Proposition 4: 1/J : G/TXT � G is a finite branched cov
ering. Let dJLa and dJLr denote the Haar measures of G and T, and let dJLa;r be the G-invariant normalized mea sure of the homogeneous space GIT. Then
Multiangles of rotation
Let us begin with the case of a rotation g E SO(n) for even n = 2 m: as in the case n=4, there are m rotation angles it1 ..., itm corresponding to the decomposition of gas di rect sum of m plane rotations: g=Rott't1 E9 ...E9 Ro�m· For odd n = 2 m + 1, there are also m angles it1, ..., itm.Calculating modulo 27T, the list (it1' itm) is an element of the m-torus rm and is unique up to the following symmetries, which define an equiva lence relation on rm: 0
0
�
the iti may be permuted; iti may be replaced by -iti, but only for an even num ber of indices i if n is even; for odd n there is no such
restriction.
Let us call the class ma(g):= [itb ...itml E T"'/� the mul tiangle of the rotation g E SO(n). Two rotations in SO(n) are col\iugate if and only if they have the same multiangle. To determine the multiangle of a rotation x E SO(n), we fix an orthonormal base of �n and consider a cof\iugate of x in the maximal torus T c SO(n) the elements of which have, with respect to the chosen base, the form _
it-
(
Rott't1 • •
•
•
0
0
0 ... Rott'tm
it E rm and g E SO(n).
+Y
=4 (cos it1 - cos it )2· 2
0
all
G/TXT
C'
Rotations in Dimension
in the case n=2 m; in the case n 2 m + 1, one has to add a first column and a first row with first element 1 and zeroes elsewhere. In either case we identify T with the standard torus rm. Obviously, it E rm has the multiangle ma(it) = [it], and this is the same for the whole cof\iugacy class:
)
1/J*dJLa = dJLa!TXJ dJLr, where J : T �
�
is the function J(it) :=det(li - Ad-'-it).
For a proof of this formula, see [2], pp. 87-95. The distribution of the multiangle
Proposition 4 may be applied in our case, with G=SO(n) and T = rm. Now 1/J((g], it)=gitg-1 has for every [g] E SO(n)/T m the same multiangle [it], i.e., ma(I/J([g], it)) = (it] E T/�.
Therefore the density of the multiangle [it], considered as a symmetric function on the torus rm, has the form f(it) =
f J(it) dJLGIT = cJ(it)
C
GIT
with a normalizing constant c. To calculate J(it) = det(ll. - Ad -'-it), we observe that in the case n=2m the elements of±-'- are the symmetric ma trices of the form A=
(
�
�
;
A im
A12 A1s ...A1 m 0 A2s ...A2m 0
0
0
0
0
)
'
0
where the AiJ are 2 X2-blocks.
VOLUME 24, NUMBER 4, 2002
9
A direct calculation shows that Ad-'-{} transforms this matrix by replacing every block Aij by the block
)
with gzm(XI. ... ,Xm)
R111J{Ai'J ) := Ro�Ai'J-Rot,&1. J I
I'
If we identify !R2x2 with IR2 0 IR2, R1Ji,1J becomes the ten i sor product of the two rotations Ro�; and Ro�I The eigen values are therefore e:!:iil;e:!:i-IJi= e i(il; :!: ili), and we obtain
:!:
(
det(ll - Ril;,1l) = 2 sin
ifi
; ifi t(2
sin
= 4(cos ifi - cos ifi?·
ifi
; ifi r
Now Ad-'-iJ is the direct sum of the R1l;,1l·Combining these i results: J(if)
=
2m(m-l)
n
l�i<.I�m
2 (cos i}i- cos i7j)
and f2m (i7)= C )
fl
l!Si<j::=m
(cos ifi- cos i7j) 2.
These formulae apply to the case of even n= 2m. In the case of odd n=2m + 1 one has J(if)
m
=2m2 n
i=l
(1 - cos ifi)
n
lS.i<jsm
(cos i}i- cos i7j)2
m
fzm+l(if)= c n i=l
(1 - cos ifi) n
lSi<jsm
fn(if)= Cgn(cos iJ-1,
10
. . . , cos ifm),
THE MATHEMATICAL INTELUGENCER
g2m+l(Xl,. ·
·
,
m= [n/2],
Xm)=gzm (Xl, ·
(xi - Xj)2
m
··, Xm)
n
i=l
(1-
Xi).
is a well-known function, namely the discriminant of the polynomial (x - x1) . . . (x - Xm). Here we consider the functionsgzm andgzm+1 on the compact simplex D = : m {x E !R ; 1;:::: x1;:::: ...;:::: Xm :2:: - 1} where they are not neg ative and must have a maximum.
g2m
·
·
Proposition 5: The global maximum of gn in D is also the
only local maximum in D..
For the maximum ofg2m,1 = x1 > ofgz m+b 1 > x1 > . . . > xm -1.
... > Xm= -1;for
that
=
Proof Let us consider the even case, i.e., the functiong2m: Obviously, one has x1=1 and Xm =-1 for any local max imum x. Fix these two coordinates and define
{
On the boundary of D' = : {1 :2:: Xz :2:: :2:: Xm-1 :2:: - 1}, has the value -oo, and this function is strictly concave in the interior: its Hessian is the matrix Hh(x)=(hij(x)) with
(cos i}i- cos i7j?·
Figure 3 illustrates the functionf5(i7) for S0(5), where the normalizing factor is C =1/(27T2): You see a "sharper" correlation between the two angles than in the case SO(4). The rotations with the pair of an gles (arccos(l/3), 1r]= [70°31'44", 180°] are the "most fre quent" ones. We shall see that the cases S0(4) and S0(5) are representative of a general phenomenon: The density of the multiangle has always a well-defined maximum with 0 :5 iJ-1 < . . . < ifm :5 7T, and for this maximum ifm= 7T, whereas iJ-1= 0 for even nand iJ-1 > 0 for odd n. To study the density functionsfn(if), observe that they may be written as
IT
l�i<js.nt
and
h
and
=
··(x)= ht]
•
m
2
-I
2 k=l (xi - xk) k*i
2
2 (xi- Xj)
•
•
fori=j fori =F j.
The diagonal elements are strictly negative, the other ele ments are strictly positive but still sufficiently small to make the sum of the elements of any row negative. Therefore, the Hessian is negative definite and h is a strictly concave func tion and has a unique local maximum in the interior of D'. As the natural logarithm is strictly increasing, the function g2m (1,x2, ... , Xm-1. -1) has also a unique local maximum. Forg2m+ 1 the reasoning is similar. 0
As a consequence of this proposition, the density fn(i7) of the multiangle of SO(n) has always one and only one maximum in (0::; iJ-1 < . . . < iftni2J ::; 1r}; for this maximum, iJ-1=0 and ifm=7T if n= 2m, whereas iJ-1 > 0 and ifm=7T in the case n=2m + 1. Here is a list of the most frequent multiangles, i.e., the [if] with maximal density fn(if), for n :5 10: 80(3):
[180°],
80(4):
[0°, 180°],
80(5):
[70°31'44", 180°],
80(6):
[0°, goo, 180°],
80(7):
[46°22'41", 106°51'07", 180°],
80(8):
[0°, 63°26'06", 116°33'54", 180°],
80(g):
[34°37'55", 7g033'46", 125°07'13", 180°],
80(10):
[0°, 64°37'23", goo, 115°22'37", 180°]
A U T HO R
SpringerMath(JJxpress Per onalized book announcement to your e-mail box. www.springer-ny.com/express
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Yo rk 1967) 2. W.
Greub, S. Halperin and R. Vanstone, Connections, Curvature and
Phy ic , and Statistics.
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Mathematical Physics
Cohomology, volume II (Academic Press, New York and London 1 973)
Scientific Computing
3. J. M . Hammersley, The distrb i ution of distances in a hypersphere,
Ann. Math. Statist. 21 (1 950), 447-452 4. B. Hostinsky, Probabi/ites relatives a Ia position d'une sphere a cen
Dynamical Sy tem.
tre fixe, J. Math. Pures et Appl. 8 (1 929), 35-43
5. A. T. James, Normal multivariate analysis and the orthogonal group,
Ann. Math. Statist. 25 (1 954), 40-75
6. R. E. Miles, On random rotations in �3• Biometrika 52 (1 965), 636-639
7. D. H. Sattinger and 0. L. Weaver, Lie Groups and Algebras with Ap plications to Physics, Geometry and Mechanics (Springer-Verlag, Berlin, Heidelberg, New York, Tokyo 1 986)
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Springer
VOLUME 24, NUMBER 4, 2002
11
Mathematically Bent
Colin Adams, Editor
body else's. Destiny waits for no one. As we stumbled toward the front line, ominous clouds hung low in the sky. We found the rest of the unit near the frontline. Pipsqueak looked like she was going to lose her breakfast, and Pops's hands were shaking. (He was a contin Colin Adams uing student.) Sarge munched noncha lantly on a toaster pastry. Was she remember that day as if it were yes really that unconcerned or was that terday. I will never forget it. Often, the impression she wanted us to have? I wake up at night, in sweat-soaked I didn't know for sure, but the toaster sheets, screaming, "Look out, Sarge, tart sure looked good. As we spread out over the lecture look out!" Often my roommate is screaming, too. "Shut up, shut up!" But hall, hunkering down in our foxholes, I can't shut up. I have to tell the story, I felt queasy myself. This was it. The the story of that fateful day. A day that real thing. No more training sessions, with dummy problems whizzing over can never be forgotten. We were fresh out of boot camp, head and a solutions manual available Leftie and me. Hardly knew an integral for cover. This would be live ammuni from a derivative. We thought the tion exploding around us. Everyone power rule was complicated. Just a else looked as frightened as I felt. Now, pair of snot-nosed calc students. But we fmd out what you're made of, I they said we were ready for Calc II. thought to myself, as the hour struck How ridiculous that sounds now. and the general down front signaled We arrived in country and were as the beginning of the battle. I gulped signed to a unit of misfits. Sarge was once and turned over the cover page. A couple of partial derivatives whis the only one of us who had seen real combat before. She had fought in tled overhead, and I thought to myself, WWWI , a web-based trig course. And I can handle this. I started firing, then there was Pipsqueak, Pops, Leftie, plugged a couple quick. Hey, no worse and me. They called me Kodowski. I than an afternoon of video games, I wanted them to call me Tootsie. But said to myself. Then I came up over the next page they refused. Before we had even fmished un and swallowed hard as I found myself packing our gear, we heard a yell. "In face-to-face with an armored series di coming!" Grunts dove for cover. Sarge vision. I didn't even stop to think. I just just kept eating her granola bar. "Re peppered them with Ratio Tests. A few lax," she said. "It's just a quiz." I stayed went up in flames. The rest rolled for low anyway. It seemed dangerous ward. I switched to Root Test, spray enough to me. But it wouldn't be long ing them indiscriminantly. A couple more went down but the rest rumbled before I understood the difference. I remember that fateful morning as forward. So I lobbed in a couple of Ba if it were yesterday. I woke to some sic Comparison Tests and a Limit Com thing dripping on my forehead. Leftie parison Test or two. Then I let loose had wet the upper bunk again. He gave with the Alternating Series Test and fol new meaning to the words math anxi lowed up with half a ton of nth Term ety. I pulled him off his bunk and we Tests. That ought to do it, I thought, as had a quick shoving match..Then we I waited for the smoke to clear. But threw on our uniforms. No time to among the littered carcasses on the field brush teeth or comb hair. Ours or any- before me, there still stood one lone se-
The Red Badge of Courage
The proof is in the pudding. Opening a copy of The Mathematical
Intelligencer you may ask yourself uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematica� it's a humor column, and it may even be harmless.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman
Science Center, Williams College, Williamstown, MA 01267 USA e-mail: [email protected]
12
I
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
ries. At first I couldn't make it out. But as it lumbered forward, I suddenly real ized what this monstrosity must be. It was the dreaded harmonic series. It looked right at me and then let out a howl that turned my bowels to ise. How do you stop the harmonic se ries? I tried desperately to remember. We had talked about this in basic train ing. My instructor's voice echoed in my head, "Pray you never see a harmonic series in battle. They are the nastiest, the ugliest series you will ever see. They diverge, but just barely. There is only one thing to do if you ever find yourself looking down the barrel of the har monic series . . . . " Yes, yes, I thought, as I waited for the voice to finish its ex planation. "Use the integral test." "See you in hell," I screamed, as I pulled the trigger. The series blew into a million pieces. I laughed maniacally. "Now that's what I call a divergent se ries." Soldiers in adjacent foxholes said, "Shhh," and the general down front gave me a concerned look. I turned the page, and took a triple integral right in the gut. I rolled out of my seat and down three steps of the auditorium stairs. A medic, must have been a TA no more than years old, rushed over. "Are you all right?," she asked, a concerned look on her face. I felt for the wound in my belly, but miraculously, my hand came out clean. "It must have hit me in the belt buckle," I said as she helped me to my feet. She handed me my helmet and gave me a strange look. She was probably won dering how anyone could survive a triple integral. But stranger things have happened. I retook my seat. There was a noise behind me and I looked around just in time to see Leftie turning tail and heading for the exit. "Leftie, get back here," I yelled. "They'll courtmartial you for sure." The neighboring soldiers shushed me again. Afraid I would attract ordi nance. I should have known Leftie wouldn't have the guts for it. Ever"since that quiz problem on improper inte grals, he had had the shakes.
22
I leaned over the exam and a word problem went off right in my face, something about length plus girth of a package at the post office. There was red ink everywhere. I waved the medic over and pointed at the problem, but she said, "I'm sorry. I can't help you." I guess there were grunts hurt worse than me. I pulled off my helmet and tied a bandana around my head. It was a sea of red ink out there. The noise was deafening. I started working on the problem in spite of the pain. At one point, I happened to glance over at the Sarge. She didn't look right. I gave her the thumbs up sign, but she didn't respond. She looked like she might be sick. She was slumped down in her seat. I couldn't see it, but I had to assume there was a pool of red ink on the exam in front of her. I realized she must have taken one in the gut. She was the one who had come up with my nickname Kodowski. Granted it was my last name, but it had meant a lot to me the first time she called me that. She had saved my ass at least a dozen times already. And now I was losing her, and there was nothing I could do about it. The frustration welled up inside me, and suddenly I roared. Something inside me snapped. I was no longer a human being. I was a calculus killing machine. I flipped the page and moved down eight partial de rivatives. I turned around and nailed three limit problems before they even saw me. I took out a triple integral in cylindrical coordinates. Nothing could stop me. Three chain rule problems turned to run, but I never gave them the chance. I flipped page after page. A man with a mission, I was singlehand edly turning the tide. Suddenly I real ized the battle was almost over. I tri umphantly flipped the last page and found myself face-to-face with the nas tiest triple integral problem I had ever seen. It was a volume inside a sphere but outside a cylinder; the famous cored apple. But it said to do it in spherical coordinates. You have to be kidding, I thought to myself. What
twisted devious mind would create such a diabolical weapon? I had no idea what to do. But then I remembered Sarge's words. "You can't come at a problem like that directly. Come at it from be low. One step at a time." "Yeah Sarge, I remember," I said out loud. I first figured out the equations for the sphere and the cylinder in spherical coordinates. One step in front of the other, Sarge. Then I looked at the intersection. "It's described by an angle, Sarge, I know." I wrote down the triple integral, Sarge's words echo ing in my ears. "Don't forget. p2 sin cp dp dcp d(} in the integrand." "Don't worry, Sarge. I won't forget that for as long as I live." And then it came down to just pulling the .trigger. The integral could essentially do itself. I circled my an swer in bright purple ink. Then I flipped the exam closed, , stood and walked down to the front of the room. The general looked at me nervously. "Are you proud of yourself?" I said. "All these young lives, wasted. Littered on the field of battle. Never again to raise a pencil for mathematics. Do you feel good about that?" He looked confused. "Here is your stinking exam", I said as I threw it down on the table. He stood open-mouthed as I turned and walked up the steps. We lost them all that day, Sarge, Pip squeak, Pops, and Leftie. They became Psych majors. I still see them in the halls sometimes, but they never meet my gaze. The math walking wounded. I was awarded a silver cross to hang on my A, making it an A+. I was pro moted, too. They made me a grader. They wanted me to go to officer' s train ing school at Princeton or maybe Berkeley. And maybe someday I will. Maybe that would make it all worth while. But I have to get over the night mares first. I have to reconcile my vic tory with the loss of my friends. I have to see mathematics as a tool for good, not a weapon of destruction. Only then, will I be able to move on.
VOLUME 24, NUMBER 4, 2002
13
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THE MATHEMATICAL INTELLIGENCER
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S O C I ETY F O R l N D U ST R I A L A N D A P P L I E D M A T H E M AT I C S
BRUCE C. BERNDT, BLAIR K. SPEARMAN, KENNETH S. WILLIAMS
Com mentary on an U n pu bl ished Lectu re by G . N . Watson on Solvi ng the Qu i ntic
he following notes are from a lecture on solving quintic equations given by the late Professor George Neville Watson (1 886-1 965) at Cambridge University in 1 948. They were discovered by the first author in 1 995 in one of two boxes of papers of Pro fessor Watson stored in the Rare Book Room of the Library at the University of
Birmingham, England. Some pages that had become sep arated from the notes were found by the third author in one of the boxes during a visit to Birmingham in 1999. "Solving the quintic" is one of the few topics in mathe matics which has been of enduring and widespread inter est for centuries. The history of this subJ"ect is beautifully iUustrated in the poster produced by MATHEMATICA. Many attempts have been made to solve quintic equations; see, for example, [6)-[14], [17)-[21), [28]-[32), [34]-[36], [58]-[60). Galois was the first mathematician to deter mine which quintic polynomials have roots expressible in terms of radicals, and in 1991 Dummit [24) gave for mulae for the roots of such solvable quintics. A quintic is solvable by means of radicals if and only if its Galois group is the cyclic group 71./571. of order 5, the dihedral group D5 of order 10, or the Frobenius group F2o of order
20.In view of the current interest (both theoretical and computational) in solvable quintic equations [24), [33), [43)-[46), it seemed to the authors to be of interest to pub lish Professor Watson's notes on his lecture, with com mentary explaining some of the ideas in more current mathematical language. For those having a practical need for solving quintic equations, Watson's step-by-step pro cedure will be especially valuable. Watson's method ap plies to any solvable quintic polynomial, that is, any quintic polynomial whose Galois group is one of 71.1571., D5 or F2o· Watson's interest in solving quintics was undoubtedly motivated by his keen interest in verifying Srinivasa Ramanujan 's determinations of class invariants, or equivalently, singular moduli. Ramanujan computed the values of over 100 class invariants, which he recorded
The first author thanks Professor Norrie Everitt of the University of Birmingham for an invitation to visit the University of Birmingham in October
1995. 1 999.
The third author thanks Carleton University for a travel grant which enabled him to travel to the University of Birmingham, England in December The authors thank the staff of the University of Birmingham Library for making the papers of Watson available to them.
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 4, 2002
15
were verified in two papers by Berndt, Chan, and Zhang [2], [3]; see also Berndt 's book [1, Part V, Chapter 34). Chan [ 16] has used class field theory to put Watson's determi nations on a firm foundation, and Zhang [61], [62] has used Kronecker's limit formula to verify Watson's calcu lations. Professor Watson held the Mason Chair of Mathemat ics at the University of Birmingham from 1918 to 1 951. He was educated at Cambridge University (1904-1908), where he was a student of Edmund Taylor Whittaker (1873-1 956). He became a Fellow of Trinity College, Cambridge, in 1 91 0. From 1 91 4 to 1918 he held aca demic positions at University College, London. Watson devoted a great deal of his research to extending and pro viding proofs for results contained in Ramanujan's Note books [39]. He wrote more than thirty papers related to Ramanujan's work, including the aforementioned papers on class invariants or singular moduli. Most mathematicians know Watson as the co-author with E. T. Whittaker of the classic book A Course of Mod em Analysis, first published in 1 915, and author of the monumental treatise Theory of Bessel Functions, first published in 1922. For more details of Watson's life, the reader may wish to consult [22], [41], [48]. We now give the text of Watson's lecture with our com mentary in italics. In the course of the text we give the contents of three sheets which presumably Watson handed out to his audience. The first of these gives the basic quan tities associated with a quintic equation, the second gives twenty-jour pentagrams used in showing that permuta tions of the suffixes of
(X1X2 + X2X3 + X3X4 + X4X5 + X5X1 - X1X3 - X3X5 - X5X2 - X2X4 - X4X1)2 G. N. Watson
without proofs in his paper [37] and at scattered places throughout his notebooks, especially in his first notebook [39]. Although many of Ramanujan's class invariants had been also calculated by Heinrich Weber [57], most had not been verified. Class invariants are certain algebraic num bers which are normally very difficult to calculate, and their determinations often require solving a polynomial equation of degree greater than 2; and, in particular, 5. Watson [52] used modular equations in calculating some of Ramanujan's class invariants; solving polyno mial equations of degree exceeding 2 was often needed. In a series ojsixjurther papers [50]-[51], [53]-[56], he de veloped an empirical process for calculating class in variants, which also depended heavily on solving poly nomial equations of high degree. He not only verified several of Ramanujan's class invariants but also found many new ones. For these reasons, Watson proclaimed in his lecture that he had solved more quintic equations than any other person. Despite Watson 's gargantuan efforts in calculating Ramanujan's class invariants, eighteen re mained unproven until recent times. The remaining ones
16
THE MATHEMATICAL INTELLIGENCER
yield six distinct expressions, and the third gives Wat son's method of solving a solvable quintic equation in radicals.
I
am going to begin by frankly admitting that my subject this evening is definitely old-fashioned and is rather stodgy; you will not fmd anything exciting or thrilling about it. When the subject of quintic equations was first seriously investigated by Lagrange it really was a "live" topic; the ex tent of the possibility of solving equations of various de grees by means of radicals was of general interest until it was realized that numbers represented by radicals and roots of algebraic equations were about what one nowa days calls algebraic numbers. It is difficult to know quite how much to assume that you already know about solutions of algebraic equations but I am going to take for granted . . .
Watson's notes do not state the prerequisites for the lecture! I cannot begin without saying how much I value the com pliment which you have paid me by inviting me to come from a provincial University to lecture to you in Cambridge; and now I am going to claim an old man's privilege of in dulging in a few reminiscences. In order to make my lee-
SHEET 1 The denumerate form of the quintic equation is PoY5 + P1Y 4 + P2 Y3 + PaY 2 + P4 Y + P5
= 0.
The standard form of the quintic equation is ax5 + 5bx4 + 10cx3 + 10dx2 + 5ex + f
=
0. (x = lOy)
The reduced form of the quintic equation is z 5 + 10Cz 3 + 10Dz2 + 5Ez + F
=
0.(z
= ax + b)
The sextic resolvent is where K = ae - 4bd + 3c2,
-2a2dj + 3a2e2 + 6abcf- 14abde- 2ac2e + 8acd2 - 4b3f + 10b2ce + 20b2d2 - 40bc2d + 15c4, M = a3cj2 - 2a3def + a3e3 - a2b2j2 - 4a2bcef + 8a2bd2f- 2a2bde2 -2a2c2dj - l la2c2e2 + 28a2cd2e - 16a2d4 + 6ab3ej -12ab2cdf + 35ab2ce2 - 40ab2d2e + 6abc3f- 70abc 2de + 80abcd3 + 35ac4e- 40ac 3d 2 - 25b4e2 + 100b3cde -50b2c 3e - 100b2c2d 2 + 100bc 4d- 25c 6.
L
=
ture effective, I must endeavour to picture to myself what is passing in the mind of John Brown who is sitting some where in the middle of this room and who came up to Trin ity last October; and I suppose that, in view of the recent decision about women's membership of the University, with the name of John Brown I must couple the name of his cousin Mary Smith who came up to Newnham at the same time. To try to read their thoughts I must cast my mind back 43 years to the Lent Term of 1905 which was in my first year. If I had then attended a lecture by a mathe matician 43 years my senior who was visiting Cambridge,
Watson Building, University of Birmingham. (Photo from 1995.)
an inspection of the Tripos lists would show that the most likely person to satisfy the requisite conditions would have been the late Lord Rayleigh, who was subsequently Chan cellor. Probably to you he seems quite prehistoric; to me he was an elderly and venerable figure whose acquaintance I made in 1912, and with whom I subsequently had some correspondence about electric waves. You cannot help re garding me as equally elderly, but I hope that, for a num ber of reasons you do not �onsider me equally venerable, and that you will believe me when I say that I still have a good deal of the mentality of the undergraduate about me. However, so far as I know, Lord Rayleigh did not visit Cam bridge in the Lent term of 1905, and so my attempt at an anal ogy rather breaks down. On the other hand a visit was paid to Cambridge at the end of that term by a much more emi nent personage, namely the Sultan of Zanzibar. For the ben efit of those of you who have not heard that story, I mention briefly that on the last day of term the Mayor of Cambridge received a telegram to the effect that the Sultan and his suite would be arriving by the mid-day train from King's Cross and would be glad if the Mayor would give them lunch and arrange for them to be shown over Cambridge in the course of the afternoon. The program was duly carried out, and during the next few weeks it gradually emerged that the so-called Sul tan was W. H. de Vere Cole, a third-year Trinity undergradu ate. It was the most successful practical joke of an age in which practical joking was more popular than it is to-day.
VOLUME 24, NUMBER 4, 2002
17
If you could be transported back to the Cambridge of 1905, you would find that it was not so very different from the Cambridge of 1948. One of the differences which would strike you most would probably be the fact that there were very few University lectures (and those mostly professor ial lectures which were not much attended by undergrad uates); other lectures were College lectures, open only to members of the College in which they were given, or, in the case of some of the smaller Colleges, they were open to members of two or three colleges which had associated themselves for that purpose. Thus most of the teaching which I received was from the four members of the Trin ity mathematical staff; the senior of them was Herman, who died prematurely twenty years ago; in addition to teaching me solid geometry, rigid dynamics and hydrodynamics, he infected me with a quality of perseverance and tenacity of purpose which I think was less uncommon in the nine teenth century than it is to-day when mathematics is tend ing to be less concrete and more abstract. Whitehead was still alive when I started collecting material for this lecture. Whittaker, who lectured on Electricity and Geometric Op tics, whose name is sometimes associated with mine, is liv ing in retirement in Edinburgh; and Barnes is Bishop of Birmingham. Outside the College I attended lectures by Baker on Theory of Functions, Berry of King's who taught me nearly all of what I know of elliptic functions, and Hob son on Spherical Harmonics and Integral Equations; also two of the Professors of that time that were Trinity men, Forsyth and Sir George Darwin, whom I remember lectur ing on curvature of surfaces and the problem of three bod ies, respectively. Two things you may have noticed, the large proportion of my teachers who are still alive, and the
insularity, if I may so describe it, of my education. If that hypothetical lecture by Lord Rayleigh had taken place, he could have given a more striking illustration of insularity which you will probably hardly credit. In his time, each Col lege tutor was responsible for the teaching of his own pupils and of nobody else; he was aided by one or two as sistant tutors, but the pupils, no matter what subject they were reading, received no official instruction except from their own tutor and his assistants. After spending something like ten minutes on these ir relevancies, it is time that I started getting to business. There is one assumption which I am going to make throughout, namely that the extent of your knowledge about the elements of the theory of equations is roughly the same as might have been expected of a similar audi ence in 1905. For instance, I am going to take for granted that you know about symmetric functions of roots in terms of coefficients and that you are at any rate vaguely famil iar with methods of obtaining algebraic solutions of qua dratic, cubic and quartic equations, and that you have heard of the theorem due to Abel that there is no such solution of the general quintic equation, i.e., a solution expressible by a number of root extractions.
In modern language, if.f(x) E iQ[x] is irreducible and of de gree 5, then the quintic equation.f(x) = 0 is solvable by rad icals if and only if the Galois group G of .f(x) is solvable. The Galois group G is solvable if and only if it is a sub group of the Frobenius group F20 of order 20, that is, it is Fzo, D5 (the dihedral group of order 10), or 7l./57L (the cyclic group of order 5); see for example [24, Theorem 2, p. 397], [25, Theorem 39, p. 609]. Thus a quintic equation .f(x) = 0 cannot have its roots expressed by a finite number of root extractions if the Galois group G of f is non-solvable, that is, if it is S5 (the symmetric group of order 120), or A5 (the alternating group of order 60). "Almost all" quintics have S5 as their Galois group, so the ''general" quintic is not solv able by radicals. It is easy to give examples of quintics which are not solvable by radicals; see for example [46]. You may or may not have encountered the theorem that any irreducible quintic which has got an algebraic solution has its roots expressible in the form
where w denotes exp(2 71i/5), r assumes the values 0, 1 , 2, 3, 4, and u Y, ut u�, u� are the roots of a quartic equation
whose coefficients are rational functions of the coefficients of the original quintic. If you are not familiar with such re sults, you will find proofs of them in the treatise by Burn side and Panton.
One can find this in Section 5 of Chapter XX of Burnside and Panton 's book [5, Vol. 2]. A modern reference for this result is [24, Theorem 2, p. 397].
Bruce Berndt with Ramanujan's Slate.
18
THE MATHEMATICAL INTELLIGENCER
When I was an undergraduate, all other knowledge about quintic equations was hidden behind what modern politi-
cians would describe as an iron curtain, and it is conve nient for me to assume that this state of affairs still per sists, for otherwise it would be a work of a supererogation for me to deliver this lecture. I might mention at this point that equations of the fifth or a higher degree which possess algebraic solutions (such equations are usually described as Abelian) are of some im portance in the theory of ellipticfunctions, apart from their intrinsic interest.
with the property that their cubes have, not six different values, but only two, namely
(a + {3E + y€2)3,
(a + {3€2 + yE)3,
and these expressions are the roots of a quadratic equation whose coefficients are rational functions of the coefficients of the cubic. When the cubic equation is
ax3 + 3bx2 + 3cx + d
=
0,
the quadratic equation is
Today such equations are called solvable. There is, for instance, a theorem, also due to Abel, that the equations satisfied by the so-called singular moduli of el liptic functions are all Abelian equations.
Singular moduli are discussed in Cox's book [23, Chapter 3) as well as in Berndt's book [1, Part V, Chapter 34]. It was these singular moduli which aroused my interest some fifteen years ago in the solutions of Abelian equa tions, not only of the fifth degree, but also of the sixth, sev enth and other degrees higher still. It consequently became necessary for me to co-ordinate the work of previous writ ers in such a way as to have handy a systematic procedure for solving Abelian quintic equations as rapidly as possible, and this is what I am going to describe tonight.
Methods for solving a general solvable quintic equation in radicals have been given in the 1990s by Dummit [24) a'Jild Kobayashi and Nakagawa [33];see also [47].
a6X2 + 27a3(a2d - 3abc + 2b3)X + 729(b2- ac)3
=
0,
and there is no difficulty in completing the solution of the cubic.
It is easily checked using MAPLE that this quadratic is correct. For the quartic equation, with roots a, {3, y, o, such ex pressions as
(a + f3 - y- o)2,
(a + X -
8-
{3)2,
(a + o ;- f3- y)2
have only three distinct values; similar but slightly simpler expressions are
af3 + yo- ay - ao- f3y- {3 o, etc., or simpler still,
af3 + yo,
ay + {3o,
a o + f3y.
When the quartic equation is taken to be
ax4 + 4bx3 + 6cx2 + 4dx + e
=
0,
the cubic equation satisfied by the last three expressions is To illustrate the nature of the problem to be solved, I am now going to use equations of degrees lower than the fifth as illustrations. A reason why such equations pos sess algebraic solutions (and it proves to be the reason) is that certain non-symmetric functions of the roots ex ist such that the values which certain powers of them can assume are fewer in number than the degree of the equa tion. Thus, in the case of the quadratic equation with roots a and {3, there are two values for the difference of the roots, namely
However the squares of both of these differences have one value only, namely
(a + {3)2 - 4a{3, and this is expressible rationally in terms of the coeffi cients. Hence the values of the differences of the roots are obtainable by the extraction of a square root, and, since the sum of the roots is known, the roots themselves are im mediately obtainable. The cubic equation, with roots a, {3, y, can be treated sim ilarly. Let €3=1 (E i=- 1).Then we can form six expressions
� + yE + ail',
f3 + y€2 + aE,
- (16b2e + 16ad2 - 24ace)=0,
and, by the substitution
aX- 2c= -48, this becomes
4&- 8(ae - 4bd + 3c2) - (ace + 2bcd - ad2 - b2e - c3)
=
0,
which is the standard reducing cubic
a- {3, f3- a.
a + {3E + y€2, a + {3€2 + yE,
a,X3 - 6a2cX2 + (16bd- 4ae)aX
y + aE + {3€2, y + ail' + {3E,
4& - I8 - J=0. This is discussed in [26, pp. 191-197; see problem 15, p. 197), where the values of I and J are given by I=ae- 4bd + 3c 2, J
=
a b c b c d c d e
I have discussed the problem of solving the quartic equa tion at some length in order to be able to point out to you the existence of a special type of quartic equation which rarely receives the attention that it merits. In general the re ducing cubic of a quartic equation has no root which is ra tional in the field of its coefficients, and any expression for the roots of the quartic involves cube roots; on the other
VOLUME 24, NUMBER 4, 2002
19
SHEET 1A The discriminant Ll o f the quintic equation i n its standard form i s equal to the product o f the squared differences of the roots multiplied by a8/3125.The value of the discriminant Ll in terms of the coefficients is
a4j4 - 20a3bcf 3 - 120a3cdf 3 + 160a3ce2j2 + 360a3d2ef 2 -640a3de3f + 256a3e5 + 160a2b2df 3- 10a2b2e2f 2 + 360a2bc2j3- 1640a2bcdej2 + 320a2bce3f - 1440a2bd3f 2 +4080a2bd2c2f- 1920a2bde4- 1440a2c3ef 2 + 2640a2c2d2f 2 +4480a2c2de2f - 2560a2c2e4 - 10080a2cd3ef + 5760a2cd2e3 + 3456a2d5f - 2160a2d4e2- 640ab3cj3 + 320ab3def 2 -180ab3c3f + 4080ab2c2ef 2 + 4480ab2cd2f 2 - 14920ab2cde2f +7200ab2ce4 + 960ab2d3ef- 600ab2d2e3 - 10080abc3df2 +960abc3e2f + 28480abc2d2ef - 16000abc2de3 - 11520abcd4j + 7200abcd3e2 + 3456ac 5f 2 - 11520ac4def + 6400ac4e3 + 5120ac3d3f- 3200ac3d2e2 + 256b 5f 3- 1920b4cej2 -2560b4d2f 2 + 7200b4de2f - 3375b4c4 + 5760b3c2df2 -600b3c2e2f- 16000b3cd2ef + 9000b3cde3 + 6400b3d4j -4000b3d3e2 - 2160b2c4j2 + 7200b2c3def- 4000b2c3e3 -3200b2c2d3f + 2000b2c2d2e2.
hand, there is no difficulty in constructing quartic equations whose reducing cubics possess at least one rational root; the roots of such quartics are obtainable in forms which involve the extraction of square roots only. Such quartics are anal ogous to Abelian equations of higher degrees, and it might be worth while to describe them either as "Abelian quartic equations" or as "biquadratic equations," the latter being an alternative to the present usage of employing the terms quar tic and biquadratic indifferently. (I once discussed this ques tion with my friend Professor Berwick, who in his lifetime was the leading authority in this countcy on algebraic equa tions, and we both rather reluctantly came to the conclusion that the existing terminology was fixed sufficiently firmly to make any alteration in it practically impossible.)
If f(x) E Q[x] is an irreducible quartic polynomial, its cubic resolvent has at least one rational root if and only if the Galois group of f(x) is the Klein 4-group V4 of or der 4, the cyclic group 7lJ47L of order 4, or the dihedral group D4 of order 8. Since D4 is not abelian, it is not ap propriate to call such quartics "abelian. " For the solution of the quartic by radicals, see for example [25, p. 548]. After this very lengthy preamble, I now reach the main topic of my discourse, namely quintic equations. Some of you may be familiar with the name of William Hepworth Thomp son, who was Regius Professor of Greek from 1853 to 1866, and subsequently Master of Trinity until 1886.A question was once put to him about Greek mathematics, and his reply was, "I know nothing about the subject. I have never even lectured upon it." Although there are large tracts of knowledge about quintic equations about which I am in complete ignorance, I have a fair amount of practical experience of them. For in stance, if my friend Mr. P. Hall of King's College is here this evening, he will probably be horrified at the ignorance which 20
THE MATHEMATICAL INTELLIGENCER
I shall show when I say anything derived from the theory of groups. On the other hand, while to the best of my knowl edge nobody else has solved more than about twenty Abelian quintics (you will be hearing later about these solvers, and I have no certain knowledge that anybody else has ever solved any), my own score is something between 100 and 120;and I must admit that I feel a certain amount of pride at having so far outdistanced my nearest rival.
Young solved several quintic equations in [58] and [59]. The notation which I use is given at the top of the first of the sheets which have been distributed. The first equa tion, namely
PoY5 + P1y4 + P2Y3 + P3Y2 + P4Y + P5 = 0, is what Cayley calls the denumerate form, while
ax5 + 5bx4 + 10cx3 + 10&2 + 5ex + f=0, is the standard form. The second is derived from the first by the substitution lOy=x, with the relations
a= Po, b=2pl, c=10p , d = lOOp , 2 3 e=2000p4, f= l0 5p5. Next we carry out the process usually described as "re moving the second term" by the substitution ax + b=z, which yields the reduced form
z5 + 10Cz3 + 10Dz2 + 5Ez + F=0, in which
C=ac - b2, D=a2d - 3abc + 2b3, E=a3e- 4a2bd + 6ab2c - 3b4, F =a4j- 5a3be + 10a2b2d - 10ab3c + 4b5. The roots of the last two quintics will be denoted by Xr and Zr respectively with r=1, 2, 3, 4, 5.
SHEET
2
4
1
2
3
1
2
5
4
5
2
2
3
5
3
2
4
5
4
3
2
5
2
4
3
3
3
1
1
1
3
4
5
2
5
5
4
3
1
1
4
2
3
2
2
5
3
1
3
4
VOLUME 24, NUMBER 4, 2002
21
SHEET
3
The roots of the quintic in its reduced form are
Zr+ l with
w
=
=
wrul + u.J.ru2 + w3ru3 + w4ru4
exp(277i/5), r = 1, 2, 3, 4, 0.
(1) U 1U4 + U2U3 -2C. (2) u1us + u�u1 + u�u4 + u�u2 = - 2D. (3) u1u� + u�u� - UJU2UsU4 - u1u2 - u�u4 - u3ul - U�Us = E. (4) u�[+ u� + u� + u� - 5(u1u4 - u2us)(uius - u�u 1 - u�u4 + u�u2) =
New unknowns, (} and
(5) (6)
T,
=
-F.
defined by
U 1U4 - U2U3 = 2 (}, u1u3 + u�u2 - u�u 1 - u�u4 = 2T. u1u4 = -c + o, u2us = - c - o. u1us + u�u2 -D + T, u�u1 + u�u4 = -D - T. u1us - u�u2 = ::±:: Y(D - T? + 4(C - 0)2 (C + 0) = Rb u�ul - u�u4 ::±:: Y(D + T)2 + 4(C + 0)2 (C - 0) R2. uru2 = (UIUs)(u�u l )/(UzUs), etc., U� = (uiusi(u�ul)/(UzUd, C(D2 - T 2) + (C2 - 02)(C2 + 302 - E) = R1R2 0. (D2 - T 2 )2 + 2C(D2 - T2)(C2 + 3 ()2) - 8Co2(D2 + T 2) +(C z o2) 2 (C 2 502) 2 + 1 6D03 T + E2 (C2 ()2) - 2CE(D 2 - T 2) - 2E(C2 - 02)(C 2 + 302) = 0. (DO + CT)(D 2 - T 2) + T(C 2 - 502) 2 - 2CDEO -ET(C2 + 02) + FO(C2 - 02) = 0. =
=
=
(7) (8) (9)
_
_
_
etc.
Young's substitutions are
T = Ot, 02 = t/1. The connexion between the (} above and the cp of Cayley's sextic resolvent is
10oV6
=
a2 ¢.
The denumerate quintic of Ramanujan's problem is y5 For this quintic,
- y 4 + y 3 - 2y 2 + 3y- 1
=
0.
C = 6, D = -156, E = 4592, F = -47328. z =
lOy - 2, (}
=
- 10V5, t
=
- 10, T = 100V5.
,------=,.-
ut u� = -13168 - 6400V5 ::±:: (2160 + 960v'5}Y79(5- 2V5),
u�, u� -13168 + 6400V5 ::±:: (2160- 960V5}Y79(5 + 2V5). =
RI, R� = 79(800 ::±:: 160V5), R1R2 = -320
We remark that Young's equations for t and t/1 in this example are: (24336 - tjJt2 ) + 12(24336 - tjJt2 )(36 + 3t/f) - 48t/1(24336 + tjJt2 ) + (36 - t/1)(36 - 5tjJ)2 - 2496tjJ2t - 581898240 - 21086464t/f + 55104tjJt2 - 9184(36- t/1)(36 + 3t/f)
22
THE MATHEMATICAL INTELLIGENCER
=
0
x
79V5.
and ( -156 + 6t)(24336 - tjJt2) + t(36 - 5tjJ)2 + 6892416 - 4592t(36 + t/1) + 47328 t/f = 0,
so that t = -10 and tjJ Watson.
=
500 in agreement with
Our next object is the determination of non-symmetric functions of the roots which can be regarded as roots of a resolvent equation. An expression which suggests itself is
(x1
+ WX"z + u?x3 + W3x4 + w4x5)5.
The result of permuting the rQots is to yield 24 values for the expression.
A permutation
u
ES
5 acts on this element by
4 5 2 3 u((x1 + WX"z + w x3 + w x4 + w x5) ) = (Xu( 1) + WX"u(2) + u?xu(3) + w3Xu(4) + w4Xu(5))5.
An easy calculation shows that a
=
Cx1 + WX"z
if and only if u = Hence
u
preserves
+ u?x3 + w3x4 + w4x5)5
(1 2 3 4 5)k for
stab85(a)
=
some k E
{0, 1, 2, 3, 4}.
(( 1 2 3 4 5)),
so that lstabs5(a)l = 5.
Thus, by the orbit-stabilizer theorem [27, p.
)I = ?
lorbs5(a
1" I
=
120 -5
=
139], we obtain
24,
so that permuting the roots yields 24 different expressions. The disadvantage of the corresponding resolvent equation is the magnitude of the degree of its coefficients when ex pressed as functions of the coefficients of the quintic; more o�er it is difficult to be greatly attracted by an equation whose degree is as high as 24 when our aim is the solution of an equation of degree as low as 5. An expression which is more amenable than the ex pression just considered was discovered just 90 years ago by two mathematicians of some eminence in their day, namely Cockle and Harley, and it was published in the
Memoirs of the Manchester Literary and Philosophical So ciety. This expression is c/>1 = X1X2 + X2X3 + X3X4 + X4X5 + X�1 - X1X3 - X3X5 - X�z - XzX4 - X4X1 ·
The quantities X1X2 + XzX3 + X3X4 + X4X5 + X�1 and X1X3 + X3X5 + x�z xzx4 + X4X1 appear in the work of Harley [29] and their difference is considered by Cayley [6]. We
+
have not located a joint paper of Cockle and Harley. When he was writing these notes, we believe Watson was read ing from Cayley [6] where the names of Cockle and Harley are linked [6, p. 3 1 1 ] .
Permutations of the suffixes give rise to 24 expressions, which may be denoted by I ± X,X8, where r and s run through the values 1, 2, 3, 4, 5 with r -=F s. The choice of the signs is most simply exhibited diagrammatically, with each of the 24 expressions represented by a separate diagram. If you tum to the second page of your sheets, you will see the 24 pentagrams with vertices numbered 1, 2, 3, 4, 5 in
all possible orders (there is no loss of generality in taking the number 1 in a special place) and the rule for determi nation of signs is that terms associated with adjacent ver tices are assigned + signs, while those associated with op posite vertices are assigned - signs. Now the pentagrams in the third and fourth columns are the optical images in a vertical line of the corresponding pentagrams in the first and second columns, and since proximity and oppositeness are invariant for the operation of taking an optical image, the number of distinct values of 4> is reduced from 24 to 12. Further, the pentagrams in the second column are de rived from those in the first column by changing adjacent vertices into opposite vertices, and vice versa, so that the values of 4> arising from pentagrams in the second column are minus the values of 4> arising from the corresponding pentagrams in the first column. It follows that the number of distinct values of cf>2 is not 12 but 6, and so our resolvent has now been reduced to a sextic equation in cf>2, with co efficients which are rational functions of the coefficients of the quintic, and a sextic equatic,m is a decided improvement on an equation of degree t20, or even on one of degree 24.
Let a = (12345) E S5 and b = (25)(34) E 85, so that a5 = b2 = e and bab = a4. As ac/>1 = 4>1 and b c/>1 = c/>1, we have stabs5( c/>1) 2::
(a, b la5 = b2 = e, bab
=
a4)
=
D5 ,
so that
l
lstabs5( c/>1) 2:: 10.
On the other hand, the first two columns of Watson's pen tagram table show that
!
lorbs5( c/>1) 2:: 12.
Hence, by the orbit-stabilizer theorem, we see that lstabs5( c/>1)
l
=
10,
lorbs5( c/>1)
l
=
12
and thus stabs5( c/>1)
=
(a, b la5
=
b2
=
e, bab
=
a4) = D5.
Now let c = (2 3 4 5), so that c2 = b. As acf>I = cf>I and ccf>I = 4>1. we have stabs5( c/>I) :::2 (a, cl a5 = c2 = e, c- 1ac = a3) = Fzo, so that lstabs5(c/>I)I 2:: 20. From thefirst column of the pen tagram table, we have lorbs5( cf>I )I 2:: 6.
Hence, by the orbit-stabilizer theorem, we deduce that lstabs5( c/>I)I
=
20,
lorbs5( c/>I) I
=
6,
and thus stabs5( c/>I)
=
(a, cla5 = c4 = e, c- 1ac = a3)
=
Fzo.
It is, however, possible to effect a further simplification; it is not, in general, possible to construct a resolvent equa tion of degree less than 6, but it is possible to construct a sextic resolvent equation in which two of the coefficients
VOLUME 24, NUMBER 4, 2002
23
are zero. We succeeded in constructing a sextic in ql- be cause the 12 values of 4> could be grouped in pairs with the members of each pair numerically equal but opposite in sign; but a different grouping is also possible, namely a se lection of one member from each of the six pairs so as to form a sestet in which the sum of the members is zero, and it is evident that those members which have not been se lected also form a sestet in which the sum of the members is zero; one of these sestets is represented by the penta grams in the first column, the other by the pentagrams in the second column.
lfr is odd, the transposition (12) changes the above equa tion to Adding these two equations, we obtain 0 = (XI - X2)(q(x2, X3, X4, X5) - q(X! , X3, X4, X5)) + r(x2, X3, X4, X5) + r(x1, X3, X4, X5).
Taking x1
= x2
we deduce that r(x2, X3, X4, X5) = 0.
Hence
A sestet is a set of six objects. Denote the values of 4> represented by the pentagrams in the first column by 4>1, c/>2, . . . , 4>6, and let 4>[ + 4> 2
+
· · ·
+ 4>6 = Er.
is then not difficult to verify that an interchange of any pair of x1, x2, . . , x5 changes the sign of Er when r is odd, but leaves it unaltered in value when r is even. It
.
By looking at the first column of the pentagram table we see that the even permutations (234), (243), (354), (235), (24)(35) send 4>1 to 1> 2, 4> 3, 4>4, 4>5, 4>6, respectively. We next show that an odd permutation cr cannot send 4>i to 4>i for any i and j. Suppose that cr(4>i) = 4>i· By the above re marks 4>i = 04>1 for some (} E A5, and 1>1 = P4>i for some p E A5. Hence so that per(} E stabs51>1
=
..
. , cr( 4>6) } � orbs54>I>
l orbs5 1>1 l
=
12,
so that Thus if T E s5 is a transposition, · · ·
+ 4> 6 ) =
c - 1>1Y +
· · ·
+ ( - 4>6Y = ( - 1Y Er.
It is now evident that each of the 10 expressions Xm - Xn (m, n = 1, 2, 3, 4, 5; m < n) is a factor of Er whenever r is an odd integer.
Clearly Er E Z[xh . . . , x5] and so can be regarded as a polynomial in x1 with coefficients in Z[x2, . . . , x5]. Di viding Er by x1 - x2 , we obtain Er = (xl - x2)q(x2, . . . , X5)
where
24
THE MATHEMATICAL INTELLIGENCER
(Xm - Xn)
divides Er when r is an odd integer. Now the degrees of E1 and E3 in the x's are respectively 2 and 6, and so, since these numbers are less than 10, both E1 and E3 must be identically zero, while E5 must be a con stant multiple of (x1 - x2)(x1 - x3)
· ·
·
(x3 - x5)(x4 - x5).
On the other hand, S2, S4, and S6 are symmetric functions of the x's, and are consequently expressible as rational functions of the coefficients in the standard form of the quintic.
( 4> - 4>I)(4> - c/>2)
and
r(Er) = r(4> [ +
II
l�m
- Xn divides Erfor
These properties ensure that the polynomial
D5 C A5.
Hence cr E A5, which is a contradiction. Now { cr( 4>!),
Thus X1 - X2 divides Er. Similarly Xm m, n = 1, 2, 3, 4, 5, m < n. Hence
+ r(x2, . . . , X5),
•
.
.
( 4> - 4>6)
has coefficients in ()I or Q(VI)), where D is the discrim inant of the quintic, with the coefficients of 4>5 and 4>3 equal to zero. Apart from the graphical representation by pentagrams (which, as the White Knight would say, is my own inven tion), all of the analysis which I have just been describing was familiar to Cayley in 1861; and he thereupon set about the construction of the sextic resolvent whose roots are 1>1 > c/>2, . .. , 4>6• The result which he obtained was the equa tion (0)
a6 4>6 - 100Ka44>4
+ 2000La2 ql-
2 - 800a 4>-v'M + 40000M = 0
in which the values of K, L, M in terms of the coefficients of the quintic are those given on the first sheet, while Ll is the discriminant of the quintic in its standard form, that is to say, it is the product of the squared differences of the roots of the quintic multiplied by a8/3 125. Its value, in terms of the coefficients occupies the lower half of the first sheet.
The work of Cayley to which Watson refers is contained in [6], where on pages 313 and 314 Cayley introduces the
pentagrams described by Watson. Note that the usual dis criminant D of the quintic is [26, p. 205]
by
a8(x1 - x2)2 (x1 - x3)2 · · · (x4 - x5) 2 = 3125Ll = 55d. There is no obvious way of constructing any simpler re solvent and so it is only natur.§ll to ask "Where do we go from here?" It seems fruitless to attempt to obtain an al gebraic solution of the general sextic equation; for, if we could solve the general sextic equation algebraically, we could solve the general quintic equation by the insertion of a factor of the first degree, so as to convert it into a sextic equation. In this connection I may mention rather a feeble joke which was once perpetrated by Ramanujan. He sent to the Journal of the Indian Mathematical Society as a problem for solution: Prove that the roots of the equation
x6 - x3 + x2 + 2x - 1 = 0 can be expressed in terms of radicals.
This problem is the first part of Question 699 in [38]. It can be found in [40, p. 331]. A solution was given by Wat son in [49]. It seems inappropriate to refer to this prob lem as a "feeble joke. " Some years later I received rather a pathetic letter from a mathematician, who was anxious to produce something worth publication, saying that he had noticed that x + 1 was a factor of the expression on the left, and that he wanted to reduce the equation still further, but did not see how to do so. My reply to his letter was that the quintic el')uation
x5 - x4 + x3 - 2x2 +
3x
-1
=
0
was satisfied by the standard singular modulus associated with the elliptic functions for which the period iK'IK was equal to v=79, and consequently it was an Abelian quin tic, and therefore it could be solved by radicals; and I told him where he would find the solution in print. I do not know why Ramanujan inserted the factor x + 1; it may have been an attempt at frivolity, or it may have been a desire to propose an equation in which the coefficients were as small as possible, or it may have been a combi nation of the two.
On pages 263 and 300 in his second Notebook [39], Ra manujan indicates that 2 114G79 is a root of the quintic equation x5 - x4 + x3 - 2x2 + 3x - 1 = 0; see [1, Part V, p. 193]. For a positive integer n, Ramanujan defined Gn by where, for any z = x + iy E C with y > 0, Weber's class invariantf(z) [57, Vol. 3, p. 1 14] is defined in terms of the Dedekind eta function 1)(Z)
=
e ;,.;z/12
00
II
m= l
(1 - e2 11'imz)
A result equivalent to Ramanujan's assertion was first proved by Russell [42] and later by Watson [53]; see also [54]. The solution of this quintic in radicals is given in [49]. In [38], Ramanujan also posed the problem offind ing the roots of another sextic polynomial which factors into x - 1 and a quintic satisfied by G47. For additional comments and references about this problem, see [4] and [40, pp. 400-401]. Both Weber and Ramanujan calculated over 100 class invariants, but for different reasons. Class invariants generate Hilbert class fields, one of Weber's primary interests. Ramanujan used class invariants to calculate explicitly certain continued fractions and prod ucts of theta functions. After this digression, let us return to the sextic resol vent·' it is the key to the solution of the quintic in terms of radicals, provided that suCh a key exists. It is possible, by accident as it were, for the sextic resolvent to have a so lution for which ¢2 is rational, and the corresponding value of ¢ is of the form p�' where p is rational. A knowledge of such a value of ¢ proves to be sufficient to enable us to express all the roots of the quintic in terms of radicals. In fact, when this happy accident occurs, the quintic is Abelian, and when it does not occur, the quintic is not Abelian. .
If ¢2 E (!) it is clear from the resolvent sextic that ¢ = p� for some p E Q. We are not aware of any rigorous direct proof in the classical literature of the equivalence of ¢2 E (!) to the original quintic being solvable. This is as far as Cayley went; he was presumably not in terested in the somewhat laborious task of completing the details of the solution of the quintic after the determina tion of a root of his sextic resolvent. The details of the solution of an Abelian quintic were worked out nearly a quarter of a century later by a con temporary of Cayley, namely George Paxton Young. I shall not describe Young as a mathematician whose name has been almost forgotten, because he was not in fact a pro fessional mathematician at all. The few details of his ca reer that are known to me are to be found in Poggendorfs biographies of authors of scientific papers. He was born in 1819, graduated M.A. at Edinburgh, and was subsequently Professor of Logic and Metaphysics at Knox College, Toronto; he was also an Inspector of Schools, and subse quently Professor of Logic, Metaphysics and Ethics in the University of Toronto. He died at Toronto on February 26, 1889. His life was thus almost coextensive with Cayley's (born August 16, 1821, died January 26, 1895). Young in the last decade of his life (and not until then) published a num ber of papers on the algebraic solution of equations, in cluding three in the American Journal of Mathematics
VOLUME 24, NUMBER 4, 2002
25
which among them contain his method of solving Abelian quintics.
These are papers [58], [59] and [60]. In style, his papers are the very antithesis of Cayley's. While Cayley could not (or at any rate frequently did not) write grammatical English, he always wrote with extreme clar ity, and, when one reads his papers, one cannot fail to be impressed by the terseness and lucidity of his style, by the mastery which he exercises over his symbols, and by the feeling which he succeeds in conveying that, although he may have frequently suppressed details of calculation, the reader would experience no real difficulty in filling in the lacunae, even though such a task might require a good deal of labour. On the other hand, when one is reading Young's work, it is difficult to decide what his aims are until one has reached the end of his work, and then one has to return to the beginning and read it again in the light of what one has discovered; his choice of symbols is often unfortunate; in
(2) u1u3 + u�u1 + u§u4 + u�u2 = - 2D, 3 3 3 2 2 + U2U32 - U1U U3U4 - U1U (3) U1U4 2 2 2 - U2U4 - U3U1 - U�U3 = E, (4) uY + u� + u� + u� - 5(ulu4 - u2u3)(uiu3 - u�u1 - u§u4 + u�u2) = -F. These coefficients were essentially given by Ramanu jan in his first Notebook [39]; see Berndt [1, Part IV, p. 38] . They also occur in [43]. We next introduce two additional unknowns, 0 and T, defined by the equations
(5) (6) in which a kind of skew symmetry will be noticed. The nat
u1, u2, u3, u4 in terms of 0, T and the coefficients of the reduced quintic by using equa tions (1), (2), (5) and (6) only. When this has been done, sub stitute the results in (3) and (4), and we have reached the ural procedure is now to determine
penultimate stage of our journey by being confronted with two simultaneous equations in the unknowns From
fact when I am reading his papers, I find it necessary to
(1) and (5)
0 and T.
we have
make out two lists of the symbols that he is using, one list of knowns and the other of unknowns; finally, his results seemed to be obtained by a sheer piece of good fortune,
while from (2) and
and not as a consequence of deliberate and systematic
u1u3 + u�u2
strategy. A comparison of the writings of Cayley and Young shows a striking contrast between the competent draughts manship of the lawyer and pure mathematician on the one hand and the obscurity of the philosopher on the other. The rest of my lecture I propose to devote to an account of a practical method of solving Abelian quintic equations. The method is in substance the method given by Young, but I hope that I have succeeded in setting it out in a more
=
(6)
we have
-D + T, u�u1 + u§u4 = -D - T;
and hence it follows that
u1u3 - u�u2 = u�u1 - u§u4 =
Y(D - T? + 4(C - 0)2(C + 0) = : R1, say; ± Y(D + T)2 + 4(C + 0) 2 (C - 0) =: R2 , say.
±
Watson makes use of the identities (uiu3 - u�u2? = (uiu3 + u�u2? - 4(u lu4i(u2u3), (u�u1 - u§u4)2 = (u�u1 + u§u4)2 - 4(u2u3i (u lu4).
intelligible, systematic and symmetrical manner. Take the reduced form of the quintic equation
z 5 + 10Cz3 + 10Dz2 + 5Ez + F = 0,
These last equations enable us to obtain simple expres sions for the various combinations of the u's which occur
and suppose that its roots are
in
(3) and (4).
Thus, in respect of
(3), we have
U2U3
where w
=
exp(2 17i/5),
r = 1, 2, 3, 4, 0.
Straightforward but somewhat tedious multiplication
with similar expressions for u�u4, u�u1, stitute these values in
u�u3. When we sub (3) and perform some quite straight
forward reductions, we obtain the equation
shows that the quintic equation with these roots is This shows incidentally that, when
termined, the signs of
R1
and
R2
0 and T have been de
cannot be assumed arbi
trarily but have to be selected so that R1R2 has a uniquely de
terminate value. The effect of changing the signs of both R1
and R2 is merely to interchange The result of rationalising
u1 with u4 and u2 with u3. (7) by squaring is the more
and a comparison of these two forms of the quintic yields
formidable equation
four equations from which
(D2 - T2 )2 + 2C(D2 - T2 )(C2 + 302 ) - 8C0 2(D2 + T2 ) (8) + (C 2 - 02 )(C2 - 502 )2 + 16D03 T + E 2(C2 - 0 2 ) -2CE(D2 - T 2 ) - 2E(C2 - 02 )(C2 + 302 ) = 0.
mined, namely
(1) 26
THE MATHEMATICAL INTELLIGENCER
u1, u2, u3, u4
are to be deter
This disposes of (3) for the time being, and we turn to (4). The formulae which now serve our purpose are
u15 _
(uiu3i (u�u l ) , etc., (U2U3)2
with three similar formulae. When these results are inserted in (4) and the equation so obtained is simplified as much as possible, we have an equati9n which I do not propose to write down, because it would be a little tedious; it has a sort of family resemblance to (7) in that it is of about the same degree of complexity and it involves the unknowns (} and T and the product R1R rationally.
2
MAPLE
gives the equation
as
(JJ2 - T2)(De2 + 2CTe + C2D) + 2(C2 - e2)(3CD()2 - Te3)
-R1R2 (Te2 + 2CD(} + C2T) + (C2 - e2)2(20T(} - F) = 0. When we substitute for this product R1R the value which 2 is supplied by (7), we obtain an equation which is worth writing out in full, namely
(De + CT)(D2 - T2) + T(C2 - 5e2)2 - 2CDEe -ET(C2 + (}2) + Fe(C2 - (}2) (9)
= 0.
We now have two simultaneous equations, (8) and (9), in which the only unknowns are (} and T. When these equations have been solved, the values of u1, u , u3, u4 are immedi 2 ately obtainable from formulae of the type giving uY in the form of fifth roots, and our quest will have reached its end.
Watson means that u1 can be given as a fifth root of an e:jipression involving the coefficients of the quintic, R1 and R2 . An inspection of this pair of equations, however, suggests that we may still have a formidable task in front of us. It has to be admitted that, to all intents and purposes, this task is shirked by Young. In place of (8) and (9), the equations to which his analysis leads him are modified forms of (8) and (9). They are obtainable from (8) and (9) by taking new unknowns in place of (} and T, the new un knowns t and 1/J being given in terms of our unknowns by the formulae
T = et,
e2
= 1/J.
Young's simultaneous equations are cubic-quartic and quadratic-cubic respectively in 1/J and t. When the original quintic equation is Abelian, they possess a rational set of solutions.
Young's pair of simultaneous equations for t and 1/J are (D2 - ljJt2)2 + 2C(D2 - 1jJt2)(C2 + 31/J) - 8CI/J(D2 + 1jJt2) + (C2 - I/J)(C2 - 51/J)2 + 16DijJ2t + E2(C2 - 1/J) - 2CE(D2 - ljJt2) - 2E(C2 - I/J)(C2 + 31/J) = 0 and (D + Ct)(D2 - 1jJt2) + t(C2 - 51/1)2 - 2CDE - Et(C2 + 1/J) + F(C2 - 1/J)
= 0.
Young goes on to suggest that, in numerical examples, his pair of simultaneous equations should be solved by in spection. He does, in fact, solve the equations by inspec tion in each of the numerical examples that he considers, and, although he says it is possible to eliminate either of the unknowns in order to obtain a single equation in the other unknown, he does not work out the eliminant. You will probably realize that the solution by inspection of a pair of simultaneous equations of so high a degree is likely to be an extremely tedious task, and you will not be mis taken in your assumption. Consequently Young's investi gations have not got the air of finality about them which could have been desired. Fortunately, however, the end of the story is implicitly told in the paper by Cayley on the sextic resolvent which I have already described to you and which had been pub lished over a quarter of a century earlier. It is, in fact, easy to establish the relations
Z1Z2 +
·
.
.
-z1Z3 -
· · ·
=
a2(x1x2 +
· · ·
-x1x3-
· · ·
)
=
a2¢1,
and also to prove that theexpression on the left is equal to
5(u lu4 - u2u3)V5 so that
Watson is using the relation Zi = axi + b (i E { 1, 2, 3, 4, 5}) to obtain the first equality. With Zr = wru l + w2ru2 + w3ru3 + w4ru4 (r E { 1, 2, 3, 4, 5}) MAPLE gives Z1Z2 + . . . -z1Z3 - . . .
= 5(u lu4 - u2u3)(w - w2 - w3 + w4)
so that z1z2 +
· · ·
- z1z3 -
· · · =
5(u lu4 - u2u3)V5
since Consequently, to obtaill a value of (} which satisfies Young's simultaneous equations, all that is necessary is to ob tain a root of Cayley's sextic resolvent; and the determina tion of a rational value of ¢2 which satisfies Cayley's sextic resolvent is a perfectly straightforward process, since any such value of a2¢2 must be an integer which is a factor of 1600000000M2 when the coefficients in the standard form of the quintic are integers, and so the number of trials which have to be made to ascertain the root is definitely limited.
The quantity M is defined on Watson's sheet 1. The con stant term of Cayley's sextic resolvent (0) is 40000M. When (} has been thus determined, Young's pair of equa tions contain one unknown T only, and there is no diffi culty at all in finding the single value of T which satisfies both of them by a series of trials exactly resembling the set of trials by which (} was determined.
VOLUME 24, NUMBER 4, 2002
27
where
Watson's metlwd of finding a real root of the solvable quintic equation: 4 ax5 + 5bx + 10c.i3 + 10dx2 + 5ex + f = 0
X = (-D + T + R1)/2, Y = (-D - T + R2)/2, Z = - C - 8.
First transform the quintic into reduced form x5 + 10Cx3 + 10Dx2 + 5Ex + F = 0.
Step 7. Determine u4 from u1u4 = -c + 8.
Watson's step-by-step procedure gives a real root of the re duced equation in the form x = u1 + u2 + u3 + U4. The other four roots of the equation have the form wlu1 + w2iu2 + w3iu3 + w4iu4 (j = 1, 2, 3, 4), where w =
Step 8. Determine u2 from u�u2 = ( -D + T - R1)/2. Step 9. Determine usfrom
U2U3 =
exp(277i/5).
INPUT: C,D,E,F
OUTPUT· A real root of the quintic is x = u1
Us + U4.
Step 1. Find a positive integer k such that
kj 16 X 108 X JJ12,
- C - 8.
The process which I have now described of solving an Abelian quintic by making use of the work of both Cayley and Young is a perfectly practical one, and, as I have al ready implied, I have used it to solve rather more than 100 Abelian quintics. If any of you would like to attempt the so lution of an Abelian quintic, you will find enough informa tion about Ramantijan's quintic given at the foot of the third sheet to enable you to complete the solution. You may re member that I mentioned that the equation was connected with the elliptic functions for which the period-quotient was v=79, and you will see the number 79 appearing some what unobtrusively in the values which I have quoted for the u's.
eVk/a is a root of (0) for E = 1 or - 1.
Step 2. Determine 8 from 8=
eaVk
10v5 ·
Step 3. Put the value of 8 into (7) and (9) and then add and subtract multiples of these equations as necessary to determine T. Step 4. Determine R1 from R1 = Y(D - Ti + 4(C - 8)2(C + 8). Step 5. Determine R2 from R1R2 = (C(D2 - T 2 ) + (C2 - 82)(C2 + 382 - E))/8.
This is the end of Watson's lecture. We have made a few corrections to the text: for example, in one place Watson wrote "cubic" when he clearly meant "quintic. " Included in this article are the three handout sheets that he refers to in his lecture. We conclude with three examples.
Step 6. Determine u1 from x2y 115 ul z2 ' _
+ u2 +
( )
Three Examples Illustrating Watson's Procedure - 5x + 12 = 0 The Galois group of x5 - 5x + 12 is D5. Here
Example 1. x5
a = 1, b = 0, c = 0, d = 0, e = - 1, J = 12, C = 0, D = 0, E = - 1, F = 12, K = - 1, L = 3, M = - 1, 11 = 5 X 2 1 2, YM = 520. Equation (0) is
4J6 + 1004J4 + 6000� - 2560004J - 40000 = 0.
Step 1
k = 10. Step 2 1
(} = V5 " Step 3
2 T = V5 "
28
THE MATHEMATICAL INTELLIGENCER
Continues on next page
Examples (continued) Step 4
Step 5
�
Rz = - Y5 - v5.
Step 6 X=
v5 + Y5 + v5 -v5 - V5 - v5 , Y= ,Z= 5 5
_
ul - -
( cv5 + V5 + v5)2 cv5 + V5 - v5) ) 115.
1
- v5 ,
25
Step 7
Step 8 _
Uz - -
Step 9
_
U3 - -
25
( cv5 + Y5 - v5)2C - v5 + V5 + v5) ) 115. 25
0 is x = u 1 + u2 + U3 Example 2. x5 + 15x + 12 = 0
A solution of x5
- 5x + 12
( cv5 - V5 - v5)2C-v5 - Y5 + v5) ) 115.
The Galois group of x5
=
+ u4
.
This agrees with [43, Example 1].
+ 15x + 12 is Fzo. Here
a = 1 , b = 0, c = 0, d = 0, e = 3, ! = 12 , C = 0, D = 0, E = 3, F = 12 , K = 3 L = 27 M = 27 11 = 2�0 x 34, 'vM = 288v5.
Equation (0) is cf>6 -
3004>4 + 54000� - 230400v54> + 1080000
=
0.
Step 1 k = 180. Step 2
Step 3
Step 4 R1 =
12VIO . 25 Continues on next page
VOLUME 24, NUMBER 4, 2002
29
Examples (continued) Step 5
6Vlo R2 - 25 .
Step 6
X=
15
- 15 + 3vl0 z = -� 6Vlo Y= ' ' 5' 25 25 /5 - - 75 - 21Vlo l u1 125
+
Step 7
u4 Step 8
)
(
( - 75
·
)
21Vlo l/5 125
+
(
)
(
)
·
225 - 72Vlo 1/5 u2 125 Step 9
·
72Vlo 1/5. U3 -- 225 +125 This agrees with [43, Example 2]. Example 3. x5
- 2fii3 + 50.1? - 25 = 0
The Galois group of x5 - 2fii3
Equation (0) is
+ 50.1? - 25 is 7L/57L. Here
a = 1, b = 0, c = -5/2, d = 5, e = O, f = -25, C = -5/2, D = 5, E = 0, F = -25, K = 75/4, L = 5375/16, M = -30625/64, il = 5 7 X 72, � = 54 X 7.
¢6 - 1875¢4 + 671875¢2 - 3500000¢ - 19140625 = 0. Step 1 k=
625.
Step 2 (} =
-v5 . 2-
Step 3
T = 0. Step 4 Step 5
R 1 = Y -25
+ IOV5.
R2 = Y - 25
-
IOV5. Concludes on next page
30
THE MATHEMATICAL INTELLIGENCER
Examples (continued) Step 6
- 5 + v -25 + 10V5 X= _ , 2
_
ul -
-5
Y=
+ v -25 - 10V5 2
(x2y) 115 _- 25 + 15V5 + 5Y- 13o - 5sV5 .
z2
, Z=
5 + V5 , 2
4
Step 7
25 + 15V5 - 5Y- 130 - 5sV5 4 Step 8
25 - 15V5 + 5Y - 13o + 5sV5 4
Step
9
- 13o + 5sV5 Us = 25 - 15V5 - 5Y 4
REFERENCES
1 . Bruce C. Berndt, Ramanujan's Notebooks, Springer-Verlag, New York, Part I (1 985), Part II (1 989), Part Ill (1 991 ), Part IV (1 994), Part
v (1 998).
der, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXIII (1 862), pp. 1 95, 1 96. [1 5, Vol. V, Pa per 316, p. 77.] 1 0. Arthur Cayley, Final remarks on Mr. Jerrard's theory of equations
2. Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ra
of the fifth order, The London, Edinburgh and Dublin Philosophical
rnanujan's class invariants and cubic continued fraction, Acta
Magazine and Journal of Science XXIV (1 862), 290. [1 5, Vol. V,
..
Arithmetica 73 (1 995), 67-85.
3. Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Ra
Paper 321 , p. 89.] 1 1 . Arthur Cayley, Note on the solvability of equations by means of rad
rnanujan's class invariants, Kronecker's limit formula, and modular
icals, The London, Edinburgh and Dublin Philosophical Magazine
equations, Transactions of the American Mathematical Society 349
and Journal of Science XXXVI (1 868), pp. 386, 387. [1 5, Vol. VII,
(1 997), 21 25-21 73.
Paper 421 , pp. 1 3-1 4.]
4. Bruce C. Berndt, Youn-Seo Choi, and Soon-Yi Kang , The prob
1 2 . Arthur Cayley, On a theorem of Abel's relating to a quintic equa
lems submitted by Rarnanujan to the Journal of the Indian Mathe
tion, Cambridge Philosophical Society Proceedings Ill (1 880),
matical Society, in Continued Fractions: From Analytic Number The
1 55-1 59. [1 5, Vol. XI, Paper 7 41 , pp. 1 32-1 35.]
ory to Constructive Approximation, B. C. Berndt and F. Gesztesy,
1 3. Arthur Cayley, A solvable case of the quintic equation, Quarterly
eds., Contemp. Math. No. 236, American Mathematical Society,
Journal of Pure and Applietl Mathematics XVIII (1 882), 1 54-1 57.
Providence, Rl, 1 999, pp. 1 5-56. 5. William S. Burnside and Arthur W. Panton , The Theory of Equa tions, 2 vols. , Dover, New York, 1 960.
(1 5, Vol. XI, Paper 777, pp. 402-404.] 1 4. Arthur Cayley, On a soluble quintic equation, American Journal of Mathematics XIII (1 891), 53-58. (15, Vol. XIII, Paper 91 4, pp. 88-92.]
6. Arthur Cayley, On a new auxiliary equation in the theory of equa
1 5. Arthur Cayley, The Collected Mathematical Papers of Arthur Cay
tions of the fifth order, Philosophical Transactions of the Royal So
ley, Cambridge University Press, Vol. I (1 889), Vol. II (1 889), Vol. Ill
ciety of London CLI (1 861 ), 263-276. [1 5, Vol. IV, Paper 268, pp.
(1 890), Vol. IV (1 891), Vol. V (1 892), Vol. VI (1 893), Vol. VII (1 894),
309-324.]
Vol. VIII (1 895), Vol. IX (1 896), Vol. X (1 896), Vol. XI (1 896), Vol. XII
7. Arthur Cayley, Note on Mr. Jerrard's researches on the equation of the fifth order, The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science XXI (1 861), 2 1 0-21 4. [1 5, Vol. V, Paper 3 1 0, pp. 5Q-54.] 8. Arthur Cayley, On a theorem of Abel's relating to equations of the
(1 897), Vol. XIII (1 897), Vol. XIV (1 898). 1 6. Heng Huat Chan, Ramanujan-Weber class invariant Gn and Wat son's empirical process, Journal of the London Mathematical So ciety 57 (1 998), 545-561 . 1 7. James Cockle, Researches in the higher algebra, Memoirs of the Lit
fifth order, The London, Edinburgh and Dublin Philosophical Mag
erary and Philosophical Society of Manchester XV (1 858), 1 31 -1 42.
azine and Journal of Science XXI (1 86 1 ) , 257-263. (1 5, Vol. V, Pa
1 8. James Cockle, Sketch of a theory of transcendental roots, The
per 31 1 , pp. 55-61 .] 9. Arthur Cayley, Note on the solution of an equation of the fifth or-
London, Edinburgh and Dublin Philosophical Magazine and Jour nal of Science XX (1 860), 1 45-1 48.
VOLUME 24, NUMBER 4, 2002
31
A U THO R S
BRUCE C. BERNDT
BLAIR K. SPEARMAN
KENNETH S. WILLIAMS
Department of Mathematics
Department of Mathematics and Statistics
School of Mathematics and Statistics
University of Illinois
Okanagan University College
Carleton University
Urbana, Illinois
Kelowna, British Columbia V1V 1V7
Ottawa, Ontario K1 S 586
U.S.A.
Canada
Canada
e-mail: [email protected]
e-mail: [email protected]
e-mail: [email protected]
Bruce C. Bemdt became acquainted with
Blair K. Spearman completed his B.Sc. and
Kenneth S. Williams did his B.Sc. degree
Ramanujan's notebooks in February 1 974,
M.Sc. degrees at Carleton University in Ot
at the University of Birmingham, England, attending lectures in the Watson Building.
while on a sabbatical year at the Institute for
tawa, Canada. He received his Ph.D. de
Advanced Study. Since then he has devoted
gree in mathematics at Pennsylvania State
He completed his Ph.D. degree at the Uni
almost all of his research efforts toward prov
University underW. C. Waterhouse in 1 98 1 .
versity of Toronto in 1 965 under the su
ing results from these notebooks and Ra
He currently teaches at Okanagan Univer
pervision of J. H. H. Chalk. After a year at
manujan's lost notebook. In 1 996 the Amer
sity College, Kelowna, BC, Canada. His re
the University of Manchester he came to
ican Mathematical Society awarded him the
search interests are in algebraic number
Carleton University in 1 966, where he has
Steele Prize for his five volumes on Ra
theory.
been ever since. He served as chair of the
manujan's Notebooks. Similar volumes on
Mathematics Department from 1 980 to
the lost notebook, to be co-authored with
1 984
George Andrews, are in preparation. He is
currently on sabbatical leave working on a
and again from 1 997 to 2000. He is
most proud of his three biological children,
book on algebraic number theory with his
Kristin, Sonja, and Brooks, his seventeen
colleague Saban Alaca.
mathematical children, his five mathemati cal children in preparation, and his current postdoc.
1 9.
cation to the finite algebraic solution of equations, Memoirs of the Ut
James Cockle, On the resolution of quintics, Quarterly Journal of
erary and Philosophical Soce i ty of Manchester XV (1 859), 1 72-2 1 9.
Pure and Applied Mathematics 4 (1 861 ), 5-7. 20.
James Cockle, Notes on the higher algebra, Quarterly Journal of
30.
21 .
James Cockle, On transcendental and algebraic solution-supple
31 .
matics 5 (1 862), 337-361 .
Magazine and Journal of Science XXIII (1 862), 1 35-139. Winifred A Cooke, George Neville Watson, Mathematc i al Gazette
32.
R. Bruce King, Beyond the Quartic Equation, Birkhauser, Boston,
David A Cox, Primes of the Form x2 + ny2 , Wiley, New York, 1 989.
33.
Sigeru Kobayashi and Hiroshi Nakagawa, Resolution of solvable
putation 57 (1 99 1 ), 387-401 .
34.
John Emory McClintock, On the resolution of equations of the fifth
49 (1 965), 256-258. 23. 24.
25.
1 996.
quintic equation, Mathematc i a Japonicae 37 (1 992), 883-886.
David S. Dum mit, Solving solvable quintics, Mathematics of Com
degree, American Journal of Mathematics 6 (1 883-1 884), 301 -
David S. Dummit and Richard M. Foote, Abstract Algebra, Pren
3 1 5.
tice Hall, New Jersey, 1 99 1 . 26.
W. L. Ferrar, Higher Algebra, Oxford University Press, Oxford, 1 950.
35.
Houghton Mifflin Co. , Boston MA, 1 998.
36.
27.
Joseph A Gallian, Contemporary Abstract Algebra, Fourth Edition,
28.
J. C. Glashan, Notes on the quintic, American Journal of Mathe
32
Robert Harley, On the method of symmetric products, and its appli-
THE MATHEMATICAL INTELUGENCER
John Emory McClintock, Analysis of quintic equations, Amerc i an
Journal of Mathematics 8 (1 885), 45-84. John Emory McClintock, Further researches in the theory of quin tic equations, American Journal of Mathematics 20 (1 898),
matics 7 (1 885), 1 78-1 79. 29.
Robert Harley, On the theory of the transcendental solution of al gebraic equations, Quarterly Journal of Pure and Applied Mathe
mentary paper, The London, Edinburgh and Dublin Philosophical 22.
Robert Harley, On the theory of quintics, Quarterly Journal of Pure
and Applied Mathematics 3 (1 859), 343-359.
Pure and Applied Mathematics 4 (1 86 1 ) , 49-57.
1 57-192. 37.
Srinivasa Ramanujan, Modular equations and approximations to
7T,
Quarterly Journal of Mathematics 45 (1 9 1 4), 350--372. (40: pp.
ory: Proceedings of a Conference in Honor of Heini Halberstam,
23-39.]
Vol. 2, B. C. Berndt, H. G. Diamond and A. J. Hildebrand, eds . ,
38. Srinivasa Ramanujan, Question 699, Journal of the Indian Mathe matical Society 7 (1 91 7), 1 99. (40: p. 331 .]
Birkhauser, Boston, 1 996, p p . 81 7-838. 62. Liang-Cheng Zhang, Ramanujan's class invariants, Kronecker's
39. Srinivasa Ramanujan, Notebooks, 2 vols., Tata Institute of Funda mental Research, Bombay, 1 957.
limit formula and modular equations (Ill), Acta Arithmetica 82 (1 997), 379-392.
40. Srinivasa Ramanujan, Collected Papers of Srinivasa Ramanujan AMS Chelsea, Providence, Rl, 2000. 41 . Robert A. Rankin, George Neville Watson, Journal of the London Mathematical Society 41 (1 966), 551 -565. 42. R. Russell, On modular equations, Proceedings of the London Mathematical Society 21 {1 889-1 890), 351 -395. 43. Blair K. Spearman and Kenneth S. Williams, Characterization of solvable q uintics x5 + ax + b, American Mathematical Monthly 1 01
(1 994), 986-992.
44. Blair K. Spearman and Kenneth S. Williams, DeMoivre's quintic and a theorem of Galois, Far East Journal of Mathematical Sciences 1 (1 999), 1 37-1 43. 45. Blair K. Spearman and Kenneth S. Williams, Dihedral quintic poly nomials and a theorem of Galois, Indian Journal of Pure and Ap plied Mathematics 30 (1 999), 839-845. 46. Blair K. Spearman and Kenneth S. Williams, Conditions for the in solvability of the quintic equation x5
+
ax + b
=
0, Far East Jour
nal of Mathematical Sciences 3 (2001 ), 209-225. 47. Blair K. Spearman and Kenneth S. Williams, Note on a paper of Kobayashi and Nakagawa, Scientiae Mathematicae Japonicae 53 (2001 ), 323-334. 48. K. L. Wardle, George Neville Watson, Mathematical Gazette 49 (1 965), 253-256. 49. George N. Watson, Solution to Question 699, Journal of the Indian Mathematical Society 1 8 (1 929-1 930), 273-275. �0. George N. Watson, Theorems stated by Ramanujan (XIV): a sin gular modulus, Journal of the London Mathematical Society 6 (1 931 ), 1 26-1 32. 51 . George N . Watson, Some singular moduli (1), Quarterly Journal of Mathematics 3 (1 932), 8 1 -98.
52. George N. Watson, Some singular moduli (II), Quarterly Journal of Mathematics 3 (1 932), 1 89-2 1 2 . 53. George N. Watson, Singular moduli (3), Proceedings o f the Lon don Mathematical Society 40 (1 936), 83-1 42. 54. George N. Watson, Singular moduli (4), Acta Arithmetica 1 (1 936), 284-323. 55. George N. Watson, Singular moduli (5), Proceedings of the Lon don Mathematical Society 42 (1 937), 377-397. 56. George N. Watson, Singular moduli (6), Proceedings of the Lon don Mathematical Society 42 (1 937), 398-409. 57. Heinrich Weber, Lehrbuch der Algebra, 3 vols. , Chelsea, New York, 1 961 . 58. George P. Young, Resolution of solvable equations of the fifth de gree, American Journal of Mathematics 6 (1 883-1 884), 1 03-1 1 4 . 5 9 . George P . Young, Solution of solvable irreducible quintic equations, without the aid of a resolvent sextic, American Journal of Mathe
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33
ip.iM$$j:J§..@hl£ili.JIIQ?-Ji
Dirk H uylebro u c k ,
Editor
Mathematics in M the Hal l of Peace Norbert Schmitz
iinster is one of the few cities fa mous not for a bloody battle but for a fruitful peace-the Peace of West phalia. In 1648, the signing of the peace treaty in Miinster and Osnabriick marked the end to the dreadful Thirty Years' War, which had caused unimag inable suffering throughout central Eu rope-in particular among the German population. An additional result of the Westphalian Peace Conference was the peace treaty between Spain and the Netherlands affirming the indepen dence of the Netherlands. Both peace treaties were ratified in the Hall of Peace, the old council cham ber of the Miinster town hall. Famous for its magnificent gable, this town hall is regarded as one of the finest exist ing examples of secular Gothic archi-
Does your lwmetown have any mathematical toumt attractions such as statues, plaques, graves, the cafe where thefamous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? Q so, we invite you to submit to this column 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]
34
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
Figure 1 . Munster town hall.
tecture. The framework of the council chamber, which is the oldest part of the town hall, was built in the second half of the 12th century. The city itself looks back upon an eventful history of more than 1200 years. Around 1577, the Hall of Peace was decorated with a rich array of Renais sance woodcarvings. In the window re cesses, one can see Moses the Legisla tor as well as the seven liberal arts (for the history of these arts and their "por traits" by sculptors and painters, see "The Liberal Arts" by B. Artmann, The Mathematical Intelligencer 20 (1998), no. 3, 40-41), in particular Ars Arith metica with tablet and stylus and Ars Geometria with tablet and compass. Like the other carvings, these fig ures are embellished with ornaments,
Figure 2. The Hall of Peace.
Figure 3. Arithmetica and Geometria.
VOLUME 24, NUMBER 4, 2002
35
arabesques, and pediments with an gels' heads. There is an abundance of objects worth seeing, but during a short journey, one could simply follow the track of the many (crowned) heads of state who celebrated here in 1998 the 350th anniversary of the Peace of Westphalia. There is a nice story about the visi tors' book in the Town Hall. In the early 1950's J.-P. Serre is said to have signed it "Bourbaki," after a visit to the Hall of Peace. Unfortunately, this story, which was told to me by P. Ullrich (Augs burg), on the authority of M. Koecher could not be verified-either by check ing hundreds of pages of the visitors'
36
THE MATHEMATICAL INTELLIGENCER
books or by a personal inquiry to J.-P. Serre himself. In Miinster, the "Arithmetica and Geometria" are not the only attractions for the mathematical tourist. The Mathematics Department of the Uni versity (Westfillische Wilhelms-Univer sitat Munster) is one of the leading departments in Germany. Here, F. Hirzebruch, H. Grauert, and R. Rem mert wrote their Ph.D. and Habilitation theses, in the 1950s, as members of the school of complex analysis around H. Behnke. During the 1970s, the main field of interest switched and the de partment again embraced Arithmetica and Geometria.
G. Faltings wrote his Ph.D. and Ha bilitation theses in this department. Since 1998 a lively Sonderforschungs bereich (Special Research Field) "Geometrische Strukturen in der Math ematik" (Geometric Structures in Mathematics) is supported by the Deutsche Forschungsgemeinschaft. Yet, the mathematics building is less in teresting than the town hall-architec turally, at least. lnstitut fOr Mathematische Statistik Universitat Munster Einsteinstr. 62 D-48149 Munster Germany
LEON GLASS
Loo ki n g at Dots he ''Prof' at the Department of Machine Intelligence and Perception at the University of Edinburgh, H. C. Longuet-Higgins, had just returned from a trip to the States where he had learned of a fascinating experiment carried out by the physicist Erich Harth. The year was 1968, and I had just completed a doctorate studying the statistical mechanics of liquids, trying to apply my craft to the study of the brain. At the time, I did not realize that the experiment would have strong impact on the rest of my career. The experiment was so simple that even a theoretician could do it. Take a blank piece of paper. Place this on a photocopy machine and make a copy of it. Now make a copy of the copy. This procedure is then iterated, always making a copy of the most recent copy. Although the naJ:ve guess might be that all copies would be blank, this was not at all the case. Small imperfections in the paper and dust on the optics of the Xerox machine introduced "noise" that arose initially as tiny specks. As the process was iterated, these tiny specks grew up-they got bigger. They did not grow to be very big, but just achieved the size of a small dot, Figure 1. The reason for this is that the optics of the photocopy machine led to a slight blur ring of each dot, so that each dot grew. On the other hand, local inhibitory fields introduced by the charge transfer un derlying the Xerography process limited the growth. These local fields also inhibited the initiation of new dots near an already existing dot; so that after a while (about 15 itera tions), there was a pretty stable pattern of dots. This analogue system mimicked lateral inhibitory fields that play a role in developmental biology and visual per ception, and I thought it would be a fine idea to study the spatial pattern of the dots. To do this, I decided to make a transparency of the dot patterns so that I could project the
dot patterns on a target pattern of concentric circles. By placing one dot at the center of the target pattern, I could count the number of dots lying in annuli a given distance away, this would give me an estimate of the spatial auto correlation function of the dots. But when I did this, I made a surprising finding. Super imposing the transparency of the dots upon the photocopy of the dots with a slight rotation, one obtained an image with an appearance of concentric circles (Figure 2). I de scribed this effect and proposed a way that the visual sys tem could process the images [1]. In 1982, David Marr called these images Glass patterns in his classic text in visual perception [2]. The effect is now well-known among visual scientists, who continue to un ravel the visual mechanisms underlying the perception of these images. But despite the underlying mathematical structure of these images and the potential utility of this effect to teach mathematics, the effect is not known at all by mathematicians, as witnessed by an early rediscovery of the effect [3] and also by the description of the effect in the Spring 2000 Mathematical Intelligencer [4]. Let me try here to give a glimpse into the mathematical underpinnings, and to describe some of the recent psychological studies of the perception of these images.
Perceiving Vector Fields Imagine a two-dimensional flow or vector field. We ran domly sprinkle dots on the plane. Next we plot the loca-
© 2002 SPRINGER-VER LAG NEW YORK, VOLUM E 24, NUMBER 4, 2002
37
_I
:
.
�
·r .
:
.
·
• .i
• •
•.
. . .·� . . .
: ·. �·· · �-. .. . . :;\···:·· ·.;. � .
_. :.
!
· : :· : . .
: : .�
. ·
.
:._._ . .
. . . . ... . •
.
Figure 1 . Original images generated in the late 1 960s by making a photocopy of a blank page and then iterating the process, always taking a photocopy of the most recent copy. Images represent the output after the 5th and 1 5th iterations.
. ·. ·· ·.·
.
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.
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.
,•
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tions of the original set of dots, and also the locations of
. ·..
.
the dots a bit later, after they have moved under the action
·. ·.:
of the flow. Provided the time interval is not too long, then when we look at the positions of both sets of dots simul taneously, we see the geometry of the vector field.
�·!:.,
Figure
�:cF·l� �
�
lar image. But other geometries can be handled [5] . First
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'.· .. �:-'-:: . -���t:�J';\:T�.'; -. ...:·!
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.. ..... . . .
Figure 2. The image generated by superimposing a copy of the 1 5th iterate on itself in a rotated position.
38
2 shows an example in which a set of dots is su
perimposed on itself in a rotated position to yield a circu
THE MATHEMAnCAL INTELLIGENCER
assume that the origin is fixed, and the transformation maps each dot to a new location by a scaling of the x coordinate by an amount
a,
a scaling of the y-coordinate
by an amount b, and a rotation of the image about the ori gin by an angle
0.Then (x,y) will be transported to the po
sition (x' ,y'), where x' y' Equation
=
=
ax cos (} - by sin ax sin (} + by cos
(1) is a linear map.
(}
(1)
(}
The properties of such maps
are well understood [6] , [7]. What is amazing is that by looking at the images of the original set of dots combined with the superimposed dots, it is possible to perceive the underlying geometry of the transformation defined by the map. The particular geome try that results is defmed by the eigenvalues of the linear transformation defined in Equation (1) . The eigenvalues are the solutions of the detenninant
I
a cos (} a sin (}
A
- b sin (}
b cos
(} - A
I
0.
(2)
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. . :--,., .�. . . . • ,•�. • ·
-1
1
.
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•
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0
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Figure 3. (a) A random pattern of 400 dots. (b) The same pattern in which the x and y coordinates are multiplied by 1 .05. (c) The same pat
tern in which the x coordinate is multiplied by 1 .05 and the y coordinate is multiplied by 0.95. A
simple computation gives those eigenvalues as
A± =
(a + b) cos (} ± V(a - b)2 - (a + b)2 sin2 (} . (3) 2
These effects can be beautifully illustrated using trans parencies of random dot patterns, and superimposing these on an overhead projector. In Figure 3a I show a random pat tern of 400 dots. The x and y coordinates of each point are multiplied by 1.05 in Figure 3(b ). In Figure 3(c), the x coor dinates of each point are multiplied by 1.05 and the y coor dinates of each point are multiplied by 0.95. The rotation of the photocopied patterns yielding circles in Figure 2 is one of the classic geometries (pure imaginary eigenvalues). An Qther geometry is provided by setting the center of the im age as fixed, and then expanding the x and y coordinates by the same constant amount (real eigenvalues greater than 1). This yields an expanding pattern, called a "node," which is illustrated in Figure 4(a) by superimposing Figure 3(a) and 3(b ). Combining expansion with rotation gives a spiral im age, called a "focus," as shown in Figure 4(b), formed by su perimposing Figures 3(a) and 3(b) in a rotated orientation (complex eigenvalues). Finally, if there is expansion in the x coordinate and contraction in the y coordinate, then there is a hyperbolic geometry called a "saddle" (the absolute value
(a) 1 • ··:...•..... "1-.,;:...• •: •.'•
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Neurophysiology of Perception Manipulation of visual images, combined with measurement of perception, or recording of electrical activity of nerve cells in the brain, provides powerlul techniques to probe the ftmc tioning of the visual system. Because of the simple structure of the random dot images, visual scientists have often used them as a starting point for their investigations. I cannot sum marize the many studies that have been carried out, but I will describe a couple and invite the reader to invent new visual effects that can be a probe of visual system ftmction.
(b)
, ... ,=,� ::• ··.• J' . · . '\ f 1aI I 11z:.,,: ::.. •• "• ._l \ U .-,: I �:!- """' � • : I I' II 1-'1 �"'J.• "' 0.5 ... ""' -� ' "••• ':' P ·'" •• • - � .. .. •
of one eigenvalue is greater than 1 and the absolute value of the other eigenvalue is between 0 and 1). A saddle (Figure 4c), can be generated by the superposition of Figures 3(a) and 3(c). Because these geometries can be easily appreci ated without using the formulae, I always use these corre lated dot images to teach about the geometry of vector fields. These geometries may even be preserved when one set of dots is one color, and the second set of dots is another color (Fig. 5a). Stan Wagon has incorporated this observation to generate colorful images ofvector fields in which local flows are represented by "tear drops." The visual system integrates the local tear drop flows to give a good representation of the geometry of the vector field [8] (Fig. 5b).
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41
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Figure 4. (a) Superposition of Figure 2(a) on Figure 2(b) to generate a node geometry. (b) Superposition of Figure 2(a) and Figure 2(b) in a ro tated position to generate a focus geometry. (c) Superposition of Figure 2(a) on Figure 2(c) to generate a saddle geometry.
VOLUME 24, NUMBER 4, 2002
39
Figure 5 (a) A random pattern of dots generated by tossing ink on paper is superimposed on itself in a ro tated position, but the two sets of dots are different colors. From a photo silkscreen print made by the au thor in the 1 970s. (b) A colorful "tear drop" represen tation of vector fields from VisuaiDSolve (Wagon and
-j
(a)
(b) 40
THE MATHEMATICAL INTELLIGENCER
Schwalbe [8]). Reproduced with permission from Wagon and Schwalbe [8].
In order to think about how the visual system might
this approach is making progress in linking the separations
process the information in the dot patterns, it is useful to
between dots in the images presented to the monkeys with
consider first the structure of the images. For each dot, there
the physiological properties of individual cells.
is a second dot that is correlated with the first dot. For ex
What about the interactions between the simple cells?
ample, for the circular images, the two dots always lie on
Zucker argues that excitatory interactions between indi
the circumference of a circle -centered at the point of rota
vidual cells in a given "clique" of cells, all of which have
tion. However, in addition, there are other dots that are also
similar orientation specificity and are located in a given col
in the vicinity of the first dot that lie in random directions
umn, might be playing an important role in contour detec
from it. In order to detect the pattern, two steps are essen
tion
tial:
(1) to detect the locally correlated dots and (2) to inte
grate the local correlations to form the global percept. Early Nobel-Prize-winning studies of the physiology of
[12]. In this formulation, a "clique" of cells is carrying
out the averaging operations that are necessary to compute the local autocorrelations. Thus, Zucker is hypothesizing that the columnar organization may play an important role
nerve cells in the visual system of the brain carried out by
in information processing.
provide a basis for hypothesizing a
This work leaves open the important question of the na
mechanism for early stages of the detection process. Rubel
ture of the interactions between columns that lead to global
Rubel and Wiesel
[9]
and Wiesel showed that some nerve cells, called "simple cells," can be excited by lines of a par ticular orientation in a given re gion of the visual field. Conse quently, two dots should also be able to stimulate a simple cell if they lie in the appropriate orien tation. In a local region, there are
percept. Psychophysical studies
Because of the moi re
carried
effect , these images
by
Wilson
and
about the nature of the inter columnar information process
can provide a powerfu l
ing. By partially rell,loving some regions of the correlated dot im
method to determ i n e a
ages, they determined that the circular image, as rn Figure
point of rotation .
many correlated dot pairs ori
out
Wilkinson pose sharp questions
2, is
easier to perceive than the other types of correlated dot images.
ented along the flow, so cells spe cific for that orientation in that local region would be pref
Because the local information was the same in the various
erentially activated compared to cells specific for other
images, the differences in ability to perceive the images
orientations. Rubel and Wiesel also found that simple cells,
must be due to the integration steps. At the moment, it ap
specific for a certain orientation but with somewhat differ
pears that these integration steps take place in a region of
i,ng receptive fields all in the same general region of the vi
the brain called area
sual field, were located in vertical columns. Further, there
V4 [11].
were cells they called "complex cells" that appeared to re
Practical Implications
ceive their input from simple cells lying in the same column
The random dot images may be useful in a variety of other
[9]. Based on these observations, I hypothesized that the sim
applications. Because of the moire effect, these images can
ple and complex cells in a column in the visual cortex could
provide a powerful method to determine a point of rotation,
provide the anatomical loci to compute the local autocorre
and to align images. Following the description of this effect
lation function of the dot patterns
in the
[ 1]. The integration of the
outputs of the local columns to form the global percept would necessarily involve inter-columnar interactions. Now, more than
30 years
after these initial hypotheses,
Scientific American,
Edward B. Seldin of Harvard
Medical School developed a method to use the moire effect to help plan maxillo-facial surgery in patients who did not have ideal alignment of the upper and lower jaws. He started
a large number of studies make it possible to refine and
out with two identical dot patterns, one fixed on the upper
modify these ideas. Movshon and colleagues have recorded
jaw and a second fixed on the lower jaw, initially in a su
[14]. By manipulating images to give
electrical activity from simple cells in the primary visual
perimposed orientation
cortex (this is called area
of macaque monkeys while
a better jaw alignment, it was possible to develop a plan for
viewing dot patterns generated by superimposing a random
the surgery. More recently, Wade Schuette of Ann Arbor,
V1)
set of dots on itself following a translation
[10].
They also
developed a mathematical model of the cortical cells, by assuming there were elongated excitatory and inhibitory regions of the receptive fields. A given cell would be ex
Michigan demonstrated a variety of ways these effects could
be used to help in alignment tasks
[13].
Similar effects also arise in color printing. Colors are of
ten represented by dots of different colors and varying
cited (or inhibited) by dots that fell in the excitatory (or in
sizes. Problems in alignment of the different colors can lead
hibitory) region of its receptive field. The good agreement
to undesirable moire effects. One way to overcome these
between the experimentally recorded activity and the the
problems is for the color screens to be stochastic images.
oretical model gives support to this conceptual model of
However, even when these images are stochastic, mis
the cortical cell. Moreover, by computing the expected ac
alignment can lead to moire effects. Such problems are be
tivity using a theoretical model, and comparing these re sults with the observed activity recorded experimentally,
ing addressed by Lau
[15], who recently rediscovered these
phenomena in the context of commercial printing.
VOLUME 24, NUMBER 4, 2002
41
Do It You rself Some of the figures in this article are generated using the Matlab programming language. Many readers are familiar with Matlab or have access to a computer t hat has Mat
pose two sets of dots with the same transformation as in
Figures 3(a) and (c), type idots(0 . 9 5 , 1 .05,0). What you obtain should look like Figure 4(c) even though the ex
lab capabilities. The following program should be called
act coordinates of the random dots will be different. (Mat
idots .m.
lab generates a different sequence of random numbers on each trial.)
function [e]=idots(a,b,theta)
Here's a problem. Superimpose two sets of random
x =ones(2, 400)-2 *rand(2, 400);
dots: an original pattern and one in which the seatings in
R= [cos(theta) -sin(theta) ;
the
sin(theta) cos(theta)J ;
However, vary the rotation angle. For exan1ple, in Figure
x
and y coordinates are 1.05 and 0.95, respectively.
6, I show the superposition of two patterns with a rota
S=[a O ;
tion of about 2.61 degrees (left panel), which gives a sad
O b] ;
dle geometry, and 5.47 degrees (right panel), which gives
xnew=R*S*x;
a spiral geometry. As the angle of rotation is varied be
x=[x xnew] ;
tween those two values, do you ever obtain a node geom
plot(x( l , : ) ,x(2 , : ) , ' . ' ) ;
etry'? With Matlab, one can explore the question numeri cally.
axis([- 1 . 1 l . l - 1 . 1 1 . 1 ]) ;
it is
However,
really better
to
compute
the
eigenvalues analytically using Eq. (3). If you do this, you
axis('square')
will find that there is a narrow range in which you must
title('Glass Pattern');
pass through the node geometry. Here is the insight. In
spiral geometry, the two eigenvalues are complex
e=eig(R*S);
the
txt= ['with eigenvalues : ' num2str(e( 1)) '
numbers with real part less than 1, and in the hyperbolic
geometry the eigenvalues are real with one eigenvalue be
and ' num2str(e(2))J ;
ing positive greater than 1, and the other eigenvalue pos
xlabel(txt) ;
itive less than 1. As (} varies, the values of the eigenval
To run this program you need to open up Matlab. Fig
ure 3(a) shows a random pattern of dots; in Figure 3(c) the x coordinate is multiplied by 1.05, the
y coordinate is
multiplied by 0.95, and there is no rotation. To superim-
ues change continuously. Both eigenvalues first become
real and less than 1, before both eigenvalues become com
plex. This is a great way to illustrate bifurcations in dy namical systems.
(b)
(a) .· . .. • • • . • .. • : .;' •• • • • ' :·. . . .: . :· . .. ... ... . ...... ,... . ... . ... .. :• • . .· . . . • , . .... . • .: ...: .• . J' ,, • . ·� "' ... , ,, . . . · · .. . , ... 0.5 . . . . . ..: · ' . .. . . : : · · ·. · ·. . . .. · , : · • • • _.,.: : • • : ,, ,, ..• # , • • ;.· ., •''• ...... . · . : · .. · ' . ' :· ....: .:···:. , ... ·. . :' ·: . •". • 0 •• • ·: ., · .:· · .. •• # -:., •• • .• .•.: • •·. · ..'• ., • . • • • , • • I '·· ;• .· ::,• • •. ,.I' • , , I. • . .· .· :- . . :- . .• . • • '• ': � N J;. # #II J t. • . •. •� , . : '"":::_:i·": -0 . 5 .· . . • ..• •• • • • •• •··� "J . 1/# I •• •• -., •. • • •.• � � ! . . : .• . , . . . . , . . .. . . . : . . . .i; :··' -i' $ . . . . . , � ··.. .· .· .· . . . I ..c
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.
.
.. -0 : .5 0 0.5 1 · 0 . 99544+0.081 1 86i, 0 . 99544-0.08 1 1 86i
-1
Figure 6. (a) Saddle geometry generated from two sets of correlated random dots by using the Matlab program with a = 1 .05, b = 0.95,
6 = 2.61 °. (b) Focus geometry generated with a = 1 .05, b = 0.95, 6 = 5.47°. For a = 1 .05, b = 0.95, is there a value of 6 in the range 2.61 ° <
6 < 5.47° that gives a node geometry?
42
THE MATHEMATICAL INTELLIGENCER
Now more than 30 years after I first observed these images composed of correlated random dots, it still seems we are just at the beginning of developing an un derstanding of how the visual system processes the in formation contained in these images. These images com bine both local and global features, which can be varied independently. Observation of experimental subjects (men, monkeys, or even pigeons! [ 16]) looking at the dot patterns is providing . a window into the physiological processes of vision.
A U THO R
REFERENCES
[1 ] L. Glass, Moire effect from random dots, Nature 223, 578-580 (1 969). [2] D. Marr, Vision, Freeman, San Francisco, 1 982. [3] J . Walker, The amateur scientist, Scientific American 242 (April
LEON GLASS
1 980).
Department of
[4] Why circles?, Mathematical lntelligencer 22, no. 2, 1 8 (2000). [5] L. Glass, R. Perez. Perception of random dot interference patterns.
Montreal , QC H3G 1 Y6
Nature 246, 360-362 (1 973).
Canada
[6] R. L. Devaney, A First Course in Chaotic Dynamical Systems.
e-mail: [email protected]
Perseus (1 992). [7] S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus (1 994). [8] S. Wagon and D. Schwalbe, Visua!OSolve: Visualizing Differential Equations with Mathematica, Springer!TELOS (1 997). However, in some cases, dots of different colors can NOT be used to capture the geometries of vector fields. L. Glass and E. Switkes, Pattern recognition in humans: correlations which cannot be perceived. Perception 5, 67-72 (1 976). [9] D. H. Hubel, T. N. Wiesel, Receptive fields, binocular interaction and functional architecture in the eat's visual cortex. J. Physiol. (Lond.) 1 60, 1 06-1 54 (1 962). [1 0] M. A. Smith, W. Bair, and J. A. Movshon, Signals in macaque V1
neurons that support the perception of Glass patterns, Journal of
Neuroscience, In Press (2002).
Physiology
McGill University
Leon Glass, after receiving a Ph.D. in Chemistry from the Uni versity of Chicago, was a postdoctoral fellow in Machine In telligence and Perception at Edinburgh; in Theoretical Biology at Chicago; and in Physics and Astronomy at Rochester. He has been at McGill since 1 975, interspersed with visits to Har vard Medical School and Boston University. Many readers will have encountered his research under many disciplinary titles; but call it physiology or theoretical biology or what you will,
it's really all mathematics. In his spare time he plays the French
hom in the I Medici Orchestra at McGill, and hikes in the
Adirondacks and other mountains.
The portrait here is a daguerrotype by Robert Shlaer; used by permission.
[1 1 ] H. R. Wilson and F. Wilkinson. Detection of global structure in Glass patterns: implications for form vision, Vision Research 38, 2933-2947 (1 998). [1 2] S. W. Zucker, Which computation runs in visual cortical columns? In: Problems in Systems Neuroscience, J. L. van Hemmen and T. J. Sejnowski (eds.) Oxford University Press, in press (2002).
[1 3] W. Schuette, Glass patterns in image alignment and analysis. United States Patent 5,61 3,0 1 3 . [1 4] J. Walker, The amateur scientist, Scientific American 243 (No vember 1 980).
· [1 5] D. L. Lau A. M. Kahn, G. R. Arce. Minimizing stochastic moire in ,
FM halftones by means of green-noise masks. Journal of the Op tical Society of America. 1 9, no. 1 1 Nov (2002). [1 6] D. M. Kelly, W. F. Bischof, D. R. Wong-Wylie, et al. Detection of Glass patterns by pigeons and humans: Implications for differences in higher-level processing, Psycho/. Sci. 1 2 , 338-342 (200 1 ) .
VOLUME 24, NUMBER 4 , 2002
43
l$@il•i§rr6'hfil@i§4fl!:l .. i§.id
Hat Tricks J. P. Buhler
Michael Kleber and Ravi Vaki l , Ed itors
In
magine that you are on a team of > 1 people on a new reality TV game show. After meeting your teammates, having the rules explained, and talking strategy with your teammates, you play the following game, once, for a possible shared prize of n million dollars. The host of the game show places black or white hats on your heads; the hat colors are chosen uniformly at ran dom (so that all 2n configurations are equally likely, which could be done, for instance, by having each hat deter mined by a fair coin flip). All players can see the color of every hat except
Hats are a This column is a place for those bits of
contagious mathematics that travel from person to person in the
common device i n mathematical puzzles .
community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributors are most welcome.
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA
their own. No communication is al lowed between teammates. Then all members of the team are required simultaneously either to pre dict their hat color, or to pass. The team loses if everyone passes, or if there are any incorrect predictions. Otherwise-i.e., if at least one person doesn't pass, and all non-pass state ments are true-the team wins n mil lion dollars. Since every non-pass prediction is, by the rules, a 50/50 guess, this seems like a difficult game for the team; e.g., if everyone guesses, the chance of suc cess is 112n. However, the value of the initial strategy session comes into sharper focus when a little thought re veals that there is a simple plan that gives the team a 509-6 chance of win ning: they appoint one person to guess, and agree that everyone else will pass. Can you devise a strategy that gives the team a better chance of winning?
The Hat Puzzle The hats problem circulated widely last year, furthered by an article in The
e-mail: [email protected]
44
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
!
New York Times and numerous dis cussions on the Internet. In full gener ality it is a fiendishly difficult puzzle; it has many variations, most not as dif ficult. It is easy to misunderstand the ques tion when you first hear it; note that af ter the initial strategy session no com munication is allowed between the players, and that the players' subse quent statements must be made simul taneously. This could be enforced by sending each player to a room con taining a computer monitor listing the names of the other players and their corresponding hat colors, and giving the player a choice of three statements: "my hat is black," "my hat is white," and "I pass." When it is necessary to distinguish this problem from the variants below, I will call it the "original" hats problem (though this is misleading: the major ity hats problem actually predates it by several years). It may seem hard to believe that any strategy could beat 50/50, since the team can't win unless someone guesses, and any guess has a 50% chance of be ing wrong. Astoundingly, the optimal strategy has a winning probability Pn that converges to 1 as n goes to infin ity. You might want to entertain your self by trying to solve this puzzle be fore reading the solution below. The case n = 3 is distinctly easier, and makes a good puzzle to pose to your friends (if they haven't already heard it). The case of general n is really quite difficult. In fact, the optimal answer is known only for n ::::; 8, n = 2k - 1, or n 2k; as will be mentioned below, the 16 is especially diabolical. So case n you should give the general case only to really good friends who can tolerate frustration, or perhaps to coding theo rists; of course, it's probably also OK if you are writing for The Mathematical Intelligencer and intend to describe (most of) the solution. Hats are a common device in math ematical puzzles. The famous "de=
=
rangement" problem asks for the prob
players will make false statements.
tial orientation of the edges of the
ability that an incompetent hat-check
Thus this strategy wins the prize with
cube: a direction on an edge tells a
n
clerk might return n hats randomly,
probability 3/4, and we will see below
player which configuration to guess,
none ending up with the rightful owner
that this is optimal.
and unoriented edges direct the player
(i.e., the probability that a random per
Somehow this surprising result is
mutation of n things has QO fixed
achieved by causing all false answers
points). The original hats puzzle first
to collide and having the true answers
appeared, in essence, in Todd Ebert's
occur
by
themselves.
This
"unex
to pass. The n
=
3 strategy given above
is illustrated in Figure 1 .
A node is a losing configuration for a given strategy if it has an outgoing
1998 computer science Ph.D. thesis,
pected power of collaboration" is a
arrow (the corresponding player will
where it arose in connection with a
theme that underlies almost all of the
make a false guess) or if it has no ar
A node is a
question in complexity theory; the
puzzles described in this article, and
rows coming into it at all.
problem was phrased (see [4]) in terms
it is really quite striking; as Ebert ob
of a warden and a prisoner. Peter Wink
serves, it seems that one draws infer
winning configuration if it has at least
ler later heard it from Peter Gacs, and
ences about a random variable
X by
one arrow coming into it, and no ar
rows going out.
immediately converted it into the hats
observing the values of random vari
Thus in Figure 1 there are two los
problem posed above. His contribution
ables that are completely independent
ing nodes: the antipodal points on the
to a recent volume in honor of Martin
of X.
Gardner [9] includes a series of puzzles
3-cube, which correspond to the
Strategies in the hats game can be
and ends with the original hats prob
viewed geometrically. For simplicity,
lem.
we name the colors
An earlier "voting" puzzle due to Steven Rudich and others [2] was also motivated by complexity theory, and can be phrased in terms of hats; it has some strong connections with the orig inal hats puzzle and will be given as the "majority hats puzzle" in the next sec
0 and 1, so that the
Good puzzles often circulate faster than gossi p
2
monochromatic configurations of the three hats. Each of the other
6 points
has one incoming arrow and no outgo ing arrow�, and is therefore a winning ' configuration. Let L denote the set of losing nodes and
W the set of winning -nodes.
Then
L is a covering code in the sense that every node is within Hamming distance 1 of an element of
L;
indeed if
v
is a
i n mathematical
winning node then it has an incoming
circles .
L . Here two nodes have "Hamming dis
remarkable in its depth and connection
configurations consist of all binary
the other by changing exactly one co ordinate. (More generally, coding the
tion. Good puzzles often circulate faster than gossip in mathematical circles, and the hats puzzle is, as we will see,
arrow that originates in an element of tance 1" if one can be obtained from
with current research . Its wide dis
n-tuples, i.e., the vertices of an n-di
semination was certainly abetted by
mensional cube. All players know the
orists consider d-covering codes in
Winkler's gregariousness, and by Sara
configuration, except for their own hat
which every node is within Hamming
Robinson's charming piece in
color; thus a player actually knows an
distance
edge on the cube joining the two con
the case
figurations possible from that player's
to hats problems.)
York Times
The New
[8].
Now we return to the puzzle itself, so if you want to solve it, e.g., in the
point of view.
case n = 3, you have to stop reading
A strategy is then a par-
If
L
d of an element of the code; d > 1 appears to be irrelevant
is any covering code, then it
gives a strategy: players seeing an edge should vote the
now.
W
LW
node, those
seeing a WW edge should pass, and
For n = 3 hats, the following strat
LL edge might as well
those seeing an guess.
egy gives the team a 75% chance of win
The probability of losing is
ning: players pass if the two hats that
ILI/2n ,
they see have different colors; if they
and a "sphere-packing bound" gives a
see two hats of the same color then
lower bound on the size of
they assert that they have the
hence an upper bound on the winning
opposite
L,
and
probability. Namely, each losing node
color.
v
There are 8 possible configurations of the three hat colors. In
"covers" the
n
+ 1 points that are at
Hamming distance at most one:
6 of them,
and the
and one of the other, and a little
a single coordinate of
thought shows that in this case two
ming spheres of radius 1 must cover
people will pass and one will make a
the n-cube, so
correct statement. In the two mono chromatic
configurations
all
three
n nodes
v itself
there are two hats of the same color
obtained by reversing
v.
These Ham
iildii;JIM
VOLUME 24, NUMBER 4, 2002
45
l£1 2: 2n!(n + 1),
strings are combined by adding corre
it is possible to come close to the
winning probability p of the strategy is
sponding bits modulo
sphere-packing bound
at most
rying.
Therefore
and
the
2,
without car
Thus players will XOR the names of
--
-
A strategy is perfect if this bound is re alized. This happens only when the covering Hamming spheres of radius
1
(n
= 2k - 1) have the special property
the players they see that have hat color
that they are linear, in the sense that
announce that they have hat color
tor space. In all likelihood, optimal
1; if the result is the 0 vector, they will 1, if
1 n . <1n+l =n+l
Pn :=::; nl(n + 1) L C FE
for large n. Hamming codes
they are vector subspaces of an F2-vec
the result is their own name, they will
codes in other dimensions will not be
announce that they have hat color
linear.
0,
It is remarkable that a purely recre
and in all other cases they will pass. It is easy to verify that the team wins
ational problem comes so close to the
are also disjoint, so that elements of L
unless the XOR of all players with hat
research frontier.
color
reference on covering codes
have outgoing arrows on every edge
tions is of course the binary Hamming
and elements of
W
have a unique in
coming arrow. In the hats game this
1
is zero. This set of losing posi
code! description will be useful later in dis
on winning configurations, and every
cussing the hats problem with more
configurations. Note that the above
n = 3 is perfect in this sense, so that 75% is indeed the opti mal probability P3·
strategy for
The problem
of fmding
theorists: this is the problem of con structing error-correcting codes. The "dual" problem of finding Hamming spheres that cover the binary n-cube hasn't received as much attention, but it does have several applications; the
2.
+ 1 is
A perfect code is both
packing and covering,
and coding
More precisely, if
L is the kernel of the linear map V*
V
VI'
r de
VI' to F2.
The
r has natural basis ele
mal strategy for the hats game (and
[v] corresponding to (character istic ftmctions of) vectors v E VI', and the map T takes [v] to v. If n = 2k, then a natural extension
construction of the Hamming codes)
of the above strategy turns out to be
when n
is as follows. At the
best possible. At the strategy session,
strategy session, team members are as
one player is chosen to play dumb: that
called (binary) Hamming codes. One explicit formulation of an opti
= 2k - 1
signed "names" that are nonzero k-bit
0/1
strings, perhaps thought of as the
binary expansions of integers from through
2k - 1.
1
During the game, each
player will then assume that the "XOR" of the players with hat color
1
is
1 of n Ln, then the density 1Lnl12 is, by defi nition, 1 - Pn. where Pn is the best over,
notes the vector space of dimension
ments
player passes, his hat color is ignored, and the remaining
2k - 1
players fol
low the above strategy (for proof that this is optimal, in the language of cov ering codes, see
[6]).
For n not of the form
a subset (not
of FE is within Hamming distance
_1_ :=::; ILnl < _2_ 2n n + l n+l
denotes the nonzero ele
of functions from
Ln is
possible probability of winning; more
V,
ments of that vector space, and F
2k - 1
In ad
cardinality of F E such that every point
is a k-dimensional vector
vector space F
n.
necessarily a subspace) of minimum
space over the field F2 of two ele ments,
but not for larger
density that come very close to the
research frontier.
T:F 2 �
= 2k,
sphere-packing bound for large n.
so close to the
codes in fact exist when n
linear codes with this property are
In other words,
finity there are nonlinear codes with
p roblem com es
theorists know that perfect !-covering
+ 1 = 2k;
= 2k - 1 players
based on the Hamming code for n
2k - 1.
for n
recreational
where
so a perfect code exists only if n
2k - 1 < n < 2k + 1 -
there is a dumb strategy, as above,
dition, it is known that as n goes to in
code
a power of
1
that a p u rely
Note that if L is a perfect code then
1 n + l'
ering codes. For
code strategy, and the remaining play
is
2n + l
has a
ers play dumb. This strategy is optimal
than two colors. Namely, the Hamming
___lfL --
[6]
that contains
are appointed to use the Hamming
defmitive reference on covering codes
[6].
[7]
It i s remarkable
disjoint
Hamming spheres is familiar to coding
companion Web site
up-to-date data on the best known cov
An equivalent, but more technical,
means that there is a unique non-pass one makes a false statement on losing
The fundamental
2k - 1
or
2k,
(the lower bound comes from the sphere-packing bound above, and the upper bound can be derived from the dumb strategy). With some work (see
[6]),
one can show that the limit, as n
goes to infinity, of (n
+ l)ILnl12n
is
1,
i.e., that for large enough n the ratio of the density of Ln to the sphere-packing
ll(n + 1) can be made arbitrar 1 (and is of course equal to 1 if and only if n = 2k - 1). As one sees on the Web site [7], al ready for n = 9 the optimal strategy is unknown: the best covering code Lg has 57 :=::; l£9 1 :=::; 62. The sphere-packing bound
ily close to
bound says that the winning probabil
ity is at most
9/10, giving p9 < 460/512;
nonzero. Here XOR is "bitwise exclu
the full story on covering codes isn't
in fact the bounds on the size of Lg im
sive-or" or "nim addition": two
known! However, as described below,
ply that
46
THE MATHEMATICAL INTELUGENCER
0/1
I I I I I .... .... .... .... I .... .... ..
Note that the value n = 16 is an es pecially diabolical value to give to someone as a puzzle: in addition to the possibility that your poor friend will be .... .... .... somehow misled by the power of 2, he .... .... .... .... will have to recover the connection ... I .... .... .... with covering codes, will have to re I 1 discover Hamming codes, and will I I have to rediscover a nontrivial theorem I I 1 / about covering codes in the case I I n = 2k . II I will go on in a moment to consider variants of the original hats puzzle. Meanwhile, there are a number of de tails and extensions that cry out for ex amination, but to save space I leave Majority Rules them as exercises for the diligent Reality TV shows fade quickly, and our reader (with hints elsewhere in this is TV game host decides that the rules must be changed in order to boost rat sue; see p. 70). ings. (Perhaps he also noticed that the 1. Using some description (as above or teams were winning too often.) The otherwise) of the Hamming code, new rules do not allow players to pass, show that in the case n = 3 = 22 - but, as compensation, a team wins if a 1 it gives the solution given earlier majority of players make true state for the hats problem, perhaps up to ments. suitable choice of labeling. In order to avoid ties this game is 2. Verify that for n = 2k - 1 the Ham played with an odd number of players. This version of the problem was ming code solution does indeed achieve the sphere-packing bound. originally stated in [2] as a voting prob :!. Show that for 2k - 1 < n < 2k+ l - lem, motivated by results on lower 1 the dumb strategy above has win bounds in computational complexity ning probability that is at least (n - arising from analyzing circuits in terms of integer polynomial "approxima 1)/(n + 1). 4. The dumb strategy for n = 5 gives a tions" to boolean functions. In addition winning probability of 3/4 = 24/32 to the important results on approxi for a 5-person team in which 2 play mating boolean functions by the signs ers play dumb. Show that the team of integer polynomials, the paper con can do slightly better by fmding a tains other variants of the puzzle (one 7-point covering code in the binary of which will be described in an exer 5-cube, giving the team a winning cise below). The majority hats problem is simi probability of 25/32. 5. Verify that the Hamming code im lar to the original hats problem in ba plicit in the XOR strategy is the sic framework, and also in that there same as the Hamming code ob is a Hamming code solution when tained as the kernel of the linear n = 2k - 1, as will follow from some of the arguments below. The optimal map described above. 6. What can the team do if the game strategies for 2k - 1 < n < 2k+ - 1 show host maliciously listens in on are not known, though they seem eas the strategy session and attempts to ier to explore than in the case of cov choose the hat colors nonranaomly? ering codes. Elwyn Berlekamp has analyzed this 7. Verify that randomization does not help in the original game, in other majority hats problem in some detail, words, no randomized strategy can and finds an amusing geometric inter do any better than a deterministic pretation of a strategy for this game. strategy. Namely, a strategy can be described by
1
giving an orientation on aU of the edges of the graph of the n-cube: again, each player sees an edge and, not being al lowed to pass, votes according to the direction of that player's edge. The op timal strategy for n = 3 can be ob tained from the optimal strategy for the original hats game above by orienting the remaining edges in a cycle. In Figure 2, the graph of the 3-cube decomposes into 2 tripods emanating from the losing set, and a cycle, indi cated as a dotted line, joining the other 6 winning configurations (the cycle can be oriented in either direction). A little thought shows that the marked points are losses in that every player votes wrong, but that the other nodes are all wins: there are two incoming and one outgoing arrows, so that the team wins by one vote. More generally, Berlekainp general izes the idea of a covering code by al lowing paths, possibly of-length more than 1, emanating from "sources" in the losing set L, that terminate in "sinks," which are the complementary winning set W. Each winning node is the terminus of exactly one such path, and the paths are all edge-disjoint. Since the number of edges at every node is odd, the graph obtained by removing the chosen paths (and the vertices in L) has even va lence. By Euler's theorem, this graph is a union of cycles. Therefore the origi nal graph can be thought of as a col lection of paths and cycles; the paths are directed, starting in L, and ending in W; each element of W is the endpoint of a uitique path. The cycles can be ori ented arbitrarily. This ensures that at each winning node all of the votes other than the decisive vote are evenly split, and at each winning node the team wins by one vote. Thus if we can fmd a set of edge-dis joint paths from losing nodes to each winning node, then this can be ex tended to an orientation of the entire graph that gives a strategy for the ma jority hats problem. Note that a node in L can have at most n outgoing paths, so that a losing node "accounts for" itself and at most n winning nodes; thus the sphere-pack ing bound still applies, i.e., (n + 1) IL I ::::::
VOLUME 24, NUMBER 4, 2002
47
2n.So far, Berlekamp cannot fmd coun terexamples (even for very large n) to the surprising conjecture that optimal size of L is as small as it can be con sistent with this bound; i.e., that the size of the optimal L is the smallest integer bigger than or equal to 2n + 1/ (n + 1). 8. Show that for n = 5 there is a strat egy for the majority hats game with only 6 losing configurations; i.e., find an orientation of the binary 5-cube in which all but 6 vertices have an ex cess of incoming arrows.(Since the best that is possible for the original hats game is a 7-node covering code, this shows, as Berlekarnp notes, that democracy is preferable to consen sus/perfection.) 9. Find an optimal strategy in the ma jority hats game for n = 9.
More Colors Again, our TV game show host wants to boost sagging ratings, and perhaps de crease winning probabilities, and de cides to start using more than 2 colors of hats.What strategies should the team use when there are q colors, q > 2? As one might guess, the team has a harder time. However, Hendrik Lenstra, Jr., and Gadiel Seroussi have shown that some of the same basic facts hold even in this case. For in stance, the winning probability is arbi trarily close to 1 for large enough n. However, perfect codes do not exist, and there are several open questions. First, let's interpret the game geo metrically. The configuration space is now a q-ary n-cube Qn , where Q is a q element set.A player sees an "edge" of the cube: the i-th player sees the con figuration v E Qn except that the i-th coordinate vi is unknown. From the point of view of this player the config uration could be any of the q configu rations that agree with v except possi bly in the i-th coordinate.A strategy is a mapping from edges e to Q U {pass}. If a player sees the edge e he passes if the label of the edge is "pass," and an nounces the corresponding color if the edge is labeled with a color. If v E Qn is a configuration of hats, let v[i] denote the set of configura-
48
THE MATHEMATICAL INTELLIGENCER
tions, not equal to v, that agree with v in all but the i-th coordinate. Thus, each element w of v[i] has Hamming distance 1 from v, and disagrees with v in precisely the i-th coordinate. If a strategy is given, then we get a partition Qn = W U L of configurations into winning and losing positions.The winning configurations have the fol lowing property: (*) For all v in W there is a coordi nate i such that v[i] c L. Conversely, if a set W satisfies (*) then it produces a strategy whose losing set is precisely the complement L of W. In order to digest this condition you might want first to convince yourself
The reader m ay h ave d rawn the conclusion that al l hats problems are i m possibly hard . that in the case q = 2 we recover the earlier analysis involving covering codes L. We will call a subset L C Qn a strongly covering code if its comple ment satisfies (*).You will also enjoy checking that in the case n = q = 3 the marked nodes in Figure 3 are a strongly covering code; to do that you have to check that for each node not in the code there is some coordinate direc tion in which the other two vectors are in the code.
As in the case of covering codes, the winning probability is of course
Thus we want to fmd small strongly covering codes. Lenstra and Seroussi generalize the sphere-packing bound to show that if L c Qn is a strongly covering code then
qn(q - 1)
IL l 2: n + q - 1 . little algebraic manipulation shows that this gives an upper bound on the winning probability p of
A
P =
1
_ lfd_ ::;; qn
n
+
_ _
q-1
which generalizes the earlier result for
q = 2.
One way to prove this is as follows. Let S be the set of ordered pairs (x, y) where x is a winning node, y is a los ing node, x and y differ in exactly one coordinate, and all other nodes differ ing from x in that coordinate are also losing nodes; i.e., S = {(x, y) : x E W, there is an i with y E x[i] and x[i]
c
L }.
We employ the usual combinatorial de vice of counting this set of ordered pairs in two ways. For each x there is (at least one) coordinate direction in which all the q - 1 other elements on that edge are in L; therefore (q 1)1*1 ::;; lSI .On the other hand, if we fix y and ask how many x's could be paired with it, we note that there are n coordinates, so that y E L can be part nered with at most n x's.Therefore
(q - 1)(qn -
��) ::;; lsi ::;; n�l -
little algebraic juggling gives the up per bound claimed above. Unfortunately, Lenstra and Seroussi also prove that for q > 2 and n > 1, perfect strongly covering codes d o not exist, so that the lower bound on L cannot actually be attained. However, they give an ingenious argument that shows that the winning probability can be made arbitrarily close to 1 by choos ing n large enough. The technique somehow intertwines the binary case with the q-nary case. A
litftll;ifi
n ___:_
_
Let n = 2k - 1. (For other n, we will use a "dumb" strategy as described above.) Let Q = ZlqZ be the q-element cyclic group. There are n = 2k - 1 nonzero k-vectors v in Qk with entries in {0, 1 ). Use them to label basis vec tors [v] whose Q-linear combinations form a group which will be identified with Qn. Then let T : Qn � Qk be the group homomorphism that maps [v] to v. So far this follows the q = 2 con struction, but we now alter it in an in teresting way: let L be the set of ele ments of Qn whose image under T is a k-tuple whose coordinates are all non zero. I claim that L is a strongly covering code. Indeed, if x E Qn is not in L, then T(x) has some coordinates equal to 0. If v denotes the 0/1 vector with 1's in the coordinates where T(x) is nonzero, then one checks that x[i] C L, where the coordinate i corresponds to the ba sis vector [v] . This shows that L is a strongly covering code, and it is easy to check that the winning probability is p = 1 - (q - 1)kfqk, which goes to 1 (albeit more slowly than one might like) as n goes to infinity. Noga Alon ([1]) subsequently gave a probabilistic construction of a strongly covering code whose winning proba bility comes much closer to the asymp totic bound. The basic idea is to make a random choice and then alter it as necessary. This is a well-known situa tion in coding theory: the best known explicit constructions fall short of what can be achieved by suitably mod ified random codes. By now, the reader may have drawn the conclusion that all hats problems are impossibly hard and that they aren't recreational in any sense of the word. In an attempt to persuade you oth erwise, here is a collection of (some what) easier hat problems. The first two can be found in Peter Winkler's.charm ing contribution Games People Don't
Play to the recent volume [9] of essays arising from a "Gathering for Gardner" in honor of Martin Gardner's contribu tions to recreational mathematics. 10. The TV game show host introduces the following more extreme varia tion of the game. The hats game is played as described originally ex cept that passes are not allowed, and players making false state ments are executed. What "worst case" strategy can the team adopt that gives them the largest number of guaranteed survivors? 1 1 . What is the team's best worst-case strategy in the following varia tion? The team members are lined up in a manner that allows play ers to see only the hats in front of them in the line, e.g., the front player sees no hat colors, and the player at the back of the line sees all colors but one-the one she is wearing. The players are required to state their hat colors, one at a time starting at the back of the line. Players making false state ments are executed. All players hear all of the statements, but not their consequences. 12. Same as the previous problem, ex cept that the game show host uses q > 2 colors of hats. 13. [2] What is the team's best strategy if the host uses the majority hat game, except that "Chicago-style" voting is allowed in which players can cast as many votes as they like? 14. (Gadiel Seroussi) What strategy would you follow if the game show host, in a fit of desperation, did not allow a strategy session, and did not turn the lights on after the hats were placed? Thus team members cannot see any hat colors; they are complete strangers. (Individual team members are allowed to as sume that their teammates are highly rational, and the rules per mit flipping coins in the dark to generate random numbers, by feel-
ing the top of a coin to see whether it is heads or tails). Acknowledgments
I thank Peter Winkler for telling me the original hats problem, Michael Kleber for encouraging me to write this article, and Elwyn Berlekamp, Hendrik Lenstra, Jr., and Gadiel Seroussi for some de lightful conversations about the puzzle. Elwyn Berlekamp, Danalee Buhler, Michael Kleber, Hendrik Lenstra, Gadiel Seroussi, Ravi Vakil, and Peter Winkler all made helpful comments on various drafts of this piece. REFERENCES
[1 ] Noga Alon, "A comment on generalized covers," note to Gadiel Seroussi , June 2001 . [2] James Aspnes, Richard Beigel, Merrick Furst, and Steven Rudich, The expressive powec.of vo'ting polynomials, Combinatorica 14 (1 994), 1 35-1 48. [3] Mira Bernstein, The Hat Pro_,blem and Ham ming Codes, in the Focus newsletter of the MAA, November, 2001 , 4-6. [4] Todd Ebert and Heribert Vollmer, On the Autoreducibility of Random Sequences, in Proc. 25th International Symposium on Math ematical Foundations of Computer · Science, Springer Lectures Notes in Computer Science, v. 1 893, 333-342, 2000. [5] Hendrik
Lenstra and
Gadiel
Seroussi,
On Hats and other Covers, preprint, 2002, www.hpl.hp.com/infotheory/hats_extsum.pdf [6] G. Cohen, I. Honkala, S. Litsyn, and A. Lob stein, Covering Codes, North-Holland, 1 997. [7] Simon Litsyn's online table of covering codes: www.eng.tau.ac.il/�litsyn/tablecr/ [8] Sara Robinson, Why Mathematicians Now Care About Their Hat Color, New York Times, Science Tuesday, p. D5, April 1 0, 2001 . On line at http://www.msri.org/activities/jir/sarar/ 01 041 ONYTArticle.html [9] Peter Winkler, Games People Don't Play, 301-3 1 3 in Puzzlers' Tribute, edited by David Wolfe and Tom Rodgers, A. K. Peters, Ltd., 2002. Department of Mathematics Reed College Portland, OR 97202 USA e-mail: [email protected]
VOLUME 24, NUMBER 4, 2002
49
l@ffli • i§rr6hlf119.1rr1rr11!.1h14J
MASS Program at Penn State Anatole Katok, Svetlana Katok, and Serge Tabachnikov
Marjorie Senec hal ,
Editor
T
he MASS program-Mathematics Advanced Study Semesters-is an intensive program for undergraduate students recruited every year from around the USA and brought to the Penn State campus for one semester. MASS belongs to a rare breed; we know of two somewhat similar mathe matics programs for American under graduates, both based abroad: Bu dapest Semesters in Mathematics, and Mathematics in Moscow; the former is in its "teens" (started in 1985) while the latter is just 1 year old. MASS at Penn State has turned 6, and this seems to be a good time to reflect on the MASS community.
How It Started All
This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department
of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail: [email protected]
50
three founders of the MASS pro gram (the first two authors of this ar ticle and the first MASS director, A Kouchnirenko) are steeped in the Russian tradition where interested stu dents are exposed to a variety of math ematical endeavors, often of nonstan dard kind, at an early age. By their senior undergraduate years such stu dents are already budding profession als. We briefly describe this tradition in the Appendix. The US educational sys tem is built on completely different principles, and interested young stu dents are routinely encouraged to progress quickly through the required curriculum. Here a typical mathemati cally gifted high school student takes courses in a local university and often is considered a nerd by his peers. The founders felt that there was a way to combine some of the best features of both traditions within the US academic environment, namely, to gather a group of mathematics majors and to expose them to a substantial amount of inter esting and challenging mathematics from the core fields of algebra, geom etry, and analysis, going way beyond the usual curriculum. The second author's first exposure to an intensive program for US under graduates was at the Mills College
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
I
Summer Mathematics Institute for mathematically gifted undergraduate women. But why not a co-educational program along the same lines, whose participants would contribute a variety of experiences and backgrounds? The number of undergraduate students in the USA, interested in mathematics and advanced enough for such a pro gram, is rather limited, and we decided not to restrict the pool of potential par ticipants. The result was the SURI (Summer Undergraduate Research Ini tiative) program at Penn State in the summer of 1993, where all three future founders of the MASS program came together. During this program it be came clear that a semester-long format would be even more productive for an intensive program organized mostly around advanced learning with ele ments of research initiation. And so we envisioned a semester long program for undergraduate stu dents from across the country. We thought it crucial for the success of the program that the cost for the partici pants should not exceed that at their home universities. It took 3 years to get the original financial commitment from the Penn State administration at various levels and to solve numerous logistic problems before the MASS pro gram could begin.
Program Description The main idea of the MASS Program, and its principal difference from vari ous honors programs, math clubs, and summer educational or research pro grams, is its comprehensive character. MASS participants are immersed in mathematical studies: since the pro gram is intensive, its full-time partici pants are not supposed to take other classes. All academic activities for a se mester are specially designed and co ordinated to enhance learning and in troduce the students to research in mathematics. This produces a quantum leap effect: the achievement and en thusiasm of MASS students increases
much more sharply than if they had
most importantly, talk mathematics
been exposed to a similar amount of
most of the time. Of course, this is ex
material over a longer time in a more
actly how "mature" mathematicians
conventional environment.
operate in their professional life! A
oped independently according to the •
interests and abilities of the student. A weekly 2-hour
interdisciplinary seminar run by the director of the
A key feature of the MASS experi
necessary condition for this environ
MASS program (the third author of
ence is an intense and productive in
ment is the gathering of a critical mass
this article), which helps to unify all
teraction among the students. The en
of dedicated and talented students,
vironment is designed to encourage
which is one of the chief accom
such interaction: a classroom is in full
plish ments of MASS.
possession of MASS (quite non-trivial to arrange in a large school such as Penn State !) and furnished to serve as a lounge and a computer lab outside of
Let us describe the main compo
Three
other activities. The MASS
colloquium, a weekly
lecture series by distinguished math
nents of MASS: •
•
ematicians, visitors, or Penn State research facuity.
core courses on topics cho
All elements of MASS
(3 courses,
class times. Each student has a key and
sen from the areas of Analysis, Al
can enter the room 24 hours a day. The
gebra/Number Theory, and Geome
16 credit hours, all listed as Honors
students live together in a contiguous
try!ropology. Each course features
classes that are transferable to MASS participants' home universities. Addi
the seminar, and the colloquium) total
block of dormitory rooms and they
three 1-hour lectures per week, a
pursue various social activities to
weekly meeting conducted by
a
tional recognition is provided through
gether. The effect is dramatic: the stu
MASS Teaching Assistant, weekly
prizes for best projects and merit fel
dents find themselves members of a
homework assignments, a written
lowships. Each student is issued a Sup
cohesive group of like-minded people
midterm exam, and an oral final ex amination/presentation.
plement to the MASS Certificate, which includes the list of MASS courses with
Individual student research proj ects ranging from theoretical math
special achievements. It also includes
study together, attack problems to
ematics to computer implementa
the descriptions of MASS courses, the
gether, debug programs together, col
tion. Most of the projects are related
list of MASS colloquia, and the de
laborate on research projects, and,
to the core courses; some are devel-
scription of MASS program exams.
sharing a special formative experience. They quickly bond, and often remain friends after the program is over. They
•
credits, grades, final presentations, and
photo © S. Katok
VOLUME 24, NUMBER 4, 2002
51
These supplements are useful for the
fundamental facts about finite groups
student's
and
question from the course and a prob
pro
lem. Then the student has an hour to
lences and enhance the student's ap
ceeded to what is often referred to as
prepare the answers, with no access to
plications to graduate schools.
"quantum
home
institution
equiva
their representations
and
of
literature or lecture notes during this
The core courses are custom de
knots and 3-dimensional manifolds as
hour. The answers to the ticket ques
signed for the program and are avail
sociated with statistical physics and
able only to its participants. Each
usual undergraduate
(and, in many
cases, even graduate) curriculum. For example, the core courses offered at MASS
2001 were
invariants
"The MASS
course addresses a fundamental topic which is not likely to be covered in the
topology":
p rog ram has been the best semester of
Mathematical Analysis of Fluid Flow by A. Belmonte, Theory of Parti tions by G. Andrews, and Geometry and Relativity: An Introduction by N.
my l ife. " the Yang-Baxter equation. The course
Higson.
tions constitute only about a third of the oral examination. Another third is a pre sentation of the research project asso ciated with the course; this presentation is prepared in advance and may involve slides, computer, etc. The last third of the exam is a discussion with the com mittee of three (the course instructor, the teaching
assistant,
and another
Penn State faculty).
A MASS colloquium is similar to a usual colloquium at a department of mathematics, with an important differ
a
was received by the students with great
ence: a speaker cannot assume much
course, an instructor is challenged to
enthusiasm and is likely to direct some
background
reach a delicate balance between cov
of them toward this active area of re
makes the speaker's task harder, we
ering the basics, with which the stu
search.
Designing
and
teaching
such
dents might be unfamiliar, and intro
material.
Although
this
find that the quality of the talks usually
The final exams (three, in total) have
benefits from this restriction. To quote
a unique format. It is quite unusual for
the opening sentences of an inspiring
a US university and represents a cre
article by
Consider, for example, a MASS 2000
ative development of a European tradi
course Finite Groups, Symmetry, and Elements of Group Representations by
good colloquium" (see at www. math.
A student draws a random "ticket"
"Most colloquia are bad. They are too
A. Ocneanu. This class started with
which typically contains a theoretical
technical and aimed at too specialized
ducing
advanced
material
typically
taught in topics courses.
tion where examinations are often oral.
J. McCarthy "How to give a
psu. edu/colloquium/go odcoll. pdf) :
photo © S. Katok
52
THE MATHEMATICAL INTELLIGENCER
an audience." This is precisely a sin that MASS colloquium is free of. As a result, along with MASS students, it is well attended by graduate students and faculty at the Department. To preserve the intellectual effort that goes into MASS colloquium talks, a group of 2 or 3 MASS students is as signed to take notes and prepare a readable exposition of the talk. We also experiment with videotaping the talks. Choosing the speakers, we always in vite mathematicians known for their ex pository skills. We also try to represent as broad a spectrum of mathematical re search as possible. We find it beneficial to combine very well-known mathe maticians with those in the early stage
photo © S. Katok
of their careers. A complete list of MASS colloquium talks can be found on the
portant function of the seminar is to
web site www.math.psu.edu/mass.
bring out elements of unity of modem
The MASS seminar plays many roles
mathematics. Often identical or similar
in the program. One of them is to in
notions appear in different courses in
troduce the students to the topics that,
various guises, and the seminar is the
otherwise, are likely to "fall between
place to explore, develop, and clarify
cracks in the floor." For example, one
these connections.
of the seminar topics in 2001 was the
ber of guest speakers, mostly Penn
State ..faculty, give expository talks at
the conference.
Here are two examples of REU stu dents' research projects.
·
"Simplices with only one integer point" (2 students; faculty mentor A. Borisov) . The students found an effective proce
jective geometry: Pappus, Desargues,
The Summer Program: REU and MASS Fest
Pascal, Brianchon, and Poncelet. Once
The Penn State Summer RED (Re
wojective geometry was a core subject
search Experiences for Undergradu
in the university curriculum, but nowa
ates) program started in 1999 as an ex
days it is perfectly possible to obtain a
tension of MASS. Unlike MASS, this
doctoral degree in mathematics without
program is not unique: currently, there
a single encounter with these facts. An
are about 50 RED programs in mathe
"New congruences for the partition
other example: the theory of evolutes
matics available to undergraduate stu
function" (1 student; faculty mentor
and involutes was a crowning achieve
dents in the USA. The Penn State RED
Ono). This project started before the
classical configuration theorems of pro
dure that allows them to describe all classes of simplices with vertices that have only integer coordinates and only one point with integer coordinates in side. Using computers they found all classes in dimensions 3 and
4. K.
ment of Calculus to be included into
is closely related to MASS: about half
RED program began. Using the theory
textbooks. Alas, a contemporary stu
of its participants stay for the MASS se
of Heeke operators for modular forms
dent is not likely to see these things any
mester in the fall. This makes it possi
of half�integral weight, the student found
more. The MASS seminar is a natural
ble to offer research projects that re
an algorithm for primes 13 :::;;
place to learn such a topic.
quire more than 7 weeks (the length of
which reveals 70,266 new congruences
Another purpose of the seminar is to prepare the students for the up
RED program) for completion.
of the form p(An
Mathematical research usually in
+ B) == 0
m :::;; 31
(mod
m),
where p(n) denotes the number of un
coming MASS colloquium talks. A col
cludes three components: study of the
restricted partitions of a non-negative in
loquium speaker is asked whether cer
subject, solving of a problem, and pres
teger n. As an example, she proved that
tain material should be covered in
entation of the result. These three com
p(3828498973n
advance so that the students get the
ponents are present in the RED pro
for every integer n. The first three con
+ 1217716)
==
0 (mod 13)
most from the talk. For example, as
gram: in addition to the traditional
gruences were found in 1 9 19 by Ra
preparation for A. Kirillov's talk on
individual/small group research proj
manujan, and after that finding new
Family Algebras in 2001, a 2-houJ sem
ects supervised by faculty members,
ones was considered a very difficult
inar was devoted to the basics of Lie
the program includes two short courses,
problem. The paper written by this stu
groups and Lie algebras. Still another
a weekly seminar, and the
dent has been accepted for publication.
MASS
Fest.
function of the seminar is to rehearse
MASS Fest is a 3-day conference at
the students' presentations of the re
the end of the REU period at which the
search projects on the
participants present
,final exam. This
their
We would like to emphasize a unique
research.
role played by the RED coordinator, M.
usually occupies the last quarter of the
This is also a MASS alumni reunion.
Guysinsky, who has been coming to
semester. Probably an even more im-
Along with the RED students, a num-
Penn State for the summer since 1999
VOLUME 24, NUMBER 4, 2002
53
as a visiting Assistant Professor sup ported by VIGRE funds.1 He organizes all the REU activities, including MASS Fest, runs the seminar, and supervises research projects, some suggested by other faculty not present during the REU period, and some by him. This re quires an unusual combination of math ematical and pedagogical talents, and we are very fortunate to have found this combination in Guysinsky.
Participants MASS participants are selected from ap plicants currently enrolled in US col leges or universities who are juniors, se niors, or sometimes sophomores. They are expected to have demonstrated a sustained interest in mathematics and a high level of mathematical ability. The required background includes a full cal culus sequence, basic linear algebra, and advanced calculus or basic real analysis. The search for participants is nation wide. Participants are selected based on academic record, recommendation let ters from faculty, and an essay. The number of MASS participants varies from year to year, with an aver age of 15 per semester. Some are Penn State students, but most are outsiders. It is interesting to analyze where they come from. For this purpose we divide American universities into four cate gories: (1) small, mostly liberal arts, schools; (2) state universities (mostly large); (3) elite private universities; (4) Penn State. The breakdown over the last 6 years is as follows: about 20% of the participants belong to the first cat egory, about 400Al to the second, only 3% to the third, and 37% to the fourth. One should take into account that some Penn State students are part-time participants (they take one or two courses), but a few of them participate in MASS more than once. These numbers are probably not very surprising (although we strongly feel even students from elite schools benefit significantly from the pro gram). Another statistic: women rep resented about 300;6 of the enrollment (with considerable deviations: in 2000, the ratio was 50/50).
About 700Al of MASS graduates have gone on to graduate programs in math ematics (one should keep in mind that some recent participants are still con tinuing their undergraduate studies). The distribution of the graduate schools is very wide. Without provid ing a comprehensive listing, we men tion some: Harvard, Cornell, Stanford, Princeton, Yale, University of Chicago, University of Michigan, University of California at Berkeley, University of Wisconsin, Indiana University, Univer sity of Utah, University of Georgia. About 15% of MASS graduates chose Penn State for graduate school. Here is what Suzanne Lynch, a MASS 96 participant who is about to receive her Ph.D. from Cornell, wrote in an unsolicited letter:
The MASS program has been the best semester of my life. I was immersed in an environment of bright moti vated students and professors and challenged as never before. I was pushed by instructors, fellow-stu dents, and something deep inside myself to work and learn about math ematics, and my place in the mathe matical world. I loved my time there, and never wanted to leave. I believe the MASS program helped to prepare me for the rigors of graduate school, academically and emotionally. . . . The MASS program has been very in strumental in opening grad school doors to me, and in giving me the courage to walk through them. Talking of MASS participants, one must mention the teaching assistants involved. TAs are chosen from among the most accomplished Ph.D. students of the Penn State Department of Math ematics. Their work is demanding but also rewarding. TAs are required to sit in the respective class and take notes; once a week they have a 1-hour meet ing with the students that is devoted to problem-solving, project discussion and, sometimes, individual tutoring. In some cases the material of a MASS course may be new for the TA as well as the students. This gives the assistant a welcome opportunity to learn a new
topic but makes the work even more challenging. Some MASS TAs are them selves MASS graduates.
Student Research During the semester, each MASS par ticipant works on three individual projects. Usually a project consists in learning a certain topic in depth, work ing on problems (ranging from routine exercises to research problems, usu ally related to the subject of the re spective course), and making a pres entation during the final examination. For many MASS participants who also attend the REU program, a project is a continuation of one started in summer. In some cases, a research project pro duced a significant piece of mathemati cal research. Here are two examples: An Nguyen, a MASS 96 student and now a graduate student in Computer Science at Stanford, rediscovered the famous value of A = 1 + Vs for the ap pearance of period-three orbits in the logistic family f(x,A) = Ax(1 x), and then went on to discover a previously unknown bifurcation point where the second period-four orbit appears: -
A
=
1
+ Y4 + 3vTo8.
James Kelley, a MASS 98 participant, now a graduate student at UC Berkeley, studied the representation of integers by quadratic forms, a classical problem in number theory. In particular, he studied a well-known problem posed by Irving Kaplansky: What integers are ofthe form x2 y2 + 7z2 where x, y, and z are in tegers? Obviously, if N is of this form, then so is Nk2• However, the converse is not necessarily true. James proved, us ing the theory of elliptic curves and mod ular forms, that every "eligible" integer N which is not a multiple of 7 and not of this form, is square-free! This result has appeared in print: J. Kelley, "Ka plansky's ternary quadratic form," Int. J. Math. Sci. 25 (2001), 289-292.
+
The research project topics may be related to the student's major, different from mathematics. For example, a biol ogy major in the 2001 course "Mathe-
1VIGRE: Grants for Vertical Integration of Research and Education, a program designed to promote educational experiences of undergraduate and graduate students in the context of ongoing mathematical research within the university.
54
THE MATHEMATICAL INTELLIGENCER
matical analysis of fluid flow" has a re search project "A mathematical analysis of fluid flow through the urinary system." MASS students present their re search projects at the Undergraduate Student Poster Sessions at thi!.January AMSIMAA joint meetings. For exam ple, N. Salvaterra and B. Wiclanan (REU and MASS 1999) were among the winners in Washington, DC, January 2000, with the poster "The Growth of Generalized Diagonals in a Polygonal Billiard" (advisors: A. Katok and M. Guysinsky). Another example: B. Chan (REU and MASS 2000) was a winner in New Orleans, January 2001 , with the poster "Estimation of the Period of a Simple Continued Fraction" (advisors: R. Vaughan and M. Guysinsky).
Funding MASS is jointly funded by Penn State and the National Science Foundation. Penn State provides fellowships for out of-state students that reduce their tu ition to the in-state level. Further sup port comes through the NSF VlGRE grant. In particular, MASS participants whose tuition in their home institution is lower than Penn State in-state tuition receive grants for the difference. The balance of the VlGRE funds are used to further decrease out-of-pocket expenses of the participants, and is distributed in dividually based on merit and need. In particular, several merit fellowships are awarded at the end of the MASS semes ter. The VlGRE grant also supports the MASS colloquium series by covering the speakers' travel expenses.
Perspectives We are confident that MASS will con tinue to grow. Here are some ideas for the program's future. •
•
One of the key issues is funding. We hope to attract private money to complement the current NSF sup port of the program. There is a con siderable interest in mathematics among private and corporate aonors, and the contribution of the MASS program to undergraduate mathe matics education is substantial. Ide ally, we would like to see the whole program endowed. ' We envision a larger, 2-level MASS
•
•
program that runs two consecutive semesters: one oriented toward freshmen and sophomores, the other, more advanced, for juniors and seniors. With a broader financial base, MASS could include a certain number of for eign students. The available NSF funds can support only US citizens and permanent residents. However, there is an interest in the program among foreign students attending American universities, and a few such students have attended MASS paying from their own funds. As a first step, we would like to extend the program to undergraduates in Canada An important issue is preservation of MASS materials. Each MASS core course developed for the program can be used elsewhere. We envision an ongoing series of small books containing course material in a lec ture notes style, detailed enough to serve as guidelines for a qualified in structor to design a similar course. As a first step, we are preparing a MASS presentation volume that will be published by the American Math ematical Society. This book will pre sent all components of the program (core courses, REU courses, MASS colloquia, students' research), and it will appear late in 2002 or early in 2003. We also hope to record MASS colloquium talks and make them available to the public, possibly on line, in the MSRI style.
Our optimism about the future of MASS is based on the enthusiasm of the students, instructors, and TAs, and on the general public interest in im proving the mathematical education in the USA.
Appendix: On the Russian Tradition of Mathematical Education Russian mathematics constitutes one of the most vital and brilliant mathe matical traditions of the 20th century. Mathematicians trained in Russia are very well represented in the top eche lon of the world mathematical com munity. Behind this flourishing stands a powerful tradition of spotting and training mathematical talent, which is
not without its downside. The subject is certainly too complex for a detailed discussion, but we will try to present a brief outline. A typical path of a mathematically talented student would start rather early. It would include participation in mathematical olympiads of various levels, from school district to the all Union one (the first Mathematical Olympiad in the Soviet Union was held in Leningrad in 1934, and Moscow fol lowed suit the next year; the first all Union Olympiad took place in 1961). Another activity for an interested school student was a kruzhok (literally, "circle"; a closer English equivalent is probably "workshop"); kruzhki also appeared in the mid-1930s. They usu ally met at the university once a week in the evening and were run by dedi eatenundergraduate or graduate stu dents with a tremendous enthusiasm for mathematics, very ·often them selves alumni of a kruzhok-a good example of "vertical integration"! The material discussed usually went well beyond the secondary school curricu lum and included challenging prob lems and nonstandard topics from ele mentary to higher mathematics. Beginning in the early 1960s, special high schools for mathematics and physics were organized in major cities. Many benefited from the help of the lo cal university faculty; for example, E. B. Dynkin and I. M. Gelfand played a prominent role in running the legendary Moscow School No. 2, whose many alumni are now professors of mathe matics in universities across the globe. Another well-known high school, the Boarding School for Mathematics No. 18 at Moscow State University, was estab lished by A N. Kolmogorov. Unlike other mathematical schools in Moscow which essentially sprang from private initiative and had no special funding, this school was a special institution af filiated with the university and specially funded by the state. Still other cele brated Moscow schools for mathemat ics were No. 7, No. 57 and No. 444 (the second and third authors are alumni of these schools, No. 7 and 2, respectively, and the first and the third authors taught in School No. 2). The mathematics cur riculum of a special school was more in-
VOLUME 24, NUMBER 4, 2002
55
Kvant, there was a
tensive and systematic than that of the
fession. Along with
kruzhki,
and this influenced our think
rich popular literature; nwnerous col
mix of undergraduates, graduates, and
ing about the structure of the MASS pro
lections of problems for all ages, and
established
books on various topics in "serious"
from the third year of the university,
mathematics. We would like to mention
every student had an advisor and was
gram. An essential part of the tradition was
seminars were usually attended by a mathematicians.
Starting
the participation of prominent mathe
some people who made a very substan
considered a member of a research
maticians of various ages in teaching
tial contribution to popularization of
community in his or her field. It was not
and popularizing mathematics. A typical
mathematics: N.
B. Vasiliev, N. Ya
unusual for the best undergraduate stu
Kvant (mean
Vilenkin, I. M. Yaglom. The third author
dents at major universities to have pa
ing "Quantum") on physics and mathe
of this article was for a nwnber of years
pers published in first-rate research
example is the magazine
matics for school students published
Kvant had 12
since 1970.
issues a year
the Head of
Kvant's
Mathematics De
partment.
journals by the end of their
5 years of
undergraduate studies. This system had multiple effects. On
and, at the peak of its popularity in the
At the university level, the emphasis
mid-1970s, boasted more than 300,000
on creative thinking continued, some
the one hand, it stimulated early de
subscribers. Among the authors were
times to the detriment of systematic
velopment of research interests and
well-known
D.
leaining. For example, the standard
mathematical precocity. On the other
B. Fuchs, I.
mandatory courses often did not fully
hand, it often led to inflated standards
M. Gelfand, S. G. Gindikin, A. A. Kirillov,
reflect the most current thinking in their
and expectations, and eventually to a
subjects, and were looked down on by
great waste of talent. A student with
mathematicians
Alexandrov, V. I. Arnold, D.
A.
A. N. Kolmogorov, M. G. Krem, Yu. V.
L. S.
the top students. A very important role
considerable talent but not very high
Pontryagin, among many others. For
was played by topics courses, offered in
self-esteem might be crushed by the system. Still, it succeeded spectacu
Matiyasevich, S. P. Novikov, and
Kvant
a wide variety of subjects and attended
opened new horizons and determined
by a mixture of undergraduate and grad
larly in producing creative and techni
their choice of mathematics as a pro-
uate
cally powerful mathematicians.
many generations of students,
students.
Similarly,
specialized
A UTH O R S
ANATOLE KATOK
Department
SVETLANA KATOK
of Mathematics
Department
Department
of Mathematics
Pennsylvania State University
Pennsylvania State University
Pennsylvania State University
University Park, PA 1 6802
University Park, PA 1 6802
University Park, PA 1 6802
USA
USA
USA
e·mail: [email protected]
e-mail: [email protected]
e-mail: [email protected]
Serge Tabachnikov wrote his thesis (1 987)
Anatole
in the
Svetlana Katok (daughter of B.A. Rosen
"Moscow mathematical school," as were
Katok was educated
feld, a "grand old man" of Moscow geom
at Moscow State University on differential
his co-authors; A.N. Kolmogorov was
etry) immigrated to the United States in
topology and homological algebra. Later his
ref
eree of his doctoral thesis. After immigrat
1 978
ing in
got her Ph.D. in
1 978
to the United States (the coun
with her husband 1 983
and
children, and
at the University of
interests shifted to symplectic geometry and Hamiltonian dynamics,
as
reflected in his
try of his birth), he taught at Maryland and
Maryland. Her research is on automorphic
book Billiards. Before coming to Penn State
Caltech before coming in
forms, dynamical systems, and hyperbolic
he taught at the University of Arkansas.
1 990
to Penn
State. Among his numerous publications
geometry. She is author of
are two books with his former student Boris
Groups
Hasselblatt:
ticle "Women in Soviet Mathematics,"
Introduction to the Modern
Theory of Dynamical Systems
forthcoming The
and the
Rrst Course in Dynamics
with a Panorama of Recent Developments.
56
SERGE TABACHNIKOV
of Mathematics
THE MATHEMATICAL INTELLIGENCER
Fuchsian
and Oointly with A. Katok) of the ar No
tices of the American Mathematical Soci ety 40 (1 993), 1 08-1 1 6.
l]¥1f9·i.(.j
David E . Rowe , Editor
Einstein's Gravitational Field Equations and the Bianchi Identities David E. Rowe
j
I
n his highly acclaimed biography of Einstein, Abraham Pais gave a fairly detailed analysis of the many difficulties his hero had to overcome in November 1915 before he finally arrived at gener ally covariant equations for gravitation ([Pais], pp. 250-261 ). This story includes the famous competition between Hilbert and Einstein, an episode that has re cently been revisited by several histori ans in the wake of newly discovered documentary evidence, first presented in [Corry, Renn, Stachel 1997]. In his earlier account, Pais empha sized that "Einstein did not know the [contracted] Bianchi identities
(
)
RJJ-V - .!gJJ-V R 0 (1) 2 ;v when he wrote his work with Gross mann." (The symbol ';' denotes covari ant differentiation, which here is used as the generalized divergence operator.) In 1913 Einstein and the mathe matician Marcel Grossmann presented their Entwurjfor a new general theory of relativity. Guided by hopes for a generally covariant theory, they never theless resolved to use a set of differ ential equations for the gravitational field that were covariant only with re spect to a more restricted group of transformations. However, when Ein stein abandoned this Entwurf theory in the fall of 1915, he once again took up the quest for generally covariant field equations. By late November he found, though in slightly different form, the famous equations: GJJ-V
=
=
- KTJJ-V, p,, IJ = 1, . . . ' 4
(2)
1 RJJ-V - - gJJ-V R 2
(3)
where GJJ-V
Send submissions to David E. Rowe, Fachbereich 1 7 - Mathematik, Johannes Gutenberg University, 055099 Mainz, Germany.
==
is the Einstein tensor. (Here TJJ-v is the energy-momentum tensor and gJJ-v the metric tensor that determines the prop erties of the space-time geometry. The contravariant Ricci tensor RJJ-v is ob tained by contracting the Riemann Christoffel tensor; contracting again R. yields the curvature scalar RJJ-vg JJ-V =
The symmetry of gJJ-v, RJJ-v, and TJJ-v means that (2) yields only 10 equations rather than 16.) Applying the covariant divergence operator to both sides of the Einstein equations (2) yields, according to (1), Gtvv
=
Ttvv
=
0.
(4)
This tells us that actually only 10 - 4 6 of the field equations (2) are indepen dent, as should be the case for generally covariant equations. Ten equations for the 10 components of the metric tensor gJJ-vwould clearly over-determine the lat ter, since general covariance requires that a.Ry smgle solution gJJ-"(xi) of (2) corresponds to a 4-parameter family of solutions obtained simply as the gJJ-v(xi) induced by arbitrary coordinate trans formations. Choosing a specific coordi nate system thus singles out a unique so lution among this family. Einstein for a long time resisted drawing this seemingly obvious conclu sion. Instead he concocted a thought experiment-his infamous hole argu ment-that purported to show how gen erally covariant field equations will lead to multiple solutions within one and the same coordinate system (see [NorJ989] and [Sta 1989]). His initial efforts there fore aimed to circumvent this paradox of his own making, for, on the one hand, physics demanded that generally co variant gravitational equations must ex ist, whereas logic (mixed with a little physics) told him that no such equations can be found (see his remarks in [Ein stein 1914], p. 574). Luckily, Einstein had the ability to suppress unpleasant con ceptual problems with relative ease. And so in November 1915 he plunged ahead in search of generally covariant equa tions, unfazed by his own arguments against their existence! Once he had them, he quickly found a way to climb out of the hole he had created (as ex plained in [Nor 1989] and [Sta 1989]). By 1916 Einstein was also quite aware that his field equations led di rectly to the conservation laws for mat 0. Nevertheless, he was ter T:Vv rather vague about the nature of this =
=
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 4, 2002
57
connection in [Einstein 1916a] , his first summary account of the new theory. There he wrote, "the field equations of gravitation contain four conditions which govern the course of material phenomena. They give the equations of material phenomena completely, if the latter are capable of being characterized by four differential equations indepen dent of one another" (ibid., p. 325). He then cited Hilbert's note [Hilbert 1915] for further details, suggesting that he
was not yet ready to make a fmal pro nouncement on these issues. Still, by early 1916 Einstein had come to realize that energy conservation can be deduced from the field equations and not the other way around. Pais remarks about this in connection with the tu multuous events of November 1915:
alize that the energy-momentum conservation laws
Einstein stiU did not know [the con tracted Bianchi identities] on No vember 25 and therefore did not re-
Pais's 20-20 hindsight no doubt identifies this particular source of Einstein's diffi culties, but it hardly helps to explain
Tf;," = 0
follow automatically from (1) and (2). Instead, he used these conser vation laws [ (5)] as a constraint on the theory! ([Pais 1982], p. 256).
Mathematische Gesel lschaft
'ula.
Hansen.
BIWDeuthal.
c. Miller.
SdUilinr.
JWIIert.
Dane7.
H1111el.
&bMidt.
1 90 2 .
H. lflllln. St:hwantelalhl.
(5)
r�hiYf.
R�t�n.
1-,l'i:�<•l:t•r.
Dlrl
llml"'tl'i•
1"'"�11>.
Felix Klein presiding over the Gottingen Mathematical Society in 1 902. The seating arrangement reflects more than just the need to have the tall men stand in back. Here, David Hilbert could affirm his undisputed position as Klein's "right-hand man," counterbalanced on the left by Klein's star applied mathematician, Karl Schwarzschild. Taking up the wings in the front row were two of Gottingen's most ambitious younger men, Max Abraham and Ernst Zermelo. Klein's attention seems to have riveted on Grace Chisholm Young, the charming Englishwoman who took her doctorate under him in 1 895. She and her husband, the mathematician W. H. Young, in fact resided in Gottingen for several years. Schwarzschild Nach/ass; courtesy of the Niedersiichsische Staats- und Universitiitsbibliothek Gottingen.
58
THE MATHEMATICAL INTELLIGENCER
what happened in November 1915. Nor does it shed much light on subsequent developments. Paging ahead, we see that Pais returns to the Bianchi identities in his discussion of energy-momentum conservation, which in 1918 was one of the most hotly debated topics in general relativity theory (GRT) (see [Cattani, De Maria 1993]). There he points out that, "from a modem point of view, the iden tities (1) and (5) are special cases of a celebrated theorem of Emmy Noether, who herself participated in the Gottin gen debates on the energy-momentum law" ([Pais 1982], p. 276). But back in No vember 1915 "neither Hilbert nor Ein stein was aware of this royal road to the conservation laws" ([Pais 1982], p. 274). Pais might have added that even in 1918 no one in Gottingen seems to have real ized the connection between Noether's results on identities derived from varia tional principles and the classical Bianchi identities. A little bit of contextualization can go a long way here. During the period 1916-1918 only a few individuals were in a position to see the connection be tween Einstein's equations and the Bianchi identities, even though the lat t�r were quite familiar to those im mersed in Italian differential geometry. Among these experts, only Levi-Civita seems to have seen the relevance of the Bianchi identities immediately. But, as we shall see below, by 1918 a handful of others began to rediscover what the Italians had already largely forgotten. Emmy Noether, however, was not among them. Her work on GRT was mainly rooted in Sophus Lie's theory of differential equations, as applied to variational problems, an area Lie left untouched. Most importantly, Noe ther's efforts came as a response to a set of problems first raised by Hilbert, who tried to synthesize Einstein's the ory of gravitation with Gustav Mie's theory of matter (see [Rowe 1999]). Hilbert's approach to energy conser vation in 1915--16 used a gener�y in variant variational principle, which (he claimed without proof) led to four dif ferential identities linking the La grangian derivatives [Hilbert 1915]. No one understood this argument at the time, and only recently have historians managed to disentangle its many threads
(see [Sauer 1999] and [Renn, Stachel 1999]). Several other related issues re mained murky, as well. The relationship between Einstein's theory and Hilbert's adaptation of Mie's matter theory, for example, was by no means clear. Nor was it easy to discern whether one could formulate conservation laws in general relativity that were fully analogous to those of classical physics. Aided by Emmy Noether, in 1918 Fe lix Klein eventually managed to fmd a simpler way to construct Hilbert's in variant energy vector in [Klein 1918a] and [Klein 1918b]. He also urged Noe ther to explore Hilbert's assertion re garding the four identities that he saw as the key to energy conservation in GRT. In July 1918 she generalized and proved this result as one of two funda mental theorems on invariant variational problems [Noether 1918]. Although fa mous today, Noether's theorems evoked very little interest at the time they were published. Moreover, unlike Pais, no contemporary writer linked Noether's results with the Bianchi identities so far as I have been able to find. IfRJrvK is the Riemann-Christoffel ten sor, then lowering the index u yields the purely covariant curvature tensor
The latter satisfies the following three properties:
RAJ.LVK = RvKAJ.L
(6)
RAJ.LVK = -RJ.LAVK = -RAJ.LKV
+RJ.LAKv (7) RAJ.LVK + RAKJ.LV + RAVKJ.L = 0 (8) =
These algebraic conditions imply that RAJ.LvK has only Cn = n2(n2 - 1) independent components. For the space-time formalism of GRT, where n = 4, the Riemann-Christoffel tensor thus depends on c4 = 20 parameters. Using its covariant form, the classical Bianchi identities read:
1�
RAJ.LVK; TJ + RAJ.LTJV,K + RAJ.LKTJ;V = 0,
(9)
where the last three indices v, K, TJ are permuted cyclically. The connection between these identities and Einstein's equations follows immediately from two basic results of the tensor calcu lus: (1) raising and lowering indices commutes with covariant differentia tion, and (2) Ricci's lemma, which as-
serts that the covariant derivative of the fundamental tensor gJ.Lv vanishes, g�v = 0. Thus, by multiplying (9) by gAv and contracting, we obtain in view of (6) and (7) :
RJ.LK;TJ - RJ.LTJ;K + Rj;_KTJ;V = 0. (10) Multiplying by gJ.LK and contracting again yields
R; TJ - R�;J.L - R �;v = 0 , (R� -
� B� R);v = 0,
which are the contracted Bianchi iden tities (1):
(R J.LV - lgJ.LV R),. v = G�';,V , = 0. 2
These conditions simply assert that the divergence of the Einstein tensor vanishes, a relation already derived by Weyl in 1917 using variational meth ods. �et neither he nor hi:; Gottingen mentors, Hilbert and Klein, recognized that these identities could be obtained directly as above using elementary ten sor calculus. As I will indicate below, disentangling these differential-theo retic threads from variational princi ples took considerable time and effort. Even as late as 1922 the Bianchi iden tities and their significance for GRT were still being "rediscovered" anew. Clearly, Pais's retrospective account skirts all the real historical difficulties, telling us more about what didn't hap pen between November 1915 and July 1918 than about what actually did. In the meantime, however, a number of new studies have cast fresh light on the early history of GRT (see especially the articles in Einstein Studies, vols. 1, 3, 5, 7). Moreover, the crucial period 1914-1918 has become more readily accessible through the publication of volumes 6 and 8 of the CoUected Pa pers of Albert Einstein ([Einstein 1996] and [Einstein 1998]; volume 7, covering his work from 1918 to 1921, is now in press). Michel Janssen's commentaries and annotations in [Ein stein 1998] are particularly helpful when it comes to contextualizing the topics I address below. This new source material has helped sustain a flurry of recent research on the early history of general relativity, some of which has filled important gaps left open by ear lier researchers. Still, no one since Abraham Pais has addressed the issues
VOLUME 24, NUMBER 4, 2002
59
be obtained from those of his own the
surrounding the interplay between the
ric tensor g�-t v. Only later, some time af
Bianchi identities and GRT, especially
ter 6 December, did he add the key pas
ory, citing a bit of folklore about second
Einstein's equations,
energy-momen
sage containing a form of (2) into the
rank tensors that he probably got from
tum conservation, and Noether's theo
page proofs. Presumably he did so with
Einstein. One finds scattered hints in
rems. So without any pretense of do
out any wish to stake a priority claim,
Einstein's published and unpublished
ingjustice to this rich topic, let me take
for he cited Einstein's paper of 25 No
papers indicating that the only possible
it up once again here.
vember. Moreover, the explicit field
second-rank tensors obtainable from the
Einstein's Field Equations
equations play no role whatsoever in the
metric tensor and its first and second de
rest of Hilbert's paper. It therefore
rivatives and linear in the latter must be
Roughly a half year after Einstein deliv
seems likely that he added this para
of the form:
ered six lectures on general relativity in
graph merely in order to make a con
Gottingen, Hilbert entered the field with
nection with Einstein's results, which
a R�-t v + b g�-tv R + c g�-t v =
0.
(11)
his famous note [Hilbert 1915] on the
were by no means clear or easily acces
Einstein knew very well that this math
foundations of physics, dated 20 No
sible at that time (only in his fourth and
ematical result was crucial when it came
vember 1915. Until quite recently, histo
final November note did Einstein pre
to narrowing down the candidates for
rians had paid little attention to the sub
sent generally covariant field equations
generally covariant field equations. And
stance
with the trace term). So the equations
without these mathematical underpin
of this paper,
which makes
horribly difficult reading. Its fame stems
(2) are rightly called "Einstein's equa
nings, his claim ([Einstein 1916a], pp.
from one brief passage in which Hilbert
tions"
318-319) that the equations (2) repre
and not
the
"Einstein-Hilbert
asserted that his gravitational equations,
sent the most natural generalization of
derived from an invariant Lagrangian,
When and why
Newton's theory would have been se
d i d physicists and
this for granted, probably because he
mathematicians
ited) expertise in the theory of differ
were identical to Einstein's equations (2), submitted to the Berlin Academy 5 days later. It was long believed that
Hilbert's "derivation" was more elegant,
and to many it appeared that he and Ein stein had found the same equations vir
tant Hermann Vermeil to give a direct proof of this fundamental result to
issues l i ke
pealed. By employing so-called normal
provi ng Einstei n ' s
mann, Vermeil was able to prove that
during November 1915, it was natural to speculate about who might have influ enced whom. Curiously, this interest dication of priority claims, as historians
ential invariants.
i nterested i n
also corresponded with one another
hoc adju
relied heavily on Grossmann's (lim
By 1917 Felix Klein asked his assis
become
tually simultaneously. Since they had
centered exclusively on a post
riously weakened. Still, he simply took
pondered who should get credit for find
which both Einstein and Hilbert had ap coordinates, as first introduced by Rie
the only absolute invariant that satisfied
eq uations?
ing and/or deriving the Einstein equa
the Riemannian curvature scalar R was the above
conditions (see
[Vermeil
1917]). In 1921 Max von Laue completed
tions: Einstein, Hilbert, or both? Proba equations." This "belated priority issue"
Vermeil's argument in [Laue 192 1 ] , pp.
was definitely put to rest in [Corry, Renn,
99-104, and Hermann Weyl gave an
1973], but until recently the balance of
Stachel 1997]. Unfortunately, some of
even more direct proof of Vermeil's re
opinion was aptly sunrmarized by Pais,
the authors' other more speculative
who wrote, "Einstein was the sole cre
claims have now been spun into a highly
sult in [Weyl 1922], Appendix II, pp.
bly Jagdish Mehra went furthest in pushing the case for Hilbert in [Mehra
ator of the physical theory of general rel
romanticized account of these events in
ativity and . . . both he and Hilbert should
God's Equation
315--317. Wolfgang Pauli also referred to
Vermeil's work in his definitive report [Pauli 192 1 ] ; but one otherwise fmds
[Acz 1999].
be credited for the discovery of the fun
As to who first derived the Einstein
damental equation" ([Pais 1982], p. 260).
equations, the answer is less clear. If
Today we know better: when Hilbert
very few references to such formal is sues in the vast literature on GRT. 1
by a derivation we mean an argument
So who first proved Einstein's equa
submitted his text on 20 November 1915,
showing that the equations (2) uniquely
tions? If this were a game show ques
it did not contain the equations (2) (see
satisfy a certain number of natural prop
tion, one might be tempted to answer:
In fact,
erties, then for Einstein and Hilbert one
Hermann Vermeil. But a more serious
text contained no explicit form for his
can only reach the conclusion "none of
response would begin by reformulating
10 gravitational field equations, which
the
the question: when and why did physi
he derived from a variational principle
416-418). Hilbert, in particular, failed to
cists and mathematicians become in
show how the Einstein equations could
terested in foundational issues like
[Rowe 2001]).
Hilbert's original
by varying the components of the met-
above"
(see
[Rowe
2001],
pp.
1An exception is the work of David Lovelock, who proved that the only divergence-free, contravariant second-rank tensor densities in dimension four are of the form . [Lovelock 1 972].
aVgG�<" + bVgg"" in
60
THE MATHEMATICAL INTELUGENCER
Making Music in Zurich. During his early struggles with general relativity, Einstein often liked to relax in the home of his colleague Adolf Hurwitz, shown here pretending to conduct his daughter Lisi and their physicist friend as they play a violin duet. Hurwitz was a pure math ematician of nearly universal breadth. Though only four years Einstein's senior, he had served as the principal mentor to both David Hilbert and Herman Minkowski during their formative years in Konigsberg. Source: George Polya, The Polya Picture Album: Encounters of a Math
ematician, ed. G. L. Alexanderson (Boston: Birkhauser, 1987), p. 24. Reprinted with permis sion of Birkhauser Publishers.
proving Einstein's equations? To an swer this, it is again helpful to look carefully at local contexts. Among the more important centers for research on GRT were Leiden, Rome, Cambridge, and Vienna. In the case of Einstein's equations, this was largely a mopping up operation, part of a communal ef fort orchestrated by Felix Klein in Got tingen. Klein's initial interest in general relativity focused on the geometrical underpinnings of the theory, including the various "degrees of curvature" in space-times (Eddington's terminology in [Eddington 1920], p. 91). By early 1918, however, Klein became even more puzzled by the various results on energy conservation in GRT that had been obtained by Einstein, Hilbert, Lorentz, Weyl, and Emmy Noether. He was not alone in this regard.
General Relativity in Gottingen As
Einstein himself conceded, energy momentum conservation was the one facet of his theory that caused virtually all the experts to shake their heads. Back in May 1916, he had struggled to
understand Hilbert's approach to this problem, the topic of a lecture he was preparing for the Berlin physics collo quium. Twice he wrote Hilbert asking him to explain various steps in his com plicated chain of reasoning (24 and 30 May, 1916, [Einstein 1998], pp. 289-290, pp. 293-294). Einstein expressed grati tude for Hilbert's illuminating replies, but to his friend Paul Ehrenfest he re marked: "Hilbert's description doesn't appeal to me. It is unnecessarily spe cialized regarding 'matter,' is unneces sarily complicated, and not straightfor ward (=Gauss-like) in set-up (feigning the super-human through concealment of the methods)" (24 May, 1916 [Ein stein 1998], p. 288). But Hilbert couldn't feign that he understood the connection between his approach to energy con servation and Einstein's. About this, he intimated to Einstein that "[m]y energy law is probably related to yours; I have already given this question to Frl. Noe ther" (27 May 1916, [Einstein 1998], p. 291). She apparently made some progress on this problem at the time, as Hilbert later acknowledged: "Emmy
Noether, whose help I called upon more than a year ago to clarify these types of analytical questions pertaining to my energy theorem, found at that time that the energy components I had set forth as well as those of Einstein-could be formally transposed by means ofthe La grangian differential equations . . . into expressions whose divergence van ished identicaUy. . . . " ([Klein 1918a], pp. 560-561). By late 1917 Klein reengaged Noe ther in a new round of efforts to crack the problem of energy conservation (see [Rowe 1999], pp. 213-228). Klein's discomfort with energy conservation in GRT had to do with his knowledge of classical mechanics in the tradition of Jacobi and Hamilton. There, conserva tion laws help to describe the equations of motion. of physical systems which would otherwise be too hopelessly com plicated to handle as an n-body prob lem. In GRT, by contrast, the conserva tion laws for matter (5) could be derived directly from the field equations (2) without any recourse to other physical principles. Hilbert's work pointed in this direction, but his ''purely axiomatic" presentation only obscured what was already a difficult problem. Klein later described [Hilbert 1915] as "completely disordered (evidently a product of great exertion and excitement)" (Lecture notes, 10 December 1920, Klein Nach lass XXII C, p. 18). In early 1918 Klein succeeded in giv ing a simplified derivation of Hilbert's invariant energy equation, which in volves. a very complicated entity ev known as Hilbert's energy vector sat isfying Div (e ") = 0. Klein emphasized that this relation should be understood as an identity rather than as an ana logue to energy conservation in classi cal mechanics. He noted that in me chanics the differential equation
d(T + U) =0 dt
(12)
cannot be derived without invoking specific physical properties, whereas in GRT the equation Div (ev) = 0 fol lows from variational methods, the principle of general covariance, and Hilbert's 14 field equations for gravity and matter.
VOLUME 24, NUMBER 4, 2002
61
Klein's article was written in the form of a letter to Hilbert. After dis cussing the main mathematical points, Klein remarked that "Fri. Noether con tinually advises me in my work and that actually it is only through her that I have delved into these matters" [Klein 1918a], p. 559. Hilbert expressed total agreement not only with Klein's de rivation but with his interpretation of it as well. He even claimed one could prove a theorem that ruled out con servation laws in GRT analogous to those that hold for physical theories based on an orthogonal group of coor dinate transformations. Klein replied that he would be very interested "to see the mathematical proof carried out that you alluded to in your answer." He then turned to Emmy Noether, who re solved the issue six months later in her fundamental paper [Noether 1918]. In the meantime, Einstein had taken notice of this little published exchange, and in March 1918 he wrote Klein, "With great pleasure I read your extra ordinarily penetrating and elegant dis cussion on Hilbert's first note. Never theless, I regard what you remark about my formulation of the conserva tion laws as incorrect" (13 March, 1918, [Einstein 1998], p. 673). Einstein ob jected to Klein's claim that his ap proach to energy-momentum conser vation could be derived from the same formal relationships that Klein had ap plied to Hilbert's theory. Instead, Ein stein insisted that "exactly analogous relationships hold [in GRT] as in the non-relativistic theories." After ex plaining the physical import of his own formalisms, he added, "I hope that this anything but complete explanation en ables you to grasp what I mean. Most of all, I hope you will alter your opin ion that I had obtained for the energy theorem an identity, that is an equation that places no conditions on the quan tities that appear in it" (ibid. , p. 674). Eight days later, Klein replied with a ten-step argument aimed at demol ishing Einstein's objections. His main point was that Einstein's approach to energy-momentum conservation ex pressed nothing beyond the informa tion deducible from the variational ap paratus and the field equations that can
62
THE MATHEMATICAL INTELLIGENCER
:!x4 {J
(T! +
}
t!)dV for
=
u =
0, 1, 2, 3, 4.
(14)
Einstein stressed to Klein that the con stancy of these four integrals with re spect to time could be regarded as anal ogous to the conservation of energy and momentum in classical mechanics.
Tullio Levi-Civita was the leading expert on the absolute differential calculus in Italy. To gether with his teacher Gregorio Ricci, he co authored an oft-cited paper on the Ricci cal culus published in Mathematische Annalen in
1 901 . In 1 91 5, Einstein confided to a friend
that Levi-Civita was probably the only one who grasped his gravitational theory com pletely: "because he is familiar with the math ematics used. But he is seeking to tamper with one of the most important proofs in an incessant exchange of correspondence. Cor responding with him is unusually interesting; it is currently my favorite pastime" (Einstein to H. Zangger, 10 April, 1915, Collected Pa pers of Albert Einstein, vol. 8, pp. 1 1 7-1 1 8). Their correspondence broke off, however,
about one month later when Italy entered the war against the Axis powers.
be derived from it. Einstein countered by asserting that his version of energy momentum conservation was not a trivial consequence of the field equa tions. Furthermore, if one has a physi cal system where the energy tensors for matter and the gravitational field, T:;and t:;, vanish on the boundary, then from the differential form of Einstein's conservations laws
I.,
a cr:; + t:;) ax.,
=
o
(13)
one could derive an integral form that was physically meaningful:
Klein eventually came to appreciate Einstein's views, though only after giv ing up on an alternative approach sug gested by his colleague Carl Runge. Sev eral experts, including Lorentz and Levi-Civita, objected to Einstein's use of the pseudo-tensor t:; to represent gravi tational energy. Klein and Runge briefly explored the possibility of dispensing with this t:;, but Noether threw cold wa ter on Runge's proposal for doing so ([Rowe 1999], pp. 217-218). By July 1918, Klein wrote Einstein that he and Runge had withdrawn their publication plans, and that he was now investigat ing Einstein's formulation of energy con servation based on T:; t:;. To this, Ein stein replied: "It is very good that you want to clarify the formal significance of the t:;.. For I must admit that the deriva tion of the energy theorem for field and matter together appears unsatisfying from the mathematical standpoint, so that one cannot characterize the t:; for mally" ([Einstein 1998], p. 834). Einstein and Klein quickly got over their initial differences regarding the status of Einstein's (13), and afterward Klein dealt with this topic and the var ious approaches to energy conserva tion adopted by Einstein, Hilbert, and Lorentz in [Klein 1918b]. Einstein re sponded with enthusiasm: "I have al ready studied your paper most thor oughly and with true amazement. You have clarified this difficult matter fully. Everything is wonderfully transparent" (A. Einstein to F. Klein, 22 October, 1918 ([Einstein 1998], p. 917). The con trast between this response and Ein stein's reaction to Hilbert's work on GRT (noted above) could hardly have been starker. Perhaps the supreme irony in this whole story lies here. For [Klein 1918b] is nothing less than a carefully crafted axiomatic argument, set forth by a strong critic of modem axiomatics largely in order to rectify
+
the flaws in Hilbert's attempt to wed GRT to Mie's theory of matter via the axiomatic method. In this paper Klein developed ideas that were closely linked with [Noether 1918] , though he mentions this.parallel work only in the concluding paragraph. Emmy Noether, on the other hand, gave several explicit references to [Klein 1918b] that make these interconnec tions very clear. In his private lecture notes, Klein later wrote that it was only "through the collaboration of Fri. Noe ther and me" that [Hilbert 1915] "was completely decoded" (Lecture notes, 10 December 1920, Klein Nachlass XXII C, p. 18). Today, this jointly undertaken work would normally appear under the names of both authors, but back in 1918 Emmy Noether wasn't even allowed to habilitate in Gottingen, despite the back ing of both Hilbert and Klein.
cation between leading protagonists in Italy and Germany proved next to im possible, as the lapse in Einstein's cor respondence with Tullio Levi-Civita demonstrated. Through his friend Adolf Hurwitz, whom he visited in Zurich in August 1917, Einstein managed to get his hands on Levi-Civita's paper [Levi Civita 1917b], which briefly reignited their earlier correspondence. Like many others, Levi-Civita found Einstein's for-
mulation of energy conservation unac ceptable due to his use of the pseudoten sor t� for gravitational energy [Cattani, De Maria 1993]. What Einstein (and pre sumably everyone else in Germany) overlooked was that in this paper Levi Civita employed the classical Bianchi identities. In [Levi-Civita 1917a] he in troduced an even more fundamental concept: parallel displacement of vec tors in Riemannian spaces, a notion
On Rediscovering the Bianchi Identities of these events, it must not be for gotten, took place against the back drop of the Great War that nearly brought European civilization to its knees. Einstein's revolutionary theory of gravitation interested almost no one prior to 1916, and before Novem ber 1919 only a handful of experts had written about it. But afterward, Ein stein and relativity emerged as two watchwords for modernity. On 6 No vember, just before the Versailles Treaty was to take effect, the British scientific world announced that Ein stein's prediction regarding the bend ing of light in the sun's gravitational field had been confirmed. Thereafter, the creator of general relativity was no longer merely a famous physicist: he emerged as one of the era's leading cultural icons. But let's now wind back the reel and look again at GRT during the Great War. Once Einstein's mature theory came out in 1916-alongside Hilbert's pa per and the pioneering work of Karl Schwarzschild containing the firSt ex act solutions of the Einstein equations many mathematicians and physicists began to take up GRT and the Ricci cal culus. Doing so in wartime, however, presented real difficulties. CommuniAll
Dirk Jan Struik, ca. 1 920, when he was working as an assistant to Jan Amoldus Schouten in Delft. Earlier Struik had studied in Leiden with Paul Ehrenfest, a close personal friend of Ein stein's who therefore realized the importance of Ricci's calculus for general relativity at an early stage. At a crucial stage, Ehrenfest arranged a meeting for Struik with Schouten, Hol land's leading differential geometer. Thus began a collaboration that led to several books and articles during the 1 920s and
30s.
VOLUME 24, NUMBER 4, 2002
63
quickly taken up by Hermann Weyl and Gerhard Hessenberg. These fast-breaking mathematical developments raised staggering diffi culties, and not just for physicists like Einstein. None of the mathematicians in Gottingen was a bona fide expert in
differential geometry, which helps ex plain why no one in the Gottingen crowd recognized the central impor tance of the Bianchi identities. Had Klein suspected that the Einstein ten sor satisfied four simple differential identities (corresponding to the four
:From the PHII.OSOPHICAT. MAGAZrNF., vol. ::dvii. 1lllarcl� 1 924.
Note on J.l[r. Harwa1·d's Paper on tlte Identical Rel(�tions in Einstein's Theory.
Tv the Edito·rs of tfte Philosophical Maga::ine .
G E�TLE¥E�,-
011
Relations in Einstein's a e '' iu the August 1922 number aS0-382) o f Ih N Theory M er pap r
" The Identical
t e Philosophical Magazine,
(pp. r . Harward proves a ge n
al
theorem which he discovered for himself, although be did riot believe it to be undiscovered before. Tht: theorem is :
(B�ve1)r + (B�ar)v + (B�,v)e1= 0.
It m ay be of i nte rest to mention that this theorem is kno wn, espe cia l ly in Germany and Italy, as " Bianchi's Iden tity/' having b ee n p u bl i s he d by L. Bianchi in the Rendic:ont-i Ace. Lincei, xi. (5) pp. 3-17 (1902) . In this paper, Bianchi already deduced from this theorem the identity Mr. Harward refers to :
G Gv - t o }J.V - o·�>l-' ·
The identity of Bianchi seems, ho·wever, to be :published for the first time by E. Padova (Rendiconti .A ce. Lincei, v. (4) pp. 174-178, 1889) , who obtained it from G. Ricci. Com pare for this the book of Struik ( Grundzuge der mehrdimen sionalen Dijfe1·entialg eometrie : B erlin, ,J. Springer, 1!j22, p. 141:l) , where the theorem is proved in a similar way as Mr. Harward do es. A similar p roof was already given by Schouten (Mathern. Zeitschr. xi. PP: 58-88, 1921). The theorem can be generalized by taking geometries with a more general parallel displacement than in ordinary Riemann geometry. We then obtain the gen eraliz ed geometries of Schouten (.1.1-latltem. Zeitschr. xiii. pp. 56-8 1 , 1922), o f which the geometries of Weyl aud Eddington (if. Eddington's ' Mathematic�\! Theory of Relativity,' Oh. vii.) are special cases. Schouten, in a recent pape-r (Mathem. Zeitscltr. xvii. pp. 111-115, 1923), proved the generalization of Bianchi's Identity for a geometry of which a geometry with a symmetrical displacement (that is, a geometry in which r�.. = r�., cf. Eddington, Zoe. cit. p . 214) is a special case. This �pecial case is treated by A. Veblen (Proc. Nat. Ac. of Sciences, July 1 922). R. .Bach already gave the generalization for the geometry o£ vVeyl (:iYiathem. Zeitsch1·. ix. pp. 110-135, 1921 ). A simple proof of Bianchi's Identity, which holds for a. Struik's copy of the open letter he and Schouten wrote to the Philosophical Magazine on 28
April, 1 923. Their account clarified several historical issues involving the Bianchi identities. It also contained a simple proof of these identities for spaces with a symmetrical connection suggested by the Prague mathematician, Ludwig Berwald. Struik presumably knew Berwald through his wife, Ruth, who studied mathematics in Prague during happier days. In 1941, Berwald and his wife were transported to the ghetto in Lodz, where they died from mal nourishment.
64
THE MATHEMATICAL INTELLIGENCER
parameters in a generally covariant system of equations), he might have turned to his old friend Aurel Voss for advice. Had he asked him, Voss likely would have remembered that he had published a version of the contracted Bianchi identities back in [Voss 1880] ! Thus, the Gottingen mathematicians clearly could have found references to the Bianchi identities, in either their general or contracted form, in the mathematical literature. They just didn't know where to look. As Pais pointed out, the name Bianchi does not appear in any of the five editions of Weyl's Raum, Zeit, Materie, nor did Wolfgang Pauli refer to it in his Ency clopii.die article [Pauli 1921]. With regard to the Bianchi identities in their full form (9), we have it on the authority of Levi-Civita that these were known to his teacher, Gregorio Ricci ([Levi-Civita 1926], p. 182). Ricci passed this information on to Emesto Padova, who published the identities without proof in [Padova 1889]. They were thereafter forgotten, even by Ricci, and then rediscovered by Luigi Bianchi, who published them in [Bianchi 1902]. Both Ricci and Bianchi had earlier studied under Felix Klein, who so licited the · now-famous paper [Ricci, Levi-Civita 1901] for Mathematische Annalen; but this classic apparently made little immediate impact. Indeed, before the work of Einstein and Gross mann, Ricci's absolute differential cal culus was barely known outside Italy [Reich 1992]. By 1918 a number of investigators outside Italy had begun to stumble upon various forms of the full or contracted Bianchi identities. Two of them, Rudolf Forster and Friedrich Kottler, even passed their findings on to Einstein in letters (see [Einstein 1998], pp. 646, 704). Forster, who published under the pseu donym Rudolf Bach, took his doctorate under Hilbert in Gottingen in 1908 and later worked as a technical assistant for the Krupp works in Essen. In explaining to Einstein that the identities (1) follow directly from (9), he noted that the lat ter "relations appear to be still com pletely unknown" (ibid.). Forster con templated publishing these results, but did so only in [Bach 1921], which dealt with Weyl's generalization of Riemann-
ian geometry. Even at this late date he presented these identities as "new" (ibid., p. 114). During the war years, the Dutch as tronomer Willem De Sitter introduced the British scientific community to Ein stein's mature theory. This helped spark Arthur Stanley Eddington's in terest in GRT and the publication of [Eddington 1918], which contains the contracted Bianchi identities. Two years later, in [Eddington 1920], he ex pressed doubt that anyone had ever verified these identities by straightfor ward calculation, and so he went ahead and carried this out himself for the the oretical supplement in the French edi tion of [Eddington 1920]. In 1922 Eddington's calculations were simplified by G. B. Jeffery, and al most immediately afterward the Eng lish physicist A. E. Harward reproved Bianchi's identities (9) and used them to derive the conservation of energy momentum in Einstein's theory in [Harward 1922]. He also cof\iectured that he was probably not the first to have discovered (9). In [Schouten and Struik 1924], an open letter to the Philosophical Magazine, dated 28 April 1923, J. A. Schouten and Dirk siruik confirmed Harward's co{\jecture, noting that (9) "is known, especially in Germany and Italy, as 'Bianchi's Iden tity.' " More importantly, they empha sized that similar identities hold in affme spaces (those that do not ad mit a Riemannian line element ds 2 = gJJ- v dxJJ-dx v). Regarding these, they re ferred explicitly to [Bach 192 1 ] for Weyl's gauge spaces as well as a 1923 paper by Schouten for non-Riemannian spaces with a symmetric connection r�v r�w These results and many more appeared soon afterward in Schouten's 1924 textbook Der Ricci Kalkiil. By this time, of course, the dust had largely cleared, as a number of leading experts-including Schouten and Struik, Veblen, Weitzenbt:ick, and Berwald-had by now shown the im portance of Bianchi-like identities in non-Riemannian geometries. What should be made of all this groping in the dark? No doubt a certain degree of confusion arose due to the importance Hilbert and' others attached to variational principles in mathemati=
cal physics. Not surprisingly, within Gt:ittingen circles there was consider able expertise in the use of sophis ticated variational methods. Emmy Noether coupled these with invariant theory to obtain her impressive results. But she and her mentors had relatively little familiarity with Italian differential geometry. Ironically, this widespread lack of fundamental knowledge of ten sor analysis had at least one important payoff. It gave the aged Felix Klein an inducement to explore the mathemati cal foundations of general relativity theory. He did so by drawing on ideas familiar from his youth, most impor tantly Sophus Lie's work on the con-
them, the early history of the general theory of relativity would not have have looked quite the same.
Acknowledgments The author is grateful to Michel Janssen and Tilman Sauer for their perceptive re marks on an earlier version of this col umn. The editor deserves a note of thanks, too, for posing questions that helped clarify some obscure points. Re maining errors and misjudgments are, of course, my own, and may even reflect a failure to heed wise counsel. REFERENCES
[Acz 1 999] Amir D. Aczel, God's Equation. Ein stein, Relativity and the Expanding Universe,
Mathematicians often prefer to fig u re out some t h i ng on thei r own rather than read someone else ' s work.
New York: Delta, 1 999. [Bach 1 92 1 ] Rudolf Bach, "Zur Weylschen Rela tivitatsthearie und der Weylschen Erweiterung .
des l
[Bianchi 1 902] Luigi Bianchi, �sui simboli a quattro indici e sulla curvature di Riemann , " Rendiconti della Reale Accademia dei Lincei
M. VOL 1 1 (11), pp. 3-7.
[Cattani, De Maria 1 993] Carlo Cattani and Michelangelo De Maria, "Conservation Laws and Gravitational Waves in General Relativ ity (191 5-1 91 8)," in [Earman, Janssen, Nor ton 1 993], pp. 63-87. [Corry, Renn, Stachel 1 997] Leo Corry, Jurgen
nection between continuous groups and systems of differential equations. Moreover, his efforts helped clarify one of the most baffling and controversial aspects of Einstein's theory: energy momentum conservation. Even Hilbert's muddled derivation of an invariant en ergy vector found its proper place in the scheme set forth in [Klein 1918b]. Through Klein, Emmy Noether became deeply immersed in these complicated problems, and she succeeded in ex tracting from them two fundamental theorems in the calculus of variations that would later provide field physi cists with an important tool for the der ivation of conservation laws. Mathematicians often prefer to fig ure out something on their own rather than read someone else's work, so we need not be surprised that the classi cal Bianchi identities escaped the no tice of such eminent mathematicians as Hilbert, Klein, Weyl, Noether, and of course Einstein himself. Had they known
Renn, and John Stachel, "Belated decision in the Hilbert-Einstein Priority Dispute," Sci ence 278 (1 997), 1 270-1 273. [Earman, Janssen, Norton 1 993] John Earman, Michel Janssen, and John D. Norton, eds. The Attraction of Gravitation: New Studies in the History of General Relativity, Boston: Birkhauser, 1 993. [Eddington 1 9 1 8] A. S. Eddington, Report on the Relativity Theory of Gravitation, London: Fleetway Press, 1 9 1 8. [Eddington 1 920] A. S. Eddington, Space, Time and Gravitation, Cambridge: Cambridge Uni versity Press, 1 920. [Einstein 1 91 4] Albert Einstein, "Prinzipielles zur verallgemeinerten
Relativitatstheorie
und
Gravitationstheorie," Physikalische Zeitschrift 1 5 (1 91 4), 1 76-1 80; reprinted in [Ein 1 996],
572-578.
[Ein 1 996] The Collected Papers of Albert Ein stein. The Swiss Years: Writings, 19 121 9 1 4 , voL 5, Martin J . Klein, A. J. Kox, Jur gen Renn, and Robert Schulmann, eds. Princeton: Princeton University Press, 1 995. [Einstein 1 91 6a] Albert Einstein, "Grundlagen
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65
der allgemeinen Relativitatstheorie," Annalen
[Laue 1 921] Max von Laue, Die Relativitats
der Physik 49; reprinted in [Einstein 1 996],
theorie, zweiter Band: Die allgemeine Rela
pp. 283-339.
tivitatstheorie und Einsteins Lehre von der
[Einstein 1 91 6b] Albert Einstein, "Hamiltonsches Prinzip und allgemeine Relativitatstheorie,"
kul zur Relativitatstheorie (Science Networks, vol. 1 1 ), Basel: Birkhauser, 1 992. [Renn, Stachel 1 999] JOrgen Renn and John
Schwerkraft, Braunschweig: Vieweg, 1 921 .
Stachel, "Hilbert's Foundation of Physics:
[Levi-Civita 1 91 7a] Tullio Levi-Civita, "Nozione
From a Theory of Everything to a Constituent
Sitzungsberichte der K6niglichen PreuBischen
di parallelismo in una varieta qualunque e
of General Relativity," Max-Pianck-lnstitut fOr
Akademie der Wissenschaften, 1 1 1 1 -1 1 1 6;
conseguente specificazione geometrica della
Wissenschaftsgeschichte, Preprint 1 1 8, 1 999. [Ricci, Levi-Civita 1 901 ] Gregorio Ricci and Tul
reprinted in [Einstein 1 996], pp. 409-41 6.
curvatura Riemanniana," Circolo Matematico
[Einstein 1 996] The Collected Papers of Albert
di Palermo. Rendiconti 42 (1 91 7), 1 73-205.
lio Levi-Civita, "Methodes de calcul differen
Einstein. The Berlin Years: Writings, 1 9 1 4-
[Levi-Civita 1 91 7b] Tullio Levi-Civita, "Sulla
tiel absolu et leurs applications," Mathema
1 9 1 7, vol. 6, A J. Kox, Martin J. Klein, and
espressione analitica spettante al tensors
Robert Schulmann, eds. Princeton: Prince ton University Press, 1 996.
[Einstein 1 998] The Collected Papers of Albert
tische Annalen 54 (1 90 1 ) , 1 25-201 .
gravitazionale nella teoria di Einstein , " Ren
[Rowe 1 999] David Rowe, "The Gottingen Re
diconti della Reale Academia dei Lincei. Atti
sponse to General Relativity and Emmy Noe
26 (1 91 7), 381 -391 .
ther's Theorems," The Symbolic Universe.
Einstein. The Berlin Years: Correspondence,
[Levi-Civita 1 926] Tullio Levi-Civita, The Ab
Geometry and Physics, 1890-1930, Jeremy
1914-1918, vol. 8, Robert Schulmann, A J.
solute Differential Calculus, trans. Marjorie
Gray, ed. (Oxford: Oxford University Press),
Kox, Michel Janssen, J6zsef lily, eds. Prince
Long, London: Blackie & Son, 1 926.
1 999, pp. 1 89-233.
[Lovelock 1 972] David Lovelock, "The Four-Di
[Rowe 200 1 ] David Rowe, "Einstein meets
[Harward 1 922] "The Identical Relations in Ein
mensionality of Space and the Einstein Ten
Hilbert: At the Crossroads of Physics and
stein's Theory," Philosophical Magazine 44
sor," Journal of Mathematical Physics 1 3
Mathematics," Physics in Perspective 3 (2001),
(1 972), 8 74-876.
379-424.
ton: Princeton University Press, 1 998.
(1 922), 380-382.
[Hilbert 1 91 5] David Hilbert, "Die Grundlagen
[Mehra 1 973] Jagdish Mehra, "Einstein, Hilbert,
der Physik (Erste Mitteilung)," Nachrichten
and the Theory of Gravitation," in The Physi
Discovery: Hilbert's First Note on the Foun
der
cist's Conception of Nature, Jagdish Mehra,
dations of Physics," Archives for History of
k6niglichen
Gesellschaft
der
Wis
senschaften zu G6ttingen, Mathematisch physikalische Klasse, 395-407.
ed. Dordrecht: Reidel, 1 973, pp. 92-1 78. [Noether 1 91 8] Emmy Noether, "Invariants Vari
[Ho-St 1 989] Don Howard and John Stachel,
ationsprobleme," Nachrichten der K6niglichen
eds. , Einstein and the History of General
Gesellschaft der Wissenschaften zu G6ttingen,
Relativity, Einstein Studies, vol. 1 , Boston:
Mathematisch-Physikalische Klasse, 1 91 8,
Birkhauser.
235-257.
[Klein 1 91 8a] Felix Klein, "Zu Hilberts erster
[Nor 1 989] John Norton, "How Einstein Found his
Note Ober die Grundlagen der Physik,"
Field Equations," in [Ho-St 1 989], pp. 1 01 -1 59.
Nachrichten der K6niglichen Gesellschaft der
[Padova 1 889] Ernesto Padova, "Sulle defor
[Sauer 1 999] Tilman Sauer, "The Relativity of
Exact Sciences 53 (1 999), 529-575.
[Schouten and Struik 1 924] J. A Schouten and Dirk Struik, "Note on Mr. Harward's Paper on
the Identical Relations in Einstein's Theory," Philosophical Magazine 47 (1 924), 584-585. [Sta 1 989] John Stachel, "Einstein's Search for General Covariance, 1 9 1 2-1 9 1 5 , " in [Ho-St 1 989], pp. 63-1 00. [Vermeil 1 9 1 7] Hermann Vermeil, "Notiz ber das
Wissenschaften zu G6ttingen, Mathema
mazioni infinitesime," Rendiconti della Reale
mittlere Krummungsmass einer n-fach aus
tisch-Physikalische Klasse; reprinted in [Klein
Accademia dei Uncei (IV), vol. 5(1), 1 889, pp.
gedehnten Riemann'schen Mannigfaltigkeit, "
1 92 1 -23], vol. 1 , 553-565. [Klein 1 91 8b] Felix Klein , "Uber die Differen
1 74-178. [Pais 1 982] Abraham Pais, 'Subtle is the Lord
tialgesetze fOr die Erhaltung von lmpuls und
. . . ' The Science and the Life of Albert Ein
Energie in der Einsteinschen Gravitations
stein, Oxford: Clarendon Press, 1 982.
K6nigliche Gesellschaft der Wissenschaften zu G6ttingen. Mathematisch-physikalische Klasse. Nachrichten 1 9 1 7, 334-344. [Voss 1 880] Aurel Voss, "Zur Theorie der Trans
theorie," Nachrichten der K6niglichen Gesell
[Pauli, 1 92 1 ] Wolfgang Pauli, "Relativitatstheo
schaff der Wissenschaften zu G6ttingen,
rie," in Encyklopadie der mathematischen
und der Krurnmung hoherer Mannigfaltigke
Mathematisch-Physikalische Klasse; reprinted
Wissenschaften, vol. 5, part 2 (1 921 ), pp.
tien," Mathematische Annalen 16 (1 880),
in [Klein 1 92 1 -23], vol. 1 , 568-584.
539-775; Theory of Relativity, G. Field, trans.
[Klein 1 921 -23] Felix Klein, Gesammelte Math ematische Abhandlungen, 3 vols. , Berlin: Julius Springer.
66
THE MATHEMATICAL INTELLIGENCER
London: Pergamon , 1 958. [Reich 1 992] Karin Reich, Die Entwicklung des Tensorkalkuls. Vom absoluten Differentialkal-
forrnation quadratischer DifferentialausdrOcke
1 29-1 78. [Weyl 1 922] Hermann Weyl, Space-Time-Mat ter, 4th ed. , trans. Henry L. Brose, London: Methuen, 1 922.
GIORGIO GOLDONI
A Visual Proof for the Sum of the Fi rst n Sq uares and for the S u m of the Fi rst n Factorial s of Order Two
It is well lrnown that the sum of the first n numbers may be
Here is a visual proof of these identities, representing the
seen as the area of a stairs-shaped polygon, and that two of
sums as volumes of certain solids.
these polygons may be arranged in a rectangle
n X (n + 1):
The sum of the first
n squares 12 + 22 + 32 + . . . + n2
may be seen as pyramid-shaped stairs:
We can arrange six of these pyramids into a parallelepiped: Step
This gives a visual proof for the identity _
1 +2+3+ . . . +n-
1:
n(n + 1) . 2
This identity can be generalized in at least two ways: •
the sum of the first
n squares:
•
the sum of the first
n factorials of order two:
(2) 1 . 2 + 2 . 3 + 3 4 + . . . .
+ n(n + 1)
=
n(n + 1)(n + 2) 3
© 2002 SPRINGER·VERLAG NEW YORK, VOLUME 24, NUMBER 4 , 2002
67
Step 2:
Step 3:
Step 4:
Step 5:
68
THE MATHEMATICAL INTELLIGENCER
We have obtained a parallelepiped n X (n + 1) x (2n + 1) and this immediately yields (1 ) For the sum of the first n factorials we need three pyra mids a little different from the ones used in the previous case, but not all three the same. One of them must be the mirror image of the other two-.
Step 3
.
Now we can arrange the pyramids in a parallelepiped: Step 1
We have obtained a parallelepiped n x (n + 1) X (n + 2), and formula (2) has been proven too.
A U T H O R
Step 2
GIORGIO GOLDONI
Civico Planetaria "F. Martino" Viale J. Barozzi ,
31
41100 Modena
Italy
e-mail: [email protected]
Giorgio Goldoni, after training in engineering and mathemat ics, has been for twenty years a high-school mathematics teacher and a member of the Centro Sperimentale per Ia Di dattica deii'Astronomia in Modena. He lives in the nearby town of
Rolo with his wife and two children. He thanks his high
school students for stimulating him to find new simple ways to
see things he thought he already knew well.
VOLUME 24, NUMBER 4 , 2002
69
Some Hints on Problems
1-14
(presented in Mathematical Entertainments, "Hat Tricks," p. 47)
1. The only solutions with winning probability 3/4 correspond to antipodal points on the 3-cube, no matter what the labeling of the cube. 2. Ifthe Hamming code is given as the kernel of the map T:F � V described in "Hat Tricks," then T is surjective and its kernel has dimension n - k 2k = 1 - k which has density 2n-k;2n 1f2k ll(n 1). 3. The density of the code L coming from the dumb strategy is 1/2k, and the desired inequality is equivalent to 1/(n + 1) ::::; 1f2k < 2/(n + 1). 4. Check that the 7 marked nodes in Figure 4 are a covering code. (The 5-cube is given as two 4-cubes; corre sponding vertices in the two halves must be joined by edges, which are left out of the figure for the sake of clarity.) 5. If l [vi] is in the kernel of T then the sum of the vi is 0 in V = F�. 6. Nothing, if the host listens in on all communication in the strategy session. However, if the team can surreptitiously generate random numbers during the strategy session, outside of earshot of the host, then they can establish a map ping between colors and 0/1 (for each player), which can be used during the game to defeat any nonrandomness that the host introduces.
r
=
=
+
1§1311;11+
70
THE MATHEMATICAL INTELLIGENCEA
l'lii!id!lllil!;l+l--lr----
7. From any player's point of view the probability of winning using a random strategy is a linear function of the avail ability parameters, which are subject to linear constraints. The optimum is attained at a vertex of the correspond ing convex polyhedron, i.e., at a deter ministic strategy. 8. Let L denote the set of six marked black points in Figure 5. Gray paths from two of the points in L end in six gray points, and the other points are all at distance 1 from L (taking into ac count the understood edges connect ing the two halves). A strategy is then
given by orienting all edges "away from on the gray paths or paths of length 1 , and decomposing the rest of the graph into cycles which can be ori ented arbitrarily. Thus L is the set of losing points for this strategy for the majority hats game for 5 people. 9. Berlekamp finds a "code" that is as small as possible, i.e., with 1 + [512110] 52 points. 10. The team can guarantee that [n/2) players would survive. One could think of couples pairing off and guessing so that one (and only one) of them would survive, or of two halves (sub-teams) wagering on opposite parities of, say, the number of black hats. 1 1. The player in back announces the parity of the hats that he sees. 12. The player in back announces the sum modulo q. 13. Players order themselves, and vote only if all prior hats are white, in which case they vote black with a large enough number of votes to swamp all earlier votes for black. 14. The only thing that can be done is to guess with probability p. The optimal probability is asymptotic to log(4)/n, and the team wins with prob ability approaching 114. L"
=
I il§i) t§i.llJ
Jet Wi m p ,
Editor
I
Geometry Civilised: History, Culture and Technique by J.
L.
welcome to submit an unsolicited
How much pleasure and surprise have expected results contained in Greek
OXFORD, CLARENDON PRESS 1 998, 309 pp.
Mathematical Intelligencer? You are
rems from initial, irrefutable axioms. since been given by the beautiful and un
Heilbron
U.S. $35, ISBN 01 9-850-6902
Feel like writing a review for The
duced the now familiar method of sys tematically deducing successive theo
texts such as Euclid's
Elements
Conics of Apollonius!
Nevertheless, to
or the
REVIEWED BY MICHAEL LONGUET-HIGGINS
day's high-school students "should not
T
he Elements, Euclid's famous trea
diately understand the point of geomet
tise on geometry, has been hailed
rical argument. Whole civilisations have
as a cornerstone of Western culture, in
become impatient if they do not imme
done the same."
review of a book ofyour choice; or, if
deed as a fme proof of idealistic phi
During the Dark Ages the West for
losophy. Nevertheless, when used as a
got Greek and soon lost all but a few
you would welcome being assigned
textbook it has been the bane of gen
scraps-of the
a book to review, please write us,
erations of high-school students, who
books of
mostly fail to see the point of Euclid's
ment) transmitted a certain amount of
telling us your expertise and your predilections.
Elements.
Roman text
agrimensura (land measure
pedantic proofs of the seemingly obvi
serviceable information from classical
ous.
times to mediaeval Europe. But "fortu
Numerous attempts have been
made to improve upon the
Elements,
nately the Arabs took a strong interest
either by introducing more natural,
Duri n g the
though less rigorous, demonstrations or by interspersing the text with im portant applications to physics and en gineering and to familiar problems in everyday
life.
Professor
Heilbron's
richly illustrated volume is perhaps the
fifteenth century the Greek texts , p reserved i n
most attractive attempt at improve ment yet. But it is more than that. In a long introductory chapter (headed "An
Byzantiu m ,
wide-ranging and readable survey of
came West .
Old Story"), the author provides a
the place of geometry in Western cul ture, with fascinating excursions into
in geometry and preserved Euclid.
the traditional geometry of Indian, Chi
When, during the twelfth century Eu
nese, Egyptian, and Babylonian civili
ropean scholars began to make useful
sations.
contacts with their better-educated Is
In both Egypt and China, accurate
land measurement became a necessity
lamic
counterparts,
the
Elements
stood ready for study, in Arabic. A few
for the equitable assessment of land
Westerners,
taxes, especially following the yearly
mastered the tongue
flooding of river valleys. Practical pre
translated Euclid into Latin." Later dur
ambitious
for learning, of Islam and
scriptions for calculating areas were not
ing the fifteenth century the original
always accurate. For example, an Egyp
Greek texts, preserved in Byzantium,
tian rule for finding the area of a given
came West at the beginning of the Re
triangle was to multiply the lengths of
naissance. After the invention of print
two adjacent sides of a triangle and di
ing the
vide by 2-not mentioning that the in
books to be published, initially in Latin
Elements
was one of the fust
1570, in English.
cluded angle must be a right angle. It was
and then, in
of Mathematics, Drexel University,
the Greeks, surely motivated by a desire
Euclid came to be a part of school
Philadelphia, PA 1 91 04 USA.
to eliminate such errors, who intro-
and college education in the West, par-
Column Editor's address: Department
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 4, 2002
71
ticularly in England, where it was pro
to higher geometries, spherical astron
moted by the Cambridge Platonists, and
omy and geodesy. A further satisfaction
subsequently also in America Ironically,
is gaining confidence in systematic rea
theory of incommensurables). A sepa
with the advent of universal education a
soning. . . . Another source of pleasure
rate chapter is devoted to Pythagoras's
reaction set in. Some reasons why geom
is the integration of the pictorial and the
theorem, which was lmown to both In
are introduced after areas, and then on to Book VI (Book
V contains Euclid's
etry puts off the average student were
verbal. . . . Finally, pursuing geometry
dian and Chinese mathematicians. Lit
neatly summarised three centuries ago
opens the mind to relationships among
tle in the book is unrelated in some way
(in 1701) by an anonymous writer:
learning, its applications, and the soci
to the material in the
eties that support them. "
a classic work is often justified by its
"The aversion of the greater part of
How
Mankind to serious attention and close
well
has
the
author
suc
Elements.
Since
later developments, some mention of
arguing; Their not comprehending the
ceeded? About as well as possible,
the elegant theorems
necessity or great usefulness of these in
given the difficulty of the task As Eu
geometry such as Desargues's theorem
other parts of Learning; an Opinion that
clid
to
on triangles in perspective would have
this study requires a particular Genius
Ptolemy I of Egypt when asked if there
been welcome. However, a theorem of
and turn of head, which few are so
is
said
to
have
answered
of projective
was a shorter way to mastering geom
Pascal's, essentially one of projective
Ele
geometry, is offered as an exercise in
happy as to be Born with: and the want
etry than working through the
of . . . able Masters." Sound familiar?
ments, "There is no royal road to geom
Chapter 3.
etry." Nevertheless the path is here
In view of the author's stated ob
eased by numerous illustrations, cho
jectives, it is surprising to find no men
began as early as 1 794 with the French
sen for artistic merit or historical in
tion
mathematician A. M. Legendre, who
terest; by applications of geometrical
which flourished especially during the
mixed
and
theorems to everyday life; by compar
trigonometry, introduced practical ex
ison of Euclidean proofs with different
18th and 19th centuries (see Japanese Temple Geometry Problems, by H. Fukagawa and D. Pedoe, Winnipeg,
The long series of attempts to im prove on the
Elements
geometry
with
as a textbook
algebra
of Japanese temple geometry,
amples,
and omitted proofs of the
methods from other cultures; by alter
obvious.
But British and American
native derivations using algebra or tri
1989). Some very remarkable theorems
schools still clung to Euclid as an in
onometry; and by posing further prob
were discovered and illustrated (usu
tellectual discipline. At Oxford Univer
lems to intrigue and test the reader. A
ally without proof) on wooden or stone
sity, a distinguished mathematics pro
number of these are drawn from a lit
tablets suspended in Japanese temples.
fessor L. C. Dodgson (who as Lewis
tle-lmown publication, the
Carroll wrote
Alice in Wonderland)
made a study of a dozen modem rivals to the
Elements
and declared, in 1883,
ary,
Ladies' Di
which prospered in England be
This at a time when Japan was quite
isolated from the rest of the world.
tween 1704 and 1840. Women took part
The book ends with an account of
in both setting and solving the prob
the Tantalus Problem, a geometrical
at present, not
lems. One editor wrote in 1718 that his
teaser involving the isosceles triangle
only unequaled, but unapproached."
women correspondents had "as clear
having angles of 20°, 80° and 80°. When
"Euclid's treatise
is,
After World War II, despite the avail
Judgements, as sprightly quick Wit, as
it was published by the
ability of some excellent texts (such as
penetrating Genius, and as discerning
Post in
N. Altschiller-Court's
try,
CoUege Geome
and sagacious Faculties as men's."
Washington
1995, the man who set the puz
zle, on being challenged for a solution,
An interesting section of Chapter 1
said he had forgotten how to do it and
proportion of American students tak
discusses the role of women in geome
could not repeat his lost performance.
ing geometry declined dramatically.
try.
Barnes and Noble 1925, 1952) the
Geometry Civilised
In Chapter 5 it is recounted how in
tion and they all gave me insights into
cludes the story of how deductive
with Princess Elizabeth of Bohemia on
the problem without being able to
geometry has been virtually banished
the problem of drawing a circle tangent
solve it," he said. With these insights
to three others, and how she astonished
and a weekend's labour he managed a
in
from the American curriculum. In Eu
rope and Canada the situation is simi lar, though the subject is still alive.
1643
Rene
Descartes
"I contacted 40 geniuses around the na
corresponded
Chapter 1 of
him with a complete solution. Another
feminine
tour de force might have been
Professor Heilbron gives as his rea
mentioned: Alicia Boole Stott, third of
son for writing this book " . . . to con
the five daughters of the mathematician
tinue to reap and in a measure to
George Boole, from 1878 onwards de
solution, which involves a clever but non-intuitive geometrical construction. The problem indeed deserves its name.
In
such a broadly conceived and
splendidly produced volume, it might
repay, the pleasure that studying geom
veloped her own method of construct
seem ungrateful, though necessary, to
etry has given me, . . . delight in finding
ing the 3-dimensional cross-sections of
a clear, tidy proof, in seeing a powerful
the regular polytopes in 4 dimensions
point out a few errors. Figure 1.1. 7 is
application of simple principles, in per
(see H. S.M. Coxeter, Regular Polytopes,
ceiving
pp. 258-259).
ships
unexpected in
buildings,
spacial in
relation
patterns
on
ceramics and textiles, in highway inter changes . . . in advancing the
72
THE MATHEMATICAL INTELLIGENCER
Elements
The book, after Chapter 1, follows roughly the order of Books I to IV of
Euclid's textbook, except that circles
clearly meant to illustrate a general sca
lene triangle, but it is drawn as equilat eral.
In Figure
1. 1.8 the lines AB and CD
are not parallel diameters as stated in the caption, though the theorem is true even if they are only parallel chords. On
258 a Freudian slip, perhaps, on the
Paulo Ribenboim is enamored of
part of the Oxford University Press, ends
Number Theory, a fact that shines
talk-to some fairly sophisticated math
through this
ematics. "Selling primes" is so light
p.
a proof with "O.E.D." On p.
260, should
flawed
but
exuberant
light entertainment for an after-dinner
5.5.21 start, "seven men bought
book The title may suggest otherwise,
weight it is almost embarrasis ng. The au
equal shares"? Near the end of the proof
but what really excites him is the the
thor assumes the reader knows nothing
294, b2/2a
b2/a. 1n the problems APS 5.2.12 on p. 215, the factor 2 should be
ory that enables us to explore and say
about the distribution, or even the in
interesting things about numbers. Cer tain areas hold a particular fascination,
finitude of the primes. 1n "What kind of a number is v2v2?", he considers it nec
on the left of the equation, not on the
and he returns to them repeatedly: pri
essary to define complex and algebraic
right. And so on.
mality, Fibonacci sequences and the
numbers with considerable care. "Gauss
more general Lucas sequences, Dio
and the class number problem" is an ex
not para
of the Tantalus Problem on p. should be
None of this detracts significantly from the author's main achievement.
phantine analysis, class numbers, irra
tended and thorough essay on the sub
Not only will the book be enjoyed by
tionality, and transcendence.
ject. It assumes the reader is familiar
mathematical specialists interested in
I particularly enjoyed his second es
with characters and quadratic recip
broadening their knowledge of other
say, "Representation of Real Numbers
rocity. "Powerless facing powers" looks
cultures, but it may serve to draw
by Means of Fibonacci Numbers." I
at powerful numbers-an integer n is k
mathematically untrained readers into
learned Kakeya's result
[ 1 ] that if (si)
powerful if prime p divides n implies
the pleasures of a subject, deductive geometry, so sadly expelled from the curriculum in some countries.
is a monotonically decreasing sequence
1
that approaches 0 and if I�= si diverges,
of theorems and conjectures, including
written as a sum of some subsequence
a detailed section on the ABC co�ecture
of the si. Ribenboim uses this as a lead
University of California, San Diego
in to Landau's theorem
La Jolla, CA 92093-0402 USA
itly evaluates I:=t
by Paulo Ribenboim NEW YORK: SPRINGER-VERLAG, 2000
US $39.95, 375 pp, paperback ISBN: 03-8798-91 1 0
REVIEWED B Y DAVID BRESSOUD
N
[2] that explic 1/F2n CFm is the mth
Fibonacci number) in terms of the
L(x) I;'=1 xn/(1 xn), and I;'=1 1/F2n -l in terms of theta Lambert series
My Numbers, My Friends: Popular Lectures on Number Theory
It is a romp through a wide assortment
then every positive real number can be
Institute for Nonlinear Science
e-rnail: [email protected]
that pk divides n-and perfect powers.
=
(also stated in two other ess�ys). One of
the few proofs in this essay assumes fa miliarity with p-adic arithmetic.
More than anything else, this is a pro fuse collection of interesting results in Number Theory, which is why repeti
functions. The proofs are almost self
tions and omissions are so frustrating.
contained.
One keeps wishing that instead of just
The only result that he
needs to quote is Jacobi's sum of two
collecting his essays, he had mined them
squares formula: Given any positive in
to put together thematic exhibits. As an
teger m, the number of pairs of inte
example, eight of the eleven essays use
gers (s,t) for which s2 + t2
m is equal
or make reference to quadratic exten
to four times the difference between
sions. Quadratic extensions are defined
the sum of the divisors of m that are
in two of the essays, but not in the first
congruent to
=
1 modulo 4 and the sum
of the divisors of m that are congruent
essay in which they are encountered. 1n
the first essay of the book, "The Fi
to 3 modulo 4. The essay concludes, as
bonacci numbers and the Arctic Ocean,"
most of them do, with references to
Ribenboim describes Lucas sequences,
some of the many related questions.
a discussion that could benefit from the language of quadratic extensions. Here
umber Theory is endlessly fasci
The omission of a proof of Jacobi's
nating. No other field of mathe
sum of two squares formula-a result
matics can match it for its range of
that is not hard to prove given the au
To cap off one's frustration with this
problems and the variety of its tech
dience that this book will draw-is
book, most page numbers listed in the index are off by one.
there is no mention of them.
niques. It has problems that can be ex
symptomatic of a serious problem with
plained to a child still struggling with
this book: It is a collection of random
Many of the essays, on their own,
the rudiments of arithmetic and prob
essays with no attempt to fmd a con
lems that can be comprehended only
sistent voice or level of detail, no con
are excellent. "What kind of a number is V2v'2?" is one of Ribenboim's best,
after years of directed post-doctoral
cern to fill in significant omissions or
ranging through continued fractions,
study. There is much to learn and ex
avoid significant repetitions. Wieferich's
measures of irrationality, and proofs of
plore at every level between these ex
proof that the first case of Fermat's
transcendence, pointing out the well
tremes. Clever amateurs can still-make
Last Theorem is true for prime p when
known as well as many obscure but in
�1 (modp2) is mentioned on page 192, again on page 220, and yet again
teresting results. But as a collection,
significant contributions, but the an swers to simple-sounding questions
2P- 1
can require results from the very fore
on page 237, occurring in three con
front of mathematical research. Riben
secutive essays.
boim's book reflects that spread.
His essays range from the trivial-
this book is disappointing. REFERENCES
[1 ] S. Kakeya. 1 941 . On the partial surn of an
VOLUME 24, NUMBER 4, 2002
73
infinite series. Science Reports Tohoku Imp.
mouse click. The manual is targeted at
Univ. ( 1 ) 3: 1 59-1 63.
both types of users. It takes you step
[2] E. Landau. 1 899. Sur Ia serie des inverses
by-step through a getting-started sec
de nombres de Fibonacci. Bull. Soc. Math.
tion, showing you how to draw and move points and lines, parallels and
France 27: 298-300. Mathematics and Computer Science
a, b, P b e the intersection point of a and b. Cinderella is able to come up with the statement that c and P are incident to each other. The man c,
and let
perpendiculars, etc. You immediately
ual describes how the software does
see the difference between free and
this.
dependent objects. For example, de
Department
point. Denote the three lines as
and
In a dynamic geometric construc
Macalester College
fine a point as the intersection of two
tion, it
1 600 Grand Avenue
previously created lines. The point
jects and visualize the effect on the en
St. Paul, MN 551 05-1 899 USA
pends
on the lines, so moving either
tire scene. Moreover, it is educational to
e-mail: [email protected]
line will change the location of the
understand the nature of the changes.
point. However, moving the point is not
This is provided by the locus feature.
de
possible, as it is the dependent object.
Cinderella: The ·Interactive Geometry Software by J. Richter-Gebert and
U.
H.
1 999 NEW YORK: SPRINGER-VERLAG US $59.95. ISBN 35-401 4-7195
REVIEWED BY GILL BAREQUET
I
happily took on myself the task of re viewing this software package be
cause I like to play with such geomet
ric toys, and also because from time to
to move around free ob
You choose a "mover" and a "road" (e.g.,
Then you get a quick overview of how
a point and line). While the mover slides
to control the appearance of objects
along the road, the construction keeps
(sizes, colors, etc.) and of the possible
changing. To visualize the dynamics of
views of a scene.
the scene, you choose a "tracer." In case
At this point you face one of the
Kortenkamp
is fun
the tracer
is
a point, the result is the
strongest features of Cinderella: its sup-
path traversed by the tracer while the
The fi rst thing that
either see the final trace or switch to an
attracted m e i n
slides slowly along the road, and you
Cinderella was its
This is, more or less, the end of your get
h u man i nterface .
mover advances along the road. You can animation mode, in which the mover
gradually see the creation of the trace. ting-started session. A very interesting section of the manual reveals information "behind the scenes." In particular, it tells you
time I have to draw geometric figures
port of different geometric views. In
for my papers. This is what
Cinderella
fact, the software supports two distinct,
how
is about. It enables you to draw dy
not to be confused with each other, fea
geometry, homogeneous coordinates,
namic geometric constructions, view
tures. The first feature is the support of
and complex numbers to maintain the
them through several types of lenses,
two non-Euclidean geometries (hyper
various geometric entities and to solve
create animations, capture a scene in a
bolic and elliptic) in addition to the reg
continuity problems. For example, as
Postscript file, and export your cre
ular Euclidean geometry. The second
sume you defme a point as one of the
ation into HTML.
feature
is the support of different views
two common points of two existing in
Some readers may be familiar with
of the geometry: Euclidean, spherical,
tersecting circles, and use that point
(manual
and hyperbolic. A good example is the
for further constructions. Now assume
available from Key Kurriculum Press).
spherical view in Euclidean geometry.
you move the circles around such that
The Geometer's Sketchpad
the
software
uses
projective
is similar in some aspects,
In this view the entire plane is mapped
they first do not intersect any more,
superior in some, and inferior in oth
to a hemisphere, shown on the screen
and then intersect again. A few inter
ers. The first thing that attracted me in
as a circle. Points at infinity lie on the
esting questions arise. For example,
was its human interface. It
boundary of that circle, so that this
when the two circles become inter
is very intuitive and easy to learn. Once
boundary is "the line at infinity." Here
secting again, which of the two inter
Cinderella
Cinderella
you have learned how to defme a line
it is very easy to draw, manipulate, and
section points would you expect to re
by two existing points and how to de
view objects at or close to infmity. The
sume the role of the intersection point
fine a point as the intersection of two
"price" is, naturally, the fast-decreasing
you
originally defmed? In addition,
given lines, you can easily guess how
resolution of details as you approach
what should happen to (not to mention
to perform many other operations, e.g.,
infmity.
how can you internally represent) those
how to create a line passing through a
Now
you experiment
with
Cin
nice theorem-proving mech
objects that depend on the temporarily disappearing intersection point?
given point and parallel to an existing
derella's
line.
anism. That is, its ability to detect geo
In a nutshell, the authors' solution
So one can play with the software
metric tautologies. For example, theo
is to represent everything with com
and learn it without any instructions.
rems that say that three seemingly in
plex numbers. Naturally, you define
Other people like to read the entire
dependent lines in some geometric
and see objects with real data, but
manual before performing the first
constructions
that's not how the software sees it. For
74
THE MATHEMATICAL INTELLIGENCER
always
meet
at
one
1.0 only points and lines can serve
example, say you define a point with
havior), but with what is missing. To
sion
coordinates (x,y), where x and y are
me it is obvious that the version I
as movers, roads, and tracers.) Another
real numbers. The software stores this
played with (version
1.0) is immature
promised feature for the next version of
as
and not ready (yet) for distribution.
the software is a scripting language for textual input, batch files, logging, etc.
(x + Oi, y + Oi). In the previous ex
ample, the intersection point is noth
For example, the only way to defme a
ing but one of the solutions Qf a sys
conic in this version is by specifying 5
Finally, I have mixed emotions about
tem of two quadratic equations. When
points. This is almost useless; normally
the manual. On the one hand, as men
the circles cease to intersect, the so
you (and I) would like to define, say,
tioned above, its contents are concise
lutions still exist, but now they are
an ellipse by its 2 focal points and the
and well organized, and they make the
complex. And so is everything that de
sum of distances from them to every
software-learning process smooth and
pends on them. We simply see on the
point on the ellipse. (Or, say, by spec
easy. On the other hand, it has (to my
screen only the real objects, or if you
ifying the lengths and positions of the
taste) two stylistic drawbacks. First, it is
like, the intersection of a 4-dimen
2 axes of the ellipse.) The authors have
a bit too self-congratulatory about the
sional space (where each of the
assured me that this is just a matter of
use of the sophisticated mathematical
x and
y axes has 2 degrees of freedom) with
providing a human interface, and that
tools in the implementation of the soft
the real plane (no imaginary compo
a more natural mechanism for defining
ware. Second, it contains spelling errors
nents).
conics is planned for a near-future re
and seems to require one more proof
lease of the software.
reading. I believe that these two draw
The continuity problem is solved by cursor tracking. Putting it simply,
I would also like to see other types of
backs can be readily fixed. Let me con
interprets how you move
entities, e.g., higher-degree polynomials
clude with a good word about the
the mouse as a guide for what the de
and even trigonometric functions. The
sired continuous move of obj ects is.
authors claim that some limited func
software installation: It's a piece of cake. ' After '3'0 seconds or so, you can play.
Again, the manual provides the de
tionality (e.g., no theorem-proving) for
Cinderella
tails.
any externally defined function (e.g., by
Faculty of Computer Science Technion - Israel Institute of Technology
vantages of Cinderella. The problem is
Mathematica) will be provided in ver sion 2.0 of CindereUa. Moreover, the au
not with what it contains (at least I was
thors promise more flexibility with the
Israel
unable to find any bug or strange be-
locus and animation features. (With Ver-
e-mail: [email protected]
Let me now refer to some disad
Haifa 32000
Erratum In his "Rediscovering a family of means"
(Mathematical Intelligencer
24 (2002), no. 2, 58-65),
Stephen R. Wassell begins his account with the use of the arithmetic mean by the ancient Babylo· nians. Readers may have been surprised, as I was, by the very early date given (and by its preci sion). One reader, Robert Davis of Southern Methodist University, wrote to query this dating. Profes sor Wassell thanks him, as do I.
Wassell was relying on an article by Maryvonne Spiesser, which he cited. Prompted by Profes
sor Davis's query, he went back to Spiesser and found he had misread her, taking an identification number for a date! He gives this corrected text for the sentence following equation (5) in his article: The arithmetic mean, the simplest of the three, was known and used by the Babylonians, per haps as early as 1900 BCE. For this he refers to Spiesser and to
0. Neugebauer and A. Sachs,
Haven, CT, 1945.
Mathematical Cuneiform Texts,
American Oriental Society, New
The Editor
VOLUME 24, NUMBER 4, 2002
75
41fi,I .MQ·h·i§i
Robin W i lson
Two Serbian Mathematicians
In this column we celebrate the work
Jovan Karamata
( 1902-1967)
is
of the Serbian mathematician
best known for his theory of "regularly
Karamata and Petrovic, both of whom
varying
Jovan of his teacher Mihailo have recently
from
the
early
that behave nicely in asymptotic rela
been commemorated on stamps.
Mihailo PetroviC (1868--1943),
functions,"
1930s. This is a class of real functions
math
ematician and philosopher, was pro
tions.
Much
later,
these
functions
found various applications in other ar
by Slobodanka Jankovic
fessor at the University of Belgrade and
eas of mathematics-especially proba
and Tatjana Ostrogorski
founder of the Belgrade mathematical school. He studied in Paris at the Ecole
bility theory, but also in number the ory, the theory of analytic functions,
Normale Superieure and obtained his
and the theory of generalized func
doctoral degree in 1894. The examin
tions.
ers for his thesis, Sur les zeros et les irifinis des integrales des equations differentieUes algebriques, were Her
University
Karamata was a professor at the of
Belgrade,
he
mite, Picard, and Painleve. He worked
ematical analysis. From 195 1 , he was a
in differential equations, · real and com
professor at the University of Geneva.
plex analysis, and algebra, and also in
He wrote many papers, mainly in clas
physics, chemistry, and astronomy, and
sical analysis, but also in number the
wrote many papers in all these areas.
ory, Fourier analysis, inequalities, and
Petrovic constructed several ma
geometry. The most important part of
chines and measuring instruments. He
his work, which included his best re
also constructed an integrator, a kind
sults, was related to the summability
of analogue computer based on hydro
theory
dynamic principles for solving first-or
Tauberian-type theorems. He became
of
divergent
series
der differential equations; for this, he
famous for his short and elegant proof
obtained a special award at the Paris
and
to
of Hardy and Littlewood's Tauberian
Exhibition in 1900, and in London in
theorem, which he published in 1930.
1907. He wrote several essays on math
This stamp commemorates the cente
ematical phenomenology, as well as
nary of his birth.
books on travel. He had many students, the most famous of whom was Jovan
Mathematical Institute SANU
Karamata. This stamp commemorates
Kneza Mihaila 35
the fiftieth anniversary of Petrovic's
1 1 000 Belgrade
death.
Yugoslavia
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,
The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]
80
where
founded an important school in math
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
Jovan Karamata
Mihailo Petrovic
