No title

1 = F(z) = In this case we have

!h(l) -

21

=

h(l)

(h(z) - h(O))Iz.

= 1 +

jh(O) - 1 1

h(O),

2::

and consequently

1 - h(O)! > 1/2.

j

But we know that ih(l) - 2 = h (l) - g( 1) < 112, and this establishes the contradiction. Therefore, the function F is nonconstant.

Strahlengrenz1nert) Intermezzo

f(ei
�

While Alice Roth taught, research in her area continued. The Swiss cheese was rediscovered by Mergelyan and was "known affectionately as Mergelyan's Swiss Cheese." [34, p. 69] Mergelyan's name was frequently found attached to the Swiss cheese up until about 1968 and occasionally even later (e.g., [32] and [6]). As E. L. Stout writes in his Math­ ematical Review of Vitushkin's 1975 [32] paper, It should be noted that the author attributes to Mergelian the first example of a nowhere dense compact set E C C for which C(E) =F R(E). This is an unfortunate though com­ mon error. Alice Roth gave an example of such a set in 1938 [24].

© 2005 Spnr1ger Sc1ence 1 Bus1ness Media, Inc , Volume 27. Number 1, 2005

49

Apparently, as Roth began working her way back into mathematics, she found Zalcman's notes [34] and recog­ nized Mergelyan's example as her own. By 1969 the error was corrected [9] , and most mathematicians felt the name "Swiss cheese" could not have been more appropriate. Roth's past as well as future work was to have a strong and lasting influence on mathematicians working in this area. Her Swiss cheese has been modified (to an entire va­ riety of cheeses); see e.g., Gaier [8, pp. 103-106] or Gar­ diner [ 10]. We now tum to Roth's fusion lemma, which ap­ peared in her 1976 paper [27] and influenced a new generation of mathematicians worldwide. Roth's Fusion Lemma

Let K1 and K2 be disjoint compact subsets of C, and as­ sume that the rational functions r1 and r2 are close on some compact set K, in the sense that 1r1 - d is small on K. Is there a rational function r that approximates r1 on K1 U K and simultaneously approximates 1·2 on K2 U K? Of course, such an r cannot approximate both r1 and r2 on K much better than r-1 and r2 approximate each other on this set. If K1 U K2 U K has a decomposition into two disjoint compact sets, with one containing K1 and the second con­ taining K2 , then this problem can be solved by Runge's the­ orem. But, if K is a "bridge" connecting K1 and Kz, as in Figure 3, then Runge's theorem will not answer this ques­ tion for us. Before turning to the solution, we consider the question from a real perspective. Consider the case in which we choose real intervals for the compact sets Ki> K2, and K and real functions r·1 and r-2, as in Figure 4. The function r that we wish to find should approximate r1 on K1 U K and rz on K2 U K, and the error bounds, lh - �IK1 uK and llr-z - �IKz uK, should both be close to lh - r2IIK· Even in this real case, it is not altogether obvi­ ous that such a rational function exists. However, the ex­ istence of such an approximation is guaranteed by Roth's fusion lemma [27]. Lemma 3 (Fusion of rational functions ). Let Ki> K2, and K be compact subsets of the extended plane with K1

and K2 d·isjoint. If r-1 and r2 ar-e any two r-ational func­ tions satisfying, for some E>0,

h Cz)

-

r-2(z)! <

E, .for z E K,

then the1·e is a positive number a, depending only on K1 and K2, and a r-ational function r- such that for- j 1, 2, =

'

r(z) - rj(z) < aE, for z E Kj I

Figure 3. K is a bridge between K1 and K2•

50

THE MATHEMATICAL INTELLIGENCER

U K.

-·----·-·

rl

- - - - - - r2 -- r

Figure 4. Fusing r1 and r2 with r.

The fusion lemma is a powerful tool that can be used to extend approximation on compact sets to obtain results on closed sets. The proof is based on techniques used in study­ ing smooth (not necessarily analytic) functions. The so­ called Pompeiu formula for differentiable functions with compact support plays an important role. This formula is similar to Cauchy's integral formula, but there is an extra term reflecting the "extent" of non-analyticity in terms of the Cauchy-Riemann operator. The miracle that allows Al­ ice Roth to obtain results concerning meromorphic func­ tions from these differentiable techniques is an implicit use of a clever trick that has become more and more fashion­ able, namely solving a non-homogeneous Cauchy-Riemann equation. In this way, she is able to get rid of the non-ana­ lytic part that arises from the Pompeiu fommla. Roth used her fusion lemma to prove Bishop's localiza­ tion theorem on compact sets. To see how this goes, let f be a continuous function defined on a compact set E. The localization theorem of Errett Bishop [4] states that f can be approximated by rational functions if and only if for each point z E E there is a closed disc Kz centered at z such that the restriction off to E n Kz can be approximated by ra­ tional functions. Of course, one direction of this theorem is obvious. In [27] , Alice Roth showed how to derive the (interesting) half of Bishop's theorem from her fusion lemma. The idea is quite simple: Having chosen the sets Kz as above, return to each point z E E and choose a disc kz that is again centered at z, but has half the radius of the disc Kz. Since the set E is com­ pact, we can find finitely many of the smaller discs, kzp . . . , kz ' such that E is contained in the union of these p sets. p We show how to obtain Bishop's localization theorem from the fusion lemma by using each of the larger sets, KzJ n E, to find an approximating rational function Tj and then fus­ ing the smaller sets kzj n E to obtain one rational function approximating f on all of E. So suppose that we have a compact subset F1 of E on which we can approximate F uniformly and we also have a disc kzJ from above. (In the first step F1 = kz1 n E and kzJ = kz2 , but in the steps thereafter, F1 is simply a compact set on which uniform approximation is possible, and kzJ is a disc we have not yet "fused" into FJ.) If F1 and kzJ hap­ pen to be disjoint, Runge's theorem will imply that there is a rational function that approximates f on F1 U (kzJ n E), and the interested reader can find the details in [8, p. 1 14].

If F1 n

kzj =F 0, then we must use the fusion lemma,

we do as follows:

which

Let SJ denote the radius of the larger disc, KzJ' and con­

l

sider the three compact sets E 1 = F1 n [w: w -

Zj l 2: 3sj l4l,

E2 = kzj n E = (w E E : w - zj s sj 12 ) and K = F1 n n [w: w - zj l s sj ). Then E1 and E2 are disjoint and, for future use, we note that F1 U (kz n E) = E 1 U j

K,j = F1 E2 U K.

Given

E > 0, by our assumption we can approximate our El4 by a rational function r1• Now

function.fon F1 to within

K,j was our bigger disc on which approximation was pos­

sible. Therefore, we can find a rational function r2 ap­

proximating/uniformly to within

E/4 on K::j n E. Now each of these rational functions is within E/4 off on F1 n K:: = j K, so r1 r2 < El2 on K. By the fusion kmma, there is a constant a (depending on E1 and E2 alone) and a rational function r such that for m = 1,2 we haVf• r - 1"111 1 < a E on E111 U K. Since f\ U (k::j n E) = E 1 U E2 U K, we have r ­ .f ! s r - 1"11,' + rm - I I < aE + E on F1 U (kzj n E). Fur­ -

themwre,

the set

F1

E

is arbitrary, so we are able to approximate .( on

U

(kzj n E),

and the fusion of F1 and

kzj n E is

complete. To finish the proof, add one disc to the set ob­

tained from the previous step and repeat the argument

above until all sets have been fused. In

1976 Roth [27],

via her fusion lemma, also extended

the Bishop localization lemma to unbounded sets to obtain Roth's localization theorem on closed sets . Namely, she

entire functions, this question was answered in Norair Arakelian

[2].

Theorem 4 (Arakelian). Let E

1964

by

be a closed set in C. The

following are equivalent: (1)

Each function continuous on E and holomorphic on the interior of E can be uniformly approximated by entire functions. (2) The complement of E in C is connected and locally connected. In

1972,

Ashot Nersessian answered this question for

meromorphic approximation

[ 19].

Theorem 5 (Nersessian). Let E be

following are equivalent:

a closed set in C. The

(1)

Each function continuous on E and holomorphic on the interior of E can be uniformly approximated by functions meromorphic on C. (2) For each closed disc K, each function continuous on E n K and holomorphic on the interior of E n K can be unUonnly approximated b:IJ rational functions. As fundamental as these two theorems are, giving a char­

acterization of those closed sets E on which the class of all

plausibly approximable functions can be approximated, a still more fundamental question is to decide when an indi­

vidual function can be approximated. Roth's localization

showed that if f is a function defined on a closed set E,

theorem does this, and is so powerful that it yields the suf­

and only if for each point

the cast> of Nersessian's theorem, this is obvious.

then f can be approximated by meromorphic functions if

p in E, there is a closed disc K1,

centered at p such that the restriction off to E n Kp can be approximated by rational functions. Again, one direc­

ficiency

(2 implies 1) in both of the preceding theorems. In

We now derive the sufficiency in Arakelian's theorem.

Suppose, then, that the complement of E in C is connected

tion is obvious. But the other direction had profound con­

and locally connected, and let f be continuous on E and

lowing section.

The complement of the compact set E n K is connected

Complete Solution for the Class of Plausibly

tion in A(E n K), in particular, the restriction off to E n

sequences as Alice Roth noted. We present these in the fol­

Approximable Functions In earlier sections we attempted to approximate all func­

tions belonging to a natural class of functions on E, namely,

the class of functions continuous on E or the class of func­ tions holomorphic on E.

Becausl:' the uniform limit of continuous functions is

holomorphic on the interior of E. Let K be any closed disc.

and so, by a famous theorem of Mergelyan

[ 18], each func­

K, can be uniformly approximated by polynomials. From the Roth localization theorem, for each

E > 0,

there is a

.f - hi < E/2 on E. By the Roth-Runge theorem (Theorem 2), there is an entire

function

h meromorphic on C such that

function g such that

inequality,

.h - gl < El2 on E. From the triangle If - g < E on E, which gives the sufficiency in

continuous and the uniform limit of holomorphic functions

the theorem of Arakelian.

lar function f can be approximated on a sl:'t E by entirl:'

sible (in the sense indicated), but for most applications the

on an open set is holomorphic, it follows that if a particu­

The theorems of Arakelian and Nersessian are best pos­

functions or by meromorphic functions having no poles on

analogous Runge-type theorems of Alice Roth suffice.

on the interior of E. Hence, the most natural class of func­

ter retirement, Alice Roth extended to arbitrary plane do­

E, then nt>cessarily f is continuous on E and holomorphic

tions to try to approximate on a set E is tlw class A(E) of

In her second period of creative mathematical work, af­

mains her thesis results on holomorphic and meromorphic

functions continuous on E and holomorphic on the interior

approximation on closed subsets of the plane. Over the past

holomorphy, of the two cla.•;;st>s we considered earlier, in

sults and in particular to Riemann surfaces. These attempts

"plausibly" approximable functions. The most natural ques­

pact sets, but attempts to approximate on closed subsets

termine those sets E for which the plausibly approximable

resolved. On the other hand, Roth's work inspired highly

of E. This class combines the attributes, continuity and

just the right dosage. We will refer to ACE) as the cla.'>s of tion on approximating a class of functions is then to dt>­

functions are in fact approximable. For approximation by

30 years,

attempts were made to further extend these re­

were successful with respect to extending fusion on com­

of Riemann surfaces encountered major obstacles still not successful research in potential theory. At the present time,

ID 2005 Spnnger Sc1ence • Bus1ness Media, Inc , Volume 27. Number 1, 2005

51

advances are also being made on such questions for solu­ tions of differential operators other than the Cauchy-Rie­ mann and Laplace operators. Looking Back at Roth's Work and Life

Alice Roth's 1938 thesis was largely overlooked, but by 1969 mathematicians were learning that the Armenian "Swiss cheese" was, in fact, discovered by a Swiss woman. As V. P. Ravin writes in his Mathematical Review of Roth's 1973 paper Meromorphe Approximationen: In 1938, the author published an article that contains im­ portant ideas of the later theory of approximation by ra­ tional functions. We mention only the first example of a nowhere dense compact set E c C on which not every continuous function can be approximated uniformly by rational ones (this set was named the "Swiss cheese" much later) or the formulation of questions that antici­ pated the later excellent work (by Arakelian, for exam­ ple) on tangential approximation (in the sense of Carle­ man). This earlier work seems to have fallen into almost complete oblivion, a common (but unfair) punishment for anyone who dares to enter a subject too early, with­ out waiting until it is accepted an1ong experts and has even become fashionable. All of Alice Roth's publications are cited in our refer­ ences [ 1 1,22,23,24,25,26,27,28]. A good account of her work and influence in approximation theory appears in Gaier's book [8]. As her friend and former student at the Humboldtianum, Prof. Peter Wilker, wrote in an obituary in the Bernese newspaper Der Bund on July 29, 1977, In Switzerland, as elsewhere, women mathematicians are few and far between . . . Alice Roth's dissertation was awarded a medal from the ETH, and appeared shortly af­ ter its completion in a Swiss mathematical journal. . . . One year later war broke out, the world had other worries than mathematics, and Alice Roth's work was simply forgotten. So completely forgotten, that around 1950 a Russian math­ ematician re-discovered sinlilar results without having the slightest idea that a young Swiss woman mathematician had published the same ideas more than a decade before he did. However, her priority was recognized. Alice Roth was hospitalized in Bern in 1977. Although she was very ill, she continued to work on mathematics. She completed her work on her final paper Uniform Ap­

proximation by Meromorphic Functions on Closed Sets with Continuous Extension into the Boundary in the In­ selspital in Bern, and Wilker helped her with the English translation as well as the typing. As Wilker continued, Alice Roth died a mathematician . . . As her last work lay finished before her, she was as pleased with it as she was 4Coined by Jean-Paul Berrut. Universitat Freiburg, Switzerland.

52

THE MATHEMATICAL I NTELLIGENCER

Fraulein Dr. Alice Roth, 1 975 (photo by A. Ballinger-Roth).

with the many flowers that decorated her room, and if someone had asked her whether or not an obituary for her were justified, she surely would have said, "Don't make a fuss about me-but my approximation [by] mero­ morphic functions on closed sets, about those you may write a story." Her last paper arrived at the Canadian Journal ofMath­ on July 1 1, 1977. Alice Roth, "die bekannteste un­ bekannte Schweizermathematikerin,"4 died on July 22, 1977. She is buried in the Roth-Landolt family grave at Friedhof Bergli, in the medieval Swiss town Zofingen.

ematics

Acknowledgments

We thank Professors Jiirg Ratz and Hans-Martin Reimann for their assistance as well as encouragement. We are also particularly grateful to Judith Fahrlander for library ser­ vices customized to our needs, and to the friends and fam­ ily of Alice Roth, on whom we depended so much for in­ formation, recollections, and photographs: Verena and Alfred Ballinger-Roth, Deli Roth, Verena de Boer, Johanna Meyer, Roland Weisskopf, and Irene Aegerter. We thank each of these people for their generosity and willingness to assist us. We are also grateful to the Universitat Bern for its support, to the Gosteli Archiv and Angela Gastl at Archiv der ETH Zurich for their assistance, and to Professor David Coyle for carefully reading t11e manuscript and for helpful suggestions. Aside from the photo of Schloss Ralligen, all photos are in the possession of the Roth family, except the

picture with her godchild, which is from Verena De Boer­ Gloor. We thank both parties for allowing us to reprint them.

[1 7] Jahresbericht der Hoheren Tochterschule der Stadt Zurich. Zurich, 1 925/26-- 1 927/28. [ 1 8] S. N. Mergelyan. On the representation of functions by series of polynomials on closed sets. Ooklady Akad. Nauk SSSR (N. S.) ,

REFERENCES

78:405-408, 1 951 . English translation: Amer. Math. Soc. Transla­

[ 1 ] G. L. Alexanderson. The random walks of George P61ya. MAA Spectrum. Mathematical Association of America, Washington, DC, [2] N. U. Arakelian. Uniform and asymptotic approximation by entire functions on unbounded closed sets (Russian). Dokl. Akad. Nauk SSSR, 1 57 :9-1 1 , 1 964. English translation: Soviet Math. Ookl. 5,

Vll:405-4 1 2 , 1 972. (Russian). [20] G. P61ya and G. Szegb. Aufgaben und Lehrsatze aus der Analy­ sis. Springer-Verlag, 1 925. 2 Bande.

[2 1 ] Protokoll des Schweizerischen Schulrates fUr das Jahr 1 938.

(1 964), 849-851 . [3] Bericht des eidgenbssischen Polytechnikums uber das Jahr 1 870.

Zurich, 1 938. Archiv der ETH Zurich, SR 2. [22] A. Roth. Ausdehnung des Weierstrass'schen Approximations­

Zurich, 1 870. Archiv der ETH Zurich, P 92899 P. [4] E. Bishop. Boundary measures of analytic differentials. Duke Math.

satzes auf das komplexe Gebiet und auf ein unendliches Interval!. Diplomarbeit, ETH , Abteilung fUr Fachlehrer in Mathematik u.

J., 27:331 -340, 1 960.

[5] T. Carleman. Sur un theoreme de Weierstrass. Ark. Mat. Astr. och Fys. , 20 B(4) : 1 -5, 1 927.

[6] R. B. Crittenden and L. G. Swanson. An elementary proof that the unit disc is a Swiss cheese. Amer. Math. Monthly, 83(7):552-554,

Physik, Zurich, 9. November 1 929. [23] A. Roth. U ber die Ausdehnung des Approximationssatzes von Weierstrass auf das komplexe Gebiet.

Verhandlungen der

Schweizer. Naturforschenden Gesel/schaft, page 304, 1 932.

[24] A. Roth. Approximationseigenschaften und Strahlengrenzwerte

1 976. [7] Dlssertationenverzeichnis 1909- 1 9 7 1 . Number 15 in Schriftenreihe

meromorpher und ganzer Funktionen. PhD thesis, Eidgen6ssische

Technische Hochschule, Zurich, 1 938. Separatdruck aus Com­

der Bibliothek. Eidg. Techn. Hochschule, Zurich, 1 972. Gaier.

[1 9] A. H. Nersessian. Uniform and tangential approximation by mero­ morphic functions. lzv. Akad. Nauk Armyan. SSR Ser. Mat . ,

2000.

[8] D.

tion 1 953 (1 953), no. 85, 8pp.

Vorlesungen uber Approximation im Komplexen.

Birkhauser Verlag, Basel, 1 980. English translation: Lectures on

ment. Math. Helv. 1 1 , 1 938, 77-1 25.

[25] A. Roth. Sur les limites radials des fonctions entieres (presentee par M. Paul Montel). Academie des Sciences, pages 479-481 , 1 4

Complex Approximation, Birkhiiuser Verlag, Boston, 1 987. [9] T. W. Gamelin. Uniform Algebras. Prentice-Hall Inc. , Englewood

Fevrier 1 938. [26] A. Roth. Meromorphe Approximationen. Comment. Math. Helv. ,

Cliffs, N. J , 1 969. [1 0] S. J. Gardiner. Harmonic approximation, volume 221 of London Mathematical Society Lecture Note Series. Cambridge University

48: 1 51 -1 76, 1 973. [27] A. Roth. Uniform and tangential approximations by meromorphic functions on closed sets. Canad. J. Math. , 28(1 ) : 1 04-1 1 1 , 1 976.

Press, Cambridge, 1 995. [1 1 ] P. M. Gauthier, A. Roth, and J. L. Walsh. Possibility of uniform ra­

[28] A Roth. Uniform approximation by meromorphic functions on

tional approximation in the spherical metric. Canad. J. Math. ,

closed sets with continuous extension into the boundary. Canad.

28(1 ) : 1 1 2-1 1 5, 1 976.

J. Math . , 30(6): 1 243-1 255, 1 978.

[1 2] M. Gosteli, editor. Vergessene Geschichte!Histoire oubliee. 11/u­ strierte Chronik der Frauenbewegung 1 9 1 4- 1 963, volume

1

and

2. Stampfli Verlag , Bern, 2000. Articles in German, French, or Ital­ ian. [1 3] G. H. Hardy. A Mathematician 's Apology. Canto. Cambridge Uni­ versity Press, Cambridge, 1 992, Reprinted 1 993. [1 4] F. Hartogs and A. Rosenthal. U ber Folgen analytischer Funktio­ nen. Math. Ann . , 1 04:606-6 1 0 , 1 931 . [1 5] H. Hopi. Korreferat uber die Dissertation von Fraulein A. Roth: Ap­ proximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen. ETH-Bibliothek Zurich, Hs 620 : 1 07 , 9. VI I . 1 938. [1 6] 75 Jahre Humboldtianum Bern: zum Geburtstag . Bern, 1 979.

[29] C. Runge. Zur Theorie der eindeutigen analytischen Funktionen. Acta Math . , 6:229-244, 1 885.

[30] 1 00 Jahre Tochterschule der Stadt Zurich. Schulamt der Stadt Zurich, Zurich, 1 975. [31 ] Verein Feministische Wissenschaft Schweiz. Ebenso neu als kuhn, 120 Jahre Frauenstudium an der Universitat Zurich . eFeF-Verlag,

Zurich, 1 988. [32] A G. Vitushkin. Uniform approximations by holomorphic functions. J. Functional Analysis, 20(2): 1 49-1 57, 1 975. [33] J . L. Walsh. U ber die Entwicklung einer Funktion einer komplexen

Veranderlichen nach Polynomen . Math. Ann . , 96:437-450, 1 926. [34] L. Zalcman. Analytic Capacity and Rational Approximation. Lec­ ture Notes in Mathematics, No. 50. Springer-Verlag, Berlin, 1 968.

© 2005 Springer SC1ence+ Bus1ness Media, Inc , Volume 27, Number 1, 2005

53

A U T H OR S

Left: Paul Gauthier (with hat) and Gerald Schmieder. Right: Pamela Gerkin and Ul­ rich Daepp. ULRICH DAEPP

PAUL M. GAUTHIER

Department of Mathematics

Department of Mathemat ics and Statistics

Bucknell University

Universite de Montreal

Lewisburg, PA 1 7837

Montreal

USA

e-mail : udaepp@bucknell .ed u

H3C 3J7

Canada

e-mail: [email protected]

Ulrich Daepp was a student at the Sttidtisches Gymnasium Bem

Paul Gauthier was born and raised in the United States of French­

and could have had Alice Roth as a teacher, had they hired her

Canadian parents and, after completing his studies, chose to re­

when she applied. He crossed the Atlantic Ocean in his earty twen­

turn to the "old country" (Quebec) for his career. He has six chil­

ties and completed his Ph.D. in algebraic geometry at Michigan

dren, and with his family has spent two years in the Soviet Union

State University. He is now at Bucknell University in Pennsylvania,

and one year in China. His hobby is singing. Alice Roth was one

where he has several joint projects with the third author, the most

of Paul's most beloved friends and (to his surprise) he was men­

important being their two children.

tioned in her will. One of his children was named after Alice and, amazingly, only in writing this biography did he learn that she has the same birthday as Alice Roth. Publishing this, his first, paper in The lntelligencer fulfills an am­ bition of many years: to catch up with another of his daughters, who published in The lntelligencer when she was still in grade school (see volume

1 8, no. 1 , p. 7).

PAMELA GORKIN

GERALD SCHMIEDER

Department of Mathematics

Falkultat V. lnstitut fOr Mathematik

Bucknell University

Universitat Oldenburg 261 1 1 Oldenburg

Lewisburg, PA 1 7837 USA e-mail: [email protected]

Germany GSchm [email protected]

Pamela Gorkin received her Ph.D. from Michigan State Univer­

Gerald Schmieder was born in Bad Pyrmont, Germany, and stud­

sity. She also studied at Indiana University for one year, and it was

ied mathematics and physics in Hannover. He works on complex

during this year that she first heard about the Swiss cheese and

approximation theory, Riemann surfaces, and geometric function

Alice Roth. Upon completing her doctorate, she began teaching

theory. His hobby is playing violin, mainly string quartets.

at Bucknell U niversity. She has been there ever since with the ex­ ception of three sabbaticals, all of which have been spent at the University of Bern, Switzerland. Her mathematical interests are pri­ marily function theory and operator theory. She enjoys watching films, learning languages, cooking, and eating.

54

e· mail :

THE MATHEMATICAL INTELLIGENCER

ll"@ii!i§j.fiii£11=tfii§#fii,j,i§.Jd

Gold bug Variations Michael Kleber

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

Please send all submissions to the Mathematical Entertainments Editor. Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380,

M i c hael Kleber and Ravi Vaki l , E d i t o rs

J

im Propp bugs me sometimes. I'm usually glad when he does. Today, Jim's bugs are trained to hop back and forth on the positive integers: place a bug at 1, and with each hop, a bug at ·i moves to either i + 1 or i 2. Of course, it might jump off the left edge; we put two bug-catching cups at 0 and - 1. Once a bug lands in a cup, we start a new bug at 1. What I haven't mentioned is how the bugs decide whether to jump left or right. We could declare it a random walk, stepping in either direction with probability 1/2, but the U. of Wisconsin professor's bugs are more orderly than that. At each location i, there is a sign­ post showing an arrow: it can point ei­ ther Inbound, toward i - 2 and the bug cups, or Outbound, toward infinity. The bugs are somewhat contrarian, so when a bug lands at i, it first .flips the arrow to point the opposite direction, and then hops that way (Fig. 1). Add an initial condition that all arrows be­ gin pointing Outbound, and we have a deterministic system. Let bugs hop till they drop (Fig. 2). Well, what happens? Or, for those who would like a more directed ques­ tion: First, show that every bug lands in a cup (as opposed to going off to in­ finity, or bouncing around in some bounded region forever). Second, find what fraction of the bugs end their journeys in the cup at - 1 , in the many­ bugs limit. Go ahead, I'll wait. Do the first ten bugs by hand and look for the beautiful pattern. You can even skip ahead to the next section and read about another related bug, one with far more inscrutable behavior, and come back and read my solution another day. I should mention, by way of a de­ laying tactic, that the analysis of this bug was done by Jim Propp and Ander Holroyd. A previous and closely related bug of theirs, in which every third visit to i leads to i 1, appears in Peter Winkler's new book Mathemal'ical -

Figure 1 . The two bug bounces.

fering from publishers A K Peters [ 14]. The book itself is a gold-mine of the type of puzzles that I expect readers of this column would enjoy immensely. Winkler's solutions are insightful, well­ written, and often leave the reader with more to think about than before. The preceding is an unpaid endorsement. Very well, enough filler; here's my answer. If you solved the problem without developing something like the

-

Stanford, CA 94305-21 25, USA

Puzzles: A Connoisseur's Collection,

Figure 2. The bounces of (a) the first bug, (b)

e-mail: [email protected] .edu

yet another delightful mathematical of-

the fourth bug.

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27. Number 1. 2005

55

theory below, please do let me know how. Before we get to the serious work, let's answer the question, can a bug hop around in a bounded area forever? It cannot: let i be the minimal place vis­ ited infinitely often by the bug-oops, half the time, that visit is followed by a trip to i - 2, a contradiction. So each bug either lands in a cup or, as far as we know now, runs off to infinity. To motivate what follows, let's have a little inspiration: some carefully cho­ sen experimental data. Perhaps you no­ ticed that of the first five bugs, the cup at - 1 catches three, while the cup at 0 catches two. If that is not sufficiently suggestive, let me mention that after bug eight the score is 5 : 3, while after bug thirteen it is 8 : 5. On the specula­ tion that this golden ratio trend con­ tinues, let us refer to the bouncing in­ sects as goldbugs. Now the details. Suppose that r.p is­ I can only imagine your surprise-a real number satisfying r.p 2 = r.p + 1; when I care t o b e specific about which root, I will write 'P± for (1 ± V5)/2. The goldbugs and signs are, in fact, a number written in base r.p. Position i has "place value" r.pi , and the digits are conveniently mnemonic: Outbound is 0, Inbound is 1, and the bug itself is the not entirely standard digit r.p, which may appear in addition to the 0 or 1 in the "r.pi 's place," where it contributes r.pi + 1 to the total. Of course, numbers do not have a unique representation with this set-up, but that's quite delib­ erate: the total value is an invariant, un­ changed by bug bounces. •

•

56

Bounce left: Suppose the bug arrives in position i and there is an Out­ bound arrow. The value of this part of the configuration is ( r.pi X r.p) + ( r.pi X 0). After the bounce, position i holds an Inbound arrow and the bug has bounced to position i - 2, for a total value of ( r.pi X 1) + (r.pi-2 X r.p). And these are the same, since 'Pi l + 'Pi = 'Pi + 1 . Bounce right: Suppose the bug ar­ rives in position i and there is an In­ bound arrow. The value before the bounce is (r.pi X r.p) + (r.pi X 1). After the bounce, the arrow points Out­ bound, value zero, and the bug is at

THE MATHEMATICAL INTELLIGENCER

i + 1, for a value of r.pi +2 , again un­ changed.

-

Now let's see what happens when we add a bug at 1, let it run its course, and then remove it after it lands in a bug cup. Placing a bug at 1 increases the system's value by r.p 2. If it eventually lands in the cup at 0 and is then removed, the value of the system drops by r.p, while if it lands in the cup at - 1 and is removed, the value of the system drops by r.p0 1. So the net effect of adding a bug at 1, running the system, and then removing the bug is that the value of the system increases by
-

=

-

•

Can a bug run off to infinity? It can­ not: if we take r.p to mean 'P+ 1.61803, then each bug can increase the net value of the system by at most 'P+. and the positions far to the right are inaccessible to the gold­ bugs, because they would make the value of the system too high. So every bug lands in a cup. What is the ratio of bugs landing in the two cups? This time for r.p, think about 'P- .61803. Between bugs, when the system consists of just In­ bound and Outbound flags, its mini­ mum possible value is r.p 1 + r.p3 + r.p5 + · · · = - 1, while its maxi­ mum possible value is
=

=

•

tion is very good: if a bugs have landed at 1 and b have landed at 0, the system's value b - a/r.p+ must lie in our interval, so lb!a 11'P+ i < 11a. Every single approximation bla is one of the two best possible given the denominator, and that denomi­ nator grows as nlr.p+. In fact, notice that the length of the interval [ - 1, 1lr.p+ ] is the sum of the two jump sizes, the smallest we could possibly hope for. If the value lies in the bottom 11r.p+ of the inter­ val, it must increase by 1, while if it lies in the top 1 of the interval, it must decrease by 11'P+ · This leaves a single point, 1 lr.p+ - 1 .38297, where the bug's destination cup isn't determined. But that value can be at­ tained only with infinitely many In­ bound signs: if we run a single bug through a system with that starting value, its ending value would be ei­ ther 11 'P+ or - l-and to attain those two bounds, we found above, you must sum the infinite series of all positive or negative place values. So for any initial configuration with only finitely many signs pointing In­ bound, the configuration's numerical value alone determines the destina­ tion cup of every single bug; you don't need to keep track of the ar­ rows at all. If you prefer integers to irrationals, consider instead the following in­ variant. Label the (bug cups and the) sites with the Fibonacci numbers (0, 1), 1,2,3,5,8, . . . , as in Figure 3. Give an Inbound arrow the value F; of its site, and give the bug there the value F; + 1 of the site to its right. This invariant, of course, is an appropri­ ate linear combination of the 'P+ and 'P- ones above. Adding a bug to the system now increases the value by 2, but what's special is that removing the bug at ei­ ther 0 or 1 subtracts 1. So the bugs implement an accumulator: after the nth bug lands in its cup, we can look

=

_

_

_

_

_

_

=

u

u

0

(0

1)

1

I

0

I

•

-

-

I

<---

----->

<---

<---

<---

2

3

5

8

13

Figure 3. The signs after bug 1 1 7 passes through the system; 1 1 7

0

I

I

21

34

55

----->

=

2 + 5 + 8 + 1 3 + 34 + 55.

Representations in base Fibonacci are not unique; ours is characterized by a lack of two con­ secutive zeros.

at the signs it has left behind, and read off the number n, written in base Fibonacci! . . . Hold the presses! Matthew Cook has pointed out to me that this point of view can be taken further. We still get an invariant if we shift all those Fibonacci labels one site to the right. Now adding a bug increases the system's total value by 1, and re­ moving it from the right bug cup de­ creases the value by 1-but remov­ ing it from the left bug cup decreases the score by 0. So after n bugs pass through, the value of the Inbound ar­ rows they leave behind counts the number of bugs that ended their jour­ neys in the left bug cup. Furthermore we can shift the la­ bels right a second time. Now when the bug lands in the left cup, its value must be the - 1st Fibonacci number, before 0--which is 1 again, if the Fi­ bonacci recurrence relation is still to hold. So with these labels, the In­ bound arrows will count the number of bugs which landed in the right cup. As an exercise, decode the ar­ rows to learn that the 1 1 7 bugs lead­ ing to Figure 3 split 72:45. Once we know that the same set of arrows simultaneously counts the total, left-cup, and right-cup bugs, it's straightforward to see that the ratios of these three quantities are the same as the ratios of three consecutive Fi­ bonacci numbers, in the limit. These arrow-directed goldbugs are doing a great job of what Jim Propp calls "derandomization." It's straight­ forward to analyze the corresponding random walk, in which bugs hop left or right, each with probability 1/z: Let Pi be the probability that a bug at place i eventually ends up in the cup at - 1, and solve the recurrence Pi = CPi- 2 + Pi + l)/2 with P - 1 = 1, Po = 0-oh, and make up for one too few initial condi­ tions by remembering that all the Pi are probabilities, so no larger than 1. Here too we get p1 = 1/'P+· So the deterministic goldbugs have the same limiting behavior as their ran­ dom-walking cousins. But if we run the experiment with n random walkers and count the number of bugs in a cup, we'll generally see variation on the or-

der of Vn around the expected num­ ber. Remember the sharp results of the 'P- invariant: n goldbugs, by contrast, simulate the expected behavior with only constant error! There are more results on these and related one-dimensional not-very-ran­ dom walks. But let's move on-these bugs long for some higher-dimensional elbow room. The Rotor-Router

This time, we will set up a system with bugs moving around the integer points in the plane. It will differ from the above in that this will be an aggrega­ tion model. Bugs still get added re­ peatedly at one source, but instead of falling into sinks (bug cups), the bugs will walk around until they find an empty lattice point, then settle down and live there forever. Generalizing the Inbound and Out­ bound arrows of the goldbugs, we de­ cree that each lattice site is equipped with an arrow, or rotor, which can be

* * *

...

..

... �

w

t

t

..

.. _., , •

... , •

- - -

- -t�

it t

.ffi_ · _

·i

- - - - ......

..�

- - - - --

..

I

Figure 4. A bug's ramble through some ro­ tors: before and after.

rotated so that it points at any one of the four neighbors. (Propp uses the word "rotor" to refer to the two-state arrows in dimension 1 as well.) The first bug to arrive at a particular site occupies it forever, and we decree that it sets the arrow there pointing to the East. Any bug arriving at an occupied site rotates the arrow one quarter turn counterclockwise, and then moves to the neighbor at which the rotor now points-where it may find an empty site to inhabit, or it may find a new ar­ row directing its next step. In short, the first bug to reach (i, J) lives there, and bugs that arrive thereafter are routed by the rotor: first to the North, then West, South, and East, in that order, and then the cycle begins with North again (Fig. 4). Once a bug finds an empty site to in­ habit, we drop a new bug at the origin, and this one too meanders through the field of rotors, both directed by the ar­ rows and changing them as a result of its visits. Every bug does indeed fmd a home eventually, and the proof is the same as for the goldbugs: the set of all sites which a particular bug visits infi­ nitely often cannot have any boundary. And so we ask, as we add more and more bugs, what does the set of occu­ pied sites look like? Let's take a look at the experimen­ tal answer. The beautiful image you see in Figure 5 is a picture of the set of oc­ cupied sites after three million bugs have found their permanent homes . The sites in black are vacant, still awaiting their first visitor. Other sites are colored according to the direction of their rotor, red/yellow for East/West and green/blue for North/South. On the cover is a close-up of a part of the boundary of the occupied region. At http://www.math. wisc.edu/-propp/ rotor-router-1.0/ you can find a Java ap­ plet by Hal Canary and Francis Wong, if you'd like to experiment yourself. As you can see, the edge of the oc­ cupied region is extraordinarily round: with three million bugs, the occupied site furthest from the origin is at dis­ tance Y956609, while the unoccupied site nearest the origin is at distance Y953461, a difference of about 1.6106. And the internal coloration puts on a spectacular display of both large-

© 2005 Springer Sc�ence+Business Media, Inc.. Vc>ume 27. Number 1 . 2005

57

Figure 5. The rotor-router blob after 3 million bugs.

scale structure and intricate local pat­ terns. When Ed Pegg featured the ro­ tor-router on his Mathpuzzle Web site, he dubbed the picture a Propp Circle, and to this day I am jealous that I didn't think of the name first: it con­ notes precisely the right mixture of aesthetic appreciation and conviction that there must be something deep and not fully understood at work Recall, for emphasis, that this was formed by bugs walking deterministi­ cally on a square lattice, not a medium known for growing perfect circles.

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THE MATHEMATICAL INTELLIGENCER

Moreover, the rule that governed the bugs' movement is inherently asym­ metric: every site's rotor begins point­ ing East, so there was no guarantee that the set of occupied sites would even appear symmetric under rotation by 90°, much less by unfriendly angles. On the other hand, while the overall shape seems to have essentially for­ gotten the underlying lattice, the inter­ nal structure revealed by the color­ coded rotors clearly remembers it. Lionel Levine, now a graduate stu­ dent at U.C. Berkeley, wrote an under-

graduate thesis with Propp on the ro­ tor-router and related models (11]. It contains the best result so far on the roundness of the rotor-router blob: af­ ter n bugs, it contains a disk whose ra­ 1 dius grows as n 14. Below I report on two remarkable theorems which do not quite prove that the rotor-router blob is round, but which at least make me feel that it ought to be. I have less to offer on the intricate internal struc­ ture, but there is a connection to some­ thing a bit better understood. Let's get to work

IDLA

Internal Diffusion Limited Aggregation is the random walk-based model which Propp de-randomized to get the rotor­ router. Most everything is as above­ the plane starts empty, add bugs to the origin one at a time, each bug occupies the first empty site it reaches. But in IDLA, a bug at an occupied site walks to a random neighbor, each with prob­ ability 1/4. The idea underlying IDLA comes from a paper by Diaconis and Fulton [6]. They define a ring structure on the vector space whose basis is labeled by the finite subsets of a set X equipped with a random walk. To calculate the product of subsets A and B, begin with the set A U B, place bugs at each point of A n B, and allow each bug to exe­ cute a random walk until it reaches an outside point, which is then added to the set. The product of A and B records the probability distribution of possible outcomes. This appears to depend on the order in which the bugs do their random walks, but in fact it does not­ we'll explore this theme soon. Consider the special case where X is the d-dimensional integer lattice with the random walk choosing uni­ formly from among the nearest neigh­ bors. Then repeatedly multiplying the singleton {0} by itself is precisely the random-walk version of the rotor­ router. A paper of Lawler, Bramson, and Griffeath [9] dubbed this Internal Diffusion Limited Aggregation, to em­ phasize similarity with the widely-stud­ ied Diffusion Limited Aggregation model of Witten and Sandler [ 13]. DLA simulates, for example, the growth of dust: successive particles wander in "from infinity" and stick when they reach a central growing blob. The re­ sulting growths appear dendritic and fractal-like, but rigorous results are hard to come by. In contrast, the growth behavior of IDLA has been rigorously established. It is intuitive that the growing blob should be generally disk-shaped, since the next particle is more likely to fill in an un­ occupied site close to the origin than one further away. But the precise state­ ment in [9] is still striking: the random walk manages to forget the anisotropy of the underlying lattice entirely!

Theorem (Lawler-Bransom-Grif­ feath). Let wd be the volume of the d­

dimensional unit sphere. Given any E>0, it is true with probability 1 that for all sufficiently large n, the d-di­ mensional IDLA blob of wdnd particles will contain every point in a ball of radius ( 1 - E)n, and no point outside of a ball of radius (1 + E)n. To be more specific, we could hope to define inner and outer error terms such that, with probability 1 , the blob lies between the balls of radius n {lj(n) and n + 80(n). In a subsequent paper [10], Lawler proved that these 1 could be taken on the order of n 13• Most recently, Blachere [3] used an in­ duction argument based on Lawler's proof to show that these error terms were even smaller, of logarithmic size. The precise form of the bound changes with dimension; when d = 2 he shows that Mn) O((ln n ln(ln n)) 112) and 80(n) = O((ln n)2). Errors on that or­ der were observed experimentally by Moore and Machta [ 12]. So how does the random walk-based IDLA relate to the deter­ ministic rotor-router? I start drawing the connection with one key fact. =

It's Abelian!

Here's a possibly unexpected property of the rotor-router model: it's Abelian. There are several senses in which this is true. Most simply, take a state of the ro­ tor-router system-a set of occupied sites and the directions all the rotors point-and add one bug at a point Po (not necessarily the origin now) and let it run around and find its home P1 . Then add another at Q0 and let it run until it stops at Q 1 . The end state is the same as the result of adding the two bugs in the opposite order. This relies on the fact that the bugs are indistinguishable. Consider the (next-to-)simplest case, in which the paths of the P and Q bugs cross at ex­ actly one point, R. If bug Q goes first instead, it travels from Q0 to R, and then follows the path the P bug would have, from R to P1 . The P bug then goes from P0 to R to Q1. At the place where their paths would first cross, the bugs effectively switch identities. For paths

whose intersections are more compli­ cated, we need to do a bit more work, but the basic idea carries us through. Taking this to an extreme, consider the "rotor-router swarm" variant, where traffic is still directed by rotors at each lattice site, but any number of bugs can pass through a site simulta­ neously. The system evolves by choos­ ing any one bug at random and moving it one step, following the usual rotor rule. Here too the final state is inde­ pendent of the order in which bugs move; read on for a proof. To create our rotor-router picture, we can place three million bugs at the origin simul­ taneously, and let them move one step at a time, following the rotors, in what­ ever order they like. In fact, even strictly following the rotors is unnecessary. The rotors con­ trol the order in which the bugs depart for the various neighbors, but in the end, we only care about how many bugs head in each direction. Imagine the following set-up: we run the original rotor-router with three mil­ lion bugs as first described, but each time a bug leaves a site, it drops a card there which reads "I went North," or whichever direction. Now forget about the bugs, and look only at the collec­ tion of cards left behind at each site. This certainly determines the final state of the system: a site ends up oc­ cupied if and only if one of its neigh­ bors has a card pointing toward it. Now we could re-run the system with no rotors at all. When a bug needs to move on, it may pick up any card from the site it's on and move in the in­ dicated direction, eating the card in the process. No bug can ever "get stuck" by arriving at an occupied site with no card to tell it a way to leave: the stack of cards at a given site is just the right size to take care of all the bugs that can possibly arrive there coming from all of the neighbors. (There is, however, no guarantee that all the cards will get used; left-{)vers must form loops.) A version of this "stacks of cards" idea appeared in Diaconis and Fulton's original paper, in the proof that the random-walk version is likewise Abelian-i.e., that their prod­ uct operation is well defined. If the bugs are so polite as to take the cards in the cyclic N-W-S-E order

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in which they were dropped, then we simulate the rotor-router exactly. If we start all the bugs at the origin at once and let them move in whatever order they want-but insist that they always use the top card from the site's stack­ we get the rotor-router swarm variant above; QED. Rotor-Roundness

Now let me outline a heuristic argu­ ment that the rotor-router blob ought to be round, letting the Lawler-Bram­ son-Griffeath paper do all the heavy lifting. I'd like to say that, for any c < 1, the n-bug rotor-router blob contains every lattice site in the disk of area en-as long as n is sufficiently large. My strat­ egy is easy to describe. Just as we did four paragraphs ago, think of each lat­ tice site as holding a giant stack of cards: one card for each time a bug de­ parted that site while the n-bug rotor­ router blob grew. Now we start run­ ning IDLA: we add bugs at the origin, one at a time, and let them execute their random walks. But each time a bug randomly decides to step in a given direction, it must first look through the stack of cards at its site, find a card with that direction written on it, and destroy it. As long as the randomly walking bugs always find the cards they look for, the IDLA blob that they generate must be a subset of the rotor-router blob whose growth is recorded in the stacks of cards. This key fact follows directly from the Abelian nature of the models. So the central question is, how long will this IDLA get to run before a bug wants to step in a particular direction and fmds that there is no correspond­ ing card available? Philosophically, we expect the IDLA to run through "almost all" the bugs without hitting such a snag: for any c < 1, we expect en bugs to ag­ gregate, as long as n is sufficiently large. If we can show this, we are certainly done: the rotor-router blob contains an IDLA blob of nearly the same area, which in turn contains a disk of nearly the same area, with probability one. To justify this intuition, we clearly need to examine the function d(iJ) which counts the number of depar-

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THE MATHEMATICAL INTELLIGENCER

tures from each site. This is a nonneg­ ative integer-valued function on the lattice which is almost harmonic, away from the origin: the number of depar­ tures from a given site is about one quarter of the total number of visits to its four neighbors.

d(iJ)

=

± (d(i + 1,J} + d(i - 1,J} + d(i,j + 1) + d(i,j - 1)) - b(iJ)

Here b(iJ) = 1 if (i,J} is occupied and 0 otherwise, to account for the site's first bug, which arrives but never departs. When (i,J} is the origin, of course, the right-hand side should be increased by the number of bugs dropped into the system. Matthew Cook calls this the "tent equation": each site is forced to be a little lower than the average of its neighbors, like the heavy fabric of a cir­ cus tent; it's all held up by the circus pole at the origin-or perhaps by a bundle of helium balloons which can each lift one unit of tent fabric, since we do not get to specify the height of the origin, but rather how much higher it is than its surroundings. For the rotor-router, the approxima­ tion sign above hides some rounding er­ ror, the precise details of which encap­ sulate the rotor-router rule. For IDLA, this is exact if we replace d by a, the ex­ pected number of departures, and re­ place b by b, the probability that a given site ends up occupied. (The results of [9] even give an approximation of d.) Now, I'd like to say that at any par­ ticular site, the mean number of de­ partures for an IDLA of en bugs (for any c < 1 and large n) should be less than the actual number of departures for a rotor-router of n bugs. If so, we'd be nearly done, with a just a bit of easy calculation to show that the Vd-sized error term at each site in the random walk is thoroughly swamped by the (1 - c)n extra bugs in the rotor-router. But this begs the question of show­ ing that the rotor-router's function d and the IDLA's function d are really the same general shape. Their difference is an everywhere almost-harmonic func­ tion with zero at the boundary-but to paraphrase Mark Twain, the difference between a harmonic function and an almost-harmonic function is the differ­ ence between lightning and a lightning bug.

Simulation with Constant Error

After I wrote the preceding section, I learned of a brand-new result of Joshua Cooper and Joel Spencer. It doesn't tum my hand-waving into a genuine proof, but it gives me hope that doing so is within reach. Their paper [4] con­ tains an amazing result on the rela­ tionship between a random walk and a rotor-router walk in the d-dimensional integer lattice ll_d. Generalizing the rotor-router bugs above, consider a lattice ll_d in which each point is equipped with a rotor­ that is to say, an arrow which points towards one of the 2d neighboring points, and which can be incremented repeatedly, causing it to point to all 2d neighbors in some fixed cyclic order. The initial states of the rotors can be set arbitrarily. Now distribute some finite number of bugs arbitrarily on the points. We can let this distribution evolve with the bugs following the rotors: one step of evolution consists of every bug incre­ menting and then following the rotor at the point it is on. (Our previous bugs were content to stay put if they were at an uninhabited site, but in this ver­ sion, every bug moves on.) Given any initial distribution of bugs and any ini­ tial configuration of the rotors, we can now talk about the result of n steps of rotor-based evolution. On the other hand, given the same initial distribution of bugs, we could just as well allow each bug to take an n-step random walk, with no rotors to influence its movement. If you believe my heuristic babbling above, then it is reasonable to hope that n steps of ro­ tor evolution and n steps of random walk would lead to similar ending dis­ tributions. With one further assumption, this turns out to be true in the strongest of senses. Call a distribution of bugs "checkered" if all bugs are on vertices of the same parity-that is, the bugs would all be on matching squares if ll_d were colored like a checkerboard. Theorem (Cooper-Spencer). There is an absolute constant bounding the divergence between the rotor and 'ran­ dom-walk evolution of checkered dis­ tributions in ll_d, depending only on

Figure 6. The greedy sand-pile with three million grains.

Figure 7. A non-greedy sand-pile. Here the dominant color is yellow, which again indi­ cates the maximal stable site, now with three grains. It is hard to see the interior pix­ els colored black, indicating sites which were once filled but are now empty, impossi­ ble in the greedy version.

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the dimension d. That is, given any checkered initial distribution of a fi­ nite number of bugs in 7l_d, the differ­ ence between the actual number of bugs at a point p after n steps of ro­ tor-based evolution, and the expected number of bugs at p after an n-step random walk, is bounded by a con­ stant. This constant is independent of the number of steps n, the initial states of the rotors, and the initial dis­ tribution of bugs. I am enchanted by the reach of this result, and at the same time intrigued by the subtle "checkered" hypothesis on distributions. (Not only initial dis­ tributions: since each bug changes par­ ity at each time step, a configuration can never escape its checkered past.) The authors tell me that without this assumption, one can cleverly arrange squadrons of off-parity bugs to reorient the rotors and steer things away from random walk simulation. Thus the rotor-router deterministi­ cally simulates a random walk process with constant error-better than a sin­ gle instance of the random process usually does in simulating the average behavior. Recall that we saw a similar outcome in one dimension, with the goldbugs. There are other results which like­ wise demonstrate that derandomizing systems can reduce the error. Lionel Levine's thesis [ 1 1 ] analyzed a type of one-dimensional derandomized aggre­ gation model, and showed that it can compute quadratic irrationals with constant error, again improving on the Vn-sized error of random trials. Joel Spencer tells me that he can use an­ other sort of derandomized one-di­ mensional system to generate binomial distributions with errors of size In n instead of Vn. Surely the rotor-router should be able to cut IDLA's already logarithmic-sized variations down to constant ones. Right? Coda: Sandpiles

All of the preceding discussion ad­ dresses the overall shape of the rotor­ router blob, but says nothing at all about the compelling internal structure that's visible when we four-color the points according to the directions of

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THE MATHEMATICAL INTELLIGENCER

the rotors. When we introduced the function d(i,J), counting the number of departures from the (i,J) lattice site, we were concerned with its approxi­ mate large-scale shape, which exhibits some sort of radial symmetry. The di­ rection of the rotor tells you the value of d(i, J) mod 4, and the symmetry of these least significant bits of d is an en­ tirely new surprise. I can't even begin to explain the fine structure-if you can, please let me know! But I can point out a surprising connection to another discrete dynam­ ical system, also with pretty pictures. Consider once again the integer points in the plane. Each point now holds a pile of sand. There's not much room, so if any pile has five or more grains of sand, it collapses, with four grains sliding off of it and getting dumped on the point's four neighbors. This may, in tum, make some neigh­ boring piles unstable and cause further topplings, and so on, until each pile has size at most four. Our question: what happens if you put, say, a million grains of sand at the origin, and wait for the resulting avalanche to stop? I won't keep you hanging; a picture of the resulting rub­ ble appears as Figure 6. Pixels are col­ ored according to the number of grains of sand there in the final configuration. The dominant blue color correspond­ ing to the largest stable pile, four grains. (This makes some sense, as the interior of such a region is stable, with each site both gaining and losing four grains, while evolution happens around the edges.) This type of evolving system now goes under the names "chip-firing model" and "abelian sandpile model"; the adjective abelian is earned because the operations of collapsing the piles at two different sites commute. In full generality, this can take place on an ar­ bitrary graph, with an excessively large sand-pile giving any number of grains of sand to each of its neighbors, and some grains possibly disappearing per­ manently from the system. Variations have been investigated by combina­ torists since about 1991 [2]; they adopted it from the mathematical physics community, which had been developing versions since around 1987

[ 1,5]. This too was a rediscovery, as it seems that the mechanism was first de­ scribed, under the name "the proba­ bilistic abacus," by Arthur Engel in 1975 in a math education journal [7,8]. I couldn't hope to survey the current state of this field here, or even give proper references. The bulk of the work appears to be on what I think of as steady-state questions, far from the effects of initial conditions: point-to­ point correlation functions, the distri­ bution of sizes of avalanches, or a mar­ velous abelian group structure on a certain set of recurrent configurations. Our question seems to have a dif­ ferent flavor. For example, in most sandpile work, one can assume with­ out loss of generality that a pile col­ lapses as soon as it has enough grains of sand to give its neighbors what they are owed, leaving itself vacant. The ver­ sion I described above is what I'll call a "greedy sandpile," in which each site hoards its first grain of sand, never let­ ting it go. The shape of the rubble in Figure 6 does depend on this detail; Figure 7 is the analogue where a pile collapses as soon as it has four grains, leaving itself empty. Most compelling to me is the fine structure of the sandpile picture. I'm amazed by the appearance of fractalish self-similarity at different scales de­ spite the single-scale evolution rule; I think this is related to what the math­ ematical physics people call "self-or­ ganizing criticality," about which I know nothing at all. But personally, in both pictures I am drawn to the eight­ petalled central rosette, the boundary of some sort of phase change in their internal structures. Bugs in the Sand

So what is the connection between the greedy sandpile and the rotor-router? Recall the swarm variant of rotor­ router evolution: we can place all the bugs at the origin simultaneously, and let them take steps following the rotor rule in any order, and still get the same final state. Since we get to choose the order, what if we repeatedly pick a site with at least four bugs waiting to move on, and tell four of them to take one step each? Regardless of its state, the rotor

directs one to each neighbor, and we mimic the evolution rule of the greedy sandpile perfectly. If we keep doing this until no such sites remain, we re­ alize the sandpile final state as one step along one path to the rotor-router blob. Note, in particular, that the n-bug rotor-router blob must contain all sites in the n-grain greedy sandpile. Surely it should therefore be possible to show that both contain a disk whose radius grows as Vn. More emphatically, the sandpile per­ forms precisely that part of the evolution of the rotor-router that can take place without asking the rotors to break sym­ metry. If we define an energy ftmction which is large when multiple bugs share a site, then the sandpile is the lowest-en­ ergy state which the rotor-router can get to in a completely symmetric way. When we invoke the rotors, we can get to a state with minimal energy but without the a priori symmetry that the sandpile evolution rule guarantees. And yet, empirically, the rotor-router final state looks much rounder than that of the sandpile, whose boundary has clear horizontal, vertical, and slope ::+:: 1 segments. At best, this only hints at why the sandpile and rotor-router internal structures seem to have something in common. For now, these hints are the best I can do.

mutative algebra and algebraic geometry, II

Acknowledgments

Thanks most of all to Jim Propp, who introduced me to this lovely material, showed me much of what appears here, and allowed and encouraged me to help spread the word. Fond thanks to Tetsuji Miwa, whose hospitality at Kyoto University gave me the time to think and write about it. Thanks also to Joshua Cooper, Joel Spencer, and Matthew Cook, for sharing helpful comments and insights.

(Turin, 1 990). Rend. Sem. Mat. Univ. Po­ litec. Torino 49 (1 991 ), no. 1 , 95-1 1 9 (1 993).

[7] Engel, Arthur. The probabilistic abacus. Ed. Stud. Math. 6 (1 975), 1 -22.

[8] Engel, Arthur. Why does the probabilistic abacus work? Ed. Stud. Math.

7

(1 976),

59-69. [9] Lawler, Gregory; Bramson, Maury; Grif­ feath, David. Internal diffusion limited ag­ gregation. Ann. Probab. 20 (1 992), no. 4, 21 1 7-21 40. 1 0. Lawler, Gregory. Subdiffusive fluctuations

REFERENCES

for internal diffusion limited aggregation.

[ 1 ] Bak, Per; Tang, Chao; Wiesenfeld, Kurt. Self-organized criticality. Phys. Rev. A (3) 38 (1 988), no. 1 , 364-374.

Ann. Probab. 23 (1 995), no. 1 , 7 1 -86.

1 1 . Levine, Lionel. The Rotor-Router Model. Harvard University senior thesis. Preprint

[2] Bjorner, Anders; Lovasz, Laszlo; Shor, Pe­

arXiv:math.C0/0409407 (September 2004).

ter. Chip-firing games on graphs. European

1 2. Moore, Christopher; Machta, Jonathan. In­

J. Combin. 1 2 (1 99 1 ) , no. 4 , 283-291 .

[3] Blachere, Sebastien. Logarithmic fluctua­ tions for the Internal Diffusion Limited Ag­ gregation. Preprint arXiv:rnath.PR/01 1 1 253 (November 2001).

ternal diffusion-limited aggregation: paral­ lel algorithms and complexity. J. Statist. Phys. 99 (2000), no. 3-4, 661 -690.

1 3. Witten, T. A . ; Sander, L. M. Diffusion­ limited aggregation. Phys. Rev. B (3)

[4] Cooper, Joshua; Spencer, Joel. Simulat­

27

(1 983), no. 9, 5686-5697.

ing a Random Walk with Constant Error.

1 4. Winkler, Peter. Mathematical Puzzles: A

Preprint arXiv:math C0/0402323 (Febru­

Connoisseur's Collection. A K Peters Ltd,

ary 2004); to appear in Combinatorics,

Natick, MA, 2003.

Probability and Computing.

[5] Dhar, Deepak. Self-organized critical state of sandpile automation models. Phys. Rev.

The Broad Institute at MIT

Lett. 64 (1 990), no. 1 4, 1 61 3-1 61 6.

320 Charles Street

[6] Diaconis, Persi; Fulton, William. A growth

Cambridge, MA 021 41

model, a garne, an algebra, Lagrange in­

USA

version, and characteristic classes. Com-

e-mail: [email protected]

© 2005 Springer Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1 , 2005

63

BRUNO DURAND, LEONID LEVI N, AN D ALEXANDER SHEN

Local R u l es an d G l o bal Ord e r , O r Aperi od i c Ti l 1 n g s

an local rules impose a global order? If yes, when and how? This is a philo­ sophical question that could be asked in many cases. How does local interaction of atoms create crystals (or quasicrystals) ? How does one living cell manage to develop into a pine cone whose seeds form spirals (and the number of spirals usually is a Fibonacci number)? Is it possible to program locally connected computers in such a way that the net­ work is still functional if a small fraction of the nodes is corrupted? Is it possible for a big team of people (or ants), each trying to reach private goals, to behave reasonably? These questions range from theology to "political sci­ ence" and are rather difficult. In mathematics the most prominent example of this kind is the so-called Berger the­ orem on aperiodic tilings (exact statement below). It was proved by Berger in 1966 [ 1 ] . 1 In 1971 the proof was sim­ plified by Robinson [7], who invented the well-known "Robinson tiles" that can tile the entire plane but only in an aperiodic way (Fig. 1 ). Since then many similar constructions have been in­ vented (see, e.g., [3, 6]); some other proofs were based on different ideas (e.g., [4]). However, we did not manage to fmd a publication which provides a short but complete proof of the theorem: Robinson tiles look simple, but when you

00 0 00 0

Fig. 1 . The Robinson tiles [reflections and rotations are allowed].

start to analyze them you have to deal with many technical details. ("This argument is a bit long and is not used in the remainder of the text, so it could be skipped on first read­ ing," says C. Radin in [6] about the proof.) It's a pity, however, to skip the proof of a nice theorem whose statement can be understood by a high school stu­ dent (unlike the Fermat Theorem, you don't even need to know anything about exponentiation). We try to fill this gap

1 1n tact, the motivation at that time was related to the undecidability of a specific class of first-order formulas, see [2] .

64

THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Sc1ence+Bus1ness Med1a, Inc.

and provide a simple construction of an aperiodic tiling with a complete proof, making the argument as simple as possible (at the cost of increasing the number of tiles). Of course, simplicity is a matter of taste, so we can only hope you will find this argument simple and nice. If not, you can look at an alternative approach in [5]. Definitions

Let A be a finite (nonempty) alphabet. A configuration is an infinite cell paper where each cell is occupied by a let­ ter from A; formally, the configuration is a mapping of type 7L2 ----> A. A local rule is an arbitrary subset L C A4 whose elements are considered as 2 X 2 squares:
� [;I;]

We say that these squares are allowed by rule L. A config­ uration T satisfies local rule L if all 2 X 2 squares in it are allowed by L. Formally this means that


=

E

L

(t1.t2) is ape­

for any x1,x2 E 7L. The Aperiodic Tilings theorem

Theorem (Berger):

There exist an alphabet A and a local

rule L such that (1) (2)

there are tilings that satisfy L; any tiling satisfying L has no period.

To prove the theorem we need some auxiliary defini­ tions. Substitution Mappings

A substitution is a mapping s of type A ----> A4 whose val­ ues are considered as 2 X 2 squares:

s: a �

s1(a)

s2 (a)

s3 (a)

s4(a)

We say that a substitution s matches local rule L if two con­ ditions are satisfied: (a) all values of s belong to L; (b) taking any square from L and replacing each of the four cells by its s-image, we get a 4 X 4 square that satis­ fies L (this means that all nine 2 X 2 squares inside it be­ long to L). Remark. Consider a square X of any size N X N (filled with letters from A) satisfying L. Apply substitution s to each letter in X and obtain a square Y of size 2N X 2N. If the substitution s matches L, then Y satisfies L. Indeed, any

2 X 2 square in Y is covered by an image of some 2 X 2 square in X. This is true also for (infinite) configurations: applying a substitution to each cell of a configuration that satisfies L, we get a new configuration that satisfies L (assuming that the substitution matches L). Proposition 1. If a substitution s matches a local rule L, there exists a configuration T that satisfies L. Proof Take any letter a E A and apply s to it. We get a 2 X 2 square s(a) that belongs to L. Then apply s to all let­ ters in s(a) and get a 4 X 4 square s(s(a)) that satisfies L. Next is an 8 X 8 square s(s(s(a))) that satisfies L, etc. Us­ ing a compactness argument, we conclude that there ex­ ists an infinite configuration that satisfies L. Here is a direct proof not referring to compactness. As­ sume that substitution s is fixed. A letter a' is a descendant of a letter a, if a' appears in the interior part of some square s(s( . . . s(a) . . . )) obtained from a. Each letter has at least one descendant, and the descendent relation is transitive (if a' is a descendant of a and a" is a descendant of a', then a" is a descendant of a). Therefore, some letter is a de­ scendant of itself (start from any letter and consider de­ scendants until you get a loop). If a appears in the interior part of sCn)(a), then sCn)(a) appears in the interior part of sC2n)(a), which appears (in its tum) in the interior part of sC3n)(a), and so on. Now we get a increasing sequence of squares that extend each other and together form a con­ figuration. (Here we use that a appears in the interior part of the square obtained from a. ) Proposition 1 is proved. Now we formulate requirements for substitution s and local rule L which guarantee that any configuration satis­ fying L is aperiodic. They can be called "self-similarity" re­ quirements, and guarantee that any configuration satisfy­ ing L can be uniquely divided (by vertical and horizontal lines) into 2 X 2 squares that are images of some letters un­ der s, and that these pre-image letters form a configuration that satisfies L. Here is the exact formulation of the re­ quirements: (a) s is injective (different letters are mapped into dif­ ferent squares); (b) the ranges of mappings s1.s2 ,s3,s : A ----> A (that cor­ 4 respond to the positions in a 2 X 2 square, see above) are disjoint; (c) any configuration satisfying L can be split by hori­ zontal and vertical lines into 2 X 2 squares that belong to the range of s, and pre-images of these squares form a con­ figuration that satisfies L. The requirement (b) guarantees that there is only one way to divide the configuration into 2 X 2 squares; the re­ quirement (a) then guarantees that each square has a unique preimage. Proposition 2. Assume that substitution s and local rule L satisfy requirements (a), (b) and (c). Then any configu­ ration satisfying L is aperiodic. Pmof Let T be a configuration satisfying L and let t = (t1,t2) be its period. Both t1 and t2 are even numbers. In­ deed, (c) guarantees that T can be split into 2 X 2 squares,

© 2005 Spnnger SC1ence+ Bus1ness Media, Inc . Volume 27, Number 1 . 2005

65

and then (b) guarantees that the t-shift preseiVes these squares (since, say, an upper left corner of a square must go to another upper left corner). Then (a) guarantees that pre-images of these 2 X 2 squares form a configuration that satisfies L and has pe­ riod t/2. Therefore, for each periodic L-configuration with period t we have found another periodic L-configuration with period t/2. An induction argument shows that there are no periodic L-configurations. Proposition 2 is proved. Using Propositions 1 and 2 we conclude that to prove the Aperiodic Tilings theorem it is enough to construct a local rule L and substitution s matching L that satisfy (a), (b) and (c). This we now do.

....... .....

..... ......

Fig. 3. A tile split i n four parts.

It is immediately clear that conditions (a) and (b) of Proposition 2 are satisfied. Indeed, to reconstruct tile x from its four parts, it is enough to erase some lines, and the position of a tile in s(x) is uniquely determined by the orientation of its central cross. The condition (c) will be checked later after the local rule is defined.

Construction: An Alphabet

Letters of A are considered as square tiles with some draw­ ings on them. We describe a local rule and substitution in terms of these drawings. Each of the four sides of a tile (1) is dark or light (has one of two possible colors); (2) has one of two possible directions, indicated by arrows; (3) has one of two possible orientations; this means that one of two possible orthogonal vectors is fixed; we say that this orthogonal vector goes "from inside to outside". (Our drawings show the orientation by a gray shading inside.) In this way we get three bits per side, i.e., 12 bits for each tile. In addition to these 12 bits, a tile carries two more bits, so the size of our alphabet is 2 14 = 8192. These two additional bits are graphically represented as follows: we draw a cross (Fig. 2) in one of four versions (which differ by a rotation).

Fig. 2. One version of cross.

It is convenient to assign color, direction, and orienta­ tion to the segments that forming a cross. Namely, two neighboring sides of a cross are dark, the other two are light. The direction arrows go from the center outward, and the orientation is shown by a gray stripe that shows the "in­ side" part as indicated in the picture (gray stripes are in­ side the dark angle). This will be important when we define the substitution. Substitution

To perform the substitutions, we cut a tile into four tiles. The middle lines of the tile become sides of the new (smaller) tiles, with the same color, direction and orientation. Before cutting we draw crosses on the small tiles in such a way that the dark angles form a square as shown (Fig. 3).

66

THE MATHEMATICAL INTELLIGENCER

Local Rule

The local rule (L) is formulated in terms of lines and their crossings. There are two types of crossings that appear when tiles meet each other. First, a crossing appears at the point where corners of four tiles meet; crossing lines are formed by the tile sides. Second, a crossing appears at the middle of tile sides, where middle lines of tiles meet the tile side. First of all, the following requirement is put: if two tiles have a common side, this shared side has the same color, direction, and orientation in both tiles. Therefore, we can speak about the color, direction and orientation of a boundary line between two tiles without specifying which of the two tiles is considered. We also require that all crossings (of both types) are either crosses or meeting points. A cross is formed by four outgoing arrows that have colors and orientation as shown in Fig. 4 (up to a rotation, so there are four types of crosses). In a meeting point, two arrows of the same color, the same orientation, but oppo­ site directions, meet "face to face," and the orthogonal line goes through this meeting point without change in color, direction, or orientation. One more restriction is put: if two dark arrows meet, then the orthogonal line goes "outward" (its direction agrees with the orientation of the arrows). Our local rule is formulated in terms of restrictions say­ ing which crossings are allowed when lines meet. Formally speaking, the local rule is a set of all quadruples of tiles where these restrictions are not violated. Fig. 4 shows the first type of allowed crossing, a cross, in one of four pos­ sible versions (which differ by a rotation). The second type

+ '

Fig. 4. A cross formed by outgoing arrows.

.... . ..... . . . . . .. . . .. . ... .

.. .. . .

t

·········

-!"

Fig. 5. Arrows meet.

t

·········

Fig. 7. Two neighbor tiles.

of allowed crossings (symbolically shown in Fig. 5) has more variations: (a) the meeting arrows can be horizontal or vertical; (b) the vertical line can have two orientations; (c) the horizontal line can have two orientations; (d) the vertical line can have one of two colors; (e) the horizontal line can have one of two colors; and finally (f ) if two light arrows meet, the perpendicular line can go in either of two directions. So we get 2 2 2 2 3 = 48 variations in this way. Remark. The Local rule ensures that the orientation of any horizontal or vertical line remains unchanged along the whole line. (Indeed, the orientation does not change at crosses or meeting points.)

groups of four tiles whose central lines form a dark square (Fig. 8).

Substitution and Local Rule

Step 2. These 2 X 2 squares are aligned.

·

·

·

·

We have to check that the substitution matches the local rule. Indeed, when tiles are split into groups of four, the old lines still form the same crossings as before, but new crossings appear. These new crossings appear (a) in the centers of new tiles (where new lines cross new ones) and (b) at the midpoints of sides of new tiles (where new lines cross old ones). In case (a) we have legal crosses by defi­ nition. In case (b) it is easy to see that two arrows meet creating a legal meeting point. See Fig. 6, which shows a tile split into four tiles, with all possible meeting places of new and old lines circled. The orientation matches because the orientation of the new crosses is fixed by s; all other requirements are fulfilled, too.

:· ··········· · ····· : ·················:

f EEl ! ..

.. . . . . ... . . .·. . ........... . ....

.. . ·

..

.... .. .

. ..

.

Fig. 8. Four adjacent tiles.

If two groups (each forming a 2 X 2 square) were wrongly aligned, as shown in Fig. 9, then the orientation of one of the lines (in our example, the horizontal line) would change along the line (recall that all crosses have fixed orientation of lines). Therefore, 2 X 2 squares are aligned.

Fig. 9. Bad placement.

Step 3. Each group has a cross in the middle.

Fig. 6. New lines meet old lines.

What can be in the group center? The middle points of the sides of the dark square are meeting points for dark arrows. Therefore, according to the local rule, an outgoing arrow should be between them. So a meeting point cannot appear in the center of a 2 X 2 group, and the only possibility is a cross. Step 4. Uniform colors on sides.

Step 1 . Tiles are grouped by fours.

To finish the proof that each group belongs to the range of the substitution, it remains to show that the color, direc­ tion, and orientation do not change at the midpoint of a side of a 2 X 2 group. This is because this midpoint is a meeting point for arrows perpendicular to the side.

Consider an arbitrary tile in this configuration and a dark arrow that goes outward. It meets another arrow from a neighboring tile, and this arrow must be dark by the local rule. These two arrows must have the same orientation, therefore we get half of a dark square (Fig. 7), not a Z­ shape. Repeating this argument, we conclude that tiles form

This is evident: the substitution adds new lines. So taking the pre-image just means that some lines are deleted, and no violation of the local rule can happen. The Aperiodic THings theorem is proved.

Self-similarity Condition

It remains to check condition (c) of Proposition 2. Assume that we have a configuration that satisfies the local rule.

Step 5. Pre-image tiles satisfy the local rule.

© 2005 Spnnger Sc1ence + Bus1ness Media, Inc , Volume 27, Number 1, 2005

67

A U T H O R S

ALEXANDER SHEN

BRUNO DURAND

LENOID LEVIN

ul. Begovaya, 1 7- 1 4

Laboratoire d'lnformatique Fondamentale de

Department of Computer Science

Moscow, 1 25284

Marseille

Boston University

RusSia

CMI, 39 rue Joliot-Curie

e-mail: [email protected]

1 3453 Marsei l le Cedex 1 3

Boston, MA 022 1 5-24 1 1 USA

France Alexander Shen has been since 1 982 at the

e-mail: [email protected]

Institute for Problems of Information Trans­

Leonid Levin has worked rncstly in theory of was one

mission in Moscow. He is known to lntelli­

Educated in Computer Science at the

compU1ation. He

gencer readers as aU1hor of several contri­

Ecole Normale Superieure de Lyon,

of the P-NP conjecture, whose monetary

butions

and

as

now

of the originators

Mathematical

Durand is now both Professor at the Uni­

Entertainments Editor. He has written text­

versite de Provence and Director of the

whose scientific value is not compU1able.

books based on courses at Moscow State

Laboratoire d'lnformatique Fondamentale

Readers curious aboU1 his more recent

University and at the Independent Univer­

de Marseille. His research is on cellular au­

thoughts are invited to his Web site, http://

tomata, tilings, and complexity. He is an

www .cs.bu.edu/fac/lnd.

sity of Moscow:

former

Bruno

see

ftp.mccme.ru/usersl

shen. Some have been translated into En­

price-tag has

grown to $1 CXXXXlO, bU1

editor of Theoretical Computer Science.

g l ish : Algorithms and Programming: Prob­

lems and Solutions, Bi rkhauser, 1 997; and

two from the American Mathematical Society.

REFERENCES

[4] J. Kari, A small aperiodic set of Wang tiles, Discrete Math.,

Amer. Math. Soc.,

66

160

(1 996), 259-264.

[ 1 ] R. Berger, The undecidability of the domino problem, Memoirs

[5] Leonid A Levin, Aperiodic Tilings: Breaking Translational Symme­

(1 966), 1 -72.

try, http:l/arxiv.org/abs/cs/0409024.

[2] E. Borger, E. Gradel, Yu. Gurevich, The Classical Decision Problem, Springer, 1 997. [Berger's theorem is considered in the Appendix

[6] C. Radin, Miles of Tiles, AMS, 1 999 (Student Mathematical Library, vol. 1 ).

written by C. Allauzen and B. Durand.] [3] B. Grunbaum, C. G. Shephard. Tilings and Patterns, Freeman, New

[7] R. Robinson, Undecidability and nonperiodicity of tilings in the plane, Invent. Math. ,

York, 1 986.

12

( 1 9 7 1 ) , 1 77-209.

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li,iM.�ffl!•i§u6'h£ili.Jiiijbl

The Home of Golden Numberism Roger Herz-Fischler

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caje where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit 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]

T

D i rk H uylebrouck, Ed itor

he expression "golden numberism" refers to the set of claims con­ cerning the purported use of the golden number (division in extreme ratio, golden section, golden ratio, . . . ) in man-made objects (art, architecture, etc.) or its purported appearance in na­ ture (human body, plants, astronomy, etc.). If we leave aside the vague state­ ments by Kepler, the early nineteenth­ century association of the golden num­ ber with phyllotaxis, and a few other extremely obscure comments, then we can state that the beginning of golden numberism is due to one person, the German intellectual Adolph Zeising (1810-1876). What is of particular interest to the mathematical tourist is that the origin of golden numberism is associated with a particular time and place. Zeis­ ing's father was a court musician in the tiny dukedom of Anhalt-Bernburg. This dukedom, as well as the other An­ halts-the various pieces of which will bring joy to any map colourist or topologist interested in non-connected surfaces-were contained in what is now the German Land of Sachsen­ Anhalt. Because of his father's occu­ pation, Zeising was born in Ballenstedt, the location of the summer palace, but a visit there made it clear that he is now completely unknown to local official­ dom. Where Zeising is known to some extent-though not in connection with the golden number-is in the city of Bernburg, which is some 75 kilometres northwest of Leipzig. Despite the rav­ ages caused by events of the last sixty years, Bernburg, situated on the Saale river, remains a charming city. There are two edifices in Bernburg that should probably be jointly designated as the birthplace of golden numberism. The first is the building formerly oc­ cupied by the Karls-Gymnasium. In 1842, after having completed a doctor­ ate with a specialty in Hegelian philos­ ophy, Zeising became a full-time pro­ fessor at that institution, and it was

I

while he taught there that a combina­ tion of readings in philosophy and other fields inspired him to think of the golden number as an inherent rule of nature. The other building of interest is the Bernburg castle. Zeising was a leader of the liberal left during the German revolution of 1848-1849, and he was elected to the first Landtag, which met in the castle. Perhaps Zeising thought about the golden number during some long-winded speeches, but more im­ portantly, the castle represented polit­ ical power. By 1851 the power was firmly in the hands of a very reac­ tionary group, and Zeising was pen­ sioned off in December 1852. Using this money he went to Leipzig to do fur­ ther research, and in 1854 he published his book, An Exposition of a New The­ ory of the Proportions of the Human Body, Based on a Previously Unrec­ ognized Fundamental Morphological Law which Permeates all of Nature and Art, Together with a Complete Comparative Overview of Previous Systems.

In the period from 1855 to 1858 Zeis­ ing continued to publish articles and a booklet dealing with the golden num­ ber. Some of these were of a popular nature, in particular his articles "Hu­ mans and Leaves" and "Face Angles," which were published in the widely read science magazine Die Natur. His book and these articles ensured that his theory became widely known, and by the time of his death in 1876 golden numberism was widespread in Ger­ many. Philosophers debated the foun­ dations of his theory, and authors sug­ gested the use of the golden number in such fields as typography and fashion. The polymath Gustave Fechner, in­ spired by Zeising's claims concerning the Sistine Madonna, started scientific investigations, which in tum laid the foundations of the field of experimen­ tal aesthetics. By the end of 1855 Zeising had moved to Munich, where he became

© 2005 Spnnger Science+Bus1ness Media, Inc., Volume 27, Number 1 , 2005

69

�but8fr .ft4rl��mufium ht b« 9unftt1Jellf 1 84 l

--

1882

Fig. 1 . The Karls-Gymnasium, Bernburg, Germany.

active in literacy circles and wrote nov­ els and plays. He also wrote on philos­ ophy-his 1855 Aesthetics, which pre­ sented an overview of the systems of Hegel and his followers, was very highly regarded-and politics. After a hiatus of ten years Zeising again wrote

on the golden number, but, aside from an article on the Cologne Cathedral, these works were of a cultural or philo­ sophical nature. A particular honour came in 1856: his admission to the Deutsche Akademie der Naturforscher. There have been many interesting

twists and turns in the development of the myth of golden numberism. Thus the association of the golden number with virtually all the pyramids of Egypt except the Great Pyramid was made by Friedrich Rober in 1855, independent of Zeising. On the other hand the first example of golden numberism in En­ glish dates from 1866 and deals with the golden number and the Great Pyra­ mid-but in a manner not equivalent to that of Rober-and again this was in­ dependent of Zeising's writings. Mter having entered France and the English-speaking world from Germany in the early part of the twentieth century, golden numberism spread rapidly. By a careful examination of sources, it is pos­ sible to trace the path travelled by the golden number myth. Aside from the topics of phyllotaxis and the Great Pyramid, virtually everything that has been written on the subject can ulti­ mately be traced back to the influence of Zeising. The next time the reader hears another "historical" claim con­ cerning the golden number he or she might care to glance at the accompa­ nying photographs of the Gymnasium and the Bernburg castle, and remem­ ber where the myth started. I consider Zeising the most intellec­ tual of authors on the subject of the golden number. Unlike others, he at­ tempted to present a true foundation­ in his case philosophical-for his argu­ ments. Not that I fmd him convincing! I am fascinated by the "sociology of mathematical myths" and the history of ideas. Thus my most recent work, Adolph Zeising, started out as a few paragraphs and then an appendix to my forthcoming book The Golden Number. In Adolph Zeising I trace the spread of golden numberism from Nees von Es­ senbeck in 1852 through 1876, the year of Zeising's death. The Golden Number will present a complete discussion of golden numberism, including the his­ torical, sociological, and philosophical aspects. Parts of the story can be found in my other writings listed below. Biographical Notes

Fig. 2. Bernburg Castle.

70

THE MATHEMATICAL INTELLIGENCER

Ebersbach's book discusses the period (1835-1852) when Zeising lived in Bernburg. In particular, there are sev-

eral references to Zeising's role in the 1848-1849 revolution. The pho­ tographs (1864 for the castle and the early part of the twentieth century for the Gymnasium) were taken from [Erfurth, p. 63) and [Schulgemeinschaft Carolinum und Friederiken-Lyzeum, p. 142] respectively. These photographs are reproduced with the kind permission of the Mittledeutsche Verlag (Halle ).

Herz-Fischler, R. "How to Find the "Golden Number' Without Really Trying . " Fibonacci Herz-Fischler, R. A Mathematical History of Division in Extreme and Mean Ratio . Wa­

Laurier University Press.

The Life and Work of a German Intellectual.

Ottawa: Mzinhigan Publishing, 2004. Herz-Fischler, R. The Golden Number. To ap­

1 987. Re-printed as A Mathematical His­

pear. Ottawa: Mzinhigan Publishing, 2005.

tory of the Golden Number. New York,

Schulgemeinschaft Carolinum und Friederiken­

Dover, 1 998.

Lyzeum. Geschichte der h6heren Schulen zu

Herz-Fischler, R. "A 'Very Pleasant' Theorem."

REFERENCES

Press, 2000. Herz-Fischler, R. Adolph Zeising (1 8 1 0-1876):

Quarterly 1 9 (1 98 1 ) , pp. 406-4 1 0.

terloo, Wilfrid

mid. Waterloo, Wilfrid Laurier University

Bernburg: Friederikenschule von 1 8 1 0 bis

College Mathematics Journal 24 (1 993), pp.

1 950,

31 8-324.

Karls-Realgymnasium von 1 853-1945. Mu­

Kar/sgymnasium von

Ebersbach, V. Geschichte der Stadt Bernburg,

Herz-Fischler, R. "The Golden number, and Di­

vol. 1 . Dessau: Anhaltische Verlagsgesell­

vision in Extreme and Mean Ratio." in Com­

schaft, 1 998.

panion Encyclopedia of the History and

340 Second Avenue

Philosophy of the Mathematical Sciences,

Ottawa, K1 S 2J2

London, Routledge, 1 994, pp. 1 576-1 584.

Canada

Erfurth, H. Gustav V61ker/ing & die altesten Fo­ tografien Anhalts. Dessau: Anhaltische Ver­

lagsgesellschaft, 1 991 .

Herz-Fischler, R. The Shape of the Great Pyra-

1 835- 1944,

nich: Wedekind, 1 980.

e-mail: [email protected]

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© 2005 Spnnger Setence+ Bus1ness Med1a. Inc . Volume 27. Number 1, 2005

71

D av i d E .

Hilbert's Early Career: Encounters with Allies and Rivals David E. Rowe

Send submissions to David E. Rowe,

R ow e , E d i t o r

I

It seems to me that the mathemati­ cians of today understand each other far too little and that they do not take an intense enough interest in one an­ other. They also seem to know-so far as I can judge-too little of our classical authors (Klassiker); many, moreover, spend much effort working on dead ends. -David Hilbert to Felix Klein, 24 July 1890

P

robably no mathematician has been quoted more often than Hilbert, whose opinions and witty re­ marks long ago entered mathematical lore along with his legendary feats. Fame gave him a captive audience, but as the opening quotation illustrates, even before he attained that fame Hilbert had no difficulty expressing his views. When he wrote those words, in fact, he had just completed [Hilbert 1890], the first in an impressive string of achievements that would vault him to the top of his profession. Initially, he made his name as an ex­ pert on invariant theory, but Hilbert's reputation as a universal mathemati­ cian grew as he left his mark on one field after another. Yet these achieve­ ments alone cannot account for his sin­ gular place in the history of mathe­ matics, as was recognized long ago by his intimates ([Weyl 1932], [Blumen­ thal 1935]). Those who belonged to Hilbert's inner circle during his first two decades in Gottingen pointed to the impact of his personality, which clearly transcended the ideas found between the covers of his collected works (see [Weyl 1944], [Reid 1970]). Hilbert's name became attached to thoughts of fame in the minds of many young mathematicians who felt in­ spired to tackle one of the twenty-three "Hilbert problems." Some of these he had merely dusted off and presented

anew at the Paris ICM in 1900, but they then acquired a special fascination. As Ben Yandell puts it in his delightful sur­ vey, The Honors Class, "solving one of Hilbert's problems has been the ro­ mantic dream of many a mathemati­ cian" [Yandell 2002, 3].1 Hilbert's ability to inspire was clearly central to Gottingen's success, even if only a part. His leadership style fostered what I have characterized as a new type of oral culture, a highly competitive mathematical community in which the spoken word often carried more weight than the information con­ veyed in written texts (see [Rowe 2003b] , [Rowe 2004]). Hilbert was an unusually social creature: outspoken, ambitious, eccentric, and above all full of passion for his calling. Moreover, he was a man of action with no patience for hollow words. Thus, when in July 1890 he con­ veyed the rather harsh views cited in the opening quotation to Klein, he was not merely bemoaning the lack of com­ munal camaraderie among Germany's mathematicians; he was expressing his hope that these circumstances would soon change. At that time plans were underway to found a national organi­ zation of German mathematicians, the

Deutsche Mathematiker-Vereinigung,

and Hilbert was delighted to learn that Klein would be present for the inau­ gural meeting, which would take place a few months later in Bremen. Both knew that much was at stake; as Hilbert expressed it, "I believe that closer personal contact between math­ ematicians would, in fact, be very de­ sirable for our science" (Hilbert to Klein, 24 July 1890 [Frei 1985, 68]). Soon after his arrival in Gottingen in 1895, Hilbert put this philosophy into practice. At the same time, his opti­ mism and self-confidence spilled over and inspired nearly all the young peo­ ple who entered his circle.

Fachbereich 1 7 -Mathematik, Johannes Gutenberg University,

1 1t was Hilbert's star pupil, Hermann Weyl, who called those who actually succeeded the "honors class"; he

055099 Mainz, Germany.

also wrote that "no mathematician of equal stature has risen from our generation" [Weyl 1 944, 1 30].

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THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Science+ Busrness Medra, Inc.

Hilbert's impact on modem mathe­ matics has been so pervasive that it takes a true leap of historical imagina­ tion to picture him as a young man struggling to find his way. Still, many of the seeds of later success were planted in his youth, just as several episodes from his early career have since be­ come familiar aspects of the Hilbert leg­ end. In [Rowe 2003a] , I describe the quiet early years he spent in Konigs­ berg, where he befriended two young mathematicians who influenced him more than any others, Adolf Hurwitz and Hermann Minkowski. By 1885 Hilbert emerged as one of Felix Klein's most promising proteges. In this role, he traveled to Paris to meet the younger generation of French mathematicians, especially Henri Poincare, reporting back all the while to Klein, who avidly awaited news about the Parisian math­ ematical scene. Here I pick up the story at the point when Hilbert returned from this first trip to Paris. Afterward he had several important encounters with the leading mathematicians in Germany. These meetings not only shed light on the contexts that motivated his work, they also reveal how he positioned him­ self within the fast-changing German mathematical community. Returning from Paris

After a rather uneventful and disap­ pointing stay in Paris during the spring of 1886, Hilbert began the long journey

home. Stopping first in Gottingen, he learned something about the contem­ porary Berlin scene when he met with Hermann Amandus Schwarz, the se­ nior mathematician on the faculty. Schwarz had long been one of the clos­ est of Karl Weierstrass's many adoring pupils in Berlin; yet much had changed since the days when Charles Hermite advised young Gosta Mittag-Leffler to leave Paris and go to hear the lectures of Weierstrass, "the master of us all" (see [Rowe 1998]). During the 1860s and 1870s, the Berliners had dominated mathematics throughout Prussia, with the single exception of Konigsberg, which remained an enclave for those with close ties to the Clebsch school and its journal, Mathematische An­ nalen (see [Rowe 2000]). However, af­ ter E. E. Kummer's departure in 1883, the harmonious atmosphere he had cultivated as Berlin's senior mathe­ matician gave way to acrimony. Weier­ strass, old, frail, and decrepit, refused to retire for fear of losing all influence to Leopold Kronecker, who remained amazingly energetic and prolific de­ spite his more than sixty years. Presumably Hilbert heard about this situation from Schwarz, who would have conveyed the essence of the situ­ ation from Weierstrass's perspective (see [Biermann 1988, 137-139]). If so, Hilbert would have heard how rela­ tions between Weierstrass and Kro­ necker had deteriorated mainly be­ cause of the latter's dogmatic views, in particular his sharp criticism of Weier­ strass's approach to the foundations of analysis. Only a few months after Hilbert passed through Berlin, Kro­ necker delivered a speech in which he uttered his most famous phrase "God made the natural numbers; all else is the work of man" ("Die ganzen Zahlen hat der liebe Gott gemacht, alles an­ dere ist Menschenwerk") [Weber 1893, 19]. Kronecker had never made a se­ cret of his views on foundations, but by the mid-1880s he was propounding these with missionary zeal. No one was more taken aback by this than H. A. Schwarz, to whom Kronecker had writ­ ten one year earlier:

Fig. 1 . Hilbert in the days when he was re­ garded as merely one of many experts on the theory of invariants.

If enough years and power remain, I will show the mathematical world

that not only geometry but also arithmetic can point the way for analysis-and certainly with more rigor. If l don't do it, then those who come after me will, and they will also recognize the invalidity of all the procedures with which the so­ called analysis now operates [Bier­ mann 1988, 138]. Weierstrass had written to Schwarz at around that time, claiming that Kro­ necker had transferred his former an­ tipathy for Georg Cantor's work to his own. And in another letter, written to Sofia Kovalevskaya, he characterized the issue dividing them as rooted in mathematical ontology: "whereas I as­ sert that a so-called irrational number has a real existence like anything else in the world of thought, according to Kronecker it is an axiom that there are only equations between whole num­ bers" (Weierstrass to Kovalevskaya, 24 March, 1885, quoted in [Biermann 1988, 137]). Whether or not Schwarz alluded to this rivalry when he spoke to Hilbert in 1886, he definitely did warn him to expect a cold reception when he pre­ sented himself to Kronecker (Hilbert to Klein, 9 July 1886, in [Frei 1985, 15]). Instead, however, Hilbert was greeted in Berlin with open arms, and his ini­ tial reaction to Kronecker was for the most part positive. Back in his native Konigsberg, Hilbert reported to the ever-curious Klein about these and other recent events. He had just completed all re­ quirements for the Habilitation except for the last, an inaugural lecture to be delivered in the main auditorium of the Albertina. His chosen theme was a propitious one: recent progress in the theory of invariants. Hilbert was pleased to be back in Konigsberg as a Privatdozent, even though this meant he was far removed from mathemati­ cians at other German universities. To compensate for this isolation, he was planning to tour various mathematical centers the following year, when he hoped to meet Professors Gordan and N oether in Erlangen. Although he had to postpone that trip until the spring of 1888, it eventually proved far more fruitful than his earlier journey to Paris. What is more, it helped him so-

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc

Volume 27, Number 1

2005

73

Fig.

2. Leopold

Kronecker

emerged

as

Berlin's leading mathematician during the 1880s when his outspoken views caused con­ siderable controversy.

lidify his relationship with Klein, who always urged young mathematicians to cultivate personal contacts with fellow researchers both at home and abroad (see for example [Hashagen 2003, 105-116, 149-162]). Within Klein's net­ work, the Erlangen mathematicians, Paul Gordan and Max Noether, played particularly important roles. The latter was Germany's foremost algebraic geometer in the tradition of Alfred Clebsch, and the former was an old­ fashioned algorist who loved to talk mathematics. Felix Klein knew from first-hand ex­ perience how stimulating collabora­ tion with Paul Gordan could be. Dur­ ing the late 1870s, when Klein taught at the Technical University in Munich, he took advantage of every opportunity to meet with his erstwhile Erlangen col­ league, who was himself enormously impressed by Klein's fertile geometric imagination. Gordan was widely re­ garded as Germany's authority on al­ gebraic invariant theory, the field that would dominate Hilbert's attention for the next five years. His principal claim to fame was Gordan's Theorem, which he proved in 1868. This states that the complete system of invariants of a bi­ nary form can always be expressed in terms of a finite number of such in-

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THE MATHEMATICAL INTELLIGENCER

variants. In 1856 Arthur Cayley had proved this for binary forms of degree 3 and 4, but Gordan was able to use the symbolical method introduced by Siegfried Aronhold to obtain the gen­ eral result. During his stay in Paris, Hilbert had briefly reported to Klein about these matters (Hilbert to Klein, 21 April 1886, in [Frei 1985, 9]). There he learned from Charles Hermite about J. J. Sylvester's recent efforts to prove Gordan's Theorem using the original British techniques he and Cayley had developed. Hilbert thus became aware that the elderly Sylvester was still trying to get back into this race (see [Parshall 1989]). Presumably Hilbert thought that progress was unlikely to come from this old-fashioned line of at­ tack, but neither had the symbolical methods employed by German alge­ braists produced any substantial new results since Gordan first unveiled his theorem. Over the next two years Hilbert had ample time to master the various com­ peting techniques. Mter becoming a

Privatdozent in Konigsberg, he was

free to develop his own research pro­ gram, and his inaugural lecture on re­ cent research in invariant theory clearly indicates the general direction in which he was moving. Still, there are no signs that he was on the path to­ ward a major breakthrough. Indeed, tucked away in Konigsberg, it seems unlikely that he even saw the need to strike out in an entirely new direction in order to make progress beyond Gor­ dan's original finiteness theorem. That goal, nevertheless, was clearly in the back of his mind when he set off in March 1888 on a tour of several lead­ ing mathematical centers in Germany, including Berlin, Leipzig, and Gottin­ gen. During the course of a month, he spoke with some twenty mathemati­ cians from whom he gained a stimu­ lating overview of current research interests throughout the country. Al­ though we can only capture glimpses of these encounters, a number of im­ pressions can be gained from Hilbert's letters to Klein, as well as from the

Felix Christian Klein Hochzeitsbilder 1 875

Fig. 3. On leaving Erlangen for the Technische Hochschule in Munich in 1 875, Felix Klein mar­ ried Anna Hegel, a granddaughter of the famous philosopher.

notes he took of his conversations with various colleagues. 2 A Second Encounter with Kronecker

In Berlin, Hilbert met once again with Kronecker, who on two separate occa­ sions afforded him a lengthy account of his general views on mathematics and much else related to Hilbert's own research. A gregarious, outspoken man, the elder Kronecker still exuded intensity, so Hilbert learned a great deal from him during the four hours they spent together. Reporting to Klein, he described the Berlin mathemati­ cian's opinions as "original, if also somewhat derogatory" (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Hilbert told Kronecker about a paper he had just written on certain positive definite forms that cannot be repre­ sented as a sum of squares. Kronecker replied that he, too, had encountered forms that cannot be so represented, but he admitted that he did not know Hilbert's main theorem, which dealt with the three cases in which a sum of squares representation is, indeed, al­ ways possible ("Bericht iiber meine Reise," Hilbert Nachlass 741). A noteworthy feature of this paper, [Hilbert 1888a], is that Hilbert actually credits Kronecker with having intro­ duced the general principle behind his investigation. This work lies at the root of Hilbert's seventeenth Paris problem, which also played an important role in Hilbert's research on foundations of geometry. Interestingly, a second Hilbert problem, the sixteenth, also crept into his conversations with Kro­ necker. It concerns the possible topo­ logical configurations among the com­ ponents of a real algebraic curve. The maximal number of such components had been established by Axel Harnack, a student of Klein's, in a celebrated the­ orem from 1876 (for a summary of sub­ sequent results, see [Yandell 2002, 276-278]). Kronecker assured Hilbert that his own theory of characteristics, as presented in a paper from 1878, enabled one to answer all questions of this type, clearly an overly optimistic assessment.

Whatever he may have thought about Kronecker's "priority claims," Hilbert stood up and took notice when his host voiced some sharp views about the significance of invariant the­ ory. Kronecker dismissed the whole theory of formal invariants as a topic that would die out just as surely as had happened with Hindenburg's combina­ torial school (which had flourished in Leipzig at the beginning of the nine­ teenth century, but by the 1880s had entered the dustbin of history). The only true invariants, in Kronecker's view, were not the "literal" ones based on algebraic forms, but rather num­ bers, such as the signature of a qua-

The o n ly true i nvariants , i n Kro necker' s view , were n u m bers , such as the signat u re of a q u ad ratic form . dratic form (Sylvester's theorem, the algebraist's version of conservation of inertia). He then proceeded to wax forth over foundational issues, begin­ ning with the assertion that "equality" only has meaning in relation to whole numbers and ratios of whole numbers. Everything beyond this, all irrational quantities, must be represented either implicitly by a finite formula (e.g., x2 = 5), or by means of approximations. Us­ ing these notions, he told Hilbert, one can establish a firm foundation for analysis that avoided the Weierstrass­ ian notions of equality and continuity. He further decried the confusion that so often resulted when mathemati­ cians treated the implicitly given irra­ tional quantity (say, x = v5) as equiv­ alent to some sequence of rational numbers that serve as an approxima­ tion for it. Not surprisingly, Hilbert took fairly

extensive notes when Kronecker be­ gan expounding these unorthodox views ("Bericht iiber meine Reise"). But he also jotted down a brief com­ ment made by Weierstrass that sheds considerable light on the differences between these two mathematical per­ sonalities. When Hilbert visited Weier­ strass shortly afterward, he informed him of Kronecker's comments regard­ ing invariant theory, including the pre­ diction that the whole field would soon be forgotten, like the work of the Leipzig combinatorial school. Weier­ strass responded by sounding a gentle warning to those who might wish to prophesy the future of a mathematical theory: "Not everything of the combi­ natorial school has perished," he said, "and much of invariant theory will pass away, too, but not from it alone. For from everything the essential must first gradually crystallize, and it is neither possible nor is it our duty to decide in advance what is significant; nor should such considerations cause us to demur in investigating such invariant-theo­ retic questions deeply" ("Bericht tiber meine Reise"). These words, with their almost fa­ talistic ring, probably left little impres­ sion on the young mathematician who recorded them. For Hilbert's intellec­ tual outlook was filled with a buoyant optimism that left no room for resig­ nation. He may not have enjoyed Kro­ necker's braggadocio, but he was clearly far more receptive to his pas­ sionate vision than to Weierstrass's more subdued outlook. Moreover, mathematically he was far closer to the algebraist than to the analyst. Even in his later work in analysis, Hilbert showed that his principal strength as a mathematician stemmed from his mas­ tery of the techniques of higher algebra (see [Toeplitz 1922]). True, Klein and Hurwitz had drawn his attention to Weierstrass's theory of periodic com­ plex-valued functions, about which he spoke in his Habilitationsvortrag shortly after returning to Konigsberg from Paris. Nevertheless, Kronecker's algebraic researches lay much closer to his heart. Soon after their encounter

2"Bericht uber meine Reise vom 9ten Marz bis ?ten April 1 888," Hilbert Nachlass 7 4 1 , Handschnftenabteilung, Niedersachs1sche Staats· und Umversitatsbibliothek Gbt· tingen.

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 27, Number 1, 2005

75

in Berlin, Hilbert would enter Leopold Kronecker's principal research do­ main, the theory of algebraic number fields. The latter's sudden death in 1891 may well have emboldened Hilbert to reconstruct this entire theory six years later in his "Zahlbericht." As Hermann Weyl later emphasized, Hilbert's am­ bivalence with respect to Kronecker's legacy emerged as a major theme throughout his career [Weyl 1944, 613]. Like Hilbert, his two closest mathe­ matical friends, Hurwitz and Minkowski, also held an1bivalent views when it came to Kronecker. No doubt these were colored by their mutual desire to step beyond the lengthy shadows that Kronecker and Richard Dedekind, the other leading algebraist of the older generation, had cast. Since Dedekind had long since withdrawn to his native Brunswick, a city well off the beaten path, it was only natural that the Konigsberg trio came to regard the powerful and opinionated Berlin math­ ematician as their single most imposing rival. In later years, Hilbert developed a deep antipathy toward Kronecker's philosophical views, and he did not hes­ itate to criticize these before public au­ diences (see [Hilbert 1922]). Yet during the early stages of his career such mis­ givings-if he had any-remained very much in the background. Indeed, all of Hilbert's work on invariant theory was deeply influenced by Kronecker's ap­ proach to algebra. Hilbert's encounters in the spring of 1888 with Berlin's two senior mathe­ maticians left a deep and lasting im­ pression. 3 Based on the notes he took of these conversations, he must have felt particularly aroused by Kro­ necker's critical views with regard to invariant theory, for he surely found no solace in Weierstrass's stoic advice. Primed for action and out to conquer, Hilbert could never have contemplated devoting his whole life to a theory that might later be judged as having no in­ trinsic significance. Whatever prob­ lems he chose to work on-even those he merely thought about but never tried to solve-he always thought of them as constituting important mathe­ matics. What makes a problem or a

theory important? Probably Hilbert carried that question within him for a long time, though anyone familiar with his career knows the answer he even­ tually came up with; one need only reread his famous Paris address to see how compelling his views on the sig­ nificance of mathematical thought could be. From the vantage point of these early, still formative years, we can begin to picture how his larger views about the character and signifi­ cance of mathematical ideas fell into place. A few strands of the story emerge from the discussions he en­ gaged in during this whirlwind 1888 tour through leading outposts of the German mathematical community. Tackling Gordan's Problem

From Berlin, Hilbert went on to Leipzig, where he finally got the chance to meet face-to-face with Paul Gordan, who came from Erlangen. Despite their mathematical differences, the two hit

It is neither possi ble nor is it our d uty to decide i n advance what is s i g n ificant . it off splendidly, as both loved nothing more than to talk about mathematics. Hermann Weyl once described Gordan as "a queer fellow, impulsive and one­ sided," with "something of the air of the eternal 'Bursche' of the 1848 type about him-an air of dressing gown, beer and tobacco, relieved however by a keen sense of humor and a strong dash of wit. . . . A great walker and talker-he liked that kind of walk to which fre­ quent stops at a beer-garden or a cafe belong" [Weyl 1935, 203]. Having heard about Hilbert's talents, Gordan longed to make the young man's acquaintance, so much so that he wished to remain incognito while in Leipzig to take full advantage of the opportunity (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Although originally an expert on

3He recalled this trip when he spoke about his life on his seventieth birthday; see [Reid 1 970, 202].

76

THE MATHEMATICAL INTELLIGENCER

Fig. 4. Paul Gordan joined Klein on the Er­ langen faculty in 1 874 and remained there un­ til his death in 1 91 2. His star student was Emmy Noether, daughter of Gordan's col­ league, Max Noether.

Abelian functions, Gordan had long since focused his attention exclusively on the theory of algebraic invariants. This field traces back to a fundamen­ tal paper published by George Boole in 1841, as it was this work that inspired young Arthur Cayley to take up the topic in earnest [Parshall 1989, 160-166]. Following an initial plunge into the field, Cayley joined forces with another professional lawyer who be­ came his life-long friend, J. J. Sylvester. Together, they effectively launched in­ variant theory as a specialized field of research. Much of its standard termi­ nology was introduced by Sylvester in a major paper from 1853. Thus, for a given binary form f(x,y), a homoge­ neous polynomial J in the coefficients off left fixed by all linear substitutions (up to a fixed power of the determinant of the substitution) is called an invari­ ant of the form f In 1868 Gordan showed that for any binary form, one can always construct m invariants h, /z, . . . , Im such that every other in­ variant can be expressed in terms of these m basis elements. Indeed, he proved that this held generally for ho­ mogeneous polynomials J in the coef­ ficients and variables ofj(x,y) with the same invariance property (Sylvester called such an expression J a con­ comitant of the given form, but the term covariant soon became stan­ dard). In 1856 Cayley published the first

finiteness results for binary forms, but in the course of doing so he committed a major blunder by arguing that the number of irreducible invariants was necessarily infinite for forms of degree five and higher [Parshall 1989, 167-179]. Paul Gordan was the first to show that Cayley's conclusion was incorrect. More importantly, in the course of do­ ing so he proved his finite basis theo­ rem for binary forms of arbitrary de­ gree by showing how to construct a complete system of invariants and co­ variants. Two years later, he was able to extend this result to any finite sys­ tem of binary forms. His proofs of these key theorems, as later presented in [Gordan 1885/1887], were purely al-

Fig. 5. Otto Blumenthal, Hilbert's first biog­ rapher, alluded to the critical meeting when Hilbert and Gordan first met.

gebraic and constructive in nature. They were also impressively compli­ cated, so that subsequent attempts, in­ cluding Gordan's own, to extend his theorem to ternary forms had pro­ duced only rather meager results. Little evidence has survived relating to Hilbert's first encounter with Gor­ dan, but it is enough to reconstruct a plausible picture of what occurred. Gordan may have been a fairly old dog, but this does not mean he was averse to learning some new tricks. Even though he and Hilbert had divergent views about many things, they never­ theless understood each other well (Hilbert to Klein, 16 March 1888 [Frei 1985, 38]). Their conversations soon fo­ cused on finiteness results, in particu­ lar a fairly recent proof of Gordan's finiteness theorem for systems of bi­ nary forms published by Franz Mertens in Grelle's Journal [Mertens 1887]. This paper broke new ground. For unlike Gordan's proof, which was based on the symbolic calculus of Clebsch and Aronhold, Mertens's proof was not strictly constructive. Gordan and Hilbert apparently discussed it in con­ siderable detail, and Hilbert immedi­ ately set about trying to improve Mertens's proof, which employed a rather complicated induction argu­ ment on the degree of the forms. After spending a good week with Gordan, he was delighted to report to Klein that "with the stimulating help of Prof. Gor­ dan an infinite series of thought vibra­ tions has been generated within me, and in particular, so we believe, I have a wonderfully short and pointed proof for the finiteness of binary systems of forms" (Hilbert to Klein, 2 1 March 1888 [Frei 1985, 39]). Hilbert had caught fire. A week later, when he met with Klein in Got­ tingen, he had already put the finishing touches on the new, streamlined proof. This paper [Hilbert 1888b] was the first in a landslide of contributions to alge­ braic invariant theory that would tum the subject upside down. Between 1888 and 1890 Hilbert pursued this theme re­ lentlessly, but with a new methodolog­ ical twist which he combined with the formal algorithmic techniques em­ ployed by Gordan. Beginning with three short notes sent to Klein for publication

in the Gottinger Nachrichten [Hilbert 1888c] , [Hilbert 1889a] , [Hilbert 1889b], he began to unveil general methods for proving finiteness relations for general systems of algebraic forms, invariants being only a quite special case, though the one of principal interest. With these general methods, combined with the al­ gorithmic techniques developed by his predecessors, Hilbert was able to ex­ tend Gordan's finiteness theorem from systems of binary forms over the real or complex numbers to forms in any number of variables and with coeffi­ cients in an arbitrary field. By the time this first flurry of activ­ ity came to an end, Hilbert had shown how these finiteness theorems for in­ variant theory could be derived from general properties of systems of alge­ braic forms. Writing to Klein in 1890, he described his culminating paper [Hilbert 1890] as a unified approach to a whole series of algebraic problems (Hilbert to Klein, 15 February 1890 [Frei 1985, 61]). He might have added that his techniques borrowed heavily from Leopold Kronecker's work on al­ gebraic forms. Yet from a broader methodological standpoint, Hilbert's approach clearly broke with Kro­ necker's constructive principles. For Hilbert's foray into the realm of alge­ braic forms revealed the power of pure existence arguments: he showed that out of sheer logical necessity a finite basis must exist for the system of in­ variants associated with any algebraic form or system of forms. Hilbert found his way forward by noticing the following general result, known today as Hilbert's basis theo­ rem for polynomial ideals. It appears as Theorem I in [Hilbert 1888c] . It states that for any sequence of alge­ braic forms in n variables c{J1, c{Jz, 4>3, . . . there exists an index m such that all the forms of the sequence can be written in terms of the first m forms, that is, cP

=

CXlcPl

+

CX2 cP2

+ ...+

CXmcPm ,

where the ai are appropriate n-ary forms. Thus, the forms cfJ1 , c{Jz, . . . cPm serve as a basis for the entire system. By appealing to Theorem I and draw­ ing on Mertens's procedure for gener­ ating systems of invariants, Hilbert

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proved that such systems were always finitely generated. There was, how­ ever, a small snag. Hilbert attempted to prove Theorem I by first noting that it held for small n. He then introduced a still more general Theorem II, from which he could prove Theorem I by induction on the number of variables. If this sounds confusing, a number of contemporary readers had a similar reaction, including a few who expressed their misgivings to Hilbert about the validity of his proof. Paul Gordan, however, was not one of them. According to Hilbert, to the best of his recollection, he and Gordan had only discussed the proof of Theorem II dur­ ing their meeting in Leipzig (Hilbert to Klein, 3 March 1890 [Frei 1985, 64]). As it turned out, Hilbert's Theorem II, as originally formulated in [Hilbert 1888c], is false. 4 Moreover, since it was conceived from the beginning as a lemma for the proof of Theorem I, Hilbert dropped Theorem II in his de­ fmitive paper [Hilbert 1890] and gave a new proof of Theorem I. Nevertheless, the latter remained controversial, as we shall soon see. Today we recognize in Hilbert's Theorem I a central fact of ideal theory, namely, that every ideal of a polynomial ring is finitely generated. Thirty years later, Emmy Noether in­ corporated Hilbert's Theorem I (from [Hilbert 1890]) as well as his Nullstel­ lensatz (from [Hilbert 1893]) into an abstract theory of ideals (see [Gilmer 1981]). Her classic paper "Idealtheorie in Ringbereichen" [Noether 1921 ] is nearly as readable today as it was when she wrote it. The same cannot be said, however, for Hilbert's papers (for En­ glish translations, see [Hilbert 1978]). Not that these are badly written; they simply reflect a far less familiar math­ ematical context. In [Hilbert 1889a, 28] Hilbert hinted that much of the inspiration for both the terminology and techniques came from Kronecker's theory of module systems. When he wrote this, he knew very well that Kronecker held very neg­ ative views about invariant theory, making it highly improbable that he

would view Hilbert's adaptation of his ideas with approval. Indeed, Kro­ necker had made it plain to Hilbert that, in his view, the only invariants of interest were the numerical invariants associated with systems of algebraic equations. Still, Hilbert quickly recog­ nized the fertility of Kronecker's con­ ceptions for invariant theory. Ac­ knowledging his debt to the Berlin algebraist, he parted company with him by adopting a radically non-con­ structive approach. Ironically, the ini­ tial impulse to do so apparently came from his conversations with Gordan. Thus, with his early work on invariant theory Hilbert sowed some of the seeds that would eventually flower into his modernist vision for mathematics, thereby preparing the way for the dra­ matic foundations debates of the 1920s (see [Hesseling 2003]). Mathematics as Theology

Kronecker seems to have simply ig­ nored Hilbert's dramatic break­ through, but others closer to the field of invariant theory obviously could not afford to do so. Paul Gordan, who had initially supported Hilbert's work en­ thusiastically, now began to express misgivings about this new and, for him, all too ethereal approach to invariant theory. His views soon made the rounds at the coffee tables and beer gardens, and more or less everyone heard what Gordan probably said on more than one occasion: Hilbert's ap­ proach to invariant theory was "theol­ ogy not mathematics" [Weyl 1944, 140].5 No doubt many mathematicians got a chuckle out of this epithet at the time, but a serious conflict briefly reared its head in February 1890 when Hilbert submitted his definitive paper [Hilbert 1890] for publication in Mathematis­ che Annalen. Klein was overjoyed, and wrote back to Hilbert a day later: "I do not doubt that this is the most impor­ tant work on general algebra that the Annalen has ever published" (Klein to Hilbert, 18 Feb. 1890, in [Frei 1985, p. 65]). He then sent the manuscript to Gordan, the Annalen's house expert on

invariant theory, asking him to report on it. Klein, having already heard some of Gordan's misgivings about Hilbert's methods in private conversations, may well have anticipated a negative reac­ tion. He certainly got one. The cantan­ kerous Gordan forcefully voiced his objections, aiming directly at Hilbert's presentation of Theorem I, which Gor­ dan claimed fell short of even the most modest standards for a mathematical proof. "The problem lies not with the form," he wrote Klein, " . . . but rather lies much deeper. Hilbert has scorned to present his thoughts following for­ mal rules; he thinks it suffices that no one contradict his proof, then every­ thing will be in order . . . he is content to think that the importance and cor­ rectness of his propositions suffice. That might be the case for the first ver­ sion, but for a comprehensive work for the Annalen this is insufficient." (Gor­ dan to Klein, 24 Feb. 1890, in [Frei 1985, p 65]. Perhaps the misgivings come down to the non-constructivity in­ volved in the [implicit] use of the Ax­ iom of Choice. Concise modem proofs like [Caruth 1996] put the latter clearly in evidence.) Klein forwarded Gordan's report to Hurwitz in Konigsberg, who then dis­ cussed its contents with Hilbert. After that the sparks really began to fly. Clearly irked by Gordan's refusal to recognize the soundness of his argu­ ments, Hilbert promptly dashed off a fierce rebuttal to Klein. He began by reminding him that Theorem I was by no means new; he had, in fact, come up with it some eighteen months ear­ lier and had afterward published a first proof in the Gottinger Nachrichten [Hilbert 1888c]. He then proceeded to describe the events that had prompted him to give a new proof in the manu­ script now under scrutiny. This came about after he had spo­ ken with numerous mathematicians about his key theorem; he had also car­ ried on correspondence with Cayley and Eugen Netto, who wanted him to clarify certain points in the proof. Tak­ ing these various reactions into ac-

4See the editorial note in [Hilbert 1 933. 1 77). 5The earliest reference to Gordan's remark- ''Das ist keine Mathematik, das ist Theologie"-appears to be [Blumenthal 1 935, 394).

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THE MATHEMATICAL INTELLIGENCER

count, Hilbert had prepared a revised proof, which he had tested out in his lecture course the previous semester. Afterward he spoke with one of the au­ ditors in order to convince himself that the argument as presented had actually been understood. Having reassured himself that this new proof was indeed clear and understandable, he wrote it up for [Hilbert 1890]. Hilbert then con­ cluded this recitation of the relevant prehistory by saying that these facts clearly refuted the ad hominem side of Gordan's attack, namely his insinua­ tion that Hilbert's new proof of Theo­ rem I was not meant to be understood and that he was content so long as no one could contradict the argument. Regarding what he took to be the substantive part of Gordan's critique, Hilbert stated that this consisted mainly of "a series of very commend­ able, but completely general rules for the composition of mathematical pa­ pers" (Hilbert to Klein, 3 March 1890 [Frei 1985, 64]). The only specific crit­ icisms Gordan made were, in Hilbert's opinion, plainly incomprehensible: "If Professor Gordan succeeds in proving my Theorem I by means of an 'order­ ing of all forms' and by passing from 'simpler to more complicated forms,' then this would just be another proof, and I would be pleased if this proof were simpler than mine, provided that each individual step is as compelling and as tightly fastened" (ibid.). Hilbert then ended this remarkable repartee with an implied threat: either his paper would be printed just as he wrote it or he would withdraw his manuscript from publication in the Annalen. "I am not prepared," he intoned, "to alter or delete anything, and regarding this pa­ per, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my rea­ soning is raised" (ibid.). Certainly Klein was not accustomed to receiving letters like this one, and especially from young Privatdozenten. Yet however impressed he may have been by Hilbert's self-assurance and pluck, he also wanted to preserve his longstanding alliance with Gordan. Moreover, in view of his older friend's irascibility, Klein knew, he had to han­ dle the squabble delicately before it be-

came a full-blown crisis. Hilbert re­ ceived no immediate reply, as Klein wanted to wait until he could confer with Gordan personally. Over a month passed, with no word from Gottingen about the fate of a paper that Klein had originally characterized as one of the most important ever to appear in the pages of Die Mathematische Annalen. Then, in early April, Gordan came to Gottingen to "negotiate" with Klein about these matters, which clearly weighed heavily on the Erlangen math­ ematician's heart. To facilitate the process, Klein asked Hurwitz to join them, knowing that Hilbert's trusted friend would do his best to help restore harmony. Gordan spent eight days in Gottin­ gen, following which Klein wrote Hilbert a brief letter summarizing the results of their "negotiations." He be­ gan by reassuring him that Gordan's opinions were by no means as uni­ formly negative as Hilbert had as­ sumed. "His general opinion,'' Klein noted, "is entirely respectful, and would exceed your every wish" (Klein to Hilbert, 14 April 1890, in [Frei 1985, p. 66]). To this he merely added that Hurwitz would be able to tell him more about the results of their meeting. But then he attached a postscript that con­ tained the message Hilbert had been waiting to hear: Gordan's criticisms would have no bearing on the present paper and should be construed merely as guidelines for future work! Thus, Hilbert got what he de­ manded; his decisive paper appeared in the Annalen exactly as he had written it. Gordan surely lost face, but at least he had been given the opportunity to vent his views. In short, Klein's diplo­ matic maneuvering carried the day. Gordan knew, of course, that he was dealing with someone who had little patience for methodological nit-pick­ ing. He also knew that Klein consis­ tently valued youthful vitality over age and experience. Hilbert represented the wave of the future, and while this conflict, in and of itself, had no imme­ diate ramifications, it foreshadowed a highly significant restructuring of the power constellations that had domi­ nated German mathematics since the late 1860s.

A Final Tour de Force

If Hilbert was scornful of Gordan's ed­ itorial pronouncements, this does not mean that he failed to see the larger is­ sue at stake. His general basis theorem proved that for algebraic forms in any number of variables there always ex­ ists a finite collection of irreducible in­ variants, but his methods of proof were of no help when it actually came to constructing such a basis. Hilbert ob­ viously realized that if he could de­ velop a new proof based on arguments that were, in principle, constructive in nature, then this would completely vi­ tiate Gordan's criticisms. Two years later, he unveiled just such an argu­ ment, one that he had in fact been seek­ ing for a long time. In an elated letter to Klein, he described this latest break­ through, which allowed him to bypass the controversial Theorem I com­ pletely. He further noted that although this route to his finiteness theorems was more complicated, it carried a ma­ jor new payoff, namely "the determi­ nation of an upper bound for the de­ gree and weights of the invariants of a basis system" (Hilbert to Klein, 5 Jan­ uary 1892 [Frei 1985, 77]) . When Hermann Minkowski, who was then in Bonn, heard about Hilbert's lat­ est triumph, he fired off a witty letter congratulating his friend back in Konigs­ berg: I had long ago thought that it could only be a matter of time before you finished off the old invariant theory to the point where there would hardly be an i left to dot. But it re­ ally gives me joy that it all went so quickly and that everything was so surprisingly simple, and I congratu­ late you on your success. Now that you've even discovered smokeless gunpowder with your last theorem, after Theorem I caused only Gor­ dan's eyes to sting anymore, it really is a good time to decimate the fortresses of the robber-knights [i.e., specialists in invariant theory]­ [Georg Emil] Stroh, Gordan, [Ky­ parisos] Stephanos, and whoever they all are-who held up the indi­ vidual traveling invariants and locked them in their dungeons, as there is a danger that new life will

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never sprout from these ruins again. [Minkowski 1973, 45]. Minkowski's opinions were a constant source of inspiration for Hilbert, so he probably took these remarks to heart. Indeed, this letter may well mark the beginning of one of the most enduring of all myths associated with Hilbert's exploits, namely that he single-hand­ edly killed off the till then flourishing field of invariant theory. As Hans Freudenthal later put it: "never has a blooming mathematical theory with­ ered away so suddenly" [Freudenthal 1971, 389]. Hilbert published his new results in another triad of papers for the Gdt­ tinger Nachrichten ([Hilbert 189 1 ] , [Hilbert 1892a], [Hilbert 1892b] ) . Nine months later he completed the manu­ script of his second classic paper on in­ variant theory [Hilbert 1893].6 He sent this along with a diplomatically worded letter to Klein, noting that he had taken pains to ensure that the presentation followed the general guidelines Prof. Gordan had recommended (Hilbert to Klein, 29 September 1892 [Frei 1985, 85]). Then, in a short postscript, Hilbert added: "I have read and thought through the manuscript carefully again, and must confess that I am very satisfied with this paper" (ibid.). Klein reassured him that "Gordan had made his peace with the newest developments," and emphasized that doing so "wasn't easy for him, and for that reason should be seen as much to his credit" (Klein to Hilbert, 7 January 1893, in [Frei 1985, p. 86]). As evidence of Gordan's change of heart, Klein men­ tioned his forthcoming paper entitled simply "Ober einen Satz von Hilbert" [Gordan 1892]. The Satz in question was, of course, Hilbert's Theorem I, which really had caused Gordan's eyes to sting, but not because he doubted its validity. Nor did he ever doubt that Hilbert's proof was correct; it was simply incomprehensible in Gordan's opinion. As he put it to Klein back in 1890: "I can only learn something that

is as clear to me as the rules of the mul­ tiplication table" (Gordan to Klein, 24 Feb. 1890, in [Frei 1985, p 65]). Hilbert had claimed that he would welcome a simpler proof of Theorem I from Gordan, and here the elderly al­ gorist delivered in a gracious manner. He began by characterizing Hilbert's proof as "entirely correct" [Gordan 1892, 132], but went on to say that he had nevertheless noticed a gap, in that Hilbert's argument merely proved the existence of a finite basis without ex­ amining the properties of the basis el­ ements. He further noted that his own proof relied essentially on Hilbert's strategy of applying the ideas of Kro­ necker, Dedekind, and Weber to in­ variant theory [Gordan 1892, 133]. Probably only a few of those who saw this conciliatory contribution by the "King of Invariants" were aware of the earlier maneuvering that had taken place behind the scenes. Nor were many likely to have anticipated that Gordan's throne would soon resemble a museum piece. 7 Not surprisingly, Hilbert put method­ ological issues at the very forefront of [Hilbert 1893], his final contribution to invariant theory. Here he called atten­ tion to the fact that his earlier results failed to give any idea of how a finite basis for a system of invariants could actually be constructed. Moreover, he noted that these methods could not even help in finding an upper bound for the number of such invariants for a given form or system of forms [Hil­ bert 1933, 319]. To show how these drawbacks could be overcome, Hilbert adopted an even more general ap­ proach than the one he had taken be­ fore. He described the guiding idea of this culminating paper as invariant the­ ory treated merely as a special case of the general theory of algebraic func­ tion fields. This viewpoint was inspired to a considerable extent by the earlier work of Kronecker and Dedekind, al­ though Hilbert mentioned this connec­ tion only obliquely in the introduction, where he underscored the close anal-

ogy with algebraic number fields [Hilbert 1933, 287] . Hilbert's introduction also contains other interesting features. In it, he set down five fundamental principles which could serve as the foundations of invariant theory. The first four of these he regarded as the "elementary propositions of invariant theory," whereas the existence of a finite basis (or in Hilbert's terminology a "full in­ variant system") constituted the fifth principle. This highly abstract formu­ lation would, of course, later come to typify much of Hilbert's work in nearly all branches of mathematics. Indeed, the only thing missing from what was to become standard Hilbertian jargon was an explicit appeal to the axiomatic method. Immediately after presenting these five propositions, he wrote that they "prompt the question, which of these properties are conditioned by the others and which can stand apart from one another in a function system." He then mentioned an example that demon­ strated the independence of property 4 from properties 2, 3, and 5. These fmd­ ings were incidental to the main thrust of Hilbert's paper, but they reveal how axiomatic ideas had already entered into his early work on algebra. 8 On September 1892, the day he sent off the manuscript of [Hilbert 1893], Hilbert wrote to Minkowski: "I shall now definitely leave the field of invari­ ants and tum to number theory" [Blu­ menthal 1935, 395]. This transition was a natural one, given that his final work on invariant theory was essentially an application of concepts from the the­ ory of algebraic number fields. One year later, Hilbert and Minkowski were charged with the task of writing a re­ port on number theory to be published by the Deutsche Mathematiker-Vereini­ gung. Minkowski eventually dropped out of the project, but he continued to offer his friend advice as Hilbert strug­ gled with his most ambitious single work, "Die Theorie der algebraischen Zahlkorper," better known simply as the

Zahlbericht. 9

6For an English translation of this and other works by Hilbert. see [Hilbert 1 978]. 7Gordan later presented a streamlined proof of Hilbert's Theorem

I

in a lecture at the 1 899 meet1ng of the DMV in Munich. Hilbert was present on that occasion (see

Jahresbericht der Deutschen Mathematiker-Vereinigung 8(1 899), 1 80. Gordan wrote up this proof soon thereafter for [Gordan 1 899] . 8For a detailed examination of Hilbert's work on the axiomatization of physics, see [Corry 2004].

80

THE MATHEMATICAL INTELLIGENCER

Killing off a Mathematical Theory

nor episode at the outset of his

Thus by 1893 Hilbert's active involve­ ment with invariant theory had ended. In that year he wrote the survey article [Hilbert 1896] in response to a request from Felix Klein, who presented it along with several other papers at the Mathematical Congress held in Chicago in 1893 as part of the World's Columbian Exposition. Hilbert's ac­ count offers an interesting partici­ pant's history of the classical theory of invariants. At the time he wrote it, in­ variant theory was a staple research field within the fledgling mathematical community in the United States, which first began to spread its wings under the tutelage of J. J. Sylvester at Johns Hopkins (see [Parshall and Rowe 1994]). Hilbert briefly alluded to the contributions of Cayley and Sylvester in his brief survey, describing these as characteristic for the "naive period" in the history of a special field like alge­ braic invariant theory. This stage, he added, was soon superseded by a "for­ mal period, " whose leading figures were his own direct predecessors, Al­ fred Clebsch and Paul Gordan. A ma­ ture mathematical theory, Hilbert went on, typically culminates in a third, "crit­ ical period," and his account made it clear that he alone was to be regarded as having inaugurated this stage. What better time to quit the field? Hilbert realized very well that many as­ pects of invariant theory had only be­ gun to unfold, but after 1893 he was content to point others in possibly fruitful directions for further research, such as the one indicated in his four­ teenth Paris problem. Although he did offer a one-semester course on invari­ ant theory in 1897 (see [Hilbert 1993]), by this time his eyes were already on other fields and new challenges. Mathematicians are constantly look­ ing ahead, not backward, and by 1893 probably no one gave much thought to the events of five years earlier. Over time, Hilbert's decisive encounter with Gordan in Leipzig was reduced to a mi-

Siegeszug through invariant theory.

Otto Blumenthal, Hilbert's first biogra­ pher, even got the city wrong, claiming that Hilbert went to Erlangen to visit Gordan in the spring of 1888 [Blumen­ thal 1935, 394]. By then forty years had passed, and presumably no one, not even Hilbert, remembered what had happened. Yet his own characteriza­ tion of this encounter could not be more telling: it had been thanks to Gordan's "stimulating help" that he left Leipzig with "an infinite series of thought vibrations" running through his brain. Scant though the evidence may be, it strongly suggests that the week he spent with the "King of In­ variants" gave Hilbert the initial im­ pulse that put him on his way. Back in Konigsberg, he adopted several of Gor­ dan's techniques in his subsequent work. Numerous citations reveal that he was thoroughly familiar with Gor­ dan's opus, especially the two volumes of his lectures edited by Georg Ker­ schensteiner [Gordan 1885/1887]. That work, the springboard for many of Hilbert's discoveries, was by 1893 prac­ tically obsolete, though no comparable compendium would take its place. His friend Hermann Minkowski saw that this presented a certain dilemma: it was all very well to blow up the cas­ tles of those robber knights of invari­ ant theory, so long as something more useful could be built on their now bar­ ren terrain. Minkowski thus expressed the hope that Hilbert would some day show the mathematical world what the new buildings might look like. In the same vein, he kidded him that it would be best if Hilbert wrote his own mono­ graph on the new modernized theory of invariants rather than waiting to find another Kerschensteiner, who would likely leave behind too many "cherry pits" (misspelled by Minkowski as "Ker­ schensteine") [Minkowski 1973, 45]. Hilbert did neither, 1 0 leaving the theory of invariants to languish on its own while the Gordan-Kerschensteiner

volumes gathered dust in local li­ braries. Invariant theory thus entered the annals of mathematics, its history already sketched by the man who wrote its epitaph. To the younger gen­ eration, Paul Gordan would mainly be remembered for having once declared Hilbert's modem methods "theology." Now that he and his mathematical regime had been deposed, classical in­ variant theory was declared a dead subject, one of those "dead ends" ("tote Strange") that Hilbert had decried in his letter to Klein from 1890.U Al­ though Leopold Kronecker had pre­ dicted this very outcome, he would hardly have approved of the execu­ tioner's methods. Yet ironically, it was Hilbert's decision to move on to "greener pastures"-even more than the wealth of new perspectives his work had opened-that hastened the fulfillment of Kronecker's prophecy. LITERATURE

[Biermann 1 988] Kurt-R. Biermann, Die Math­ ematik und ihre Dozenten an der Berliner Uni­ versitat, 1 8 1 0- 1 933, Berlin: Akademie-Ver­

lag, 1 988. [Blumenthal 1 935] Otto Blumenthal, "Lebens­ geschichte," in [Hilbert 1 935, pp. 388-429]. [Caruth 1 996] A Caruth, "A concise proof of Hilbert's basistheorem." Amer. Math. Monthly 1 03 (1 996), 1 60-1 61 . [Corry 2004] Leo Corry, Hilbert and the Ax­ iomatization of Physics (1898- 1 9 1 8): From "Grundlagen der Geometrie" to "Grundlagen der Physik", to appear in Archimedes: New Studies in the History and Philosophy of Sci­ ence and Technology, Dordrecht: Kluwer

Academic, 2004. [Fisher 1 966] Charles S. Fisher, "The Death of a Mathematical theory: A Study in the Soci­ ology of Knowledge, " Archive for History of exact Sciences, 3 (1 966), 1 37-1 59.

[Frei 1 985] Gunther Frei, Der Briefwechsel David Hilbert-Felix Klein (1 886- 1 9 1 8), Ar­

beiten aus der Niedersachsischen Staats­ und Universitatsbibliothek Gbttingen, Bd. 1 9, Gbttingen: Vandenhoeck & Ruprecht, 1 985. [Freudenthal 1 98 1 ] Hans Freudenthal, "David Hilbert," Dictionary of Scientific Biography,

9For a brief account of the work and its historical reception, see the Introduction by Franz Lemmermeyer and Norbert Schappacher to the English edition [Hilbert

1 998,

xxiii-xxxvi]. 1 0Perhaps the closest he came to fulfilling Minkowski's w1sh was the

Ausarbeitung

of his

1 897 lecture course. now available in English translation in [Hilbert 1 993]

1 1 Historical verdicts with regard to the sudden demise of invariant theory have varied considerably (see [Fisher

1 966] and [Parshall 1 989]). The merits of classical in­

variant theory were later debated in print by Eduard Study and Hermann Weyl. For an excellent account of this and other subsequent developments in algebra, see [Hawkins 2000].

© 2005 Spnnger SC1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1 , 2005

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I 6 vols. , ed. Charles C. Gillispie, vol. 6, pp.

[Hilbert 1 890] --, " U ber die Theorie der al­

Volume 1 : Ideas and their Reception, ed.

388-395, New York: Charles Scribner's

gebraischen Formen," Mathematische An­

David E. Rowe and John McCleary, Boston:

nalen 36 (1 890), 473-534. , " U ber die Theorie der al­

(Parshall and Rowe 1 994] Karen H. Parshall

Sons, 1 97 1 . [Gilmer 1 98 1 ] Robert Gilmer, "Commutative

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--

Academic Press, 1 989, pp. 1 57-206.

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gebraischen lnvarianten 1," Nachrichten der

and David E. Rowe, The Emergence of the

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Gesel/schaft der Wissenschaften zu G6ttin­

American Mathematical Research Commu­

Martha K. Smith, New York: Marcel Dekker,

gen 1 891 , 232-242. [Hilbert 1 892a] -- , " U ber die Theorie der al­

nity, 1 8 76-1900. J.J. Sylvester, Felix Klein,

[Gordan 1 885/1 887] Paul Gordan, Vorlesungen

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vol. 8, Providence, Rhode Island: American

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Gesellschaft der Wissenschaften zu G6ttin­

Kerschensteiner,

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Leipzig:

Teubner,

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1 887.

[Hilbert 1 892b]

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, " U ber einen Satz von

and E.H. Moore, History of Mathematics,

Mathematical Society, 1 994. (Reid 1 970] Constance Reid, Hilbert. New York:

--

, " U ber die Theorie der al­

Springer Verlag, 1 970.

gebraischen lnvarianten Ill," Nachrichten der

[Rowe 1 998] --, "Mathematics in Berlin,

Hilbert," Mathematische Annalen 42 (1 892),

Gesel/schaft der Wissenschaften zu G6ttin­

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--

[Gordan 1 899]

--

, "Neuer Beweis des

[Hilbert 1 893]

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H.G.W. Begehr, H. Koch, J . Kramer, N.

, " U ber die vollen lnvari­

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antensysteme," Mathematische Annalen 42

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242.

287-344] . [Hilbert 1 896]

Schappacher,

and

E.-J.

Thiele,

Basel:

Birkhauser, 1 998, pp. 9-26. (Rowe 2000]

--

, "Episodes in the Berlin­

Gbttingen Rivalry, 1 870-1 930," Mathemati­ --

, " U ber die Theorie der al­

cal lntelligencer, 22(1 ) (2000), 60-69.

[Hashagen 2003] Ulf Hashagen, Walther von

gebraischen lnvarianten," Mathematical Pa­

Oyck (1856-1934). Mathematik, Technik und

pers Read at the International Mathematical

tingen: A Sketch of Hilbert's Early Career,"

TH

Congress Chicago 1893, New York: Macmil­

Mathematical lntelligencer,

Munchen, Boethius, Band 47, Stuttgart:

lan, 1 896, 1 1 6-1 24; reprinted in [Hilbert

44-50.

Wissenschaftsorganisation

an

der

Franz Steiner, 2003.

1 933, 376-383].

[Hawkins 2000] Thomas Hawkins, Emergence

[Hilbert 1 922]

[Rowe 2003a]

--

[Rowe 2003b]

--

, "Neubegrundung der

, "From Konigsberg to Gbt­

--

25(2)

(2003),

. "Mathematical Schools,

Communities, and Networks," in Cambridge

of the Theory of Lie Groups. An Essay in the

Mathematik. Erste Mitteilung," Abhandlun­

History of Science, val. 5, Modern Physical

History of Mathematics, 1 869-1 926. New

gen aus dem Mathematischen Seminar der

and Mathematical Sciences, ed. Mary Jo

York: Springer-Verlag, 2000.

Hamburgischen Universitat,

[Hesseling 2003] Dennis E. Hesseling, Gnomes in the Fog. The Reception of Brouwer's In­ tuitionism in the 1920's, Basel: Birkhauser,

1:

1 57-1 77;

reprinted in [Hilbert 1 935, 1 57-1 77]. [Hilbert 1 932/1 933/1 935]

--

, Gesammelte

Abhandlungen, 3 vols., Berlin: Springer, 1 932-

1 935.

2003. [Hilbert 1 888a] David Hilbert, " U ber die Darstel­

[Hilbert 1 978]

Nye, Cambridge: Cambridge University Press, pp. 1 1 3-1 32. [Rowe 2004]

--

, "Making Mathematics in an

Oral Culture: Gbttingen in the Era of Klein and Hilbert," to appear in Science in Con­

--

, Hilbert's Invariant Theory

text, 2004.

lung definiter Formen als Summe von For­

Papers, trans. Michael Ackermann, in Lie

(Toeplitz 1 922] Otto Toeplitz, "Der Alge­

menquadraten," Mathematische Annalen 32

Groups: History, Frontiers, and Applications,

braiker Hilbert," Die Naturwissenschaften 1 0,

(1 888), 342-350; reprinted in [Hilbert 1 933,

ed. Robert Hermann, Brookline, Mass . : Math

1 54-1 6 1 ] .

Sci Press, 1 978. , " U ber die Endlichkeit des

[Hilbert 1 993] --, Theory of Algebraic Invari­

lnvariantensystems fUr binare Grundformen,"

ants, trans. Reinhard C. Lauenbacher, Cam­

[Hilbert 1 888b]

--

Mathematische Annalen 33 (1 889), 223-

226; reprinted in [Hilbert 1 933, 1 62-1 64]. [Hilbert 1 888c] braischen

--

, "Zur Theorie der alge­

Gebilde

1,"

Nachrichten

der

bridge: Cambridge University Press, 1 993. [Hilbert 1 998]

--

(1 922) 73-77 . (Weber 1 893] Heinrich Weber, "Leopold Kro­

, The Theory of Algebraic

Number Fields, trans. lain T. Adamson, New

York: Springer-Verlag, 1 998.

necker," Jahresbericht der Deutschen Math­ ematiker-Vereinigung 2(1 893), 5-1 3.

[Weyl 1 932] Hermann Weyl, "Zu David Hilberts siebzigsten

Geburtstag, "

Die

Naturwis­

senschaften 20 (1 932), 57-58; reprinted in [Weyl 1 968, vol. 3, 346-347].

Gesellschaft der Wissenschaften zu G6ttin­

[Mertens 1 887] Franz Mertens, Journal fUr die

(Weyl 1 935] --, "Emmy Noether," Scripta

gen 1 888, 450-457; reprinted in (Hilbert

reine und angewandte Mathematik 1 00(1 887),

Mathematica, 3 (1 935), 201-220; reprinted

1 933, 1 76-183].

223-230.

[Hilbert 1 889a] braischen

--

Gebilde

, "Zur Theorie der alge­ II,"

in [Weyl 1 968, vol. 3, 435-444].

[Minkowski 1 973] Hermann Minkowski, Briefe

[Weyl 1 944] -- , "David Hilbert and his Math­

der

an David Hilbert, eds. L. ROdenberg und H.

ematical Work," Bulletin of the American

Gesellschaft der Wissenschaften zu G6ttin­

Zassenhaus, New York: Springer-Verlag,

Mathematical Society, 50, 61 2-654; reprinted

gen 1 889, 25-34; reprinted in [Hilbert 1 933,

1 973.

Nachrichten

1 84-1 9 1 ] .

[Noether 1 92 1 ] Emmy Noether, "ldealtheorie in

[Hilbert 1 889b] braischen

--

Gebilde

, "Zur Theorie der alge­ Ill," Nachrichten der

Gesellschaft der Wissenschaften zu G6ttin­

Ringbereichen," Mathematische Annalen 83 (1 921 ), 24-66.

in [Weyl 1 968, vol. 4, 1 30-1 72] . (Weyl 1 968]

--

, Gesammelte Abhandlun­

gen, 4 vols . , ed. K. Chandrasekharan, Berlin:

Springer-Verlag, 1 968.

[Parshall 1 989], Karen H. Parshall, "Toward a

(Yandell 2002] Ben H. Yandell, The Honors

gen 1 889, 423-430; reprinted in [Hilbert

History of Nineteenth-Century Invariant The­

Class. Hilbert's Problems and their Solvers,

1 933, 1 92-1 98].

ory," in The History of Modern Mathematics:

Natick, Mass. : A K Peters, 2002.

82

THE MATHEMATICAL INTELLIGENCER

i;i§lh§l,'tJ

O s m o Peko n e n , Ed itor

I

Statistics on the Table: The History of Statistical Concepts and Methods Stephen M. Stigler CAMBRIDGE, MASS , HARVARD UNIVERSITY PRESS PAPERBACK 2002 (first pnnllng 1 999) 5 1 0 PAGES US $ 1 9.95 ISBN 0-674-00979-7

REVIEWED BY IVO SCHNEIDER

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

T

he book consists of 22 chapters, all of which except the first were pre­ viously printed as articles in reviewed journals or books. The chapters are distributed in five parts with the titles: I. Statistics and Social Science, II. Gal­ tonian Ideas, III. Some Seventeenth­ Century Explorers, IV. Questions of Discovery, and V. Questions of Stan­ dards. One could argue about the choice of these titles, their order, or their interrelations. In a strictly histor­ ical account one would begin with part III; part I is as much concerned with economic as with social questions, and Galtonian ideas are very much related to social science. In short, the selection arrangement of these 22 articles is less stringent than in a book which treats a topic systematically or strictly chrono­ logically. Trying to get the original publica­ tions in order to find out about the changes made for the sake of this book (which according to the sample I could check are small), I learned that many of these articles are not easily avail­ able. The book contains 21 of the 38 ar­ ticles and books concerning the history of statistics that Stigler published be­ tween 1973 and 1997, including his monograph from 1986, The History of

Statistics-the Measurement of Un­ certainty before 1 900. So one advan­ Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: [email protected]

tage of the book is to make available some of Stigler's publications which are otherwise not easy to get. From a historical point of view the most interesting question is: How does

Statistics on the Table relate to Stigler's History of Statistics, which appears to

claim to cover the history of statistics, at least for the time before 1900? First, Stigler justifies the new book with the argument that, because sta­ tistical thinking and so statistical con­ cepts permeate the whole range of hu­ man thought, statistics in historical accounts is practically "never covered completely." Statistics on the Table represents, according to Stigler, "only a small selection of the possible themes and topics," but it treats, however in­ completely, one of the most important aspects of statistical work: statistical evidence in the form of data and their interpretation for the solution of a problem. For many, this project in such a general formulation represents the whole of statistical science, so it is not surprising that Statistics on the Table and The History of Statistics have sev­ eral things in common. Chapter 1 deals with the Karl Pearson of 1910/1 1 and so its material is not contained in The History of Statistics, which ends with 1900. It deals with the effect or non-effect of parental alcoholism on the alcoholism of the offspring. Pear­ son's vote for non-effect in the light of data collected in the Galton Laboratory met with considerable resistance and criticism due to different factors and interests, one of them being the tem­ perance movement of the time. What Pearson expected, or rather requested, from his critics was "statistics on the table," data which could confirm the position of his opponents and so dis­ prove him. Interestingly, Pearson's re­ quest was not seen by economists like Keynes, one of his opponents, as the appropriate method to deal with the controversy. Having shown in this way that "statistics on the table" was still far from being generally accepted as the method of handling questions of this kind, Stigler goes back in time in order to reconstruct the way in which collections of data were used before

1910.

© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number I , 2005

83

Starting with an evaluation of Quetelet's statistical work in chapter 2, which is less detailed than the chapter on Quetelet in his History, Stigler de­ votes the next two chapters to the sta­ tistical work of the economist Jevons, who is mentioned several times in the History but without any further evalu­ ation of his statistics. Stigler's interest in Jevons is motivated by the effect of his statistical work in overcoming the typical mid-nineteenth-century separa­ tion of statistics, understood as data collection, from the interpretation of the data, especially in the social and economic domain. A chapter on the work of the econ­ omist and statistician Francis Ysidro Edgeworth ends the first of the five parts of Statistics on the Table. Edge­ worth does not figure prominently in the history of statistics, because his statistical ideas were dispersed over a great many not easily digestible arti­ cles. So Stigler, like a Robin Hood of the history of statistics, takes away from the rich in reputation in order to give to the historically neglected Edge­ worth, whom he had honoured already in his History with a full chapter, indi­ cating that in his eyes Edgeworth was as important in the development of sta­ tistics as Galton and Karl Pearson. The five chapters devoted to "Gal­ tonian ideas" in part II. differ from the Galton chapter in the History by em­ phasizing different topics and the im­ pact of Galton and his methods. So whereas Galton's work on fingerprints is only mentioned without any further analysis in the History, Stigler devotes the whole of chapter 6 of Statistics on the Table to it, including the accep­ tance of fingerprints as evidence in court. Galton's and his contempo­ raries' contribution to "stochastic sim­ ulation," with a set of special dice used for the generation of half-normal vari­ ants, plays no role whatsoever in the History. Regression is the topic of the next two chapters, of which only the second deals with Galton's contribu­ tion to it, whereas chapter 8, "The his­ tory of statistics in 1933," hints at the more subtle aspects of regression visi­ ble in Hotelling's devastating review of Horace Secrit's book The Triumph of Mediocrity in Business from 1933, the

84

THE MATHEMATICAL INTELLIGENCER

year which Stigler likes to consider as the proper starting point of mathemat­ ical statistics. Stigler's inclination to act occasionally as an agent provoca­ teur might in this case be seen at odds with his History, which should be con­ sequently " The history of non-mathe­ matical statistics. Chapter 10 then discusses the relatively early use (com­ pared with other social sciences) of statistical techniques in psychology, which according to Stigler is due to ex­ perimental design. The following three chapters deal with publications of the second half of the 17th century. It is not clear to me why the first two of them belong in Sta­ tistics on the Table. Nor do I under­ stand the use of the word "probability" in the context of Huygens's tract on the treatment of games of chance or the correspondence between Pascal and Fermat common to these chapters. Pascal, Fermat, and Huygens were all well aware of contemporary concepts of probability, though none of these concepts appears in their treatment of games of chance. In chapter 13, how­ ever, a contemporary concept of prob­ ability is treated, as used by John Craig in order to determine the trustworthi­ ness of statements concerning histori­ cal events, or in Craig's term the "his­ torical probability," which increases in proportion to the number of witnesses in favour of it and decreases in pro­ portion to the time elapsed since the event. A function representing the de­ pendence of Craig's historical proba­ bility on time and distance is tested by putting on the table the data of Laplace's birth and death found in 65 books of the 19th century. Of the six articles making up "Ques­ tions of discovery," chapter 18, which treats the history of the so-called Cauchy distribution, has no relation to any concrete set of data put on the table, whereas the other five chapters do. The first chapter of this part is de­ voted to eponymy, the practice in the scientific community of affixing the name of a scientist to a discovery, the­ ory, etc. as a reward for scientific ex­ cellence in the relevant field. Such a re­ ward presupposes distance in time and place from the work honoured by eponymy. Accordingly, it cannot be ex"

pected that scientific discoveries are named after their original discoverer, or, to formulate it more aphoristically as a law of eponymy, "no scientific dis­ covery is named after its original dis­ coverer." Since Stigler sees the sociol­ ogist Robert K. Merton as the originator of this so-called law, and since Stigler does not want to begin with a counterexample for the validity of this law, the title-giving eponymy of this law is "Stigler's law of eponymy." For the joke's sake it does not matter that this is not an eponymy proper, which would have demanded that the community of sociologists after an ap­ propriate period of time ought to have accepted it. A proper eponymy­ namely the affixing of the names of Gauss and sometimes Laplace, but not the name of de Moivre, the real "dis­ coverer," to the normal distribution­ is discussed on the basis of 80 books published between 1816 and 1976. The discussion showed that at least in this case, the eponymy was awarded only after considerable time by the scien­ tific community. Seen with eyes ac­ customed to eponymic practice, the next chapter (15), which answers the question "Who discovered Bayes's the­ orem" with Nicholas Saunderson as the most probable candidate, appears as Stigler's next attempt to avoid a coun­ terexample to the law of eponymy, be­ cause most people believe that Bayes discovered Bayes's theorem. A prob­ lem not touched so far when dealing with discoveries was treated by Thomas S. Kuhn, who pointed to si­ multaneous discovery of the "same" thing by several people. In his discus­ sion of Kuhn, Yehuda Elkana pointed to difficulties inherent in the concept of "sameness. " Difficulties of this kind are the topic of chapter 16, which de­ scribes the first steps made by Daniel Bernoulli and Euler concerning the theme of maximum likelihood. Sameness plays no important role in the next chapter, dealing with the claims of Legendre and Gauss con­ cerning the method of least squares. The data of the French meridian arc measurements and their interpolation by Gauss in 1 799 are interpreted as in­ conclusive for Gauss's claim to have devised and applied the method of

least squares at that time or even be­ fore. Again Stigler's social attitude as the Robin Hood of the history of sta­ tistics becomes evident when he states that, despite Gauss's undisputed mer­ its in developing algorithms for the computation of estimates, it was Le­ gendre "who first put the method within the reach of the common man." However, Gauss's contribution to the method of least squares is seen much more positively in this article than in Stigler's History. The last chapter (19) in this part is mainly concerned with a paper of Karl Pearson and his collaborators from 1913, in which Pearson fitted a quasi-in­ dependent model to the data of incom­ plete contingency tables, testing the fit by a chi-square test, which, as was rec­ ognized by R. A Fisher, used the wrong number of degrees of freedom. But the concept of degrees of freedom had been introduced only in 1922 by Fisher, who had not seen that for the special class of tables considered by Pearson the use of the correct number of degrees of free­ dom would not have changed Pearson's conclusions. The last three chapters are sub­ sumed under the title "Statistics and Standards." In the first the observation that many of the most powerful statis­ tical methods, like the method of least squares, are originally connected with the determination of standards like the standard meter, is interpreted not as ac­ cidental but as a consequence of the purpose of a standard to measure, count, or compare as accurately as pos­ sible. In a second part Stigler describes how the fact that in experimental sci­ ence no absolute accuracy can be achieved, that every measurement is in­ evitably connected with error and so with uncertainty, eventually led to the creation of standards of uncertainty, standards in statistics like standard er­ ror curves or standard deviations. The last chapter (22), written to­ gether with William H. Kruskal, is re­ lated to standards in statistics in that the first part of it answers the question when and why the normal distribution was called "normal." The other parts of the chapter are concerned with the am­ biguity of the words "normal" and "nor­ mality," exemplified in paragraphs de-

voted to the terms "normal equations" in connection with the method of least squares, "normality in medicine," and "normal schools" in the educational system. Stigler maintains that there is a mutual dependence between the use of "normal" in science and in the realm of public discourse. Before this last chapter I found my favourite "The trial of the Pyx," which is a test to control the quality and cor­ rectness of the coin production at the Royal Mint for more than seven cen­ turies. Stigler interprets the trial of the Pyx as "a marvellous example of a sam­ pling inspection scheme for the main­ tenance of quality." It is amusing to read his report of the most famous master of the Royal Mint, Isaac New­ ton. Stigler amasses arguments for scepticism concerning Newton's hon­ esty as master of the Mint, thus dis­ qualifying Stigler forever as a member of the invisible college of Newtonians. However, perhaps concerned about his good relations to highly regarded British institutions, he finds on the ba­ sis of the research work of others "no grounds for believing that he
Theory of Bergman Spaces Boris Korenblum, Haakan Hedenmalm, and Kehe Zhu HEIDELBERG, SPRINGER-VERLAG 2000. PAGES, 2 ILLUS €59 50

286

REVIEWED BY DAVID BEKOLLE

L

et D denote the unit disk of the complex plane. For 0 < p :s; oc and - 1 < a :S oo, the Bergman space A€ of

D is the closed subspace of the Lebesgue space Lg : = LP(D, (1 - lzl2)"'dxdy (z = x + iy) consisting of holomorphic functions. For a = 0, we write AP = A{;. Intensive research on the theory of Bergman spaces has been carried on since the early 1970s. The start followed the "essential" completion of the theory of Hardy spaces on D. In particular, A� is a closed subspace of the Hilbert space a. We call Bergman projector and we denote by Pa the orthogonal projector of the Hilbert space L� onto its closed subspace A�. It is well known that Pa is the integral operator defined on L� by the Bergman kernal Ba(z, w) : = Ca( l - ZW)-(l+a). One of the first fundamental results in the theory of Bergman spaces was established in 1984 independently by F. Forelli and W. Rudin [7] and E. M. Stein [10]. According to this result, for 1 :S p < oc and - 1 < a < oc, the Bergman projector Pa extends to a bounded pro­ jector of L€ to A€ if and only if p > 1. For an account of the results obtained in the 1970s and 1980s, see the book of K. Zhu Operator Theory in Bergman Spaces [ 1 1 ] . The aim o f the present book by Korenblum, Hedenmalm, and Zhu is to present some deep results obtained in the 1990s in function theory and in op­ erator theory in Bergman spaces on the unit disk, namely: 1. K. Seip's geometric characteriza­ tions of interpolation and sampling sequences for A€; 2. the discovery by H. Hedenmalm of contractive zero divisors for A� and its implementation for A€ (p =t- 2) by P. Duren, D. Khavinson, H. S. Shapiro, and C. Sundberg; 3. outstanding results related to the "curiously resistant" characteriza­ tion problem of zero sequences of A€ functions; 4. other striking results on the bihar­ monic Green function due to H. Hedenmalm, P. Duren, D. Khavin­ son, H. S. Shapiro, and C. Sundberg, and on invariant subspaces of A€ due to A Aleman, A. Borichev, H. Hedenmalm, S. Richter, S. M. Shimorin, and C. Sundberg. The book under review is welcome and will be very useful-among many

© 2005 Spnnger Sc1ence+ Bus1ness Med1a, Inc , Volume 27, Number 1, 2005

85

reasons, because it includes a self-con­ tained proof of K. Seip's geometric characterizations of interpolation and sampling sequences for Bergman spaces A g. Recall that a sequence r = {zj}j of distinct points of D is an inter­ polation sequence for AR (0 < p < oo) if, for every sequence {w1)1 of complex numbers satisfying the condition

L Cl - lzJI2? +ajwjf < 00, j

there exists a function! E Ag such that = w1 for all j. A geometric char­ acterization of interpolation sequences for Hardy spaces was obtained in 1958 by L. Carleson [5] for p = oo. For gen­ eral p E (O,oo] see, e.g., the books [6] and [8]. Following the theorem of Forelli­ Rudin and Stein stated above, interpo­ lation sequences for Ag were first stud­ ied by Eric Amar [ 1], and it is unfair that his name is not quoted in this book regarding results of Chapter 4. To prove his theorems, K. Seip re­ lies heavily

f(z1)

on a fundamental paper of B. Karen­ blum [9] and on earlier results of A. Beurling [3] on interpolation for the Banach space of functions of exponential type :S a and bounded on the real line.

•

•

Seip's characterizations use notions of density inspired by Beurling and Karen­ blum. One of these notions of density, denoted D1 (f)), is defmed as follows. Let r = {z1)1 be a separated (with re­ spect to the Bergman distance) se­ quence in D, and let r E Cf,1 ) . We set

D(f,r) For every quence

L. l logjzJ I J

_!_ 2

< lzJ I < r)

1 log (1-r )

z E D, rz : =

:

{

we form a new se­

z

-_ _Z..L i_ __ 1 - ZjZ

}j·

The upper Seip density of r is defined by

D1(f)

=

limsupr--> 1 SUPz ErJJ(fz,r).

Finally, K. Seip's Theorem states the following: Let 0 < p < oo and - 1 < a <

86

THE MATHEMATICAL INTELLIGENCER

oo, let r be a sequence of distinct points of D. Then the following conditions are equivalent: 1. r is an interpolating sequence for A&'; 2. r is separated and D 1(f) < ; 1 • "

The proof of this theorem sheds light on Korenblum's difficult paper [9]. The book under review is directed at graduate students and new re­ searchers in the field. It will be very useful for senior researchers as well. At the end of each chapter, various ex­ ercises are proposed to the reader. Many open problems are also stated. For a more recent report on open prob­ lems on zero sequences, invariant sub­ spaces, and factorization of functions in A 2, the interested reader may con­ sult the report by Aleman, Hedenmalm, and Richter [2]. As a minor point (compared to the quality and quantity of the book as a whole), the reviewer mentions the fal­ sity of the proof of Theorem 1.21, page 21, which identifies the dual space of A&' (0 < p :S 1, - 1 < a < oo) with the Bloch space. The statement is correct, but its proof should be corrected in the second edition of the book. For a cor­ rect proof, see [4] . REFERENCES

[1 ] Amar, E. Suites d'interpolation pour les

classes de Bergman de Ia boule et du polydisque de en. Can. J. Math. 30 (1 978), 71 1 -737. [2] Aleman,

A,

H.

Hedenmalm, and S.

Richter. Recent progress and open prob­ lems in the Bergman space (preprint). [3] Beurling, A The Collected Works of Arne Beurling by L. Carleson, P. Malliavin, J.

Neuberger, and J. Wermer. Vol. 2, Har­ monic Analysis, Boston, Birkhauser (1 989), 341 -365. [4] Bekolle, D. Bergman spaces with small ex­

ponents. Indiana Univ. Math. J. 49 (3) (2000), 973-993. [5] Carleson, L. An interpolation problem for

bounded analytic functions. Amer. J. Math. 80 (1 958), 921 -930. [6] Duren, P. Theory of HP spaces, Academic

Press, New York (1 970). [7] Forelli, F. and W. Rudin. Projections on

spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1 974), 593-602. [8] Garnett, J. B. BoundedAnalytic Functions,

Academic Press, New York (1 981).

[9] Korenblum, B. An extension of the Nevan­

linna theory. Acta. Math. 1 35 (1 975), 1 87219. [1 0] Stein, E. M. Singular integrals and estimates

for the Cauchy-Riemann equations. Bull. Amer. Math. Soc. 79 (1 973), 440-445. [1 1 ] Zhu,

K. Operator Theory in Function

Spaces. Marcel Dekker, New York (1 990).

Faculte des Sciences Universite de Yaounde I B.P. 81 2 Yaounde Cameroon e-mail: [email protected]

Gamma: Exploring Euler's Constant Julian Havil PRINCETON, PRINCETON UNIVERSITY PRESS 2003 XXIII + 266 PAGES. US $29.95. ISBN 0-691 -09983-9

REVIEWED BY GERALD L. ALEXANDERSON

T

his is a Golden Age-well, at least it is for students and those of us who love to read about mathematics outside our own area of expertise. In my youth we could choose from the books of E. T. Bell, What Is Mathe­ matics? by Courant and Robbins, some of P6lya's books, and Rademacher and Toeplitz. There were others, but the list was short. Today the catalogues of Springer, Cambridge, Princeton, Ox­ ford, the AMS, and the MAA overflow with general books, accessible to stu­ dents and mathematical amateurs, and on a wide variety of subjects. Now we even see books coming out on specific numbers, notably Eli Maor's e: The Story of a Number (Princeton Univer­ sity Press, 1998), Paul Nahin's An

Imaginary Tale/The Story of v=I (Princeton University Press, 1998), David Blatner's The Joy ofPi (Walker, 1999), Charles Seife's Zero: The Biog­ raphy of a Dangerous Idea (Penguin, 2000), Hans Walser's The Golden Sec­ tion (MAA, 2002), and Mario Livia's The Golden Ratio: The Story of ¢, the

World's Most Astonishing Number

(Broadway, 2003). And here we have Havil's book on the Euler-Mascheroni constant. Given the plethora of inter-

esting numbers, this series could go on for some time. Of course, numbers don't get very much more interesting than Euler's number 'Y· I was dubious when I read G. J. Chaitin's article, "Thoughts on the Riemann Hypothesis" in the Winter, 2004, issue of this magazine (vol. 26, no. 1) in which he included Havil's book in his list of recent books on the Riemann Hypothesis. I checked Harold Edwards's review of the RH books by Derbyshire, du Sautoy, and Sabbagh in the same issue and noted that he did not see fit to include Gamma in his re­ view. At the time I was familiar with the three reviewed by Edwards and didn't see the relevance of Havil's book in this context. But I see that a case can clearly be made for its inclusion, as Chaitin does. Havil is obviously enthusiastic about his subject and remarkably eru­ dite. There are many references to de­ velopments in mathematics that are re­ lated to Euler's constant, often quite recent results. He obviously watches the literature. Most of the connections have to do with problems that at some level involve either natural logarithms or partial sums of the harmonic series. He looks at ways of calculating 'Y to great accuracy. The problem is non­ trivial, for y is defined as limn__,x (Hn ln n), where Hn 1 + + + + . . . Both Hn and ln n grow without + _1_, n bound, but they grow very slowly, so just calculating the difference for larger and larger n is not efficient. What do we know about y? First we learn what we don't know-whether it is irrational, for example. Unlike con­ stants such as 1T and e, where questions of whether they are irrational and tran­ scendental were settled before the 20th century, at the beginning of the 2 1st we still don't have this information about y. We do learn that from an Euler­ Maclaurin expansion we get

t f i

=

n

r=I

k�l

1

- - ln n k

1 2n

-

1

+

1 12n2

--

1 252n6

-- + -- + 120n4

One might ask how Lorenzo Mascheroni (better known for his proof that a geometric construction

possible with straightedge and com­ pass can be carried out with the com­ pass alone) got his name attached to y. He approximated it to 32 decimal places, only the first 19 of which were correct! In 1962 Donald Knuth com­ puted 'Y to 1271 decimal places, and in 1999 it was calculated to 108,000,000 decimal places. Writing 'Y as a contin­ ued fraction, we find that the conver­ gent 323007/559595 differs from 'Y by 1.025 X 10- 12. Again using contin­ ued fractions, Thomas Papanikolaou showed that if 'Y were to be a fraction, its denominator would be greater than 10242080, perhaps providing a hint that 'Y is irrational. In demonstrating these and many other facts about y, we're led on an il­ luminating tour of Bernoulli numbers; the Basel problem; f(x), Euler's gamma function; Stirling's approxima-

N u m bers don ' t get m uch more i nterest i n g than E u l e r ' s n u m ber y. tion formula; and much, much else. And if that weren't enough, in a series of appendices we find a concise intro­ duction to complex function theory. Sometimes the mathematical state­ ments are sturming and leave one won­ dering how things seemingly so dis­ parate can be related. In the chapter about appearances of harmonic series, the author talks about musical tones, of course, then describes surprising re­ sults on the infrequency of record rain­ falls, for example. Next he provides an economical test for destruction of beams to check their strength and breaking points. Then there's a question of sending Jeeps across the desert, and problems of card sorting, Hoare's Quicksort algorithm, the maximum pos­ sible overhang of playing cards placed on the edge of a table, and so on and so on. Some of these are fairly well known, though others were to me quite new and surprising. On logarithms, he describes clearly Benford's now well-known but surprisingly recent law on the lack of uniform distribution of digits in collec-

tions of data (like baseball statistics, ge­ ographic areas, street addresses, death rates, and such). Just as Benford's Law says that l's ap­ pear more often than 2's as leading dig­ its, 2's more often than 3's, and so on, for a descending curve of frequencies, for me the chapters of the book were rather the opposite, increasing in inter­ est as I went along. I found the begin­ ning material on the history of loga­ rithms rather heavy going. Perhaps it's just too familiar, but there's also the problem that however valuable loga­ rithms were in their infancy, the calcu­ lations are not likely to be exciting to the modern reader. Havil quotes Laplace, however: " . . . by shortening the labors, [logarithms] doubled the life of the as­ tronomer." The middle chapters have many cormections to interesting prob­ lems; the latter ones make cormections to number theory and succeed in mak­ ing these quite clear in spite of the de­ tails being considerably more mathe­ matically challenging to the reader. The historical references are charm­ ing, and many of the quotes are fasci­ nating and, to me, unfamiliar. Here are a few examples: •

•

•

•

•

Leo Tolstoy: "A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the de­ nominator the smaller the fraction." Heinrich Hertz: "One cannot escape the feeling that these mathematical formulas have an independent exis­ tence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them." Krzysztof Maslanka: "We may­ paraphrasing the famous sentence of George Orwell-say that 'all mathe­ matics is beautiful, yet some is more beautiful than the other. ' But the most beautiful in all mathematics is the Zeta function. There is no doubt about it." Paul Erdos (paraphrasing Einstein): "God may not play dice with the Uni­ verse, but there's something strange going on with the prime numbers!" David Hilbert (in response to a ques­ tion of which mathematical problem

© 2005 Springer Science+ Bus1ness Med1a, Inc., Volume 27, Number 1 , 2005

87

•

was the most important): "The prob­ lem of the zeros of the Zeta function, not only in mathematics, but ab­ solutely most important!" This is followed by the following from Morris Kline: "If I could come back after five hundred years and find that the Riemann Hypothesis or Fermat's last 'theorem' was proved, I would be disappointed, because I would be pretty sure, in view of the history of attempts to prove these conjectures, that an enormous amount of time had been spent on proving theorems that are unimpor­ tant to the life of man."

The last is identified as the response in an interview. The book in which it appeared is identified correctly and the date is correct, but the interviewer and editors of the book are not identified anywhere. The interviewer happened to be this reviewer. These and many other sections raise a question. There's a fine line between a book meant largely for entertaining and educating the generally informed but non-specialist reader, and a schol­ arly book that carefully documents statements made in the text. The quo­ tations above appear without specific references. Even more frustrating to this reader is the lack of specific ref­ erences to theorems and articles cited. On page 1 13, the author cites a re­ markable result of de la Vallee-Poussin of 1898, which he calls "baffling," that "if we divide an integer n by all inte­ gers less than it and average the deficits of each quotient to the integer above it, the answer approaches 'Y as n ---> oo. " As an example he points out 1 _ Ig�gg cllOOOO l - 10000 ) gives that _ 10000 r� 1 I r r 0.577216. . . . He then adds that the result remains true if the divisors are those in any arithmetic sequence or if they are only the prime divisors. Some­ one interested in pursuing this further is in for a literature search, for no spe­ cific references are given, and de la Val­ lee-Poussin, though listed in the index, is not listed in the set of references at the end. Because of the age of the work, Mathematical Reviews and MathSciNet will be of no help. And even Zentralblatt would fail if it hadn't

88

THE MATHEMATICAL INTELLIGENCER

recently incorporated the earlier mate­ rial from the Jahrbuch. So tracking down something like this is not trivial. On page 1 15, Havil cites a 1990 result of "Maier and Pommerance," in a well­ known paper on gaps between primes

(Trans. Amer. Math. Soc. 322 (1) (1990) 201-237), but he does not give

this reference, and a reader not know­ ing something of the result would have a problem: there are lots of Maiers in mathematics-this one happens to be Helmut Maier-and Havil misspells the name of the second author. (The sec­ ond author is, of course, Carl Pomer­ ance.) The result, which Havil justifi­ ably calls stupendous, is that [ CPn + 1 - Pn) (log log log PnY 7 nlim ->x [(log Pn)(log log Pn)(log log log log Pn)] 2:

4ey!c,

where c = 3 + e - c and Pn is the nth prime. (Incidentally, as stated in the book, the right-hand side is incorrect; it should be 4e�'/c.) I conclude that even in a general book of this sort, some endnotes pointing the reader to the sources would be helpful. Now, at the risk of appearing petty I'll bring up something even less im­ portant but still something of a prob­ lem. Recently I reviewed another Princeton University Press book and commented that the use of both serif and sans serif type faces gives "the pages rather a strange look" Another reviewer was less generous and said it looked like an explosion in a font fac­ tory. There's no such problem here. The type is dignified and carefully set, to the point where the typesetter has fastidi­ ously made sure that in every differen­ tial, the "d" is in roman and the "x" or " " y is in italics! But one could have hoped that the Press would bring on board a sufficient number of proof­ readers and fact checkers that we would not find formulas misstated, mis­ leading, or not corresponding to the ac­ companying figure (see pages 44, 74, 94, 100). The errors cited are easily cor­ rected, but when one fmds these slips from time to time, it makes a person wonder what might be wrong in the ma­ terial that is not so easily checked and with which one is not familiar. Diacrit-

ical marks are only sometimes present (never, apparently, on Erdos and, in the case of the oft-cited de la Vallee­ Poussin, almost always with a grave ac­ cent instead of the correct acute ac­ cent). Authors are not the best people for fmding these things; they have spent too much time with the manuscript. It is clear from the text that this author regards Euler as a mathematical hero, yet in citing Euler's Introductio, prob­ ably his masterpiece, the title is mis­ spelled and the date of publication is wrong (page 15). Other authors' names are misspelled-Montucla, Bernoulli (and why does Jacob Bernoulli have separate entries in the index?). There are the usual typographical errors and little slips in grammar. Those one ex­ pects. But some of the errors I have cited are disturbing to the careful reader, and a press with the prestige of Princeton should not be placing some­ thing like this in print without more careful checking. For all that, this is a most enjoyable book-full of good historical asides, truly beautiful bits of mathematics, and clear exposition. I highly recommend it-but I hope that subsequent editions (including the paperback) can include some corrections and more references. Department of Mathematics and Computer Science Santa Clara University Santa Clara, CA 95053-0290 USA e-mail: [email protected]

Schliisseltechnologie Mathematik Einblicke in aktuelle Anwendungen der Mathematik Hans Josef Pesch STUITGART/LEIPZIG/WIESBADEN: B.G. TEUBNER VERLAG. 1

AUFLAGE 2002

1n the series Mathematik fur lngenieure und

Naturwissenschaftler. 1 85 PAGES.

€ 22

90 ISBN 3-51 9-02389-X

REVIEWED BY GERHARD BETSCH

T

he title means what the author claims: Mathematics is a key technology of our future. It permeates more and more our everyday life. Nevertheless, the number of students of mathematics and of subjects with a strong mathematical background is decreasing-at least in Germany. How can we change this develop­ ment? The intended readership consists of high school graduates, high school teachers, and freshmen in subjects with a strong mathematical "flavour" or "vein." But people from all profes­ sions with a sufficient interest in math­ ematics will profit considerably from this book I give a sketch of the contents. •

•

•

•

•

John Bernoulli's problem of the brachistochrone (1696) and the Fer­ mat Principle. The isoperimetric problem and its roots in antiquity. More generally: Calculus of varia­ tions, its origin and development. Problems of optimal control. Pontr­ jagin's Maximum Principle. Differen­ tial Games. The key role of modem numerical mathematics in solving control problems. Problems and methods of Optimal Control and Numerical Analysis in connection with the development of the ISS (International Space Sta­ tion). Return of a space shuttle into the at­ mosphere. Experiments in space; the Genesis mission. Lagrange points in gravita­ tional fields. Automatic steering of aircraft. Com­ pensation, or avoiding of "micro­ bursts." Optimal planning of robots. Optimal control of chemical processes. Con­ trol problems in economics. Computation in real time.

Features of this book are an abun­ dance of solid historical information, and very informative sections on tech­ nical problems involved. The text is supported by very instructive high­ quality illustrations. For readers who do not skip the for­ mulas, the author offers carefully se­ lected (non-trivial) problems.

The author is a professor of engi­ neering mathematics in the University of Bayreuth (Germany). The book be­ longs to a series of textbooks for fu­ ture engineers and scientists. A translation of this work into Eng­ lish would be desirable. Furtbrunnen 1 7

7 1 093 Wei I im Sch6nbuch Germany e-mail: Gerhard . Betsch@t -online.de

Sync-How Order Emerges from Chaos in the Universe, Nature, and Daily Life Steven Str-ogatz QG,

NEW YORK, HYPERION BOOKS, 2004, $US

1 4 95

ISBN 07868872 1 4

REVIEWED BY A . W . F . EDWARDS

''

B

eing obliged to stay in my room for several days," wrote the feverish Christiaan Huygens in Febru­ ary 1665, "I have noticed an admirable effect which no-one could ever have imagined. It is that my two newly-made [pendulum] clocks hanging next to each other and separated by one or two feet keep an agreement so exact that the pendulums always oscillate to­ gether without variation. After admir­ ing this for a while I finally realised that it occurs through a kind of sympathy: mixing up the swings of the pendulums I found that within half an hour they al­ ways return to consonance." A third of a millennium later the de­ signers of London's new Millennium Bridge for pedestrians-claimed as the world's flattest suspension bridge­ were treated to another "admirable ef­ fect" which perhaps they should have imagined. As the enthusiastic crowd crossed it on opening day its impercep­ tible swaying motion caused walkers to adopt an unconscious sailor's roll so as to keep their balance. But of course they did so all together and, as with a child on a swing, positive feedback did the rest. The bridge was closed. Brian Josephson, Nobel Laureate in Physics in 1973, was first with the explanation.

Both these stories, and many more, are relayed in Sync, an eloquent grand tour of synchronous behaviour in physical, biological, and human sys­ tems. Divided into three parts, Living Sync, Discovering Sync, and Explor­ ing Sync, it would have been easier reading if Discovering Sync had come first. Sympathetic pendulums and swaying bridges are more accessible images than firing brain cells and cou­ pled oscillators. The physical chapters in Discover­ ing Sync are particularly rewarding, partly because they are lighter on au­ tobiographical detail. Strogatz almost makes quantum theory and Josephson junctions comprehensible. Living Sync is burdened with a som­ niferous chapter "Sleep and the daily struggle for sync" about circadian rhythms and experiments in which people spent long periods isolated from any information about the pas­ sage of time (but no mention of the Po­ lar Eskimos and their dayless winters). In Exploring Sync the author dwells on the inappropriateness of linear models in human affairs-hardly a new thought, but one which needs constant repetition at a time when universities, at least in Europe, are increasingly af­ flicted by intervention based on the bureaucratic assumption that each is merely the sum of its parts. A penultimate chapter, "Small world networks," is an interesting account of the properties of communication net­ works and how their structure-vary­ ing between regular and random-in­ fluences the path-lengths between typical nodes. One almost expects a further digression into population ge­ nealogies and Bayesian probability net­ works, but it is hard to fit network sto­ ries into a book on synchrony. There is no mathematics. "To con­ vey the vitality of mathematics to a broad spectrum of readers, I've avoided equations altogether, and rely instead on metaphors and images from everyday life to illustrate the key ideas." With considerable success. Gonville and Caius College Cambridge CB2 1 TA, U . K. e-mail: awfe@cam .ac.uk

© 2005 Spnnger Sc1ence+Bus1ness Medta, Inc . Volume 27, Number 1 , 2005

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Mathematics Unlimited2001 and Beyond Bjorn Engquist and Wilfried Schmid, editors NEW YORK, SPRINGER-VERLAG, 2001 1 237 PAGES, HARDCOVER US $59.95 ISBN: 3-540-669 1 3-2

REVIEWED BY PETER W. MICHOR

E

xpectations were high at the turn of the century for publications that would describe the state of mathemat­ ics by looking into the future. Hilbert's problems at the Paris congress set the example. This book is the contribution from Springer-Verlag. This is an anthology of 63 articles in­ cluding five interviews, by a range of well-known mathematicians and other scientists. Among these articles one finds overviews over particular fields stressing open problems, descriptions of neglected themes, and essays on the relation between mathematics and soci­ ety. Two themes that are interconnected in many ways in many of the articles are "applications" and "computing." Let me start by quoting. 1

The authors remember from their high school days how they had to learn computing sines and cosines from ta­ bles. The calculator was there, actu­ ally we all had our own, but the edu­ cational programs had not yet adapted to the new technology. Basi­ cally, the same effect hits the univer­ sities when we ignore the existence of Matlab, Mathematica, Maple, and similar software, which solves virtu­ ally any exercise in basic calculus and linear algebra. Sometimes it ap­ pears that many teachers in mathe­ matics regard such software as a threat towards their profession. That is in our view a tragic misunder­ standing; proper application of soft­ ware would allow us to increase the level of calculus, by enabling the stu-

dents to learn more andfocus on what are really the difficult issues rather than wasting their time on repetitive trivialities. [ . . . j It is the author's opinion that the undergraduate edu­ cation at most universities is out of phase with the modern professional application of mathematical models. [ . . . } Not only pure mathematicians seem to neglect the importance of do­ ing computer based mathematics and the need to adapt the education ac­ cordingly. Also in classical subjects, like physics and the geosciences, the role of mathematics and computers are kept at a moderate level with lit­ tle impact on the culture or courses. Some trivial observations explain the slow progress at incorporating mod­ ern computing tools. Students are sent like ping-pong balls between univer­ sity buildings. Each building has its own traditional culture and theo­ ries-and its own budget that must be protected. Each building gets its share of courses in a program, and the pro­ fessors in the building put in much effort to preserve the traditions of their particular subject. The result is a set of 'pure' subjects and strong con­ servatism-two characteristics that are not well correlated with a multi­ disciplinary and rapidly developing technological world. In basic mathematics, it seems that we really teach our students something that can be compared to doing astron­ omy without telescopes, or doing biol­ ogy without microscopes. Nobody should come out of basic mathematics instruction at universities without flu­ ency in at least one general-purpose computer algebra program. The fol­ lowing quotation reinforces this opin­ ion2 :

Do you have a message that you would like university mathematicians to hear? Of course, there is a very clear message. I am worried that mathe­ maticians are moving further and

1 Langtanger, Tveito: How should we prepare the students of Science and Technology, p.

812.

2Mathematics: From the outside looking i n . Achim Sachem interviewed b y V.A. Schmidt. 3After the "Golden Age": What next? Lennart Carleson interviewed by Bjorn Enquist. 4Bailey, Borwein: Experimental mathematics: recent developments and future outlook, p. 55. 5H. Cohen: Computational aspects of number theory, pp. 301 -330.

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THE MATHEMATICAL INTELLIGENCER

further away from the relevant prob­ lems in the natural sciences and society. In our core scientific engineering areas, we are continuously involved with mathematics, but there are very few mathematicians working at the labs. When we go to the universities, we find that even there the mathe­ maticians are not dealing with the questions we confront. Why? A math­ ematician's response to a very prac­ tical problem is often that it is too complicated. A mathematician will often want to ignore a particular con­ straint and limit the discussion to a specific problem that captures the essence. This approach is not helpful at all to the engineers. Furthermore, mathematicians seem to feel that there are enough beautiful problems within mathematics. Also Lennart Carleson3 stresses the importance of maintaining "all the con­ tacts with the neighboring applica­ tions, not only with computer science but also with physics and scientific computation and chemistry and biol­ ogy and so on." Research in pure mathematics can benefit a lot from using computers. Consider the following remarkable for­ mula4 "whose formal proof requires nothing more sophisticated that fresh­ man calculus:

"'

1T =

:;?;0

1 16k

(

4 2 8k + 1 8k + 4

1 1 ) ----8k + 5 8k + 6

This formula was found using months of PSQL computations, after corre­ sponding but simpler n-th digit formu­ las were identified for several other constants, including log(2)." This theme is taken up in another article, 5 which describes in 22 gems the uses of computational techniques and of computer experiments in number theory, in particular in class field the­ ory, and in arithmetic geometry, e.g.,

for the construction of tables of ellip­ tic cuiVes of given conductors. This ar­ ticle concludes with 12 challenges for the twenty-first century, including the Birch and Swinnerton-Dyer conjec­ ture, which is also one of the seven mil­ lennium problems. Only 1 1 of the 63 articles are de­ voted to pure mathematics alone. One of them6 describes an intriguing new countable class (ff of complex numbers called periods which contains all alge­ braic numbers. The elementary defini­ tion of a period says that its real and imaginary parts are given by the values of absolutely convergent integrals of rational functions with rational coeffi­ cients, over domains in !Rn given by polynomial inequalities with rational coefficients. The number 7T, logarithms of algebraic numbers, and values of Riemann's zeta function at integers ::::: 2 are periods, whereas e, 117T, and Euler's constant y are conjectured not to be periods. It is still an open problem to exhibit at least one number which is not a period. The question is to find an algorithm whether or not two periods are equal, and the conjecture is that one may pass between two integral representations of a period by using only additivity of the integrals, the change of variables formula, and the fundamental theorem of calculus. Pe­ riods have also an important role in the theory of £-functions and motives. This book gives a wide overview of different aspects of the possible future development of pure mathematics, also by posing conjectures, on wide­ open fields in applied mathematics and other sciences where mathematics plays or should play an important role, and on questions of education and the usefulness of mathematicians for an­ swering questions in engineering and natural, economic, and social sciences. It is very interesting reading. I hope that it will have some impact on the way in which we educate young math­ ematicians so that they will be able not only to push the frontiers in mathe­ matics itself in the future (where the accomplishments and prospects are bright enough) but also to answer ex­ pectations from outside mathematics. 6M.

Kontsevich, D. Zagier: Periods,

Fakultat fUr Mathematik Universitat Wien Nordbergstrasse 1 5 1 090 Vienna Austria e-mail: [email protected]

The Mathematical Century by Piergiorgio Odifreddi Arturo Sangalli, translator; Foreword by Freeman Dyson PRINCETON, PRINCETON UNIVERSITY PRESS 2004.

xx + 204 pages.

US $27.95

ISBN 0-691 -09294-X

REVIEWED BY GERALD L. ALEXANDERSON

T

o choose and explain to a general audience the thirty most important mathematical problems solved in the twentieth century takes courage. Pro­ fessor Odifreddi has done it here with remarkable clarity and elegance. He recognizes the difficulty of the task, particularly if one tries to do it in 180 pages. The challenges are: (1) the ab­ straction of modem mathematics and the difficulty of explaining the mean­ ing of the theories to the non-special­ ist; (2) the vast amount of mathe­ matics produced, particularly in the second half of the century; and (3) the fragmentation of mathematics into subfields. These difficulties do not de­ ter him, and he exhibits an amazing grasp of the various streams of modem mathematics. Further, he largely suc­ ceeds in showing how the various problems are related. The author is never without opin­ ions. On computers: "As is often the case with technology, many changes are for the worse, and the mathemati­ cal applications of the computer are no exception. Such is, for example, the case when the computer is used as an idiot savant, in the anxious and futile search for ever larger prime numbers. The record holder at the end of the twentieth century was 26,972.593 1, a number that is approximately 2 million digits long." On mathematics: " [A ma­ jority of the subfields of mathematics] are no more than dry and atrophied -

twigs, of limited development in both time and space, and which die a nat­ ural death." He claims that the disci­ pline "has clearly adopted the typical features of the prevailing industrial so­ ciety, in which the overproduction of low-quality goods at low cost often takes place by inertia, according to mechanisms that pollute and saturate, and which are harmful for the envi­ ronment and the consumer. The main problem with any exposition of twen­ tieth-century mathematics is, therefore . . . to separate the wheat from the chaff, burning up the latter and storing the former away in the barn." When readers recover from trying to decide whether their own contribu­ tions would be burned or stored, they can go on to Chapter 1, a brilliant es­ say on the foundations of mathematics, before they explore the thirty prob­ lems. Here the author shows his own mathematical predilection: mathemati­ cal logic. In the seventeen pages of this introductory chapter the author takes us from Pythagoras to Leibniz, to Frege and Cantor, Russell, Zermelo, and Fraenkel, then on to Grothendieck, Godel, Bourbaki, Eilenberg and Mac Lane, Church and Rosser, Kleene and Scott. It's a fast tour but leaves the reader with a good idea of how to re­ late sets, functions, categories, and lambda calculus, a treatment that even someone not much interested in foun­ dations can enjoy. This chapter is something of a tour de force. Now we come to his choice of prob­ lems. With such a list one can never please everyone. He has relied strongly on Hilbert's famous list of twenty-three problems described at the 1900 Paris Congress, as well as those problems solved by Fields Medalists or by win­ ners of the Wolf Prize. This is probably as rational a plan as any in searching for the "top 30" problems, but it also raises questions. Wiles's solution of the Fermat problem is included in spite of the fact that Wiles did not receive a Fields Medal because his age exceeded by a year the traditional cut-off age of 40. There is another question we could raise about using the Fields Medals as a criterion: some have suggested that

pp. 771-808.

© 2005 Spnnger Science+ Bus1ness Media, Inc , Volume 27, Number 1 , 2005

91

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certain branches of mathematics have been favored by Fields committees and some others have been ignored. Choosing the most important thirty inevitably raises the "41st chair" ques­ tion. L'Academie Fran<;aise chooses for membership the "40 immortals," and that makes people wonder who would occupy the 4 1st chair if there were one. From each of the past four centuries there were intellectual lumi­ naries who did not make it: Descartes, Rousseau, Zola, and Proust are in that august company. So what are the prob­ lems that might have occupied the 31st or 32nd slots? Or should others replace some of those already on the list? That could provide mathematical dinner party conversation for years. Let's look at the problems that did make the author's list. They're divided into three categories: Pure Mathematics

1. Analysis: Lebesgue Measure (1902); 2. Algebra: Steinitz Classification of Fields (1910); 3. Topology: Brouwer's Fixed-Point Theorem (1910); 4. Number Theory: Gelfond Trans­ cendental Numbers (1929); 5. Logic: Godel's Incompleteness Theorem (1931); 6. The Calculus of Variations: Doug­ las's Minimal Surfaces (1931); 7. Analysis: Schwartz's Theory of Dis­ tributions ( 1945); 8. Differential Topology: Milnor's Ex­ otic Structures (1956); 9. Model Theory: Robinson's Hyper­ real Numbers (1961); 10. Set Theory: Cohen's Independence Theorem (1963); 1 1. Singularity Theory: Thorn's Classi­ fication of Catastrophes (1964); 12. Algebra: Gorenstein's Classifica­ tion of Finite Groups (1972); 13. Topology: Thurston's Classification of 3-Dimensional Surfaces (1982); 14. Number Theory: Wiles's Proof of Fermat's Last Theorem (1995); and 15. Discrete Geometry: Hales's Solu­ tion of Kepler's Problem (1998). Applied Mathematics

1. Crystallography: Bieberbach's Sym­ metry Groups ( 19 10); 2. Tensor Calculus: Einstein's Gen-

eral Theory of Relativity (1915); 3. Game Theory: Von Neumann's Minimax Theorem (1928); 4. Functional Analysis: Von Neu­ mann's Axiomatization of Quan­ tum Mechanics (1932); 5. Probability Theory: Kolmogorov's Axiomatization (1933); 6. Optimization Theory: Dantzig's Simplex Method (1947); 7. General Equilibrium Theory: the Arrow-Debreu Existence Theorem (1954); 8. The Theory of Formal Languages: Chomsky's Classification (1957); 9. Dynamical Systems Theory: The KAM Theorem (1962); and 10. Knot Theory: Jones Invariants (1984). Mathematics and the Computer

1. The Theory of Algorithms: Turing's Characterization (1936); 2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950); 3. Chaos Theory: Lorenz's Strange At­ tractor (1963); 4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976); and 5. Fractals: The Mandelbrot Set (1980). One could easily argue about which problems belong where in the list (knot theory as part of applied mathematics?), but the author describes the distinction he makes between pure and applied mathematics thus: "Mathematics, like the Roman god Janus, has two faces. One is turned inward, toward the hu­ man world of ideas and abstractions, while the other looks outward, at the physical world of objects and material things. The first face represents the pu­ rity of mathematics, where the attention is unselfishly focused on the discipline's creations, seeking to know and under­ stand them for what they are. The sec­ ond face constitutes the applied side of mathematics, where the motives are in­ terested, and the aim is to use those same creations for what they can do." Overall the exposition is extraordi­ narily fine. The context of a problem is set and the result is explained in terms as simple as the subject allows. Con­ tributors at all levels of the solution are introduced, and if any were awarded a

Fields Medal or a Wolf Prize, that infor­ mation is included. The language at times is almost poetic. Not having avail­ able to me the original Italian edition, I cannot say whether the elegance of the language is primarily the contribution of the author or of the translator. Of course, occasionally one becomes aware that it is a translation. For exam­ ple, Bishop Berkeley's classic descrip­ tion of infinitesimals as "ghosts of de­ parted quantities" when passed from English to Italian and back to English be­ comes "ghosts of deceased quantities." It's not as good. Nor is it even correct! There are a few hints that English is not the native language of the author or the translator, but they are rare: "a regular polyhedra" (page 72), "the best of the two" (page 88). The description of the Mobius strip with a top spinning on the surface (page 78) is a delightful device for explaining orientation, but it could be expressed with less ambigu­ ity. It could have been made more clear by inserting parenthetically what is meant by "traveling once along the strip" or even by drawing a suitable figure. The Mordell Conjecture is cred­ ited to Leo Mordell in various refer­ ences and in the index when surely "Louis Mordell" was intended. The au­ thor consistently attributes A. 0. Gel­ fond's work to Gelfand. Hassler Whit­ ney's name is usually spelled correctly but is misspelled on page 70. David Rodney Heath-Brown may be called "Roger" by his friends, but for the rest of us it looks odd. I would have re­ ferred to I. R. Shafarevich, not "Igor." Perhaps the author is closer to some of these giants in mathematics than this reader. But this is quibbling when so much of the text is full of interesting insights and so eloquently expressed. A person could ask about the in­ tended audience for the book The au­ thor divides the references into two parts: "for general readers" and "for advanced readers." When he says "gen­ eral readers" he doesn't mean some­ one without any mathematical back­ ground. Though the author does a masterful job of describing difficult mathematics in accessible terms, still, sentences like the following are not for the faint of heart: " . . . Weil proposed his own conjecture, a version of the

© 2005 Spnnger Sc1ence+ Business Med1a, Inc , Volume 27, Number 1 , 2005

93

Riemann hypothesis for multidimen­ sional algebraic manifolds over finite fields, which became known as the Weil conjecture. It was proved in 1973 by Pierre Deligne [whose] proof was the first significant result obtained through the use of an arsenal of ex­ tremely abstract techniques in algebraic geometry (such as schemas and l-actic cohomology) introduced in the 1960s by Alexandre Grothendieck. . . . " As a lagniappe, the author includes four open problems for the 21st century:

(4) Complexity Theory: The P Problem ( 1972).

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plete lists, if only to see what the au­ thor has not found to be worthy of in­ clusion. For example, the first Fields Medalist on the list is Jesse Douglas, who won in 1936 and is cited earlier in the text as the first Fields Medalist, when Lars Ahlfors, who also won that year, was probably the first, at least al­ phabetically. His contributions just didn't make it into the book We should, however, be thankful for what we get, a truly gripping account of big problems of the twentieth century.

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The last three are on the Clay Insti­ tute's list of million dollar Millennium Prize Problems. In addition to the other Clay Institute problems, there are a few others one might have expected to see-the Goldbach Conjecture, the Twin Primes Conjecture, for example. They're famous and they're still at­ tracting people to work on them. In addition to the short list of sug­ gested readings and a very helpful in­ dex, the author includes chronological lists in a concluding chapter: Hilbert Problems, Fields Medalists, Wolf Prize winners, Turing Awardees, Nobel Lau­ reates-but only those cited in the text. It would be interesting to see com-

(1) Arithmetic: The Perfect Numbers Problem (300 Be); (2) Complex Analysis: The Riemann Hypothesis (1859); (3) Algebraic Topology: The Poincare Conjecture ( 1904); and

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alois: Abel's work on the unsolv­ ability of the general quintic equa­ tion was continued by the brilliant young French mathematician Evariste Galois (181 1-1832), who determined (in terms of the so-called Galois group) which equations can be solved by radicals. Galois had a short and tur­ bulent life, being sent to jail for politi­ cal activism. He died tragically in a duel at the age of 20, having sat up the pre­ vious night writing out his mathemati­ cal achievements for posterity. Gauss: Carl Friedrich Gauss ( 17771855) presented the first satisfactory proof of the fundamental theorem of al-

I

gebra (that every polynomial equation has a complex root) and made the first systematic study of the convergence of series. In number theory he initiated the study of congruences and proved the law of quadratic reciprocity. He also showed that a regular n-sided polygon can be constructed with straight-edge and compasses whenever n is a Fermat prime (such as 17, as shown on the stamp). Gazeta mathematica: The Roman­ ian monthly Gazeta mathematica was first published in 1895. With its aim of developing the mathematical knowl­ edge of high-school students, it has had an enormous influence on mathemati­ cal life in Romania for many decades. Gerbert: Gerbert of Aurillac (9381003) trained in Catalonia and was probably the first to introduce the Hindu-Arabic numerals to Christian Europe, using an abacus that he had designed for the purpose. He was crowned Pope Sylvester II in 999. Goldbach's conjecture: In 1742 Christian Goldbach wrote to Leonhard

Euler conjecturing that every even in­ teger (> 2) can be written as the sum of two prime numbers. Although this remains unresolved, a partial result of Chen Jing-Run ( 1966), shown on the stamp, implies that every sufficiently large even integer can be written as the sum of a prime number and a number with at most two factors. Gregorian calendar: The Julian cal­ endar of 45 BC had 3651/4 days, which was eleven minutes too long. In 1582, Pope Gregory XIII issued an edict that corrected the over-long year by re­ moving three leap days every 400 years, so that 2000 was a leap year but 2 100, 2200 and 2300 are not. The Gre­ gorian calendar was quickly adopted by the Catholic world, and other coun­ tries eventually followed suit: Germany in 1700, Britain and the American colonies in 1752, Russia in 19 17, and China in 1949.

Galois Gazeta matematica Goldbach's conjecture

Gauss

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Gerbert

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Gregorian calendar