No title

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

More Visible Sums

2un

1

·

2 + 2

·

3 +

·· ·

+ n(n + 1)

simultaneously. I recently gave the

Dr. Giorgio Goldoni: A visual proof for

following at an on-site teacher train­

n squares and for the sum of the first n factorials of or­ der two, Math. Intell., vol. 24, no. 4 (2002) 67-69.

the sum of the first

Here I would like to propose an al­ ternative proof to deduce the formu­ las for

Sn

Figure 1. Continues on next page.

4

=

in the article by

I was much interested

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

=

12 + 2 2 +

· ··

+ n2, and

ing course in Aomori (in northern Japan).

1 +2+

···

+n

1

= 2 · n(n

+ 1) tn (triangular number). It is easy to t1 + see that tn + tn-l = n2. Let Un t2 + · · · + tn, i.e., 2un = 1 2 + 2 · 3 + · · + n(n + 1). Un may be called a Put

=

=

·

·

Figure 1. (Continued)

tetrahedral number. Then we see eas­

a couple of opposite edges. This means

ily that

that

Sn

=

1 2 + 22 + · · · + n2 = t1 + t1 + t2 + t2 + t3 + · · + tn-2 + tn-1 2(t1 + t2 + · · · + tn) + tn-1 + tn

=

Un

n

L

k�l

k · (n + 1

- tn

= 2un -

t · n(n

+ 1)

(1)

On the other hand, an arrangement of

1

= 2 · n(n +

1

1)2 +

1

2 · n(n +

1

= 2 · n(n + 1 ) (n +

k)

= 2 · n(n +

·

=

-

3un

1 )2 - Sn

(2)

2un

=

1) 2), or

(1/3) n(n + 1)(n + 2). ·

Thanks to my colleague H. Yamai

(1) and (2) as simulta­ neous equations in Sn and Un, we have

for the figures. Note the related letter

finally

1999, no. 4, 106.

Considering

of I. Konstantinov in

Nauka i Zizn',

balls in a regular tetrahedron of side­ length

n

is decomposed into the sum

1 X n, 2 X (n- 1), . , (n- 1) X 2, n X 1,

of rectangles of sides

3 x (n- 2), .

.

when we cut it along planes parallel to

3sn

= n(n +

1) 2 1 2

1

2 · n(n +

= - · n(n +

1)

1)(2n + 1) and

Sin Hitotumatu, Prof. Emeritus, Kyoto University c/o Department of Information Sciences, Tokyo Denki University, Hatoyama Campus, Saitama, Japan 350-0394

VOLUME 25, NUMBER 3, 2003

5

FRI EDRICH L. BAUER

Why Legendre Made a Wrong Guess About 1r(x), and How Laguerre's Continued Fraction for the Logarithmic Integral Improved It

�.......

arl Friedrich Gau5 j , in 1792, when he was 15, found by numerical evi­ dence that 1r(x), the number of primes p such that p

<

x, goes roughly with

xlln x (letter to Encke, 1849). This was, as can be seen from Table 1, a very weak approximation with an error of about 100/o. Adrien-Marie Legendre (1752-1833) conjectured that for

1r(x) the approximation

taken seriously, since Legendre had certainly observed the locally irregular, "slightly chaotic" character of the number­

7T(X) = ln X

X -

theoretic function 1r(x). Chebyshev, however, had at his disposition the tables of L. Chemac (1811), going up to 106,

A(x)

and of J. C. Burckhard (1814/1817), going up to 3

limx--.oo A(x)

So

become suspicious about the reversed trend. Being an ex­

= 1.08366 . . .

cellent mathematician, Chebyshev succeeded, as men­

Forty years later, Pafnuty Lvovich Chebyshev (1821-1894) showed that Legendre's conjecture was wrong. He proved that if tfie liillif liriix--.oo A(x) exists, it must be equal to 1.

tioned above, in clarifying the situation. Moreover, Chebyshev showed around 1850 that for suf­ ficiently large x:

Legendre's mistake can be explained easily: When he

0.92129 :::;

made it, the largest tables of primes were those of J. H . Lambert (1770), going up to 105, and o f G . Vega (1796), go­ ·

· 106.

he could e_asily fmd the values shown in Table 2 and could

holds, with

ing up to 4

In 1798 and again in 1808,

105. Legendre had found 1r(x) by simple counts,

resulting in the values given in Table 1 for A(x)

-xhr(x) (rounded to 7 digits).

=

ln

x

The oscillations occurring for x 2:: 105 were not to be

7T(X)I1:X :::; 1.10555.

This result is theoretically remarkable, but has limited prac­ tical value, for it gives a relative error of several percent, reflecting the aforementioned weakness of the approxima­ tion by _!:____. Of course, the Chebyshev inequality was later Inx

© 2003 SPRINGER-VERLAG

NEW YORK, VOLUME 25, NUMBER 3, 2003

7

Table 1

X ln x

X

10

X

1T(x) 4

4.34

A(x)

In x

1T(x)

2.500000

1T(

x)

/ 1:X

2.302585

-0. 1 974 1 5

0.921 03

1 . 1 9829

20

6.68

8

2.500000

2.995732

0.495732

50

1 2.78

15

3.333333

3.91 2023

0.578690

1 .17361

1 00

21.71

25

4.000000

4.605170

0.605 1 70

1.151 29

200

37.75

46

4.347826

5.2983 1 7

0.950491

1 .2 1 86 1

500

80.46

95

5.263158

6.21 4608

0.951450

1 .18078

1 . 103

1 44.76

168

5.952381

6.907755

0.955374

1.1 6054

2. 1 03

263.1 3

303

6.500660

7.600902

1.002424

1.1 5152

5. 1 03

587.05

669

7.473842

8.5171 93

1 .043352

1 .13960

1 . 104

1 085.74

1229

8.136697

9.21 0340

1 .073644

1. 1 3 1 95

2. 1 04

2019.49

2262

8.84 1 733

9.903488

1.061 755

1. 1 2008

5 . 1 04

4621 .1 7

5 1 33

9.740892

1 0.819778

1 .078886

1 .11076

1. 105

8685.89

9592

1 0.425354

11 .51 2925

1 .087571

1 .10432

2. 1 05

1 6385.29

1 7984

1 1.120996

12.206073

1 .085076

1 .09757

3 . 1 05

23787.74

25997

11 .539793

1 2.899220

1.085871

1 .09267

4. 105

31009.63

33860

1 1 .813349

1 2.611538

1.071 745

1 .091 92

improved, e.g., by James Joseph Sylvester in 1892: for suf­

value is 50847534. Technical improvements in the organi­

ficiently large x

zation of Meissel's method were made by D. N. Lehmer

(1958), by David C. Mapes (1963), then by Jan Bohman

'jI1n X

0.95695 s; 1T(X) ___3!_ s; 1.04423.

(1972), who calculated some isolated values as high as 11'(4 · 1012), and more recently by J. C. Lagarias, A M.

This is better, but still has a relative error of a few percent.

Odlyzko (1985), and V. S. Miller (1985), who computed se­

Besides, while the last column in Table 1 shows that for

lected values up to 11'(4 · 1016). Further efforts followed;

the Chebyshev upper bound to hold will require x 2:: 105,

values up to 7T(102D) were calculated by M. Deh�glise;

tually, in 1962, using very delicate analysis, Rosser and

cember 2000 by Xavier Gourdon.

for the Sylvester bound to hold will require x

2::

10n. Ac­

Schoenfeld showed rigorously that for x 2:: 17

At present, a grid of values has been calculated up to

11'(4

. '/I___3!_ ln x

1 s; 1T(X)

ln x - _!_

· 1022) = 783964159852157952242,

A(x)

to

a

value A(4 · 1022)

ln x-�

for x

2::

67.

1T(X)

2

2

The other line that Chebyshev had opened would have been more interesting. After the prime number tables were extended by Z. Dase (1862), J. Glaisher (1883), and D. N.

C.

�

without building up the whole table was found in 1871 by the astronomer Daniel Friedrich Ernst Meissel (1826-1895), who gave 11'(107), 11'(108), and in 1885 11'(109), the latter as

= 1,

and

i.e.,

X 1n X_ 1

li(x)

=

for

X--'>

oo.

Lx _!!±_ _ 1n t 2

are asymptotically equal: li(x)

�

1r(x) for x - -'>

oo.

From the

asymptotic expansion of the logarithmic-integral function li(x)

=

___3!_ 1n x

+

1! . x

(1n x)2

+

5084 7478 with a computational error, while the correct

Table 2

(Table 4)

F. Gauss conjectured in 1792 that the prime-counting

cated a decrease in A(x) with growing x. Further exten­ computers. More important, a method of calculating A(x)

1.020426

function 1T(x) and the logarithmic integral

Lehmer (1914) up to 107, the values given in Table 3 indi­ sions of the tables up to 109 took place after the advent of

=

strongly supports the conjecture, implicitly made by Cheby­ shev, that limx-w' A(x)

X

the world record as

of March 2001. It shows for growing x a clear decrease of

Another result of Rosser and Schoenfeld is very useful:

X -----,- :s 1T(X) :s

11'(1021) and 11'(1022) were calculated in October and De­

� (ln x)3 +

Table 3

+

...

kl· ·X (1n x)k+1

+ 0

(

X (1n x)k+2

)

5 . 105

38102.89

41 538

12.037171

13.1 22363

1.085 1 93

1 .0901 5

1 . 1 06

72382.41

78 498

1 2.739178

1 3.81 55 1 1

1.076332

1 .08449

xh(x)

148 933

1 3.428857

1 4.508658

1 .079800

1. 08041

1T(x)

lnx

137848.73

5 . 1 06

x/ln x

2. 1 06

324150.19

3485 1 3

1 4.346667

15.424948

1 .078281

3 . 106

201151.62

216 816

13.836617

14.914123

1.077506

1.07787

1 . 107

620420.69

664579

15.047120

16.118096

1.070976

8

THE MATHEMATICAL INTELLIGENCER

X

A(x)

comes li(x)

xlln x for x �

imply the fundamental �

1r(x) (and also for x�

oo,

by Richard Crandall and Carl Pomerance, Springer, New

and the conjecture would

oo,

prime number theorem

York,2001,and the Internet home page of Chris K. Caldwell,

for x �

much better approximation to the prime-counting function

X

ln x

�

--

1r(x)

=

_x_, In x- 1

([email protected])) fail to mention that x/(ln x-1) is a

oo

1r(x) than xlln x,and a much simpler one-suitable for pocket

calculators -compared to li(x) if the higher accuracy li(x) of­

the first term of the

fers is not needed. In fact,the accuracy of xl(ln x-1) is im­

Laguerre continued fraction; see below).

proved for growing values of x; 1022/(ln 1022-1) has a rela­

Chebyshev, around 1850,came very close to a proof of

tive error of about (ln 1022r2

the prime number theorem,and Georg Friedrich Bernhard

Riemann (1826-1866), in 1859 studying the ?-function, made important further contributions, but it took almost

50 more years, until 1896 when Charles-Jean de la Vallee­

li(x)

+ O(x ·e:xp(- AVlllX ))

term was improved to O(x a =

=

1 5• Meanwhile,the error

1n x -2

=

li(x)

+ O(x112

•

ln x). In 1976,

Schoenfeld showed that under the Riemann hypothesis 1 -87T

=

·x112 ln x •

for x

2:

·

n-1

-

which is the contraction of the continued fraction of

showed that the Riemann hypothesis is equivalent to the far

:-s

4 J 1 II���

�

· · ·-l

t' f3 -i by Korobov and Vinogradov in 1958. But in

l1r(x) -li(x)l

=

·e:xp(-A(ln x)"'(ln ln x)f3)) with

1901,the Danish mathematician Helge von Koch (1870-1924) tighter error estimate 1r(x)

further improvement possible.

xl(ln x- 1) is the first term of the continued fraction that

li(x)

for some positive constant A; a suitable 1 value was determined in 1963 by Arnold Walfisz to be A

·10-4. Table 5 shows that

1886):

independently gave a rigorous proof,together with a rather =

However, there is a

=

was given in 1885 by Edmond Nicolas Laguerre (1834-

Poussin (1866-1962) and Jacques Hadamard (1865-1963)

weak error estimation 1r(x)

3.9

the values for 104 and above are lower bounds.

Nielsen (1906) for the logarithmic-integral function li(x)

=

2657.

:xl_ �-lln1xl_ �-l1n2 xl_ � -l:xl

ll

-

Many eminent mathematicians,among them Edmund Lan­

·

·.

Table 5 shows that

dau (1877-1938), Atle Selberg (*1917), and Paul Erdos

(1913-1996), dealt with the prime number theorem,finally

X

reducing the proof to an elementary level free of function

ln x-1---­ lnx-3

1

theory, but much more complicated.

-�

-

n ll ln x

'

the second term of the Laguerre continued fraction,has for

Under the influence of the glory of Analytic Number The­

x

ory,there was not too much interest left in simple numerical

=

1022 a relative error of about (1n 1022)-3

201467286691248261498 still gives much better accuracy,

with

k

=

X

1 . 1 08

x/ln x 542868 1 .02

in many practical =

the relative error being about 10-11. But computation of

1. But most books dealing with the subject

Table 4

small

cases. Computation is simple. (No question: li(1022)

�)),is obviously of the same order as

recent literature,an exception is the book

sufficiently

·10-6,

the asymptotic expansion of the logarithmic-integral function

·(1/(1-

be

7.7

which

as (xlln x)

should

=

questions. In fact,the approximation xl(ln x-1),rewritten

(in the Prime Numbers

li(1022) is much more cumbersome.) Table 5 shows fur-

1T(x)

5 761 455

x/'lT(x)

In x

1 7.356727

A (x)

1 8.420681

1 .063954

1 . 1 09

48254942.43

50 847 534

1 9.666637

20.723266

1 .056629

1 . 1 010

43429448 1 .90

455 052 51 1

2 1 .975486

23.025851

1 .050365

1 . 1 Q11

3948131653.67

4 1 1 8 054 8 1 3

24.28331 0

25.328436

1 .045126

1 . 1 Q12

361 91 206825.27

37 607 912 01 8

26.5901 49

27.631021

1 .040872

1 . 1 013

334072678387.1 2

346 065 536 839

28.896261

29.933606

1 .037345

. 1 014

3 1 02 1 034421 66.08

3 204 941 750 802

31.201815

32.236191

1 .034376

1 . 1 015

28952965460216.79

29 844 570 422 669

33.506932

34.538776

1 .031 844

1 . 1 016

271 43405 1 1 89532.39

279 238 341 033 925

35.811701

36.84 1 36 1

1 .029660

1 . 1017

2554673422960304.87

2 623 557 157 654 233

38.1 1 61 89

39.143947

1.027758

1

1 . 1018

241 2747121 6847323.76

24 739 954 287 740 860

40.420447

41.446532

1.026085

1 . 1 019

228576043 1 06974646.1 3

234 057 667 276 344 607

42.72451 4

43.7491 1 7

1.024603

1 . 1 020

2 1 7 1 47240951 62591 38.26

2 220 8 1 9 602 560 9 1 8 840

45.028421

46.051 702

1.023281

1 . 1 021

20680689614440563221 .48

21 1 27 269 486 01 8 731 928

47.332193

48.354287

1.022094

1 . 1 022

197406582683296285295.97

201 467 286 689 3 1 5 906 290

49.635850

50.656872

1.021022

VOLUME 25, NUMBER 3, 2003

9

Table

IS. Continued fraction approximations to the prime-counting function n(x) x

X

__ _ _

1

In x-

'll{x)

ln x -

1

X

-

-

1 lnx-3

-0.35

3.68

1 01

- 'll{x)

li(x) -

'll{x)

'll{x) 2

4 25

102

2.74

8. 53

5

1 03

1 .27

8.93

10

1 68 1 229

1 04

- 1 1 .02

1 3.34

17

1 05

-79.90

27.59

38

9 592 78 498

106

-467.55

99.50

1 30

107

-3120.03

232.18

339

664 579

108

-211 5 1 . 1 9

296.76

754

5 761 455

1 09

-145991 .55

-531 .83

1 701

50 847 534

1 010

-1 040539.71

-8896.92

3 1 04

455 052 511

1011

-763851 2.27

-57738.46

1 1 588

4 1 1 8 054 813

1012

-5771 8368.01

-385385.30

38263

37 607 9 1 2 018

1013

-44667661 8.38

-2599887.97

1 08971

346 065 536 839

3 1 4890

3 204 941 750 802

1014

-35271 1 5021 .36

-1 7666487.47

1 Q15

-28336573668.95

- 1 221 29383.48

1 052619

29 844 570 422 669

3214632

279 238 341 033 925

1 Q1 6

-231 0828031 05.06

-863688021 .83

1017

-1 9091 90842201 .98

-6236796467. 1 2

7956589

2 623 557 1 57 654 233

21949555

24 739 954 287 740 860

1 018

-15955501 820884.84

-45888167744.56

1019

-134 70567387421 6.1 7

-343541 1 53401.90

99877775

234 057 667 276 344 607

222744643

2 220 8 1 9 602 560 91 8 840

1020

-1 1 476285471 86596.81

-261 3363726855.1 4

1021

-9857223 1 08746375.71

-201 63286970669.57

597394254

21 1 27 269 486 0 1 8 731 928

-8529084465601 2772.24

- 1 57576742975045.01

1 932355208

201 467 286 689 3 1 5 906 290

1022

thermore, that the values forx/(lnx- 1) are upper bounds forx

=

101 up tox

=

103, lower bounds forx = 104 up to

X= 1Q22.

183-256, 281-397 (1896). Paul Erdos, On a new method in eleme ntary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat/.

Likewise, the values for

(

1 /1l lnx-1- lnx-3

x

Charles-Jean de Ia Vallee-Poussin, Ann. Soc. Sci. Bruxelles 20,

)

Acad. Sci. U.S.A. 35, 374-384 (1949). A U THOR

are upper bounds forx = 102 up tox = 108, lower bounds forx

=

109 up tox = 1022. Note: the error changes sign be­

tweenx = 3

· 108 andx=

4

·

108; in this range the error is

pretty small, smaller than the error in li(x). Conclusion

For many applications, xl(lnx-1), which produces with little additional computational effort a much better ap­ proximation than xlln x, can be recommended-it gives three correct decimals for x 2:: 1012. With some extra ef­ fort, xl(lnx -1-

1-) may be used-it gives five corlnx- 3

rect decimals forx 2:: 109•

Besides, it would be highly interesting to obtain rigor­ ous Chebyshev-type bounds C', C" for C'

:::;

I '/ lnx-1

1T(X)

X

:::; C".

FRIEDRICH L. BAUER

Nordliehe Villenstrasse 19

D-82288 Kottgeisering Germany

Friedrich L. Bauer, born in Regensburg in 1 924, studied math­ ematics, physics, astronomy, and logic at Munich University

REFERENCES

after the War. He became internationally known as an inno­

Pafnuty Lvovich Chebyshev, Sur Ia fonction qui determine Ia totalite des

vator in computer hardware and software and numerical analy­

nombres

premiers, Oeuvres I, 27-48 (1851).

Pafnuty Lvovich Chebyshev, Memoire sur les nombres premiers, Oeu­ vres I, 49-70 (1854).

10

THE MATHEMATICAL INTELUGENCER

sis. In particular, he played a key role in creating ALGOL 60. He set up the collection on Computer Science at the Deutsches Museum in Munich. He is now Professor Emeritus.

Jaques Hadamard, Oeuvres I, 189-210 (1896). Helge von Koch, Math. Annalen 55 (1902), 441-464. Edmond Nicolas Laguerre, Sur Ia reduction en fractions continues d'une

Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen GroBe, Werke, 136-144. J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for

fonction que satisfait a une equation ditterentielle lineaire du premier

some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.

ordre dont les coefficients sont rationelles, J. Math. Pures et Appl.

Lowell Schoenfeld, Sharper bounds for the Chebyshev functions O(x)

(4) 1 (1885).

Edmund Landau, Vorlesungen Ober Zahlentheorie, S. Hirzel, Leipzig

and 1/J(x). II. Math. Camp. 30 (1976), 337-360. Atle Selberg, An elementary proof of the prime number theorem for arithmetic progressions, Ann. Math. (2) 50, 305-313 (1949).

1927. Adrien-Marie Legendre, Th8orie des nombres. 2nd edition, 1798, No.

James Joseph Sylvester, On arithmetical series, Collected Works Ill, 573-587 (1892).

394-401.

cKichan

SOHWA�f.IIIC.

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VOLUME 25, NUMBER 3, 2003

11

Opinion

Mathematics and War: An Invitation to Revisit Bernheim Booss-Bavnbek and Jens H0yrup

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion should be submitted to the editor-in­ chief, Chandler Davis.

P

hysicists, chemists, and biologists have a tradition of discussing meta-aspects of their subject, including the military use and misuse of the knowledge they produce. Concerns of the latter kind are rare among mathe­ maticians. No rule without exceptions. During the Vietnam war, a number of appeals were circulated among US mathemati­ cians (with reverbations in particular in France and Japan and at the ICM in Moscow in 1966 and Nice in 1970) not to engage in war-related work One such appeal was published in the Notices of the AMS in January 1968. Grothendieck's resigning from mathe­ matics fell in the context of this debate. [Godement 1978], not really debate but politico-economical analysis, was writ­ ten from a mathematician's perspec­ tive even though it dealt with scientific research in general. [Gross 1978] was shorter but concentrated on mathe­ matics. In the new context of the euro-mis­ sile controversy of the early 1980s, mil­ itary research came into the focus of debate in universities of West Ger­ many. [BooB & H0yrup 1984] was a product of this new discussion con­ centrated on mathematics; the broader discussion is reflected in [Tschimer & Gobel (eds.) 1990]. The "Forum on Mil­ itary Funding of Mathematics" pub­ lished in the Mathematical Intelli­ gencer (1987), no. 4, reflects problems arising for the US mathematical com­ munity from the "Strategic Defense Ini­ tiative" in the same phase. See also [Davis 1989]. Some more publications

followed, mainly with historical em­ phasis. As warfare is now again becoming an all-too-obvious aspect of our world and of "Western" policies, the time seems ripe for taking up the issue anew. Just after the Kosovo war, Zen­ tralblattjur Didaktik der Mathematik dedicated an issue to it: vol. 98, no. 3 (June 1998). On August 29-31, 2002, 42 mathematicians, historians of mathe­ matics, military historians and ana­ lysts, and philosophers gathered in the historic military port of Karlskrona, to discuss four questions:1 • To what extent has the military played an active part throughout his­ tory, and in particular since World War II, in shaping modem mathe­ matics and the careers of mathe­ maticians? • Are mathematical thinking, mathe­ matical methods, and mathemati­ cally supported technology2 about to change the character and perfor­ mance of modem warfare, and if so, how does this influence the public and the military? • What were, in times of war, the eth­ ical choices of outstanding individu­ als like the physicist Niels Bohr and the mathematician Alan Turing? To what extent can general ethical dis­ cussions provide guidance for work­ ing mathematicians? • What was the role of mathematical thinking in shaping the modem in­ ternational law of war and peace? Can mathematical arguments sup­ port actual conflict resolution?

A shorter version of the present paper has appeared as "Feature - M athematics and War" in the European Mathematical Society Newsletter 46

(December 2002), 20-22.

1We use the opportunity to thank Maurice and Charlyne de Gosson and the Blekinge Institute of Technology and its Mathematics Department for organizing this conference, supported by Stig Andur Pedersen of The Dan­ ish Network for History and Philosophy of Mathematics (MATH NET) and Reiner Braun of The International Net­ work of Engineers and Scientists for Global Responsibility (INES). From the conference, a kind of enlarged pro­ ceedings will appear as Bernheim BooB-Bavnbek & Jens Hoyrup (eds.), Mathematics and War. Basel & Boston:

Birkhauser, 2003. M uch of what is said in the following draws on this volume. On the theme in general, see

also [BooB & Hoyrup 1 984], [Epple & Remmert 2000], [Godement 1 994 and 2001], [Meigs 2002], and [The

AMRC Papers].

2This "broad concept" of mathematics is the one that serves in the following; it also embraces computers and computer science.

12

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

"Noli turbare circulos meos": When Archimedes's city was conquered in spite of his astounding mathematical engineering, he pretended that he had only done pure mathematics. This anecdote has remained popular since Roman Antiquity. The mosaic, "Death of Archimedes," is now known to be a seventeenth-century forgery. [Source: Stiidtische Galerie, Frankfurt-am-Main, Germany.]

Perspectives from Mathematics All mathe!l_laticians know the ta1es, re­

in logistics, which is likely to have been

law, that origin had a1ready been left

much more important from the military

behind, and the theory was linked in­

point of view. Mathematics served as a

stead to the philosophica1 discussion

have- heard about. early modem ballis­

toolbox, and military officers may have

of loca1 motion-and was largely irrel­

tics and fortification mathematics and

been the largest group that received genera] mathematica1 training; but the

evant for the firing of guns because of the influence of air resistance, as

the iiitportance of trigonometry for

involvement of mathematics as a gen­

pointed out explicitly by Ga1ileo.

navigation. All these cases of mathe­ matics. being -rrnplicated. in conquest,

era] endeavour with the military in­

Even to this rule there is an excep­

stitution was not very intimate, and

tion. That part of the Sumero-Babylon­

liaf>Ie or not, about Archimedes and his defence of Syracuse. They may a1so

·

warfare, or preparation for war have one thing in common: that which was

combined with technica1 and military

specifica1ly military applications had

ian legacy which is most spoken of in

no independent role as a shaping force

genera1 histories

for mathematics. Tartaglia's composi­

namely the invention and implementa­

knack was a1most exclusively already

tion of straight lines and circles in ba1-

existing mathematics. In this respect

listics was clearly inspired from gun­

such examples do not differ from the

nery and the war against the Turks. When Galileo introduced the parabolic

use of simple accounting mathematics

of mathematics­

tion of the place va1ue system-may be

a child of war. In c. 207 4 BCE, King

Shulgi organized a military reform in the Sumerian Empire, and the next

VOLUME 25, NUMBER 3, 2003

13

There is no known picture of Turing during the wartime period, but this photograph shows Alan Turing (at left) with his athletic club in 1946. At this point he was engaged in designing a digital computer at the National Physical Laboratory, London. This design used his wartime knowledge of electronic technology to put his 1936 theory of the universal machine into a practical form. The codebreaking machinery at Bletchley Park, although very advanced, had never actually used Turing's fundamental idea of the universal machine and the stored program, but as soon as the war ended Turing set to work to bring it to reality.

year an administrative reform (seem­ ingly organized under the pretext of a state of emergency but soon made per­ manent) enrolled the larger part of the working population in quasi-servile labour crews and made overseer scribes accountable for the perfor­ mance of their crews calculated in ab­ stract units worth %0 of a working day (12 minutes) and according to fixed norms. In the ensuing bookkeeping, all work and output therefore had to be calculated precisely and converted into these abstract units, which asked for multiplications and division en masse. Therefore, a place value system with base 60 was introduced for inter­ mediate calculations. 3 Its functioning presupposed the use of tables of mul-

tiplication, reciprocals, and technical constants, widely taught in school; though the basic idea had been "in the air" for some centuries, implementa­ tion awaited decisions made at the level of the state and firmly enforced. Then, as in many later situations, only war provided the opportunity for such social willpower. Apart from that, the conclusion stands that the military had little influ­ ence on mathematics. As already men­ tioned, the employment of forti:fication mathematicians and the teaching of naval and artillery officers did serve mathematics by providing job oppor­ tunities and a market for mathematics text books (copiously decorated with military symbols).

All this changed in the twentieth century. The new relationship can be said to have started around the :First World War, and to reach full develop­ ment during the Second World War. During World War I, two important new military technologies depended on mathematics in the making: sonar, and aerodynamics. They were so im­ pressive that Emile Picard, in spite of his own patriotism (which non-French cannot help seeing as pure chauvin­ ism), feared that young mathemati­ cians might opt in future for applied mathematics only [Proc. . . . 1920: xxviii]. In general, however, the imme­ diate role of the pure sciences, mathe­ matical and otherwise, was that of providing manpower that could be con-

3Because it was a floating-point system with no indication of absolute place, it could be used only for intermediate calculations-just like the slide rule of engineers in quite recent times. Since intermediate calculations have not survived, the exact dating of the implementation can only be inferred by indirect arguments. See, e.g., [Hoyrup 2002: 314].

14

THE MATHEMATICAL INTELLIGENCER

When listening to a music CD, we enjoy and do not think of the military origin of the coding involved. Similarly, mathematicians going to the wonderful Mathematical Research Institute Obetwolfach enjoy the ambience and have no reason to worry about the fact that the institute originated as a m ilitary research institution in 1944-apparently well planned for the purpose though too late to have effect.

verted into first-class creative engi­

Aerodynamics of course survived, but

without counting costs and benefits,

neers-not restricted to applying a set

only as one current among others.

made it possible to boost a development

of standard rules but able to implement

In World War II, the organization of

theoretical knowledge and make it

science intended to support the war ef­

which

otherwise

might

have

taken

decades5-and perhaps, in cases like

function in practice; this was also the

fort was a major concern of both Axis

DDT and atomic reactors, might have

role of most of the mathematicians ac­

and Allied powers; mathematical tech­

been stopped at an early stage when as­

tually involved in the war effort (if they

nologies

sociated problems became visible.

were not, as in France, sent into the

During WWII, mathematicians

trenches). Nobody will claim that math­

computer, the Bomb) can be argued to have been war-decisive; computers, nu­

large numbers were recruited, many of

ematics was in any way decisive for the

clear energy, jet propulsion-all math­

them to teach sailors and air-crew

outcome of the war, nor that WWI ap­

ematically constructed and computed

members basic trigonometry (etc.), but

plications of mathematics left impor­

for the war-have changed our world

many also to serve as top-level creative

tant traces in the post-war world (civil

beyond recognition since 1945. Admit­

engineers. Afterwards, the latter have

aviation still belonged to the future).

tedly, all of these build on pre-war theo­

often tended to regard what they did

retical insights4; some of them (comput­

dismissively ("I did not write one line that was publishable," as J. Barkley

Picard's wonies proved unfounded.

(radar, sonar, the decipher

in

Mainstream mathematics soon reverted

ers, jet engines) had pre-war prototypes

to the pre-War model, even more swiftly

for the machines which reached com­

Rosser [1982; 509f] summarizes one re­

than the precariously erected organiza­

pletion during the war; but in all cases

action), perhaps because puzzle-solv­

tion of planned science was dismantled.

the war, by making available huge means

ing with no further theoretical impact

4At times these had been obtained in contexts fully detached from every technical application. N. Wiener and E. Hopf had calculated the radiation equilibrium at stel­

lar surfaces, but their theory could be applied to the expanding surface of the exploding bomb [Wiener 1964: 142�. A. A. Markov had investigated his eponymous processes as pure mathematics and illustrated the applicability of the concept on linguistic material [Youschkevitch 1974: 129]; in the Manhattan Project they turned out to be relevant for solving diffusion equations and for describing nuclear branching processes.

5The parallel to the invention of the place value system in Sumer is striking. In that case, parallel processes not furthered by a military government indeed took a much longer time: in China the unfolding took more than a millennium, in India it never really took place before the "Indian" system was brought back from abroad.

VOLUME 25, NUMBER 3, 2003

15

did not look important in the mathe­

matical tools (ballistic computation,

matician's hindsight; this assessment

modelling, . .

Target selection and order of battle, scheme

) from the creation

of modern air raid build-up. In World War II,

notwithstanding, what was done de­

of new mathematical insights (se­

the destruction of a major composite target

pended critically on mathematical in­

quence analysis, Monte Carlo simula­

might require the deployment of a thousand

genuity and training. A striking exam­

tion, . . . ). As a rule but not consis­

bombers. Nowadays a similar task may be ef­

ple is

0. R. Frisch and R. Peierls's

.

tently, the former type is the chore of

fected by, say, 29 heavy bombers (lower level

mathematicians who are paid by or

of the above schematized front view of an at­

sential questions surrounding the con­

connected to the military institution

tack-heights are indicated in kilometres).

struction of a uranium bomb in March

itself; new insights directed toward

But these have to be supported by another

1940 and their "back-of-an-envelope"

military goals are more likely to come

set of 275 fighters and ground attack fight­

discovery that its critical mass was so

from mathematicians inspired by the

ers ("SEAD package") to suppress enemy air

military's problems but bound less

defence (bottom). Higher up, 24 "Intelligence­

closely to the military institutions.

Surveillance-Reconnaissance" (ISR) aircraft

mathematical formulation of the es­

small that military use was feasible. 6 In some cases, of course, the solv­ ing of problems defined by the war did

•

Second, we should remember that

guide the action of the lower levels; and on

have important theoretical impact: we

mathematical research

top, dozens of ISR spacecrafts participate.

all know about the emergence of com­

more than the production of theorems

Much more than informatics is thus involved

consists in

theory,

of presumed military use. Several in­

in the support of the mission itself. As shown

Monte Carlo simulation, operations re­

stitutions (Suss's original plan for the

in the table at the bottom of the diagram, the

search, and statistical quality control.

German

puter

science,

information

Oberwolfach

Institute

in

average precision of bombing and firing is

This time, nothing was dismantled

1944, the American Mathematics Re­

commonly characterized by the Circular Er­

after the war (though many mathe­

search Center in Wisconsin) exem­

ror Probable (CEP), that is, the radius of a disc

maticians hurried to get out of military

age) 50% of the shots hit while 50% fall out­

on. In the slightly longer run (a decade

a two-way chain, which grosso modo works as follows:7 A core group of highly

side. (Kolmogorov's approach was more so­

or so), civil re-application of the new

skilled mathematicians familiar with

phisticated). The table shows how CEP has

mathematical war techniques trans­

the direct problems of the military em­

decreased dramatically in aerial bombing

formed them and accelerated their de­

ployer (efficiency of bombing, con­

over the last 60 years and how the efficiency

velopment: only the war effort had al­

trolled

bacteriological

of a bomber increased correspondingly. The

lowed the creation of the first costly

agents, better radar detection and

table gives the calculated number of bombs

computers, but only commercial use

avoidance of enemy detection,

research)-the Cold War was already

allowed mass production, open com­

plify an efficient model,

spread

of

or

around the goal point within which (on aver­

required for "destroying" (i.e., hitting once) a

whatever) find out which of these can

20m

x

30m object. So, the dramatic decrease

ef­

be approached mathematically, un­

of CEP has to be paid for by dramatic in­

forts, and reduction of costs. (Actually,

dertake an initial translation, and di­

crease of support craft. However, air raids

stored-program

rect the translated problems to other

are still the cheapest way of waging punitive

ENIAC reached the working stage only

experienced mathematicians who are

war (forbidden by international law, but prac­

after the war, though at first in military

well informed and centrally located

tised), inflicting huge economic losses on the

contexts). We may add that getting rid

within the mathematical milieu; these

enemy at extremely low operational costs.

of insistence on immediate applicabil­

parcel out the questions into problems

petition,

intensive

development

computers

like

the

ity ("better a fairly satisfactory answer

which colleagues may take up as

now than the really good answer two

mathematically interesting,

years after defeat") gave space for

even without knowing that they enter

was implemented too late to become

fruitful interaction between theoretical

into a network of military relevance;

efficient; we know that it has func­

understanding and applications in (for

once such questions have been an­

tioned in the US. We know less about

instance) computer science.

swered, the same chain functions

the organization of military mathemat­

backwards, reassembling the answers

ical research in the late Soviet Union,

and channelling the global solution to

but it appears that there, as in produc­

In discussing mathematical research for

military

purposes,

both

perhaps

during

the employer. (Evidently, mathemat­

tion and research in general, the civil

World War II and in recent decades, we

ics of civilian relevance can be created

should differentiate several situations

this way too, if the funding is there.)

and

and problems. •

First, we must distinguish the appli­

military

domains

were

more

sharply separated than in the West.

This is only one among several mod­

We may sum up in some general ob­

some­

els. We know that it was planned to

servations on "the perspective from

times repetitive) of existing mathe-

function in World War II Germany but

mathematics":

cation (sometimes creative,

6See [Gowing 1964; 40-43, 389-393] and [Dalitz & Peierls 1997: 277-282]. This latter volume presents Peierls as a physicist, but his actual chair was in "applied math­ ematics"; even a "broad concept" of mathematics does not free us from delimitation problems. 7Concerning the Oberwolfach Institute, this structure follows (conventional whitewashing notwithstanding) from analysis of the material presented by H. Gericke [1984], ct. [H0yrup 1986]; on the same institution, see also [Remmert 1999]. For the Wisconsin Institute, see [The AMRC Papers].

16

THE MATHEMATICAL INTELLIGENCER

ISR spacecraft

3 6000

-

(24)

GPS:

Key to the scheme of modern air raid build-up- Precision Bombing

COM: 10+

Acronyms AWACS

Airborne warning and control system for air

IR-NRO: 3

1000

and combat control B-<>

Long-range bomber with weapon payload

850

of more than 1 0 tons

MET: 10

".,-

COM

Military communication I signals intelli-

LM-NRO: 600

2

gence spacecraft E-8

Joint surveillance and targeting attack radar system JSTARS

EA-68

"Prowler" carrierborne radar jammer

EC-130

"Compass Call" communication jammer

F-<>

100

Fighter and fighter ground attack aircraft

ISR aircraft

GPS

U2: 20

Global positioning system navigation

5

satellite IR-NRO

AWACS: 12

Infra-red (US} National Reconnaissance Office space-

10

craft

RC-135: 5 E-8:

2

ISR

Intelligence, surveillance, reconnaissance package

10

LM-NRO

SEAD package

F-15: 51

National Reconnais-

EC-130: 6

sance Office space-

EA-68: 37 F-16: 157 F-117:

5

Imaging radar (US}

craft MET

Weather satellite

RC-135

"Rivet Joint" signals intelligence gathering

24

aircraft SEAD

Suppressing enemy air defence package

U-2

- •

u

ll

..

�I

Mathematical war research has re­ sulted in certain fundamental theo­ retical innovations. It is striking, how­ ever, that all of these appear to depend on an exceptional mathe-

Circular Error (CEP) Table War

WWII

Korea Vietnam

� •

w

Gulf Kosovo

•

matician. The names of Turing, von Neumann, Shannon, Wald, and Pon­ tryagin may suffice to make the point. However, the utility of mathematics for the treatment of military prob-

Optical spy plane

CEP[ m ]

1100 330 130 70 13

#bombs

9140 823 128 38 2

lems does not depend critically on the presence of an exceptional math­ ematician. Mathematicians in large numbers have proved themselves unexpectedly able to function as ere-

VOLUME 25, NUMBER 3, 2003

17

•

•

ative mathematical engineers, in the

bombs provided with guidance sys­

of fragmentation bombs on human

sense explained above.

tems); delivery systems (including for

bodies was to be predicted but hu­

This ability has depended in large part on their capacity to become fa­

instance aeroplanes provided with

manitarian concerns prohibited test­

electronic countermeasure circuitry);

ing on pigs, mathematical simulation

miliar with methods and approaches

the reconnaissance, control, and com­

of various mathematical disciplines

munication interface ("to ensure that

and to synthesize these. The survival

the right forces are at the right spot at

of the unity of mathematics is thus

the right moment, and with the right

made more acceptable to the public by the presentation of warfare as

information

enemy"­

precise and hence "more rational

mathematical journals then in tech­

Svend Bergstein); and, across all of

and clean." Although that aspect of the matter is not much discussed in

about

the

nical application.

these, high-speed cryptography. The

It should not be forgotten that the

improvement

data-transmission

the public sphere, this increased

traditional application of the toolbox

technologies is of general importance

precision of weapons (which is real)

of already existing mathematics goes

strongly made by Colonel Svend Berg­ lated, no more today than in the times

•

of the Prussian military thinker Carl

preached German invincibility by presenting the Wehrmacht as "Fast as German greyhounds, tough as

even more sophisticated mathematics

German lederhosen, hard as Krupp

than the transmission. 9 Similarly, the strategic planning of

steel," mathematics presents mod­ em warfare as "fast by avionics, pre­ cise by GPS, safe by optimized op­ erations planning."

depends on mathematical calcula­ tion; even the dismantling of weapons

many unpredictable external factors involved, but also we see those aspects of human behaviour which are most atavistic and contrary to reason-in Bergstein's view due especially to the prevalence of stress and sleep depri­ vation during combat. Nevertheless, mathematics has be­ come an integral and even essential part of modem warfare. (This does not major expense of the military appara­

presupposition for their transmission

the possible use of weapons systems

von Clausewitz: not only are there too

mean that mathematics has become a

tion of mathematics. Whereas Hitler

nowadays often asks for the use of

the point was

stein that actual war cannot be calcu­

depends essentially on the applica­

creation of data is not only an evident

but is, in itself, something which

Military Perspectives

At the conference,

of

for many of these questions, but the

cent mathematical research.

•

ical representations of the task to be performed may serve to make the

negotiations was analyzed mathe­

soldier see it as mere manipulation

matically. Fortunately, nobody im­

of symbols and thus to eliminate the

plemented the strategy suggested by

need for appeals to atavistic in­

the naive versions of such planning

stincts-say, seeing a village to be

games-to make a nuclear first strike

bombed as triangles in a computer

and promise help to the SOo/o-annihi­

game may facilitate the killing. (Evi­

lated enemy if no counterattack were made. l0

metres already has much the same

Perhaps unexpected by civilians

effect.)

ics performed by mathematically

We list various aspects of this role

trained independent personnel and not by the active warriors is manda­

of mathematics as discussed at the conference and elsewhere. 8

tory if strategic gains and losses are to be assessed realistically; leading

Mathematics serves in managing the

Similarly, certain uses of mathemat­

lizing disequilibrium in the SALT

alysts, simple accounting mathemat­

use costly resources more efficiently.)

•

systems without the risk of destabi­

but emphasized by some military an­

tus-mathematics is a cheap way to

•

Ideologically, the waging of war is

demonstrated ad oculos, if not in the

on, now at the level created by re­

•

was employed. •

dently, being at a height of 5 kilo­

Utility is one thing, backfiring an­

other. First, seeing war as "more ratio­ nal and clean" may affect (and often ap­ pears to affect) not only the public but also the political planners. The planners may be unmoved by the devastating ef­ fect on the victims; still, they don't want

institution. Purchases of weapons sys­

officers, like all of us, are easy vic­

tems are planned, war-games and lo­

tims of self-deceiving optimism and

to be misled into recklessly engaging

gistics are calculated.

pessimism

their armed forces in operations and wars that are less easily won than pre­

Weapons and weapons systems are optimized and their efficiency during

action is enhanced. This regards munitions (including missiles and

•

according

to

circum­

stances. At the opposite end of the scale, mathematics may also be an indis­ pensable tool. Thus, when the effect

dicted by the machine-rational percep­ tion of the character of war. Less dangerous for planners but just

8Evidently it is difficult to find any technology created during the last decades which is not somehow driven by mathematics. The list discusses such facets of the mat­ ter as go beyond what holds for any practice that involves computers or microelectronics.

91nterestingly, the analysis of damage to the intestine of a wounded soldier by magnetic resonance imaging (MRI) and the localization of enemy ground forces by syn­ thetic aperture radar (SAR) build on the same mathematics- both, indeed, by cleverly arranged rapid repetition squeeze out of a "short antenna" as much information as could be gained from an extended antenna without advanced mathematics [Schempp 1998: 44 and passim]. 1 0This does not disprove the utility of game-theoretical modelling, only the belief that human behaviour is always adequately described by the "rational economic man." Actually, sociobiological models of the same mathematical type show that the survival of the species is better guaranteed if egoistic suboptimizing is punished. The fear that the enemy might not accept the kind offer but take "irrational" revenge was exactly what made nuclear deterrence work, thus saving our species during the Cold War.

18

THE MATHEMATICAL INTELLIGENCER

Demolished Varadin Danube River Bridge in Novi Sad, Yugoslavia. The destruction of the bridges across the important international water­ way Danube, some of them in the North of Yugoslavia and hence far away from Kosovo where the Yugoslav military operational capability was to be hindered, was unlawful by The Geneva Protocol I. The justification given is that this kind of warfare is, after all, cost-efficient in human lives, even for the target population-as illustrated by the undamaged blocks of flats standing near to the crushed bridge (carpet bombing in Europe being unacceptable). The mysterious health problems of NATO soldiers who participated in the Gulf and Kosovo wars and the dramatic increase in cancer rates in Iraq tell us that other damage that does not show up on photographs may turn up in medical statistics. An even greater cost of this high-precision warfare supported by mathematics is the very introduction of the concept of justified punitive wars without bloodshed. This creates invincibility illusions and talks people into accepting war. [Source: NATO Crimes in Yugoslavia. Ministry of Foreign Affairs, Belgrade, 1 999.]

as thr�atening to victims is the relative inexpensiveness of present-day mathe­ matically supported asymmetric war­ fare for the attackers-if the subjuga­ tion of Serbia in the Kosovo war cost only $7 billion, that is, $700 per Yu­ goslav capita, the temptation is great to solve all similar problems in a similar way. The moment such a war turns out to . involve the use of ground forces, costs of course explode, and we are brought back to the situation discussed in the previous paragraph. Another feature of the mathemati­ zation of warfare is the transformation of the "Krupp model" into an "infinite Krupp model"; this feature regards sym­ metric situations rather than the field

of easy asymmetric wars and updated "gunboat diplomacy." War and pre­ pared war is always between two (pos­ sibly more) parts-Clausewitz would speak of a Zweikampj, a duel, which has now become a "duel of systems." In the nineteenth century, Friedrich Krupp would first develop nickel-steel armour that could resist existing shells, then chrome-steel shells that could pierce this armour, then high-carbon armour plate that resisted these, then cap-shot shells that could break this plate-and that was the end of it. In the duel between ground-to-air missiles and aeroplanes, no physical limit pre­ vents an ever-ongoing sophistication and an ensuing arms race iterating ad

infinitum. Capshot shells were and re­ main extremely expensive; so are stealth bombers and fighters-but such measures as depend solely on so­ phistication of software and hardware have neither budgetary nor intellectual definitive bounds. Processes depending on physics and chemistry may have natural boundaries. Those depending solely on mathematics seem to have none. The virtual absence of limits en­ hances the stress on both sides, and thus the speed and instability of such a race. Ethics

Mathematics, according to a familiar view, is a neutral tool. As once formu-

VOLUME 25, NUMBER 3, 2003

19

lated by Jerzy Neyman, "I prove theo­ rems, they are published, and after that I don't know what happens to them." This is certainly an important fea­ ture of the mathematical endeavour, and not only for theorems and theorem production. Also the teaching of math­ ematics, the production of high-level general mathematical competence in the population, is a precondition not only for the waging of modem war but also for the functioning of our whole technological society (quite apart from the cultural value we suppose it to pos­ sess). But the title "mathematics and war"

implies ethical dilemmas. We start by looking at the actual ethical choices of some well-known figures.

Kolmogorov's Theory of Firing, four front pages 1 942, 1 945, 1 948, 2002. The work by Kolmogorov and collaborators on Firing The­ ory was interesting enough to be translated

•

•

Laurent Schwartz used his high aca­ demic prestige to make his resis­ tance to the French and American wars in Algeria and Vietnam more ef­ fective; he saw no connection be­ tween his work in mathematics and his political commitment. Niels Bohr, on becoming aware of the German nuclear bomb project, sup­ ported the competing Anglo-Ameri­ can project; later discovering the dangers that were to arise from

by the RAND Corporation in 1 948 and is still regarded as fundamental in quite recent US military education. It is one of the apparent paradoxes of the relation between mathe­ matics and warfare that an earlier paper by Kolmogorov on the same topic was published in 1 942 for everybody to read (including the enemy). Could it be that this early paper (in contrast to the 1 945 book) was too mathe­ matical and too general to inform military practice directly?

Leading Polish army officers were present at a ceremony in 2001 when a memorial plaque was unveiled at the tomb of the cryptologist Mar­ ian Rejewski (1 905-1980). The photo shows some generals, together with Rajewski's daughter and the President of the Polish Mathematical Society. Not many mathematicians have experienced similar honours in life or posthumously. As a mathematics student, Rejewski had been recruited in 1 929 by the Cipher Bureau of the General Staff of the Polish Army. Rejewski then created a mathematical method for breaking the German Enigma code of that time. Long before their competitors, the Polish Cipher Bureau officers realized the potential of mathemat­ ics in cryptological research. [Source: Polish Mathematical Society, c/o Prof. K. Ciesielski, Jagiellonian University, Krakow.]

20

THE MATHEMATICAL INTELLIGENCER

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VOLUME 25, NUMBER 3, 2003

21

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the success of the latter, he issued warnings to responsible politicians (Churchill, Roosevelt) and to the public (the "Open letter")-using his prestige as an originator of the un­ derlying theory and as a collaborator (and arguably overrating the impact his interventions might have). Alan Turing, quite sceptical of British society (for political as well as per­ sonal reasons), put his outstanding abilities in the service of war with total loyalty when he felt it was needed; unlike Bohr, he did so with­ out ever claiming any special au­ thority. Kinnosuke Ogura had been a strong promoter of (Marxist-inspired) de­ mocratic modernization of Japan, and had opposed Japanese policies as being parallel to German and Ital­ ian Fascism. After the beginning of the aggression against China in 1937, however, patriotism and the prospect of war as a way to modernization moved him to play a central role in the organization of Japanese mathe­ matics in the service of the military state. After the war he expressed re­ gret, without specifying what he had done. John von Neumann, like Turing, ap­ plied his outstanding abilities in war research. He did so both during World War II and early in the Cold War. Whereas Turing had been a loyal participant about whose per­ sonal attitudes in the matter we know nothing, Neumann made the creation of the H-bomb a personal project which (well served by Stanislaw illam and Edward Teller) he carried out wholeheartedly-his aim being to make possible a pre­ emptive first strike. Lev S. Pontryagin gave up an ex­ tremely fruitful research line in al­ gebraic topology and created control theory. In hindsight this appears to have been caused by a will to serve his socialist country by solving the problems of guiding intercontinental ballistic missiles-thus making im­ possible the first strike. Decades before, G. H. Hardy had tried to avoid that usefulness of his science which consists in "accen­ tuat[ing] the existing inequalities in

22

THE MATHEMATICAL INTELLIGENCER

Vostok launch-rocket of the first manned spacecraft (launched on April 1 2, 1 961). According to information received from Samara Aerospace University, the launchings of Sputnik 1 and Sputnik 2 in 1 957 were made without recourse to the Pontryagin Maximum Principle. Later tasks- bringing down cosmonauts safely, and guaranteeing that intercontinental missiles sur­ viving a first strike would not miss New York by more than the radius of efficiency of a hy­ drogen bomb-did use the Maximum Principle, which was in the public domain well before that. For Sputnik 1 and 2, Pontryagin assisted in finding the correct weight of the spacecraft. [Courtesy of Samara State Aerospace Museum.]

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the distribution of wealth, or more directly promot[ing] the destruction of human life," by concentrating on supposedly useless number theory. Ironically, he repeated this phrase in 1940, when number theory was about to become a cryptographic resource. The radical pacifist Lewis Fry Richardson published his path­ breaking Weather Prediction by Nu­ merical Process in 1922 after having made sure that 64000 "computers"

(human beings furnished with desk calculators) would need more than one day to predict the weather one day ahead. This he saw as a guaran­ tee that numerical weather predic­ tion could not be put to military use. To what extent can these figures serve as exemplars and role models? First, they show that two fundamen­ tally different situations must be dis­ tinguished. One is that of Laurent

Schwartz, Hardy, and Richardson: deep highly stratified or a more egalitarian in its periphery, knowingly or not, or scepticism towards their own society, education system. One may organize perhaps be wholly outside it. In each or aspects of that society as a warring the research of an institution, be a pres­ situation, the scope of ethical choices power. The other situation is that of the tigious researcher, or be the newly ap­ is different, and no general ethical remaining characters: they accepted pointed young colleague. One may be rules or advice can be issued. A some­ their own society and its warfare or ar­ in the top of the "AMRC chain," or be what less abstract discussion of the mament policies, either in general or under actual circumstances-certainly with different degrees of identification. In the second situation, the ethical t.. , -e l: "' /!. l: 0 t> �.. ... dilemmas are few. Obviously, one will ..... ' � � � a a � see no objections to doing his best. •• tf .!: t � IC It . .. lit b 1"0 .. .. • • IC Dilemmas, it is true, are not totally ab­ � � ": q) ft b • • sent: one may, like von Neumann, give 0 1t � l: "' 'Ill t It • . lf . tD • . at "' � 1: · an extra push; one may, like Turing, be ..... � ; 11•' � A 'ill.. � � . • il• fully loyal but leave the political deci­ ·n. .b0 · � l: 0 lv 1t' ' l: tt· .it- . • sions to those who are officially enti­ i! ID . : . .. -; co a · ;;l "' � : It · fl. • • tled to take them (whether politicians, .. ... 1: b · � 1J • 1t: . . voting citizens in general, or military • � • � 0 '61 I> t II � :1' a! .a> • a � • men); or one may, like Bohr, use one's 0 "t'. • • * t ., � � particular standing and insight to mod­ IC 1" "1: � • � ftt t .. ' � 1"o • .&. tt· • 4IIJ . t erate, warn, or point to alternatives. . IC � • ' * .. = It lv J:. . The situation of the sceptic is less ft 111 ..... 111.. -.,.- " . ' D -e clear-cut. Very few of us are in a situ­ 0 .. ' * . ' .... . fi . Ill ation (the situation, say, of von Neu­ l, "' · t · • �· 0 • a ' mann and Pontryagin) where nobody "' 0 • • \.... a ' :JL 1: . i'f i -; "1: l: ., • • else could do what we are doing; these 1\ fl � - N � .. • few may influence matters directly by t � J,: 11• JJ I! 1'.> co ' 0 � 1fi *' b ft! Je deciding to cooperate or not to coop­ II It· 0 If 1: 11• .,, � I 1: IC erate, but they remain exceptions. l:: l:: � '· � :'if � � ; Most mathematicians, if they choose --. "' :,... It 1l: li. t.' ·"' .. 1t: 't' . ,. .. not to cooperate with the military in 0 t[) "' 81 • ' b 111 mathematics research and teaching, :X: · -r It"' � * 0 .a> 1C . '1: " "ft. • will have little effect, and little of what ' 11.. .. . . • • � . L "' most mathematicians do in research or "l' "" ;. "-' I> JM � teaching is directed toward a specific application. Deciding to abstain from working with a particular discipline be­ cause it seems "corrupt" is most futile. Giving up mathematics is giving up not Translation from the Japanese of the Ogura text: only military applications but anything mathematics can be used for-and Today is a critical time to fight the Greater East Asia War with all whatever cultural value we may as­ the might of our whole nation. In particular, contemporary war be­ cribe to mathematics. ing a war of science, the responsibility of those w ho study science However, the practice of the math­ and technology is very heavy. What should mathematical research and ematician consists in more than the ab­ education be, to meet Japan's grave needs today? I have tried in this stract production and dissemination of book to discover a correct guiding principle for this serious task in theorems. Any mathematician is in a historical perspective. particular situation, and in any partic­ ular situation there are specific condi­ tions and specific room for decisions. The first page of Ogura's 1 944 book on Mathematics in Wartime One may, for instance, widen one's Kinnosuke Ogura (1 885-1 962) was an excellent Japanese mathematician, struggling for the own insight and global understanding modernization and democratization of his home country. However, during the Greater East of the role of mathematics, and try to Asia War he was trapped in the mobilization of the Tennoist aggresson against China. Above, share it with students, colleagues, and the first page of Ogura's 1 944 book on Mathematics in Wartime. As the translation below it the public-or one may choose to re­ shows, he is writing a bellicose appeal to mobilize mathematics for the Tennoist victory. Af­ main (and leave others) blissfully ig­ ter 1 945, lending of extant library copies was banned, and the book was silently excluded norant. One may be a teacher within a from Ogura's Collected Works in 8 volumes.

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VOLUME 25, NUMBER 3, 2003

23

matter can be found, however, in [BooB & H0yrup 1984]. What can be said in general is that the supposed neutrality of mathemat­ ics per se does not entail the neutral­ ity of these ethical choices. An Enlightenment Perspective?

The Enlightenment believed that rea­ son might serve general progress; Jean­ Jacques Rousseau and Jonathan Swift pointed out that too often reason is used in the service of purely technical rationality, and for purposes of subop­ timization, with morally and physically disfiguring effects. According to Daniel Defoe's Robinson Crusoe, "Reason is the Substance and Original of the Math­ ematicks." Where does mathematics stand with regard to disfigurement and progress today? Much of what was said above points to disfigurement. Most alarming are probably not the actual uses but the ideological veil of rationality, clean­ ness, and "surgical accuracy" which is derived from the mathematization of warfare. By generalization one might claim that this applies not only to the military aspects of modem technical society but to the technically rational society as a whole. However, one of the ways in which mathematics serves the military points in the opposite direction: the sober­ minded elimination of self-deceiving optimism and pessimism which can be provided by mathematical reasoning and calculation. Mathematics-based reason at its best should allow us also in larger scale to unlearn conventional wisdom, to undermine facile indoctri­ nation, to distinguish the possible from glib promises. It might help us, if not to fmd any absolutely best way-this is too much to expect from reasoned analysis-then at least to evade the worst. If reason is really "the Substance and Original of the Mathematicks," mathematics might serve to make clear to us that war is fundamentally irra­ tional and unreasonable not only in commonplace ideological generality but in specific detail. If mathematics is not able to do such things, then its presumed cultural value might be noth­ ing but a convenient excuse for ruth­ less technical suboptimization.

24

THE MATHEMATICAL INTELLIGENCER

HVGO NIS

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I V R E B E L L I A

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T

'B ._ 1> A M t ,

Apud I 0 H A N N B M B L A E V.. M D

C X L V t.

Title page of the second edition of Grotius's De Jure Belli Ac Pacis. As indicated by the armil­ lary sphere, the publisher was also engaged in mathematical publishing. Historians of inter­ national law credit Hugo Grotius with the creation of modern international law, as in partic­ ular established in the Peace Treaty of Westphalia of 1 648 and the Charter of the United Nations of 1 945, and trace the origins of it back to patterns of mathematical thinking of strik­ ing public appeal in Grotius's time. Military analysts of our own time blame the striking pub­ lic appeal of mathematics-supported modern warfare for undermining international law.

Admittedly, technical rationality prevails over reason for the moment, both in the general political arena and in the uses to which mathematics is put. Mathematical theories are ethically neutral, it has been argued. Mathematics as a social undertaking is ethically am­ biguous: responsibility, whether they acknowledge it or not, remains with

its practitioners, disseminators, and users. REFERENCES

BooB, B ern h eim , &

Jens H0yrup, 1984. Von

Mathematik und Krieg. Ober die Bedeutung von ROstung und milita rischen Anforderun­ gen fO r die Entwicklung der Mathematik in Geschichte und Gegenwart. (Schriftenreihe

A U T H O R

-- , 2001. "Postface. Science, technologie, armament", pp. 377-465 in idem, Analyse mathematique, vol. II. Berlin etc. : Springer.

Gowing, Margaret, 1964. Britain and Atomic Energy 1 939- 1945. London: Macmillan & Co

Ltd. Gross, Horst -Eckart, 1978. "Das sich wandel­ nde Verhii.ltnis von Mathematik und Produk­ tion," pp. 226-269 in P. Plath & H. J. Sand­ kOhler (eds), Theorie und Labor. Dialektik als Programm

Koln:

der Naturwissenschaft.

Pahi-Rugensetin Verlag. H0yrup, Jens, 1986. Review of H. Gericke

BERNHELM BOOSS·BAVNBEK

JENS HSYRUP

Department of Mathematics and Physics

SectiOn for Philosophy and Science

1984. Zentralblatt fOr Mathematik 565, 7.

Roskilde University

Studies

H0yrup, Jens, 2002. Lengths, Widths, Sur­

Postboks 260

Roskilde University

faces: A Portrait of Old Babylonian A lgebra

Postboks 260

and Its Kin . (Studies and Sources in the His­

Denmark

4000 Roskilde

e-mail: [email protected]

Denmark

4000 Roskilde

tory of Mathematics and Physical Sciences). New York: Springer. Meigs, Montgomery C. , 2002. Slide Rules and

e-mail: [email protected] Bernheim Booss- Bavnbek has written pro­ lifically on mathematics, especially global

Submarines. American Scientists and Sub­ Trained first as a physicist, Jens H0yrup

surface Warfare in World War II. Honolulu,

analysis of partial-differential equations,

has been since 1 973 a specialist in con­

Hawaii:

and applications. A recent publication i s

ceptual and cultural history of mathemat­

Reprinted from the 1990 edition.

University Press of

the

Pacific.

Proc. . . . 1920: Comptes Rendus du Congres

Elliptic Boundary Problems for Dirac Op­

ics, and a faculty member at Roskilde Uni­

erators (Birkhauser, 1 993). He also has a

versity. He has recently published Human

International des Mathematiciens (Strasbourg,

Sciences: Reappraising the Humanities

22-30 Septembre 1920). Toulouse, 1921.

continuing professional interest in the so­ cial context of mathematics. Readers may recall his contentious article "Mem­ ories and Memorials" in The lntelligencer 1 7 (1 995), no. 2, 1 5-20.

through History and Philosophy (SUNY

Remmert, Volker, 1999. "Vom Umgang mit der

Surfaces: A Portrait of Old Babylonian Al­

tut im 'Dritten Reich. ' " 1 999. Zeitschrift fUr

Press , 2000), and Lenghts - Widths­

Macht. Das Freiburger Mathernatische lnsti­

gebra and its IVn (Springer, 2002).

Sozialgeschichte des 20. und 2 1 . Jahrhun­ derts 14, 56-85.

Rosser, J. Barkley, 1 982. "Mathematics and Mathematicians in World War II." Notices of the American Mathematical Society 29: 6,

Wissenschaft und Frieden, Nr. I ). Marburg:

angewandter Mathematik': Kriegsrelevante

Bund demokratischer Wissenschaftler. Some­

mathematische Forschung in Deutschland

what updated English translation as pp.

wahrend des II. Weltkrieges", vol. I, pp.

Resonance Imaging. Mathematical Founda­

225-278, 343-349 in Jens H0yrup, In Mea­

258-295

tions and Applications. New York, Wiley.

in

Doris

Kaufmann

(ed.) ,

509-515. Schempp, Walter Johannes, 1998. Magnetic

sure, Number, and Weight. Studies in Math­

Geschichte der Kaiser-Wilhelm-Gesellschaft

ematics and Culture . New York: State Uni­

im Nationalsozialismus. Bestandsaufnahme

Madison Wisconsin Collective.

versity of New York Press, 1994.

und Perspektiven der Forschung. Two vol­

Wisconsin: Science for the People, 1973.

Dalitz, Richard H., & Sir Rudolf Ernst Peierls

umes, Gbttingen: Wallstein Verlag.

The AMRC Papers. By Science for the People,

Madison,

Tschirner, Martina, & Heinz-Werner Gobel

(eds), 1997. Selected Scientific Papers of Sir

Gericke, Helmuth, 1984. "Das Mathematische

Rudolf Peierls. With Commentary. Singapore

Forschungsinstitut Oberwolfach," pp. 2 3-39

and London: World Scientific Publishing, Im­

in Perspectives in Mathematics. Anniversary

Phillips-Universitat Marburg. 50 Jahre nach

perial College Press.

of Oberwolfach 1984. Basel: Birkhii.user.

Beginn des II. Weltkrieges. Marburg: Eigen­

Davis, Chandler, 1989. "A Hippocratic oath for

Godernent, Roger, 1978. "Aux sources du

mathematicians?", pp. 44-47 in Christine

modele scientifique americain 1-111". La Pen­

Keitel (ed.), Mathematics, Education, and

see 201 (Octobre 1978), 33-69, 203 (Fevrier

Society. Science and Technology Education.

1979), 95-1 22, 204 (Avril 1979), 86-110.

Document Series No. 35, Paris: UNESCO. Epple, Moritz, & Volker Remmert, 2000. " 'Eine ungeahnte Synthese zwischen reiner und

--

, 1994. "Science et defense. Une breve

histoire du sujet 1 . " Gazette des Mathemati­

ciens 61, 2-60. (Part II has not appeared).

(eds), 1 990. Wissenschaft im Krieg- Krieg in

der Wissenschaft. Ein Symposium an der

verlag AMW. Wiener, Norbert, 1964. I Am a Mathematician: The Later Life of a Prodigy. Cambridge,

Mass.: M.I.T. Press. Youschkevitch, Alexander A , 1974. "Markov," pp. 124-130 in Dictionary of Scientific Biog­

raphy, vol IX. New York: Scribner.

VOLUME 25, NUMBER 3, 2003

25

M a thern ati c a l l y B e n t

Co l i n Adam s ,

E d itor

The Three Little P igs The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself

uneasily, "What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta­ tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267 USA e-mail: [email protected]

Colin Adams

O

nce upon a time there were three little pigs who lived with their mother, a mathematician at Nebraska. The first little pig adored number the­ ory. His goal in life was to prove the Swine-erton-Dyer Conjecture. The sec­ ond little pig was fascinated by ques­ tions about complexity. Her goal in life was to prove that Pig = NPig. The third little pig loved algebraic geometry. He wanted nothing more than to prove the Hog Conjecture. One day, their mother said, "I have taught you everything I know. It is now time for you to go to graduate school." So each pig packed a small bag filled with favorite math books and slung it on a stick, and all three set off to make their way in the mathematical world. The first little pig did a Ph.D. at the University of Illinois and obtained a post-doc at Columbia. In his first year there, he announced a proof of the Swine-erton-Dyer Conjecture. Hugo C. Wolfe was visiting Columbia from Berkeley at the time, and after devour­ ing the preprint, he rushed down the hall and banged on the little pig's door. "Little pig, little pig," he roared. "It is I, Hugo Wolfe, from Berkeley. Open the door and let me in." The little pig, who had recently grown a fashionable goatee, replied "Not by the hair on my chinny chin chin," as he cowered inside his office. For he was afraid of Hugo C. Wolfe, as was everybody else in mathematics. "Then I will huff and puff and fmd a hole in your proof. " And Wolfe huffed and he puffed and he came up with a hole in the first lemma.

"But there can't be a hole in the first lemma," squealed the little pig. "That is a folk lemma that has been around for­ ever." Wolfe laughed viciously. "You should know better than to rely on a folk lemma. " And the little pig's proof fell apart like a house made of straw, and the little pig's career did likewise. The second little pig did her Ph.D. at Wisconsin, and she received a post­ doc at Rice University. In her second year there, she announced a proof that Pig NPig. After reading the preprint posted on the Web, Hugo C. Wolfe im­ mediately flew to Houston, and arrived at the little pig's office door. "Little pig, little pig, it is I, Hugo C. Wolfe," cried Wolfe in his most intimi­ dating manner. "Open the door and let me in." "Not by the hair on my chinny chin chin," replied the second pig. Although she didn't have any hair on her chin, her brother, who now worked in kitchen supplies, had told her it was tradition to say this. "Then I will huff and puff and find a hole in your argument." And Wolfe huffed and he puffed and he found a hole in the second lemma. "That lemma can't be wrong," said the second pig through the door. "I found it on the Web. I'll fix the proof." But the hole could not be fixed, and her entire proof came tumbling down like a house made of sticks, and the second little pig's mathematical career did likewise. Discouraged, she opened a barbecue joint in South Houston, rev­ elling in the irony of it all. The third little pig received his Ph.D at Northwestern University, and took a post-doc at Michigan. After three years of work, the little pig announced a proof of the Hog Conjecture. Hugo C. Wolfe didn't even wait to read the preprint. He rushed to Michi­ gan, relishing the opportunity to de­ stroy the third pig's career as well. =

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25,

NUMBER 3, 2003

27

Wolfe stood outside the door and

"Little pig, little pig, open the door

stroked his long nose, a malevolent

and let me in." "I go by Herbert T. Boar," replied the pig through his closed door.

added, "You don't know that the iso­ metry group will have a subgroup of or­

smile creasing his face. Then he stalked

der p. " He was grasping at straws now,

off down the corridor.

but he was desperate. Everyone was

"Open the door, Boar, or I will huff

The little pig was awarded the Fields

and puff and create a counterexample

Medal, the award to be given at the In­

that will destroy this entire department."

"As any undergradute mathematics

ternational Congress of Mathematics,

major can tell you," replied the pig, "the

"Listen Wolfe, I'm not scared of you.

taking place in Porkugal. Before the cer­

Sylow subgroup theorems imply that

I know my proof is airtight. The lem­

emony, each ofthe awardees was to give

since p divides the order of the isome­ try group, there must be a subgroup of

looking at him, some smiling.

mas are all from refereed journals and

a talk As the little pig fmished his

I checked each one of their proofs my­

remarks,

self. So huff and puff away."

opened the floor to questions. Hugo C.

Wolfe's collar felt much too tight. He

Now Wolfe was greatly angered that

Wolfe stood up. "Little Pig," he said. "I

found he couldn't swallow. He felt like

the master of ceremonies

order p. "

the pig wasn't afraid of him. So he

have a question." A hush fell over the

he was on fire. Someone laughed. Wolfe

huffed and he puffed, but no coun­

audience, for they knew what it meant

tried to disappear, to sink beneath the

terexample came to him. So he huffed

when Hugo C. Wolfe had a question.

and puffed some more. He tried to

"It appears to me that in Lemma 3.4,

show that the tangent bundle did not

you Dehn fill the Whitehead link com­

lift

to

the universal cover.

He

chairs around him, but there was no es­ cape.

Everyone began laughing

and

pointing. Wolfe let out a strangled cry

at­

plement to obtain a hyperbolic orb­

and ran out of the auditorium, never to

tempted to prove that the metric was

ifold. But the result is not in fact an orb­

be seen in mathematical circles again.

not in fact Riemannian, but still to no

ifold at all. It is a manifold." Wolfe

Eventually, the little pig became chair

avail. Finally, he banged on the pig's

grinned wickedly. The little pig smiled

of the department at Michigan, which

door once more.

back confidently. For Hugo C. Wolfe

was housed in a beautiful brick build­

had fallen into his trap.

ing designed by the pig himself. He in­

"Little pig, little pig," he roared. "I don't have a counterexample yet, but it's only a matter of time."

"Yes," said the pig, "but as any grad­

vited his brother and sister to come live

uate student knows, a manifold is an

with him in Ann Arbor, where they cre­

The little pig said, "I'm not afraid of

orbifold, just with trivial singular set."

you. Even the weakest link in my ar­

Wolfe began to turn red. "Well, yes,

gument, that Dehn filling the cusped

3-

manifold yields a hyperbolic orbifold,

is safe from you."

ated the first dinner theater serving up hickory smoked barbecue with lec­

I suppose that is true," he mumbled. He

tures on projective algebraic varieties.

began to sweat.

And the dinner choices were chicken,

It suddenly felt much

too hot in the room. "But anyway," he

beef, and soy meat substitute.

E XPAND YOUR MIND Four Colors Suffice

How the Map Problem

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Robin Wilson

On October 23, 1 852,

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Robin W:il

h istory: What is t h e least

� as Solved

possible n u m ber of colors needed to fill in any map

n

( real or invented) so that neighboring counties are always colored d ifferently?

Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly i nnocuous question baffled g reat minds and stimu lated exciting mathe­ matics with far-flu n g applications.

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Exploring Euler's

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Julian Havil With a foreword by Freeman Dyson Among the myriad of con­ stants that appear in mathe­ matics,

n,

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most famil iar. Following

closely behind is g, or

gamma, a constant that arises in many mathemati­ cal areas yet maintains a profound sense of mystery.

Gamma travels through

countries, centuries, lives and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.

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•

READ EXCERPTS AT MATH.PUPRESS.PRINCETO N . EDU

i,f,@j!i§ufihi¥119-'I,J,IIijihtiJ

Sem inar· Workshop in Mathematics, Yaounde, Cameroon, Decem ber

1 0- 1 5, 2001

T

Marjorie Sen ech a l ,

Ed itor

he Republic of Cameroon, a cen­

ering was "Classical Analysis, Partial­

tral African nation situated be­

Differential Equations, and Applica­

tween Nigeria, Chad, Central African

tions." The international character of

Republic, Republic of Congo, Gabon,

the meeting is indicated by the list of

Equatorial Guinea, and the Bight of Bi­ afra, achieved independence in 1960.

speakers from many countries of Eu­ rope and Africa, including:

Its capital, Yaounde, a city on seven

hills, is also the capital of the Center province (one of ten). The University of Yaounde, founded in 1961, was the

Prof. Fulvio Ricci, Scuola Normale

only one in Cameroon for 32 years.

Superiore di Pisa, Italy

Naturally, Yaounde has long been a point of convergence for all the stu­ dents from the high schools and lycees who knocked on the university's doors. In 1993, because the university could not handle the growing number of stu­

throughout the world, and through all

Prof. Norbert Noutchegueme, Uni­ versite de Yaounde I, Cameroon Prof. David Bekolle, Universite de

dents, the public authorities created six universities in Cameroon, two of

Prof. Hamidou Toure, Universite de

them

Ouagadougou, Burkina-Faso

in

Yaounde.

University

of

Yaounde I, on the site of the former University of Yaounde, is composed of

ulty of Sciences, which includes the

of mathematical communities

Prof. Marco Peloso, Politecnico de Torino, Italy

Yaounde I, Cameroon

ters, and Social Sciences, and the Fac­

This column is a forum for discussion

Prof. Aline Bonarni, Universite d'Or­ leans, France

two faculties: the Faculty of Arts, Let­

Cyrille Nana

I

Department of Mathematics.

Gustavo

Garrig6s,

Universidad

Aut6noma de Madrid, Spain The workshop featured

a short

course in classical analysis, "Continu­ ity of Bergman projectors on tube do­

The Department of Mathematics,

mains over cones, from the analytic and

headed by Professor David Bekolle, of­

geometric points of view," with lectures

fers the basic specialties,

by David Bekolle, Aline Bonarni, Gus­

algebra,

time. Our definition of "mathematical

analysis, and geometry. It is notewor­

tavo Garrig6s, Fulvio Ricci, and Marco

community" is the broadest. We include

thy that the department has organized

Peloso. One is especially interested

"schools" of mathematics, circles of correspondence, mathematical societies,

at

here in the symmetric cones and most

Yaounde, including one on computer

several mathematics

workshops

particularly in Lorentz cones; the tech­

tools in complex dynamical systems

niques used by the speakers were orig­

student organizations, and informal

organized by CIMPA (Centre Interna­

inal and their way of presenting this ab­

communities of cardinality greater

tional de Mathematiques Pures et Ap­

struse subject led us to understand

pliquees) in 1999, a workshop on math­

what Bergman spaces are, the impor­

than one. What we say about the communities is just

as

unrestricted.

We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

ematics and fil�aria_irl �QOO, a.rHl most

tance of the Paley-Wiener theorem for

recently a GIRAGA (African Research

calculating the formula of the Bergman

Group in Algebra, Geometry, and Ap­

kernel or to associate to the dyadic de­

plications) workshop in September

2002. Here I describe the seminar­

composition of the cone and extend

workshop held in Yaounde from De­

the continuity of the Bergman projec­ tor, of which one of the applications is

cember 10 to December 15, 2001, in

the theorem of atomic decomposition

connection with the opening of the

of functions in the Bergman space of a

African Center for Research in Mathe­

tube above the Lorentz cone. The iden­

matics and Computer Science (CARIM),

tification that they established be­

established in September 2000 by a de­

tween the Lorentz cone and the cone

of Mathematics, Smith College,

cree of Professor Henri Hogbe Nlend, Minister of Cameroon Scientific Re­

of positive-definite symmetric matrices in dimension 3 is also of interest.

Northampton, MA 01 063 USA

search.

Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal,

Department

e-mail: [email protected]

It is important to note that in sev­

The theme of this international gath-

eral of these lectures, the speakers es-

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25,

NUMBER 3, 2003

29

tablished connections between classi­ cal analysis and signal-processing and

to general relativity; Gustavo Garrig6s,

ter his first lecture, "I have found the

and Hamidou Toure, the group on nu­

exposition, all the time." The zeal of the

image-processing, or between classical

and applications to signal-processing;

tions. To this effect, Bonami showed,

merical analysis of partial-differential

analysis and partial-differential equa­ starting with Fourier analysis, the con­ nection which exists between the sam­

pling theorem used in the atomic de­

demand sources." Fulvia Ricci said af­

the group on the theory of wavelets

equations (homogenization, numerical

solutions) and applications to hydrol­

ogy and petroleum research.

participants very attentive during my auditors attracted the attention of the

lecturers who were interested in the

participants. Thus Gustavo Garrig6s af­

ter his first lecture declared, "I appreci­

In the different working groups,

ate the hospitality of the people here.

rem, well known in signal-processing.

students, researchers, engineers, and

having to give it in English, my French

method of wavelets in image compres­

ated debates and exchanges among the

composition of functions in Bergman

space and Shannon's sampling theo­

Gustavo

Garrig6s

spoke

about

the

sion. Fulvio Ricci showed that Fourier

analysis permits the solution of certain

partial-differential equations; he gave some

there were presentations by doctoral

students. All the presentations gener­ participants. What interested us most

was the possibility for our students to

discuss mathematics with high-level re­

My course went very well; I truly regret

is very bad, I can't even write in French.

If I tried to write phonetically, everyone would have been scandalized."

At the end of the work, there was a

round-table,

a

discussion

forum

be­

a priori estimates for elliptic op­

searchers. It was clear from these dis­

tween mathematicians and engineers on

ities to the Schrodinger equation. In

different participants was complete. As

to petroleum research, to hydrology, and

erators and applied Strichartz inequal­

cussions that the contact between the

addition, Norbert Noutchegueme talked

Aline Bonami said, "We are not isolated

ory of general relativity, and Hamidou

people pose many questions, and we are

about Einstein's equations and the the­

here, not at all; the contact is good, the

Toure discussed applications to hy­

getting to know them. When we were in

To give a larger number of par­

the thesis students. Here it is different,

drology and petroleum research. ticipants an

opportunity to speak,

Poland, we could not even meet with

the people greet you and get to know

the organizers set up four parallel

you. " Also, the dedication with which

the classical analysis group; Norbert

was noted. As Gustavo Garrig6s af­

working groups. Fulvio Ricci directed

Noutchegueme, the group on partial­ differential equations with applications

the participants followed the course

firmed, "After a half-hour course, the

people are interested in wavelets; they

the theme, "Application of mathematics

to signal-processing and image-process­ ing," presided over by Minister Henri

Hogbe

Nlend,

internationally

Cameroon mathematician.

known

We received impressions and advice

from several speakers, and the global

point of view of a participant from

Burkina-Faso.

Interview with Marco Peloso

CYRILLE NANA: Is this the first time you

have been in Africa?

Group shot of the participants. Jean-Luc Dimi (Congo-Brazzaville} and Gustavo Garrig6s (Spain} are squatting in the front. Behind Garrig6s is David Bekolle (Cameroon}; continuing to the right are Hamidou Toure (Burkina Faso}, Henri Hogbe Nlend (Minister of Research, Cameroon}, Aline Bonami (France}, Blaise Some (Burkina Faso}; behind Hamidou and Bonami are Marco Peloso and Fulvio Ricci (Italy}.

30

Tl-IE MATHEMATICAL INTELLIGENCER

Workshop participants with three of the lecturers: first, third, and sixth from the left are Gustavo Garrig6s, Marco Peloso, and Fulvio Ricci. MARCO PELoso: I have never been in Africa before; this is the first time I have been here. c. N.: Are you surprised to see Africans doing mathematics? M. P.: No, not at all. c. N.: What idea did you have of Africa in general? M. P. : A very poor continent where the people struggle to live and survive. Everything has gone well so far. c. N.: What do you think of the way the Seminar has been organized? M. P.: I think the organization is excel­ lent: 100 participants, the environment and the hall very suitable, we had the facilities that we needed; there was much discussion in the working groups; the organization was perfect. c. N.: I noticed that each of you devel­ oped part of a single theme: how did you arrange to do that? M. P.: I Inlls_t s�y_tha,t this_was done on purpose, but I think that there was a problem of notation. The subjects were developed in a logical order and there was coordination between the speak­ ers, but I think that it would have been advantageous for each speaker to write out his lecture in detail. Unfortunately, we didn't have the time to do that. c. N . : What do you think of the fact that we included applied mathematics in the seminar? M. P.: It was important to organize the

sessions for applied mathematics be­ cause Africa must try to develop in the applied sciences. I think also that the association of pure and applied math­ ematics is very useful and beneficial for both groups of researchers. Per­ sonally, I would like to keep up with the applications to, say, hydrology. c. N.: A propos of the young re­ searchers, how have you found them and what do you advise? M. P.: As far as I'm concerned, it was in­ teresting to listen to the young African mathematicians. I hope that the young can find this conference useful, both for learning and for meeting people. Thus you can have the chance perhaps to travel in Europe if you wish. c. N . : What areas of mathematics inter­ est you the most? M. P.: Harmonic analysis, real variables, Hardy spaces, convergence of singular integrals, BJ\1:0 d:ual of H1, holomor­ phic functions, the study of the behav­ ior of the atomic decomposition at the boundary, functions of several complex variables in general, and other areas. Interview with Fulvio Ricci CYRILLE NANA:

before?

Have you been in Africa

FULVIO RICCI: I was in Egypt once, but it is different here. c. N . : Do you know any African mathe­ maticians?

F. R.:

I know African mathematicians such as Noel Lohoue. c. N . : What do you think of the organi­ zation of this gathering? F. R.: The organization was done with much care, it was generous, the at­ mosphere very pleasant, there were enough people who followed the talks, and I see that there are many activities in this part of Africa. Moreover, I have seen young people who are very open to listening about areas in which they do not work. We had a large enough room, but perhaps the blackboard was defective; certainly this room was not made for lectures. It is good to have thought of a room with Internet access; the hotel is reasonable. Good organization! I understand that the hours were set in relation to local activities, but one gets used to it. I had feared that it would be very warm, but it is very pleasant and the air-condi­ tioning of the room is good too. c. N.: What areas of mathematics inter­ est you the most? F. R.: Harmonic analysis; I work in Eu­ clidean analysis, classical Fourier analysis, singular integrals, analysis on Lie groups (in particular nilpotent Lie groups, the Heisenberg group), the prob­ lem of resolvability of differential oper­ ators, questions of harmonic analysis that are related to complex analysis.

VOLUME 25, NUMBER 3, 2003

31

c. N . : To hear you speak, you do har­ monic analysis; how is it that here you have been speaking about Bergman spaces? F. R.: I would first like to remark that there is a part of complex analysis which is geometric, for example, func­ tions of several complex variables; also, Lie groups, it is analysis on prob­ lems having geometric aspects. I began to work on Bergman spaces in 1995, I had heard Aline Bonami and David Bekolle speak about them, and I was interested in collaborating with them. c. N.: How long have you been doing harmonic analysis? F. R.: I began in harmonic analysis when I was in America (USA). In Italy, the field was not developed in the 1970s; I received a Ph.D. in the United States and then returned to Italy. c. N.: I suppose you have had many stu­ dents? F. R. : I have had about a dozen students who are now professors, researchers, etc. c. N.: What advice would you give to a young person beginning research? F. R. : It's necessary that the researcher engage in the study on the intellectual level, seek out contacts with the sub­ ject, and imitate what is available in the literature and what others have done. Researchers must be clever enough to pursue the problem posed, and to go on from there to enlarge the field of in­ terest, to find connections with related problems. c. N.: In general, did the seminars in clas­ sical analysis interest many people? F. R. : There are relatively few partici­ pants who are interested in classical analysis, I understand well that there are many people in applied math for obvious reasons. c. N.: What do you think of CARIM? F. R.: CARIM is very important not only for Cameroon, but for the entire region; it would be important to make con­ nections with other institutes, in Eu­ rope in particular. c. N.: Thank you.

Brief Discussion with Gustavo Garrigus CYRILLE NANA:

Do you teach in Madrid? I don't have a per-

GUSTAVO GARRIGOS:

32

THE MATHEMATICAL INTELLIGENCER

A U THOR

manent teaching post. From time to time I go to Orleans and Italy to work. N. c.: What are the domains in which you work. G. G.: I have worked on Bergman spaces for two years; wavelets, since my the­ sis in 1995. It was Bonami who en­ couraged me to study Bergman spaces and also signal-processing. c. N.: How is it that in such a short time you have been able to do so many dif­ ferent things? G. G.: In fact, after my thesis I had op­ portunities to do something else as a post-doc. I have collaborated with en­ gineers; the best moment, it is to do something new with other people. It is necessary to note that it is easier to talk with an analyst than with an engineer; in fact, it takes time to understand what an engineer is saying. c. N.: Thank you.

CYRILLE NANA

Departement de Mathematiques Universite de Yaounde I

B.P. 8 1 2 , Yaounde , Cameroon e-mail: [email protected]

Cyrille Nana, a student at the Univer­ sity of Yaounde I, was a participant in the Seminar-Workshop. He is writing his thesis on the continuity of the Bergman projector in tube domains,

Interview with Prof. Blaise Some,

and recently made a preliminary re­

Universite de Ouagadougou,

port on his results on a research visit

Burkina-Faso

to the laboratory MAPMO in Orleans ,

CYRILLE NANA:

What are your impres­ sions of the results of this seminar? BLAISE SOME: I believe that the organi­ zation is good; on the level of recep­ tion, it was good; the lodging, good; the restaurant was good; the first day, the service was slow. The lectures concerning applica­ tions were good. Prof. Noutchegueme introduced things well. As a whole, the course was very good. I much appreci­ ated the round-table, for it permitted the public to see that Mathematics is well. The graphics were good, but there were transparencies written by hand, and that was not good. On the scientific side, it was a total success; the time was very short, how­ ever, so one could not go into depth on some points. c. N.: How would you advise the students who participated in the colloquium? B. s. I heard several students' work; there was one that I noticed particu­ larly, on finite elements. He showed that he had mastered his domain of re­ search. The advice for students is first to learn the literature in his domain, to have a large view of all those who have worked in that area, and to know the specialized journals. N. c.: One word about CARIM?

France, directed by Aline Bonami.

B.

s.: CARIM is a good idea as such; now it is necessary to see the texts that will defme its functioning. c. N.: Thank you.

Summary

The Seminar-Workshop, according to the opinions expressed in the inter­ views, went very well, on the scientific level as in all other respects. At the end, the organizing committee convened an evaluation session which was coordi­ nated by Professor Jean Luc Dirni of the Universite Marien Ngouabi de Braz­ zaville in Congo. During this session the participants brought out some weaknesses that, while not negligible, did not hinder at all the progress of the work. I remark in conclusion that during the academic year 2001-2002, David Bekolle taught a course for DEA (Diplome d'Etudes approfondies) stu­ dents on the theme developed in the seminar. Course notes are now in preparation which spell out in detail the results that the participants es­ tablished during the workshop, writ­ ten by themselves.

@,i,Fftj.J§!:bhifli=tft%§[email protected]§•id

The Card Game SET

Benjamin Lent Davis and Diane Maclagan

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.

M i c h ael Kleber a n d Ravi Vak i l ,

T

he card game called SET1 is an ex­ tremely addictive, fast-paced game found in toy stores nationwide. Al­ though children often beat adults, the game has a rich mathematical struc­ ture linking it to the combinatorics of finite affine and projective spaces and the theory of error-correcting codes. Last year an unexpected connection to Fourier analysis was used to settle a basic question directly related to the game of SET, and many related ques­ tions remain open. The game of SET was invented by population geneticist Marsha Jean Falco in 1974. She was studying epilepsy in German Shepherds and began repre­ senting genetic data on the dogs by drawing symbols on cards and then searching for patterns in the data. Af­ ter realizing the potential as a chal­ lenging puzzle, with encouragement from friends and family she developed and marketed the card game. Since then, SET has become a huge hit both inside and outside the mathematical community. SET is played with a special deck of cards (Fig. 1 ). Each SET card displays a design with four attributes-number, shading, color, and shape-and each attribute assumes one of three possi­ ble values, given in Table 1 . Table 1

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

Number:

{One, Two, Three}

Shading:

{Solid, Striped, Open}

Color:

{Red, Green, Purple}

Shape:

{Ovals, Squiggles. Diamonds}

A SET deck has eighty-one cards, one for each possible combination of at­ tributes. The goal of the game is to find collections of cards satisfying the fol­ lowing rule.

Ed itors

The SET rule: Three cards are called a SET if, with respect to each of the four attributes, the cards are either all the same or all different.

For example, Figure 2 illustrates a green SET. All cards have the same shape (ovals), the same color (green), and the same shading (solid), and each card has a different number of ovals. On the other hand, Figure 3, also green, fails to be a SET, because there are two oval cards and one squiggle card. Thus the cards are neither all the same nor all different with respect to the shape attribute. To play the game, the SET deck is shuffled and twelve cards are dealt to a table face-up (Fig. 4). All players si­ multaneously search for SETs. The first player to locate a SET removes it, and three new cards are dealt. The player with the most SETs after all the cards have been dealt is the winner. Occasionally, there will not be any SETS among the twelve cards initially dealt. To remedy this, three extra cards are dealt. This is repeated until a SET makes an appearance. This prompts the following SET-theoretic question. Question. How many cards must be dealt to guarantee the presence of a SET?

Figure 5 shows a collection of twenty cards containing no SETs. A brute-force computer search shows that this is as large as possible, as any collection of twenty-one cards must contain a SET. There is a wonderful geometric re­ formulation of this Question as fol­ lows. Let IF3 be the field with three el­ ements, and consider the vector space

Stanford University,

Department of Mathematics, Bldg. 380,

SET

is a trademark of SET Enterprises, Inc. The

Stanford, CA 94305-2125, USA

cards are depicted here with permission.

e-mail: [email protected]

play is protected intellectual property.

SET

SET

game

Figure 1 . Typical SET cards.

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25, NUMBER 3. 2003

33

Figure 2. A SET.

1Ft A point of IF§ is a 4-tuple of the form (x1, x2, x3, x4), where each coordinate assumes one of three possible values. Using the table of SET attributes (Table 1), SET cards correspond to points of IF§, and vice-versa. (2, 1 , 3, 2)

�

[ill .

Two Solid Purple

Squiggles

�

Under this correspondence, three cards form a SET if and only if the three associated points of IF§ are collinear. To see this, notice that if a, {3, y are three elements of IF3, then a + f3 + y = 0 if and only if a = f3 = y or { a, {3, y} { 0, 1 , 2 } . This means that the vectors a, b, and c are either all the same or all different with respect to each coordi­ nate exactly when a + b + c = 0. Now a + b + c = 0 in IF§ means that a - b = b - c, so the three points are collinear. Note that this argument works when IF§ is replaced by IF � for any d. From this point of view, players of SET are search­ ing for lines contained in a subset of IF§. We summarize this rule as follows. =

The Affine Collinearity Rule. Three points a, b, c E IF � represent collinear points if and only if a + b + c = 0.

We define a d-cap to be a subset of

IF� not containing any lines, and ask the following. Equivalent Question. What is the maximum possible size of a cap in IF§?

In this form the question was first answered, without using computers, by Giuseppe Pellegrino [ 19] in 1971. Note that this was three years before the game of SET was invented! He ac­ tually answered a more general ques­ tion about "projective SET, " which we explain in the last section.

Figure 3. Not a SET.

34

THE MATHEMATICAL INTELLIGENCER

Figure 4. Can you find all five SETS? (Or all eight for those readers with black-and-white pho­ tocopies.)

Although SET cards are described by four attributes, from a mathematical perspective there is nothing sacred about the number four. We can play a three-attribute version of SET, for ex­ ample by playing with only the red cards. Or we can play a five-attribute version of SET by using scratch-and­ sniff SET cards with three different odors. In general, we define an affine SET game of dimension d to be a card game with one card for each point of IF�, where three cards form a SET if the corresponding points are collinear. A cap of the maximum possible size is called a maximal cap. It is natural to ask for the size of a maximal cap in IF�, as a function of the dimension d. We denote this number by ad, and the known values are given in Table 2.

6 1 1 2 ,;

a6

,; 1 1 4

The values of ad in dimensions four and below can be found by exhaustive com­ puter search. The search space be­ comes unmanageably large starting in dimension five. Yves Edel, Sandy Fer­ ret, Ivan Landjev, and Leo Storme re-

[]] [!!] [JJ ill] []] [!!] [JJ [)I]

Figure 5. Twenty cards without a SET.

cently created quite a stir by announc­ ing the solution in dimension five [6]. We shall spend some time working our way up to their solution. There are many other possible gen­ eralizations of the game of SET. For ex­ ample, we could add another color, shape, form of shading, and number to the cards, to make the cards corre­ spond to points of IF 4. Here, however, several choices need to be made about the SET rule. Is a SET a collection of cards where every attribute is all the same or all different, or is it a collec­ tion of collinear points? In IF i, there are four points on a line, so do we require three or four collinear points to form a SET ? Furthermore, if we choose the collinearity criterion, then collinearity of SET cards is sensitive to the choice of which color, shape, etc. corresponds to which element of IF 4. Because of these complications we will restrict our attention here to caps (line-free collections) in IF�. We can exhibit caps graphically us­ ing the following scheme. Let us con­ sider the case of dimension d = 2. A two-attribute version of SET may be re­ alized by playing with only the red ovals. The vector space IF § can be

[IT!] [ill] I D D D I l l l l l [ill OJ I � HI ill] [I] [B] � � � � l lo o o l

[[] [}] []]

5-cap

H� # #x ### ��� it= * # * �� � * # # lj * # * ### # * xl:lx * # # ### **# xj:jx * #x H� H� =#= l-eap

X I IX 2-cap

Figure 6. The correspondence between 2-at­ tribute SET and

IF3.

graphically represented as a tic-tac-toe board as in Figure 6. We indicate a sub­ set S of IF § by drawing an "X" in each square of the tic-tac-toe board corre­ sponding to a point of S. The lines con­ tained in S are almost plain to see: most of them appear as winning tic-tac-toes, while a few meet an edge of the board and "loop around" to the opposite edge. Check that the two lines in Fig­ ure 7 correspond to SETS in Figure 4. Figure 8 contains pictures of some low-dimensional maximal caps. In di­ mensions one through four, the caps are visibly symmetrical, and each cap contains embedded copies of the max­ imal caps in lower dimensions. No such pattern is visible in the diagram of the 5-cap. It is natural to ask if the maximal caps in Figure 8 are the only ones in each dimension. In a trivial sense, the answer is 'no', since we can make a new cap by permuting the col­ ors of an old cap. There are many other permutations of IF:� guaranteed to pro­ duce new caps from old. Permutations of IF� taking caps to caps are exactly

,'

, '

,

'

X , ,

'

'

V: V': ,

'

.,'-��----+----+-""x ·-, " : ,

, I

I ., '

'

... "" 1 � , .. .. ... .... (

'

-, '

I

>( >< '

,

..

'

'

,

-

' I

I

I

I I I

-

I

I '

Figure 7. This collection of points contains two lines which are indicated by dashed curves.

X

x

3-cap

x

x

X

x

4-cap

X

X

Figure 8. Low-dimensional maximal caps.

those taking lines to lines, and such a permutation is called an affine trans­ formation. Another characterization of affine transfonnations is that they are the permutations of IF� of the form

a(v)

=

Av + b,

where A is an invertible d X d-matrix with entries in 1Fa, b is an arbitrary vec­ tor of IF�, and v is a vector in IF �. We say that two caps are of the same type if there is an affine transformation taking one to the other. For example, consider the affine transformation a(x, y) = ( -x - y, - x + y - 1) taking a vector (x, y) E IF§ to another vector in IF�. Applying this to a 2-cap gives an­ other 2-cap of the same type. This is il­ lustrated in Figure 9, where we have

declared the center square of the tic­ tac-toe board to be the origin of 1Fl It is known that in dimensions five and below there is exactly one type of maximal cap. An affine transformation taking a cap to itself is called a sym­ metry of the cap. Although it is not ob­ vious from Figure 8, the maximal 5-cap does have some symmetries. In fact, its symmetry group is transitive, meaning that, given two points of the 5-cap, there is always a symmetry taking one

Figure 9. Two 2-caps of the same type.

VOLUME 25. NUMBER 3 . 2003

35

to the other. Michael Kleber reports that the stabilizer of a point in the 5-cap is the semidihedral group of or­ der 16. The symmetry group is useful for re­ ducing the number of cases that need to be checked in exhaustive computer searches for maximal caps, thus greatly speeding up run times. To see this idea in action, check out Donald Knuth's SET-theoretic computer pro­ grams [17] .

We can make some progress on com­ puting the size of maximal caps using only counting arguments. Proposition 1. A maximal 2-cap has four points.

Proof We have exhibited a 2-cap with four points. The proof proceeds by contradiction. Suppose that there ex­ ists a 2-cap with five points, x1, x2, Xs, X4, X5. The plane rF§ can be decomposed as the union of three horizontal paral­ lel lines as in Figure 10. Each line contains at most two points of the cap. Thus, there are two horizontal lines that contain two points of the cap, and one line, H, that con­ tains exactly one point of the cap. With­ out loss of generality, let x5 be this point. There are exactly four lines in the plane containing the point x5, which we denote H, L1, L2, Ls. This is illustrated in Figure 1 1 . Since the line H contains none o f the points x1, . . . , x4, by the pigeon-hole principle two of these points Xr and Xs must lie on one line Li. This shows that the line Li contains the points Xr, Xs and x5, which contradicts the hypothesis that the five points are a cap. 0 We can apply the method of Propo­ sition 1 to compute the size of a max­ imal cap in three dimensions.

� ><=i =><� : : : : : ;

Figure 10.

�x3

X4 = ;

�====

=X s= ;

IF�

decomposed as the union of

three parallel lines.

36

I

/

, ' ' ,X 3 ' ,

'

,

.... ,

,

, I I I I

...

'

'

'

{lc n H11, jc n H21, c n Hs! l,

'

H .- - - - -

Figure 1 1 . The four lines containing

Xs.

Proposition 2. A maximal 3-cap has nine points.

Combinatorics

H

decomposed as the union of three par­ allel planes, H1, H2, Hs in many differ­ ent ways. Given such a decomposition, we obtain a triple of numbers,

L3

... I I I I

THE MATHEMATICAL INTELLIGENCER

Proof We have exhibited a 3-cap with nine points. The proof proceeds by contradiction. Suppose that there ex­ ists a 3-cap with ten points. The space rF § can be decomposed as the union of three parallel planes. Since the inter­ section of any plane with the 3-cap is a 2-cap, Proposition 1 implies that no plane can contain more than four points of the cap. This means that the plane containing the fewest number of points must contain either two or three points, for if it contained four points we would need twelve points total, and one or zero points would mean at most nine points total. Call this plane H, and note that there are at least seven points of the cap, x1. . . , x7, not contained in

called the (unordered) hyperplane triple, where 1C n Hil is the size of C n H;. Since a 2-cap has at most a2 = 4 points, the only possible values for a hyperplane triple are {4, 4, 2 } or {4, 3, 3}. Let a = the number of { 4, 4, 2 } hyperplane triples, b = the number of {4, 3, 3 } hyperplane triples. How many different ways are there to decompose rF§ as the union of three hy­ perplanes? On the one hand, there are a + b ways. On the other hand, there is a unique line through the origin of rF§ perpendicular to each family of three parallel hyperplanes, and we can count these lines as follows: Any nonzero point determines a line through the ori­ gin, and there are 3s 1 = 26 nonzero points. Since each line contains two nonzero points, there must be 26/2 = 13 lines through the origin. Thus, -

a+b

.

H.

Let a and b be two points of the cap on the plane H. There are exactly four planes in the space IF§ containing both a and b, which we denote H, M1, M2, Ms. Since H does not contain the points x1 , . . . , x7, by the pigeon-hole princi­ ple one of the planes M; must contain three of these points x,, X8, x1• This shows that the plane Mi contains a to­ tal of five points of the cap, which con­ tradicts Proposition 1 . D Unfortunately, this method is not strong enough to prove that a4 20. To do this, we employ another time­ honored combinatorial technique, namely, counting the same thing in two different ways. By way of introduction, we will give another proof that as = 9. =

Proposition 3. A maximal 3-cap has nine points.

Proof The proof is again by contradic­ tion. Suppose that there exists a 3-cap C with ten points. The space IF§ can be

=

13.

To obtain another equation in a and b, we will count 2-marked planes, which are pairs of the form (H, {x, y} c H n C), where H is a plane. It can be checked that there are exactly four planes containing any pair of distinct points. This is a special case of Propo­ sition 4 which follows. Thus, there are 1 4( �) = 180 2-marked planes. On the other hand, for each {4, 4, 2} hyperplane triple we count ( � ) + (�) + (�) = 13 2-marked planes, and for each {4, 3, 3} hyperplane triple we count ( � ) + (�) + (�) = 12 2-marked planes. Hence, 13a + 12b = 180. The only solution to these equations is a = 24, b 1 1 . This is a contradic­ tion since a and b can only take non­ negative values. D =

-

In the proof above we needed to count the number of hyperplanes con­ taining a fixed pair of points, or in other words, containing a fixed line. To apply this method to maximal 4-caps,

we will need to solve a generalization of this problem. Define a k-jlat to be a k-dimensional affine subspace of a vec­ tor space. Proposition 4. The number of hyper­ planes containing a fixed k-flat in IF� is given by

sd-k - 1 2 Proof Let K be a k-flat containing the origin. Then the natural map

are 13el) = 2730 2-marked hyper­ planes. As in the proof of Proposition 3, there are

l (�) (�) (�)J X993 l(;) (;) (;)J +

+

+ · · · +

+

+

xm

2-marked hyperplanes. Explicitly com­ puting each coefficient above yields the formula

IF�-k � IFg /K = IF � -k

(2) 75Xggs + 70x9s4 + 67x975 + 66x966 + 66xsss + 64xs76 + 63x777 = 2730.

gives a bijection between hyperplanes of IFg containing K and hyperplanes of IF� -k containing the origin. Each hyperplane containing the ori­ gin is determined by a nonzero normal vector, and there are exactly two nonzero nonnal vectors determining each hyperplane. Thus, there are half as many hyperplanes as there are nonzero vectors. Since there are 3d -k - 1 non­ zero vectors, there must be (3d -k - 1)/2 hyperplanes containing the origin. D

To obtain yet another equation in Xijk, let us count 3-marked hyper­ planes, which are pairs of the form (H, {x, y, z} c H n C), where H is a hy­ perplane. Notice that, since {x, y, z) C C, the points x, y, and z cannot be collinear. There are 4 hyperplanes con­ taining 3 distinct non-collinear points, thus, there are 4(�1) = 5320 3-marked hyperplanes. Imitating our count of 2marked hyperplanes above, we find that

This lets us apply the ideas of Proposition 3 to calculate a4 .

(3) 169Xggs + 1 44xgs4 + 129x975 + 124x966 + 122x8s5 + 1 1 1xs76 + l06x777 = 5320.

Proposition 5. A maximal 4-cap has twenty points.

Proof We have exhibited a 4-cap with 20 points. The proof proceeds by con­ tradiction. Suppose that there exists a 4-cap C with 2 1 points. Let Xijk be the number of ( i, j, k } hyperplane triples of C. Since a 3-cap has at most a3 = 9 points, there are only 7 possible hy­ perplane triples: [i, j, k J = [9, 9, 3 }, (9, 8 , 4 }, [ 9, 7, 5 }, [ 9 , 6, 6 }, { 8 , 8 , 5 }, { 8 , 7, 6}, { 7, 7, 7}. The number of ways to decompose IF§ a s a union of three parallel hyperplanes is equal to the number of lines through the origin in IF§, which is (34 - 1)/2 = 40. Thus,

We now have three equations in seven variables, and so in principle there could be infinitely many solu­ tions. Fortunately we are only inter­ ested in the nonnegative integer solu­ tions. Adding 693 times equation (1) to three times equation (3), and then sub­ tracting off 6 times equation (2), gives 5xgs4 + 8x975 + 9x966 + 3xsss + 2x876 = 0. The only nonnegative solution to this equation is Xgs4 = X97s = Xg66 = Xsss = xs76 0. But equation (2) minus 63 times equation (1) is =

1 2xggs + 7Xgs4 + 4x975 + 3x966 + 3xss5 + Xs76

=

2 10.

( 1) X993 + X984 + X975 + X966 + Xss5 + Xs76 + X777 = 40.

This reduces to 12x993 = 2 10, which contradicts x993 being an integer. D

To obtain another equation in Xijk, let us count 2-marked hyperplanes, which are pairs of the form (H, {x, y } c H n C), where H is a hyperplane. Using Proposition 4 above, we find that the number of hyperplanes containing a distinct pair of points is 13. Thus, there

This proof was improved from a pre­ vious version by conversations with Yves Edel. The method of counting marked hyperplanes via hyperplane triples gives the shortest known proof of a4 20 that does not use an exhaus­ tive computer search. Unfortunately, a =

straightforward application of this method fails to show that as = 45. Part of the problem is that the new equations counting 4-marked hyperplanes require an additional variable to distinguish be­ tween the cases when four points are affmely dependent or independent. In the next section we describe another approach which computes as. The Fourier Transform

The Fourier transform is an immensely useful tool for analyzing problems with associated symmetry groups. It is a natural construction in representation theory, and we refer the reader to the book of Fulton and Harris [ 7] for more about this fascinating subject. In this section we describe a Fourier trans­ form method originated by Roy Meshu­ lam [18] which was later used by Jiir­ gen Bierbrauer and Yves Edel [ 1 ] . The following bound appears in these pa­ pers: Proposition 6. Let C C IF� be a d-cap such that any hyperplane intersects C in at most h points. Then

1 + 3h h ' 1 + 3d - l

p s

where p is the size of C. In particular, any hyperplane inter­ sects a d-cap in a (d - 1)-cap. Starting with the fact that a1 = 2, we can in­ ductively apply Proposition 6 to obtain a3 s 9 , The bound a6 s 1 14 comes from ap­ plying Proposition 6 using h = a5 = 45 and d = 6. Thus, for low-dimensional caps, Proposition 6 gives nearly sharp bounds. In contrast to other methods, Proposition 6 does not become more difficult to apply as the dimension grows larger. Given a function f : IF� � 0::: , define the Fourier transform off to be a new function] : IFg � ([ defined by the for­ mula

� f(x)gz·x, xEIF� 3 where g e27Ti1 . Given a set S c IF�, the characteristic function of S is de­ fined by the formula (4)

](z) = =

) X(X =

{

1 if X E S, 0 if x � S.

VOLUME 25. NUMBER 3. 2003

37

Knowing the characteristic function of S is exactly the same as knowing the set S. The Fourier transform of the characteristic function, T(z) = xcz) =

L

.:cEIF�{

L

xcx) gz x =

cES

e·c,

has a natural geometric interpretation as follows. Notice first that T(O) is sim­ ply the size of the set S. Next, let z be a nonzero vector, and consider the three parallel hyperplanes H0, H1 , H2 normal to z, where Hi = {x E

z · x = j} .

IF�,

To each nonzero vector z we associate an (ordered) hyperplane triple (ho, h 1 , hz) = cs n Ho . .

's n

H1 � .

's n

Hz ').

Proposition 7. The complex number T(z) encodes the same data as the or­ dered hyperplane triple (h0, h 1 , h2) as­ sociated to z. In particular,

and

Fourier bound of Proposition 6. We re­ fer the reader to their paper for more details. We now summarize the proof of Edel, Ferret, Landjev, and Storme [6] that a5 = 45. Proposition 9. A maximal 5-cap has 45 points.

Proof Figure 8 contains a 5-cap with 45 points, so we only need to show that there is none with 46 points. Suppose for a contradiction that C is a 5-cap with 46 points. By the Fourier analysis bound of Proposition 6, if every hyperplane inter­ sects C in at most 18 points, then C can have at most 45 points. Thus, there must be a hyperplane H intersecting C in 19 or 20 points. Deleting a point of C not on H produces a 5-cap with 45 points such that H is a hyperplane intersecting in 19 or 20 points. However, in [6] it is shown that every 5-cap with 45 points has no hyperplanes intersecting in 19 or 20 points. The proof exploits an inge­ nious identity in the equations for count­ ing marked hyperplanes, together with an exhaustive computer search. D

some low d there is a d-cap with high solidity, but for all larger d every d-cap has a substantially smaller solidity. This would make the name "asymp­ totic solidity" rather questionable, but the following proposition shows that this never happens. Proposition 10. Asymptotic solidity is the limit as d ____. x of the solidity of max1·ma1 d-caps.

Proof This would be a very short proof if we knew that the solidity of maximal caps ad was an increasing sequence. Unfortunately, this is not known, so we will take a more roundabout approach. We first note that given a d-cap C, we can construct a 2d-cap C' with the same solidity. We do this by taking the product of C with itself: each point of C' has as its first d coordinates a point of C, and as its last d coordinates an­ other point of C, so C' , = C1 2 . Then a(C') = = a(C). In fact we can also apply this construction to take the product of a drcap C1 and a d2-cap C2 to get a (d1 + d2)-cap C' with solidity

\fiCi2

a(C') "'-V1C1 1 Cz1• For example, tak­ ing the repeated product of a d-cap with itself n times gives an (nd)-cap with the same solidity. So we can re­ place sup by lim sup in the definition of asymptotic solidity, justifying the "asymptotic" in the name. We now note that this product construction shows that the function f : N ____. N defined by settingj(d) to be the size ad of a maximal d-cap satisfies =

Solidity In this section we discuss what is known

1

h1 =

3 CP

hz =

3 CP

1

1

-

u) +

v3

-

u)

v3

-

1

v, v,

where T(z) = u + iv and p = T(O) is the size of S. We call T the (ordered) hyperplane triple junction of S. In the previous section our interest in hyperplane triples was ad hoc; we studied them be­ cause, in the end, it paid to do so. We now see that hyperplane triples arise naturally via the Fourier transform. There is an amazing formula count­ ing the number of lines contained in a set S. Proposition 8. Let S be a subset of IF� that contains p points and l lines. Then P +

6l =

L

1 T3(z), 3 d zE IF:{

where T is the hyperplane triple junc­ tion of S. In [ 1 ] , Bierbrauer and Edel use the formula above together with some clever estimates of I T3(z)1 to prove the

38

THE MATHEMATICAL INTELLIGENCER

about high-dimensional maxinlal caps. In [3], A Robert Calderbank and Peter Fishburn create very large high-dimen­ sional caps via product constructions based on large low-dimensional caps. As a measure of the "largeness" of a cap, define the solidity of a d-cap C to be a(C) =

\IC,

and define the asymptotic solidi ty of maximal caps to be the supremum of the solidities of maximal caps,

Thus, asymptotic solidity is at least the solidity of any particular cap. Since every d-cap has fewer than 3d points, the asymptotic solidity is at most 3. On the other hand, the cap consisting of all 2d points with all components 0 or 1 shows that the solidity is at least 2. The central open question is the following. Question. Is the asymptotic solid­ ity less than 3?

The definition of asymptotic solid­ ity leaves open the possibility that for

f(m + n) ?. f(m)f(n). Then Fekete's Lemma (see, for exam­ ple [20, Lemma 1 1.6]) implies that lim11 -.x f(n) 11" exists. Since solidity is the lim sup of f(n) 11", this limit must equal a, completing the proof. D Calderbank and Fisbum use this product construction to show a > 2.210 147. In [3], they explicitly give two 6-caps, each with 1 12 points. They ex­ ploit these caps in a refined version of the above product construction to get a 13,500-cap with the required solidity. This result has been improved, with a simpler cap, by Yves Edel, who has constructed a 62-cap with solidity 2.21478 1 , and a 480-cap with solidity 2.21739 [5].

Projective Caps

A basic property of an affine SET game is that each pair of cards is contained in a unique SET. Are there other SET-like card games with this property? Yes! In fact, there is a non-affine SET-like game with only seven cards. Consider the Fano plane in Figure 12. The seven points of the Fano plane are indicated by the dots in the figure. Each line of the Fano plane consists ei­ ther of the three dots lying on a line seg­ ment of the diagram, or the three dots lying on the circle. In the Fano plane, any pair of points determines a unique line, and every line has precisely three points. We define the Fano SET game to be a card game with one card for each point of the Fano plane, where three cards form a SET if the corresponding points of the Fano plane are collinear. There is a natural projective geo­ metric construction of the Fano plane. Given a vector space V= \ the pro­ jective space of V, lflld [Fq, is an object tailored to encode the incidence struc­ ture of linear subspaces of V. In par­ ticular, the elements of the set lflld [Fq are just the one-dimensional subspaces of V. These are called the projective points of [pld[F q· Given two distinct one­ dimensional subspaces, there is a unique two-dimensional subspace con­ taining them. Thus, if we call the two­ dimensional subspaces of V the pro­ jective lines, then we have the nice fact that any two projective points deter­ mine a unique projective line. When q = 2, then each projective line contains exactly three projective points. To see this, notice that since the underlying field is rF2 , any one-dimen­ sional subspace contains exactly two points: the zero vector, and the nonzero basis vector. Thus, there is a bijection between nonzero vectors and projective points. Let L be a two-di­ mensional subspace of [F � + l represent­ ing a projective line, and let {e, j] be a vector space basis of L. Then L con-

rFg+

tains exactly four vectors: 0, e, j, and e + f The nonzero vectors represent the three projective points of L. Amaz­ ingly, this gives rise to the same test for collinearity as in the affine case: The Projective Collinearity Rule: Three non-zero vectors a, b, c E [F�+ 1 represent collinear projective points if and only if a + b + c= 0. The vector space [F� has eight vec­ tors, and so the projective space IP2 rF2 has seven projective points. In fact, IP2rF2 is the Fano plane. We define a projective SET game of dimension d to be a card game with one card for each projective point in lflld fF2, where three cards form a SET if the corresponding projective points are collinear. Then the Fano SET game is just the projec­ tive SET game of dimension two. H. Tracy Hall [8] has devised for himself a deck of cards for a playable projective SET game of dimension five. The key step of his construction is to group the components of a vector of [F � into three pairs,

a = (x1 , x2, y " Y2, z 1 , z2) = ((x 1x2), (YlY2), (z 1 z2)) , and then to interpret each pair as the binary expansion of one of the integers 0 , 1 , 2 , or 3. For example,

a = ( 1 , 0, 0, 1, 0, 0) ((10), (01), (00)) = (2, 1 , 0). =

To further encode the vector as a design on a card, we associate a symbol to each of the three integers 1, 2, or 3 , and use the blank symbol for the integer 0. We print three such symbols on each card, one for each coordinate, and distinguish the symbols by printing them in three different fonts. In Figure 13 we do this using different families of symbols for the different coordinates: { �. •, * ] , { <, = , > ], and {Z', IR, Q}. Hall has a much cuter way to do this using the charac­ ters from a popular children's game. With respect to this method of en­ coding vectors, the projective collinear­ ity rule has the following translation. The Projective SET Rule: Three cards are called a SET, if each font appears in exactly one of the fol­ lowing three ways:

Figure 12. The Fano plane.

•

Not at all.

Figure 1 3. Can you find all four projective SETs? •

•

As

the same symbol exactly twice, and not as any other symbol. As all three symbols.

Hall reports teaching, and then losing at, this game to his nine-year-old niece. Are there still other SET-like games be­ yond affine and projective SET? Yes and no. A Steiner triple system is a set X to­ gether with a collection S of three-ele­ ment subsets of X, such that, given any two elements x, y E X, there is a unique triple {x, y, z ] E S. Interpreting elements of X as cards, and triples in S as SETs, then we obtain a SET-like game from any Steiner triple system. The affine and pro­ jective SET card games are examples of Steiner triple systems, but it turns out that there are many more exotic Steiner triple systems. Their study is a very rich subject, and the interested reader should look at the book of Charles Colbourn and Alexander Rosa [4]. A natural invariant attached to any Steiner triple system is its symmetry group. This is defmed in precisely the same way we defined symmetry groups of affine SET games, namely, as the per­ mutations of the points of X taking triples in S to triples in S. One way of studying Steiner triple systems is via their symmetry groups. A notable prop­ erty of the symmetry groups of affme and projective SET games is that their symmetry groups are 2-transitive on cards; that is, there is a symmetry tak­ ing any ordered pair of cards to any other ordered pair of cards. In particu­ lar, this means that, up to symmetry, there is only one type of SET. To cap­ ture this, let us defme an abstract SET game to be any Steiner triple system where the symmetry group acts 2-tran­ sitively on points. We have the follow­ ing deep theorem classifying abstract SET games, first conjectured in 1960 by Marshall Hall, Jr. [9].

VOLUME 2 5 , NUMBER 3 , 2003

39

Theorem 1 1 . The only abstract SET games are affine and projective SET games, in IF� and [P>diF2, respectively.

This result is due to Jennifer Key and Ernest Shult [ 16], Hall [ 10], and William Kantor [ 15]. Interestingly, the proofs use part of the classification of finite simple groups. If we actually play projective SET we want to lmow how many cards need to be dealt to guarantee a SET. Just as in the affin e case, we call a collection of points in [P>diF2 containing no three points on a projective line a cap. The problem of fmding maximal caps for projective SET was solved in 1947 by Raj Chandra Bose. In [2], he showed that the maximal caps d of [P>diF2 have 2 points. Bose's interest in this problem certainly didn't stem from SET, as the game was not to be invented for another 27 years. Rather, he was coming at it from quite another direc­ tion, namely, the theory of error-cor­ recting codes, which is the study of the flawless transmission of messages over noisy communication lines. As detailed in the book of Raymond Hill [ 14], there is a correspondence between projective caps in [P>diF2 and families of efficient codes. Specifically, if we form the ma­ trix whose columns are vectors repre­ senting the projective points of the cap, then the kernel of this matrix is a linear code with Hamming weight four. The more points the cap contains, the more "code-words" the corresponding code has, and so this naturally motivates the problem of finding maximal projective caps. Bose completely solved this prob­ lem when q = 2, but, as in the affine case, things become much more difficult when q 3. We denote by bd the size of a maximal projective cap in [P>diF3. The lmown values of bd are given in Table 3. =

bd

2

Acknowledgments

We have been fascinated with SET since first spending many many hours playing it when we started graduate school to­ gether, and have had many helpful con­ versations with other SET-enthusiasts. We would never have spent so many hours thinking about the "sET-Problem" without Joe Buhler's goading remark that it was shocking that two Berkeley students couldn't work out the answer. Josh Levenberg helped then by writing our original brute-force program, before we came up with a non-computer proof of "20." Galen Huntington got us think­ ing along the right lines with the proof of Proposition 1. Thanks also to Michael Kleber for many interesting conversa­ tions on this topic, only a fraction of which have occurred during the writing of this article, and for informing us of Proposition 10. More recently Mike Zabrocki's computer demonstration at FPSAC 2001 reinspired us, and Sunny Fawcett has been invaluable with inte­ ger programming assistance and other ideas. Finally, we are indebted to Joe Buhler, Juergen Bierbrauer, and Yves Edel for extensive comments on an ear­ lier draft of this paper. REFERENCES

[1] J. Bierbrauer and Y. Edel. Bounds on affine

Table 3 d

game with cards given by points of IP'51F3, there is still some interesting SET­ theory associated with the study of maximal projective caps in this space. In particular, the 45-point affine cap in Figure 8 was constructed by deleting a hyperplane from the 56-point projec­ tive 5-cap given by Hill in Figure 4 of [ 13]. Uniqueness of this affine cap was shown in [6] to be a consequence of the uniqueness of the projective cap, which in tum was demonstrated by Hill in [ 12] by means of a code-theoretic argument.

2

3

4

5

6

4

10

20

56

?

caps. J. Combin. Des . , 1 0(2): 1 1 1-1 1 5, 2002. [2] R. C. Bose. Mathematical theory of the

[5] Y. Edel. Extensions of generalized product caps. Preprint available from http://www. mathi. uni-heidel berg. de/ �yves/. [6] Y. Edel, S. Ferret, I. Landjev, and L. Storme. The classification of the largest caps in AG(5, 3). Journal of Combinator­ ial Theory, Series A , 99:95-1 1 0, 2002.

[7] W. Fulton and J. Harris. Representation theory, A first course. Readings in Mathe­

matics. Springer-Verlag, New York, 1 99 1 . [8] H. T. Hall. Personal communication. [9] M. Hall, Jr. Automorphisms of Steiner triple systems. IBM J. Res. Develop , 4 :460-472, 1 960. [ 1 0] M. Hall, Jr. Steiner triple systems with a dou­ bly transitive automorphism group. J. Com­ bin. Theory Ser. A, 38{2) : 1 92-202, 1 985.

[1 1 ] R. Hill. On the largest size of cap in 55,3. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 54:378-384 (1 974), 1 973.

[1 2] R. Hill. Caps and codes. Discrete Math . , 22(2) : 1 1 1 -1 37, 1 978. [13] R. Hill. On Pellegrino's 20-caps in 54,3. In Combinatorics '8 1 (Rome, 198 1) , pages

433-447. North-Holland, Amsterdam, 1 983. [1 4] R. Hill. A first course in coding theory. The Clarendon Press Oxford University Press, New York, 1986. [1 5] W. M . Kantor. Homogeneous designs and geometric lattices. J. Combin. Theory Ser. A , 38( 1 ):66-74, 1985.

[1 6] J. D. Key and E. E. Shult. Steiner triple sys­ tems with doubly transitive automorphism groups: a corollary to the classification the­ orem for finite simple groups. J. Combin. Theory Ser. A , 36(1 ) : 1 05-110, 1 984.

[ 1 7] D. Knuth. Programs setset, setset-all, set­ set-random. Available from http://sunburn. stanford.edu/�knuth/programs.html. [1 8] Roy Meshulam. On subsets of finite abelian groups with no 3-term arithmetic progressions. J. Combin. Theory Ser. A, 71 ( 1 ) : 168-172, 1995. [1 9] G. Pellegrino. Sui massimo ordine delle calotte in 54,3. Matematiche (Catania) , 25:149-157 (1 97 1 ) , 1970. [20] J. H. van Lint and R. M. Wilson. A course in combinatorics . Cambridge University

Press, Cambridge, second edition, 2001.

symmetrical factorial design. Sankhya, 8:

The sizes in dimensions 2 and 3 are due to Bose [2], dimension 4 is due to Pel­ legrino [19], and dimension 5 is due to Hill [ 1 1, 12]. We note that we always have a :S b , since there is a copy of d d IF� inside [P>d1F3, so Pellegrino's result is the first proof that a4 :S 20. Even though there is no abstract SET

40

THE MATHEMATICAL INTELLIGENCER

107- 1 66, 1 947. [3] A R. Calderbank and P. C. Fishburn. Max­

Department of Mathematics, and Computer Science, Saint Mary's College, Moraga, CA

imal three-independent subsets of {0, 1 , 2}n

94575, USA

Des. Codes Cryptogr. , 4(3):203-21 1, 1994.

e-mail address: [email protected]

(4] C . J. Colbourn and A Rosa. Triple sys­ tems. Oxford Mathematical Monographs.

The Clarendon Press Oxford University Press, New York, 1 999.

Department

of

Mathematics,

Stanford

University, Stanford, CA 94305, USA e-mail address: [email protected]

HANS-JOACHIM ALBINUS

Pyth ag o ras ' s Oxe n Revi s ited e t is said that when Pythagoras discovered his famous theorem, in a right-angled

/) (/

triangle the squares of the smaller sides sum up to the square of the hypo teneuse, he sacrificed a hundred oxen to thank the gods.

There is a poem in German describing this sacrifice:

Yom Pythagoraischen Lehrsatze

Die Wahrheit, sie besteht in Ewigkeit, Wenn erst die blade Welt ihr Licht erkannt; Der Lehrsatz nach Pythagoras benannt Gilt heute, wie er galt zu seiner Zeit. Ein Opfer hat Pythagoras geweiht Den Gotten, die den Lichtstrahl ihm gesandt Es taten kund geschlachtet und verbrannt Ein hundert Ochsen seine Dankbarkeit. Die Ochsen, seit dem Tage, wenn sie wittern DaB eine neue Wahrheit sich enthtille Erheben ein unmenschliches Gebrtille; Pythagoras erftillt sie mit Entsetzen; Und machtlos sich dem Licht zu widersetzen VerschlieBen sie die Augen und erzittern. In the German vernacular there is a double meaning to the word Ochse, which constitutes the poem's essential irony:

1. an ox 2. a blockhead. 1 He

Readers unfamiliar with German perhaps are aware of one of the two known English translations, one by the American biophysicist and biologist Max Delbrtick (19061981) , 1 the other by Robert Edouard Moritz (1868-1940), an American mathematician, who had studied in Gottingen: Delbrtick's translation can be found in [7] , p. 53; Moritz's, in his book On Mathematics and Mathematicians2 ([6] , p. 309). Both translations are weak in various respects. Moritz's, for example, has lost the structure of rhyme in the first two verses, but it is in words and rhythm closer to the German text than Delbrtick's. Of course, there is no adequate trans­ lation for Ochsen covering both senses. It is often believed (see for example remarks by Olga Taussky ([7] , p. 53) or Mark Kac ([5] , p. 98-99) that the poem is by the German writer Heinrich Heine. But this is definitely not true. The author of the sonnet is Adelbert von Chamisso (30.01 . 1781-2 1.08. 1 838) , French by birth, whose family fled to Prussia during the French Revolution. There he grew up to become an officer in the Prussian army and later joined a circumnavigation of the globe as a natural scientist. Af­ ter his return to Germany he worked at the botanical gar­ dens in Berlin, where he became head of the herbarium. From his literary writings the most famous perhaps is the novel Peter Schlemihls wundersame Geschichte (pub­ lished 1814) about the fate of a man who sells his shadow.

was the son of the German historian Hans Delbruck. In 1 969 Max Delbruck (together w1th Salvador Edward Luria and Alfred Day Hershey) received the Nobel Prize

for Physiology and Medicine for his bas1c research in molecular biology and bacteriological genetics. 2Prev1ous editions titled Memorabilia Mathematlca or the Philomath's Quotation-Book.

© 2003 SPRINGER-VERLAG NEW YORK. VOLUME 25, NUMBER 3, 2003

41

Delbriick's translation

Moritz's translation

The truth: her hallmark is eternity soon as stupid world has seen her light Pythagoras' theorem today is just as right As when it first was shown to the fraternity.

Truth lasts throughout eternity, When once the stupid world its light discerns: The theorem, coupled with Pythagoras' name, Holds true today, as't did in olden times.

The Gods who sent to him his ray of light To them Pythagoras a token sacrificed: One hundred oxen, roasted, cut, and sliced Expressed his thanks to them, to their delight.

A splendid sacrifice Pythagoras brought The gods, who blessed him with this ray divine; A great burnt offering of a hundred kine, Proclaimed afar the sage's gratitude.

The oxen, since that day, when they surmise That a new truth may be unveiling Forthwith burst forth in fiendish railing.

Now since that day, all cattle [blockheads] when they scent New truth about to see the light of day, In frightful bellowings manifest their dismay;

Pythagoras forever gives them jittersQuite powerless to stem the thrust of such emitters Of light, they tremble and they close their eyes.

Pythagoras fills them all with terror; And powerless to shut out light by error, In sheer despair they shut their eyes and tremble.

As

The sonnet Vom Pythagordischen Lehrsatze was writ­ ten in August 1835 and first published in 1836 ([3] , p. 765). Cited above is the version found in a modern critical edition of Chamisso's complete works (for example [3] , p. 536; there are identical editions by Carl Hanser Verlag, Mi.inchen, 1982, and Insel-Verlag, Leipzig, 1980); see [3], p. 679-682, for remarks concerning the punctuation and or­ thography, 3 which was left as close as possible to the orig­ inal text. By the way, the source from which Chamisso was in­ spired to write the poem is believed to be an aphorism (no. 268, to be exact) by Ludwig Borne ( 1 786-- 1 837), who had already made use of the two meanings of Ochsen ( [3], p. 765):

Als Pythagoras seinen bekannten Lehrsatz entdeckte, brachte er den Gottern eine Hekatombe dar. Seitdem zit­ tern die Ochsen, sooft eine neue Wahrheit an das L icht kommt. These sentences can be found in Borne's complete works4 (for example [ 2 ] , p. 3 18); and again Moritz gives an English translation ([6] , p. 308):

After Pythagoras discovered his fundamental theorem he sacrificed a hecatomb of oxen. Since that time all dunces tremble whenever a new truth is discovered. Here he decided to use the second meaning of Ochsen. Now, how has the belief arisen that the poem Vom Pythagordischen Lehrsatze was written by Heinrich Heine

(1 797-1856), as stated in the editor's note5 to Taussky's ar­ ticle ([7], p. 53) and in Mark Kac's (1914-1984) autobiog­ raphy Enigmas of Chance ([5], p. 98-99), too? One hint could be Borne, who was at first a close friend of Heine and later, in the days of their exile in Paris, became his in­ timate enemy (the so-called Heine-Borne controversy). Could this be the reason that Taussky and Kac and others remembered Heine's name instead of Borne's? Heine was more popular than Chamisso or Borne. And very likely they all remembered a passage in a text Heine wrote during a visit to the Frisian island of Norderney (see for example (4], p. 71), where he mentions Pythagoras and his hundred oxen in the context of the transmigration of souls (both human and animal) and possible funny inci­ dents caused thereby:

Wer weijS! wer weijS! die Seele des Pythagoras ist viel­ leicht in einen armen Candidaten gefahren, der durch das Examen fdllt, weil er den pythagordischen Lehrsatz nicht beweisen konnte, wdhrend in seinen Herren Ex­ aminatoren die Seelen jener Ochsen wohnen, die einst Pythagoras, aus Freude iiber die Entdeckung seines Satzes, den ewigen Gottern geopfert hatte. A translation may read as follows:

Who knows! Who knows! Perhaps Pythagoras 's soul is now in a poor candidate who fails his examination because he is unable to prove the Pythagorean theorem, while in his examiners live the souls of those oxen which once, in the joy of the discovery of his them·em, Pythagoras had sacrificed to the eternal gods.

3A more modern spelling of the poem can be found in [1]. p. 72, of which Moritz was aware. 4Be aware that some other editions of Borne's works use a different numbering of the aphorisms and miscellanea. 5The editor's question in The Mathematica/ lntelligencer for a hint to the poem's authorship or to a reference went unanswered at the time.

42

THE MATHEMATICAL INTELLIGENCER

Thus, Pythagoras appears in the works of at least three prominent contemporary poets of the early 19th century: Ludwig Borne, Adelbert von Chamisso, and Heinrich Heine! REFERENCES

[1] Ahrens, W . : Scherz und Ernst in der Mathematik. Geflugelte und ungef/Qgelte Worte. Teubner, Leipzig, 1904.

[2] Borne, Ludwig: Sarntliche Schriften , vol. 2. Melzer, Dusseldorf, 1964. [3] Charnisso, Adelbert von: Sarntliche Werke in zwei Banden , vol. 1: Gedichte- Oramatisches. Wissenschaftliche Buchgesellschaft, Darm­

stadt, 1982.

Cauchy Product of Series MAURICE MACHOVER Abel, Mertens, Cauchy, Hardy Proved some theorems very arty: Sum up A,., sum up Bn,

[4] Heine, Heinrich: Sakularausgabe. Werke- Briefwechsei- Lebens­ zeugnisse, vo l . 5: Reisebilder / 1 824- 1 828. Akademie, Berlin, 1970.

[5] Kac, Mark: Enigmas of Chance. An Autobiography. Harper & Row,

Take the Cauchy product Cn. If the fll'St two sums converge, Will the third to limit surge'?

New York, 1985. [6] Moritz, Robert Edouard: On Mathematics and Mathematicians. Dover, New York, 1958.

Let us these four men consult.

Abel proved the first result:

[7] Taussky, Olga: From Pythagoras' Theorem Via Sums of Squares to Celestial Mechanics. The Mathematical lntelligencer, 10 (1988) , no. 1' p. 52-55.

Let the Cn sum to C,

An to A, Bn to B; Abel showed accordingly That A times B will equal C. But we would like to sum Cn

ot in the "IF" but in the "THE

A U THO R

"

Along comes Mertens with a beaut About convergence absolute:

If absolu tely sums up one

(Each still converging to its sum),

Convergence of Cn comes free, And A times B still equals C.

And furthermore great Cauchy says, Two absolute sum-ups more pays, For then the summing up to C Will also go absolutely.

HANS-JOACHIM ALBINUS

lnnenministerium Baden-WOrttemberg

And last comes Hardy with his gift.

Dorotheenstrasse 6

This time we have a little shift:

D-701 73 Stuttgart

We still assume the first two sum,

Germany

But as to Cn, we

e-mail: [email protected]

are

mum;

Instead, assume that bounds we ken Hans-Joachim Albinus ought perhaps to have been named for his famous ancestor Decimus Clodius Albinus, claimant to the Roman imperial throne until he was killed in 1 97

A.D.

by

his rival Septimus Severus. H.-J. Albinus did a master's degree in mathematics and

On nAn and nBn. With these assumptions, Hardy says, Convergence of the Cn stays; It needn't go absolutely,

But A times B still equals C.

geography at Ruhr-Universitat Bochum, then worl<ed in com­ puting centres. Progressively doing more and more manage­ ment and less and less mathematics, he became deputy head of electronic communication and management science in the provincial lnnenministerium. His interest in history of mathematics and computing sur­

Department of Mathematics St. John's University Jamaica,

NY 1 1 439 USA

e-mail: cherokeezip @ webtv .net

vived. He also loves contemporary arts, fashion design, haute cuisine, and motorboats. He lives in Leonberg, one of the "Kepler towns" he invited us to in his Mathematical Tourist ar­

ticle, vol . 24, no. 3, 50-58.

VOLUME 25, NUMBER 3 . 2003

43

ip,i$•VMj.l§rr@ih$ili.IIIQt11

Regiomontanus Ottomar Gotz

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? .(f 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]

44

THE MATHEMATICAL INTELLIGENCER

© 2003

''

D i rk H uylebro u c k ,

K

Ed itor

onigsberg in Bavaria" is a picturesque market town situated in southern Germany about 100 kilo­ meters northwest of Nuremberg. Whilst Konigsberg was a busy trading centre on an important trade route during the Middle Ages, today the town appears as a tranquil place with a well­ preserved medieval town-centre amidst the lovely hills of the "HaBberge." It is worth visiting for its characteristic atmosphere with timbered medieval houses still in use, a city wall fortified by towers, and the ruins of an 800-year­ old castle founded by the Emperor Bar­ barossa. It was in this lovely town on June 6th in 1436 that the most famous as­ tronomer and mathematician of the 1 5th century, Johannes Muller-in la­ tinized form called Regiomontanus­ was born. Living at the end of the Mid­ dle Ages, he was embedded in the traditional thought of a geocentric cos­ mology based on the authority of Aris­ totle's Metaphysics and the conviction of mankind's unique central place in the cosmos-in agreement with the Holy Scripture. Regiomontanus died in Rome at the age of only 40. There he had been invited to the Court of Pope Si�:tus IV for the purposes of a calen­ dar reform. Copernicus, who was to revolutionize cosmology, was then just 3 years old. Regiomontanus was a versatile ge­ nius. His scientific life spans ancient and medieval mathematics completely, including the sophisticated methods of Ptolemy in modeling the geocentric system. In addition to his theoretical abilities, he was gifted with a distinct practical sense. Nobody of the 1 5th century exploited the possibilities of book printing in science more effec­ tively than Regiomontanus. He had the gigantic ambition to pub­ lish the works of ancient mathematics and astronomy as well as his own rich work He also designed astronomical instruments. His most significant work in mathematics is a geometry of plane

SPRINGER-VERLAG NEW YORK

I

and spherical triangles supplemented by a table of sines to 7 decimal places, the "De Triangulis omnimodis," which must be regarded as the first textbook of trigonometry ever. In this work Re­ giomontanus introduced for the first time in the Western world the theorems of sine and cosine for spherical trian­ gles, setting a standard in trigonometry for centuries. Another work of his came to have extraordinary significance and wide use. It was an improved and essentially revised edition of the monumental compendium of ancient astronomy, the Almagest of Ptolemy. In his Epitoma in Almagestum Regiomontanus added new data and advanced alternative mathematical methods, which un­ doubtedly gave Copernicus's studies further impetus. The Epitoma became the most important astronomical text published in the 1 5th century. It was used as a textbook as well as a manual in practical astronomy for more than 100 years. Most of his works resulted from commissions he received from various eminent persons. Emperor Friedrich III requested a birth reading for his bride-at that time an honorable and serious request. For the King of Hun­ gary and his chancellor, the archbishop of Esztergom, he developed tables for the positions of planets and stars with an appendix containing numerical ta­ bles of sine and tangent. It is known that Columbus used Regiomontanus's tables on his famous voyages of dis­ covery. His most creative period, how­ ever, appears to have been the time when he was in the service of Cardinal Bessarion. He followed the Cardinal to Rome after his studies at the university of Vienna, and was a scientific advisor at his court for some years. At that time he finished the successful astronomi­ cal work Epitoma and cultivated a rich correspondence, from which a collec­ tion of 400 mathematical problems has been preserved. He gave lectures at the University of Padova. Last but not

The house where Regiomontanus was born, located on the "salt-mar­

A monument in the marketplace of Konigsberg includes a statue of

ketplace" in Konigsberg in Bavaria (photographs by Ottomar Gotz}.

Regiomontanus. In the background the town hall.

Title page and last page of his sine table (from Stadtarchiv Schweinfurt}.

VOLUME 25, NUMBER 3. 2003

45

of his great plans short. He was buried in the "Gottesacker," which is the Campo Santo Teutonico, the cemetry for German citizens at St. Peter's. Vis­ itors will find this memorial tablet:

•

In memoriam Johannes Miiller genannt Regiomontanus Astronom-Mathematiker­ Wegbereiter des neuen Weltbildes * 6. VI. 1436 zu Konigsberg in Franken + 6. VII. 1476 in Rom

N u re m berg

B udapest

Venice Padova

The m a p shows the main stations in Regiomontanus's life.

least, he rediscovered an ancient man­ uscript of 6 books of Diophantus's fa­ mous Arithmetica in Venice and trans­ lated it from Greek into Latin. In 1471, only 5 years before his tragic death in Rome, Regiomontanus started a career as a scientific pub-

lisher and practical astronomer in Nuremberg. He established a print shop and set up an observatory, whose equipment Regiomontanus described in a letter to the mathematician Chris­ tian Roder of Erfurt. His sudden death cut the realization

Die Biirger seiner Vaterstadt AD. MCMLXXVI

Apart from his birthplace Konigsberg, where there is a statue of him on the marketplace, the house where he was born, and the parish church where he was baptized, the Mathematical Tourist can fmd places outside of Germany where he is memoralized. The above­ mentioned tablet in the Vatican's Campo Santo Teutonico, a memorial in the castle of Budapest, and a memorial tablet on the wall of the one-time castle of the archbishop of Esztergom. Finally, there is yet another memorial that can be viewed periodically by anybody-a moon crater named Regiomontanus. And everyone can find it easily by look­ ing at the site http://www.lunarrepub­ lic.com/atlas/sections/f4.shtml. Birkenstrasse 40 D-97422 Schweinfurt Germany e-mail: [email protected]

MOVING?

We need your new address so that you do not miss any issues of THE MATHEMATICAL INTELLIGENCER. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., Journal Fulfillment Services P.O.

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46

THE MATHEMATICAL INTELLIGENCER

The

M a the m atic a l

Tourist

The Christoffel Plaque in M onschau Helena Alexandra van der Waall and Robert Willem van der Waall

r rk H u y l e b ro u c k ,

Editor

,.�ile on holiday in Monschau WW (Germany), the authors found

themselves in front of a commemora­ tive plaque marking the 1 50th birthday of the mathematician Elwin Bruno Christoffel (1829-1900). That occasion stimulated them to prepare an over­ view of his work; of some places where he lived, worked, and taught; and of the discovery of some less known but very conclusive sources of information on that score. Introduction

The mathematics of Elwin Bruno Christoffel (born 1829, Montjoie; died 1900, StraBburg) is well known. Many people are aware of the Christoffel­ Darboux-Einstein summation formu­ lae, of Christoffel symbols, or of the Christoffel-Schwarz mapping theorem. Besides some unpublished lecture notes, Christoffel left thirty-five printed papers dealing with subjects like function the­ ory including conformal mapping, prop­ agation of electricity, Gaussian quad­ rature, continued fractions, dispersion of light, movements of points in peri­ ods, continuity conditions for differen­ tial equations, minimal surfaces, the­ ory of invariants, geodetic triangles, geometry and tensor analysis, orthog-

The memorial.

I

onal polynomials, shock waves, poten­ tial theory, Riemann integrals, Jacobi's theta-sequences, irrational numbers­ altogether, a beautiful collection of 19th-century mathematics; here one could consult his Collected Papers [8). There also exist retrospective sur­ veys on the mathematical works of Christoffel. To start with, see [9) and [12), about a century old. Additional material of a much later date about Christoffel's geometry can be found in [ 1 1 ) ; the "Riemann example" of a con­ tinuous non-differentiable function is scrutinized in depth in [4). Some mod­ em, very informative contributions are provided in [ 13] and [ 14). At first sight, less seems to be known about Christoffel as a person acting in his society and milieu, or about his social, mathematical, and intellec­ tual acquaintances in Berlin, ZUrich, and StraBburg (the places where he gave his lectures). However, as we will see in the next section, there are some illuminating sources.

The only known photograph of Christoffel.

© 2003 SPRINGER-VERLAG NEW YORK, VOLUME 25, NUMBER 3. 2003

47

The (German) caption in [6] to this photo-

View from dormer window of the building at the RurstraBe 1 , in 1 889, at the place where

graph indicates, on physiognomical grounds,

Christoffel was born.

that it might be another picture of Christoffel, an at older age.

I.

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View of the house.

48

THE MATHEMATICAL INTELLIGENCER

JmL. ••

REROU.'\1

G l" I T A V I

S C H £ 11 1<.

Title page of E.B. Christoffel's doctoral thesis.

[11 De motu permanent1 e!ectnc1tatls 1n corpor1bus homogene1s Inaugural­ Dissertation zu Berlin, 1856, 64 Se1ten.

[ I I ] Ober die Gau6ische Quadratur und eine Verat1geme1nerung derselben. J

Re1neAngew Math. 55 ( t 858), 6 1 -82 [ I l l ] Ober d1e lineare Abhl!ng1gkelt von

nane e Ia rappresentaz1one di una data supert1c1e. Ann Mat Pura Appl. Sene I I , 1 ( 1 867) . 89- 1 03. [XIII] Ober einige allgeme1ne E1genschaften der MintmumsfiB.chen. J Re1ne Angew

chen. J. Reine Angew. Math. 55 ( 1 858), 281 -299.

variantentheone.

von Kettenbri..ichen« 1m 57 Band des

Reme

[XVII]

Math

1 1 9- 1 76. lJber die Transformation ganzer homo­

( 1 86 1 1 . 906-921 und 997-999 Ann

Kgl preuss Akad W1ss Berlin ( 1 869),

des Herrn Weierstra6. J. Reme Angew

[XVIII] Uber dO! TransformatiOn der homoge­ nen D1HerenliaJausdn.icKe zwetten Gra­

[VII] Ober d1e kle1nen Schw1ngungen emes penod1tch e1ngenchteten Sy<;tems ma­

teneller P unkte J. Re1ne Argew Math.

S3 ( 1 854), 273-268

1 -6.

des J Reme Angew Math. 70 ( 1869),

e1nfach zusammenh8.ngenden, ebenen

[X] Zur Theone der e1nwerthu�en Potentla­

Flache auf e1nem Kre1se. Nachr. Kgl.

Appl Sene I I , 10 ( 1 883), 8 1 - 100

Uber den E1nflu6 von Real1tats- und Stet1gke1tSbedingungen auf d1e Losung

Fl!iche auf e1nem Kreis. Nachr. Kgl

Pura

D1fferent1algletchungen.

Nachr

Appl

Sene

II,

( 1 687).

Ges

Zuncn

Ober d1e Vollwer1h1gke1t und d1e Stetlg· ke1t

[XXXIV]

15

D1e Convergenz der Jacob1'scnen Tne­ ta-Relhe mtt den Moduln R1emanns VlerteiJSChr Naturtorsch 4 1 ,2 ( 1 896), 3-6

[XXXI II]

analyt1scner

Ausdrucke.

Ann 53 ( 1 900) . 465-492.

Math

Vollst!nd1ge Theone der R,emann­ schen Theta�Funct1on. Math Ann. 5-4 ( 1 90 1 ) , 347-399

Ges. Wiss. Gott1ngen ( 1 870), 359-369. [XXIII] Uber die I ntegration von zwet partiellen

Math. Ann. 19 ( 1 682), 280-290

253-276 [XXXII]

Ges Wiss Goningen ( 1 870), 283-298. [XXII] l.iber die Abbildung e1ner n- blanngen, e1nfach zusammenhangenden, ebenen

[XII] S u i problema delle temperature staz1o-

der Anzahl der linear unabhlngtgen ln­ tegrale erster Gattung. Ann Mat. Purl\

Angew Math 70 ( 1 869), 241 -245.

( 1 87 1 ), 1 -9. [XXI] Uber die Abb•ldL·ng e1ner e1nbl8nngen.

gewOhnl1cher Differentialglelchungen. J Re1ne Angew Math . 66 ( 1 866), 1 - 1 4 .

Ann. Mat. Pura Appl Ser�e I I , i ( 1 879).

Lehrsatte uber anthmet1sche e,gen­ schaften der lrrat,onalzatllen. Ann Mat

des betreffendes Theorem. J . Re1ne

Ober d1e DISpers,on des L1chtes. Ann.

Reme Angew Math. 54 ( 1 865).

mann'schen lntegrale erster Ganung

Bemerkungen zur tnvanantenttleor'e

Phys1k u Chem1e. 1 24 ( t 665). 53-60. le. J

[XXIX]

240-30 1 . AlgebraiScher Bewe1s des Sattes von

[XXXI]

let. Ann. Mat. Pura Appl. Sene I I , 4

3 2 1 -368

[XXVIII]

ner Different1alausdrUcke zwe•ten Gra­

[XX] Sopra un problema proposto da Dlnch­

[XI]

Pura Appl. Sene I I . 8 ( 1877),

1 93-243 Llber die canon,sche Form der Ale­

[XXX]

Messungen auf derselben. J Re1ne An­

gew. Mam 54 ( 1 865). 1 93-209

Mat

46-70. [XIX] Uber e1n d1e Transformatton homoge­

[Yilt] l.iber d1e Best1mmung der Gestalt e1ner krummen O�rfllcl'le durcl'l lokale

[IX]

( 1 877), 8 1 - 1 1 2 Llber d1e For1pflanzung von St6Ben

durch elastiSChe teste KOrper. Ann

Phys. Tech. ( 1 868),

gener 01fferent1alausdrUcke. Monatsb

Math 82 ( 1 864), 255-272.

gle•chungen vertrlgi1Chen Unstetlgkel­

[XVI] Allgeme1ne Theorie der geodatischen Dre1ecke. Abh. Deutsch. Akad. Wiss. Berlin Kl

Untersuchungen uber d1e 1111t dem Fort­ bestehen 11nearer. partieller 0Jfferentlal­ ten. Ann. Mat. Pura Appl. Sene I I , I,

[XXV I I ]

253-272.

natsb. Kgl. preuss. Akad. Wiss. Berlin Pl'lys1 k und Chem"' 1 1 7 ( 1 8621. 27-45. [VI] Verallgeme1nerung em1ger Theoreme

[XXVI]

Angew.

Math 68 ( 1868), 246-252.

Mathematik. J Re1ne Angew Math 58 ( 1 86 1 ), 90-92. [V ] l.iber die D1spersion des Lichts Mo­

J

{XV] Theone der b1lmearen Funct1onen J. Reme Angew. Math. 68 ( 1 868),

ler und Nenner der NB.herungswerthe Journals fUr die re1ne und angewandte

Mat. PuraAppl Sene i i . 8 ( 1 B77), 1 - 1 0

Math. 67 ( 1 867), 2 1 8-228.

[XIV] Bewe1s des Fundamentalsatzes der In·

Functionen einer einzigen Verttnderll­

[IV] Zur Abhandlung von He1ne· ..uber Zah­

Ges W1ss GOft1ngen ( 1 B71 ) , •35-•53 Observat10 Anttl�t!CI Ann. Mat Pura Appl Sene I I , 8 ( 1 875), t
[XXIV]

[XXXV] Querschn ,nstheorO!.

Kgl.

Math.

Ann

55

( 1 902), 497-5 1 5

The complete list of the printed mathematical publications of E.B. Christoffel.

A Memorial and a Hidden Source

In the neighbourhood of the so-called Drielandenpunt (this is Dutch for the point where three countries meet, the countries being The Netherlands, Bel­ gium, Germany) is a colourful and pic­ turesque city called Monschau (until 1918: Montjoie). A plaque on one of the walls of the building at the RurstrafSe 1 carries the following inscription: Dr. Elwin Bruno CHRISTOFFEL Professor der Mathematik in Zurich, Berlin, StrafSburg * 10. 1 1 . 1829 in Monschau + 5.3. 1900 in Stra.Bburg GEBURTSHAUS It shows the location where Christof­ fel was born, though the original house burned down in 1835. The plaque con­ tains a sculpture portrait en profil, too, based on the only known photograph that definitely represents Christoffel; he was then about 40 years of age. The building now houses the Par­ fumerie-Foto-Drogerie "Servaes" on street level. Its proprietor Mrs. Ingrid Hermanns told us that the plaque was unveiled in 1979 in commemoration of Christoffel's 150th birthday. She also

The authors with the memorial plaque.

VOLUME 25, NUMBER 3, 2003

49

tvO�

.

•

_.. .. .

��· _.

.

t"'�'IIJ'

I

0'

Map of the Drielandenpunt, the meeting point of The Netherlands, Belgium and Germany. The city of Monschau i s situated at the lower right

corner of the map. [Adapted from the map of Belgium & Luxemburg, 25th edition, Suurland-Falkplan, Wegener-Suurland, Eindhoven, The Netherlands.}

showed us an invaluable item in con­ nection with all aspects of the life and work of Christoffel: a three-issues-in­ one volume of a journal dealing with the local history of the city of Aachen and its surroundings [6]. That unfamil­ iar source started off our investigations of Christoffel. Its eighty pages are filled with all sorts of profound and important information. It provides photographs of Lejeune-Dirichlet, Geiser, Prym, Rost, Hilbert, Riemann, Einstein, Weier­ strass, Klein, von Laue, Reuleaux, Ricci­ Curbastro (to mention a few), as well as a description of the social environ­ ment of Monschau. Christoffel's ge­ nealogy can be found in it, as well as a history of his life and work, his school

50

THE MATHEMATICAL INTELLIGENCER

years in Cologne, some mathematics, and politics combined with intrigues in Zurich, Berlin, and StraBburg. The vol­ ume also contains a bibliography of all his printed p ublications too. This beau­ tiful and enlightening material is in German and is accompanied with an enormous amount of annotation. Additional Information and Conclusion

An English version of some of the con­ tributions in [6] is presented in [7] , al­ though in a slightly more condensed form. Reference [7] was published in November 1979, on the occasion of a conference on the influences and vari­ ous aspects of the work of Christoffel

in Aachen and Monschau. Much is to be learned from the twelve lectures presented at that conference and from forty-five invited papers; see [7]. Its in­ troduction provides a good impression of the time and circumstances in which Christoffel worked. A non-mathemati­ cal yet handy source, [ 1] contains a short overview of the conference. In addition one discovers from [3] and [5] the story of the lively mathematical ac­ tivity in the region of the Drielanden­ punt during the last 1200 years; math­ ematics in Zurich and Berlin over the ages has been described in [ 10] (see the contribution of G. Frei about Zurich and that of E. Knobloch about Berlin). In conclusion, we fmd that an au-

thoritative overview of the life and work of Christoffel and its impact up to our times has been given in ( 1 ) , [3], [5], [6], [7], and [ 1 1 ) , while his mathematics is presented in the greater part of (7] and (of course) in [8]. Besides that, modem electronic data include [ 14) and, above all, the unsurpassed source [ 13]. Hun­ dreds of papers have been published since 1980 in which Christoffel's math­ ematics play the leading part. Acknowledgments

We are indebted to Mrs. Ingrid Her­ manns and her husband for providing an original issue of [6). Prof. P.L. Butzer gave his permission to use the pho­ tographs in Figures 2 and 3; the pho­ tographer Pejo WeiB gave his approval to reproduce Figure 4. The list of publi­ cations and the title page of E. B. Christoffel's thesis are to be found in [6]; the reproduction of any material from [6) is by courtesy of Mr. Heinz Frenz on behalf of the board of the Heimatblatter des Kreises Aachen. We thank the Heimatblatter for its full cooperation.

Professor der Mathematik in Zurich, Berlin und

Straf3burg - G edenkschrift zur

1 50 .

Wiederkehr des Geburtstages, in: Heimat­

blatter des Kreises Aachen, Jahrgange 34/35 (1 978, 3/4 und 1 97 9 , 1 ) - 80 Seiten

9. C . F . Geiser und L. Maure r - Eiwin Bruno Christoffel, in: Mathematische Annalen , 54 (1 901 ), 329-341 . 1 0 . Jahrbuch 1 994; S.

Ueberblicke

der

Mathematik

D. Chatterji et al. (ed .), Vieweg­

Braunschweig; ISBN 3-526-06578-8; VII +

(in einem Band). 7. E. B. Christoffel- The influence of his Work

on Mathematics and the Physical Sci­

265 pages. 1 1 . K.E.B.

Leichtwe i f3 - Christoffels

Einfluf3

ences ; edited by Paul Leo Butzer und

auf

Franziska Feher (International Christoffel

Sitzungsberichte der Berliner Mathematis­

Symposium in honor of E . B . Christoffel on

chen Gesellschaft, Jahrgange 1 972-1 987

die

Geometrie,

Seiten

93-1 03

in:

the 1 50 Anniversary of his Birth, 1 979,

(1 . 1 0 . 1 97 1 -3 1 . 1 0. 1 987); Berlin 1 987, 1 85

Aachen and Monschau (Germany)); 1 98 1 -

Seiten.

Birkhauser Verlag , Basel-Boston-Stuttgart; ISBN 3-7643- 1 1 62-2; YXI/ + 761 pages.

8. E.B. Chri stoffei - Gesammelte Mathema­ tische Abhandlungen ;

unter

Mitwirkung

von A. Krazer und G. Faber herausgegeben von L. Maurer; zwei Sande, 1 9 1 0 . Teub­

1 2 . W. Windelban d - Zu r Gedachtnis

E.

B.

Christoffel ; i n : Mathematische Annalen 54 (1 901 ), 341 -344. 1 3. http://www-groups.dcs.st-and.ac. ukl -history/Mathematicians/Christoffel.html. 1 4 . http://www .stetson .edu/-efriedma/ periodictable/htmi/C F . html.

ner Verlag, Leipzig und Berlin.

A U T H O R S

REFERENCES

1 . Bericht uber das lnternationale Christoffel­ Symposium

in Aachen

und

Monschau.

8.-1 1 . November 1 979; von P . L. Butzer and F. Feher; in: Berichte zur Wissenschaft­

geschichte 3 (1 980), 1 93-20 1 . 2. P . L . Butzer-An outline of the life and work of E . B . Christoffel ( 1 829-1 900); i n : Histo­

ria Mathematica 8 (1 982), 243-276.

HELENA ALEXANDRA

ROBERT WILLEM

VAN DER WAALL

VAN DER WAALL

School of Mathematics

Korteweg-de Vries

3. P . L . Butzer- Mathematics in the region

University of Sydney

Institute for Mathematics

Aachen-Liege-Maastricht from Carolingian

Sydney, NSW 2006

University of Amsterdam,

times to the 1 9th century; in: Bull. Soc.

Australia

4. PL

1 0 1 8 TV Amsterdam

The Nether1ands

Roy. Liege 5 1 (1 982), 5-30. Butzer and E . L . Stark - ' Riemann's

example'

of a continuous

non-differen­

tiable function in the light of two letters

Helena Alexandra van der Waall studied mathematics at the University of Amsterdam

( 1 865) of Christoffel to Pryrn; i n : Bull. Soc.

up until 1 994. Her dissertation "Lame equations with finite monodromy" was accepted

Math. Beige, Ser. A 38 ( 1 986), 45-73.

at Utrecht, in 2002. Her mathematical interests are algebraic number theory, computer

5. P . L . Butzer- Scholars of the Mathematical

algebra, differential Galois theory; her hobbies include architecture, art, design, skiing,

Sciences in the Aachen-Liege- Maastricht

and travel. She was a researcher at Simon Fraser U ni versity, Burnaby, Canada, and i n

region during the past 1 200 years, an

2003 she went to the University of Sydney, Australia, as a v isit ing scholar.

overview (with the assistance of Helga

Her father, Robert Willem van der Waall, studied mathematics at the University of

P.L.

Leiden up u nt i l 1 966, and subsequently at the University of Mainz and the College de

Butzer e t a l . (ed .), Karl der GroBe und sein

France. His dissertation "On monomial groups" was accepted at Leiden in 1 97 2 . H i s

Butzer Felleisen), pages 43-90 in:

Nachwirken: 1200 Jahre Kultur und Wis­

mathematical interests are number theory, group theory, coding theory, geometry, his­

senschaft in Europa. Band 2: Mathematis­

tory of science; his hobbies include chess, genealogy, and old scripts like Hieroglyphs,

ches Wissen, Turnhout Brepols (1 998);

Bruno

Christoffel

lntelligencer from lntel/igencer author.

He has been a faithful reader of The

ISBN 2-503-50674-7. 6. Elwin

Unear 8, Maya.

(1 O.November

is his first appearance as an

the very beginning, but this

1 829, Monschau-5. Marz 1 900, Straf3burg),

VOLUME 25, NUMBER 3, 2003

51

ALEXAN DER GEORGE

A Free Lu n c h C h ess an d Log i c ?

A cafe in Amsterdam. Claus and Connie are seated at the bar, finishing thei1· drinks. They have been squabbling for much of the evening. Claus has taken a hard-boiled egg from a carousel and is spinning it distractedly on the counter. He places it in his coat pocket. CLAUS: Shall we go?

CONNIE:

What are you doing with that egg? I'll eat it later. CONNIE: But you haven't paid for it. CLAUS: I thought it was complimentary! CONNIE: You always think you can get something for noth­ ing! Well, not that egg. CLAuS: (Replacing the egg, resentfully) I'm glad you're here to keep me honest. They walk toward the door but stop before a table w'ith a chessboard and pieces on it. CLAUS: Look, an abandoned game. Who's better off here, White or Black? CONNIE: We don't know whose move it is. CLAUS: Let's assume it's White's. They ponder the position (Diagram 1) for a while. CLAUS: No wonder they abandoned this: it's a forced mate in two for White. CONNIE: How so? CLAUS: Well, we can reason as follows: Either the history of this game is such that Black can castle (viz., neither its king nor its rook has moved) or it is such that Black cannot castle. If Black can castle, then Black's last move must have been Pg7-g5 1 (for Black has moved neither king CLAUS:

' Conventionally, the columns are labeled "a" to "h," moving from left to right. And the rows are labeled "1" to "8," moving from bottom to top; thus, in Diagram 1 the Black k1ng is standing on e8 and the White king on f5.

© 2003 SPRINGER·VERLAG NEW YORK. VOLUME 25. NUMBER 3. 2003

53

nor rook, and Pg6-g5 would have placed White in an im­ possible checked position). But in that case, White can move l.Ph5xPg5 en passant (e.p. ), leading to mate the fol­ lowing move (either 2.Rd8 or, if Black castles, 2.Ph7). On the other hand, if Black cannot castle, then White can move l. Ke6 (but not l . Ph5xPg5 e.p. since we cannot now show that Black's last move must have been Pg7-g5), which leads to mate next move (namely, 2.Rd8). Summa­ rizing: either Black can castle or Black cannot castle; if the first, then l .Ph5xPg5 e.p. leads to mate; if the second, then l . Ke6 leads to mate; therefore, either l .Ph5xPg5 e.p. leads to mate or l . Ke6 leads to mate. We can't know which leads to mate, though we do know that either one does or the other does. CONNIE: There's something about the way you're arguing that I don't like. I think that in order to prove a disjunction "X or Y, " you would need a proof of at least one of its dis­ juncts, viz. either a proof of X or a proof of Y; or at the minimum, you would need a method that would in princi­ ple yield a proof of one of the two disjuncts. You shouldn't be asserting a disjunction unless you're confident that you know, or could in principle find out, which of the two dis­ j uncts actually holds. CLAUS: It's true that I don't now know that l.Ph5xPg5 e.p. leads to mate; nor do I now know that l.Ke6 leads to mate. And it's also true that I do not have any means for ascer­ taining which does lead to mate. But I can prove that one or the other must! CONNIE: You are confident about that conclusion only be­ cause you helped yourself to the assumption that either Black can castle or Black cannot. But you have no right to that disjunctive assumption, because you neither know that Black can castle, nor know that Black cannot castle, nor have a method that would eventually tell us which it is. CLAUS: But I don't need to defend my assertion of that dis­ j unction! It just follows as an instance of The Law of the Excluded Middle: for every statement X, "X or not-X'' is true. CONNIE: I reject your "law"! Nothing's on the house in logic. If you want to assert "X or not-X'' then you must either possess a proof of X or possess a proof of not-X, or show how one could arrive at one or the other. You'll

2The next day,

cLAus

questioned

CONNIE

about why she had been willing to say

that if Black can castle then 1. Ph5xPg5 e.p. leads to mate. For this appears to depend on the assumption that either Black will castle (in which case 2 . Ph7 mates) or Black will not castle (in which case. 2.Rd8 mates). And this disjunctive as­ sumption is really another application of the Law of the Excluded Middle, which presumably

CONNIE

will likewise judge problematic: for we are neither

in

a posi­

tion to establish that Black will castle, nor in a position to establish that Black will not castle, nor 1n possession of a method that will settle the matter. CONNIE

replied that the appearance was deceptive, for her reasoning need not

be viewed as resting on such an assumption. To say that 1. Ph5xPg5 e.p. leads to mate just means that for any given next move by Black there is a mating move by White. And,

CONNIE

continued, she had no difficulty accepting this: for if we

are given any following move by Black, we can examine it to determine whether it is a castling move or not and on this basis provide the mating move by White.

54

THE MATHEMATICAL INTELLIGENCER

have a license to assert that either Black can castle in this position or Black cannot once you've established that Black can castle or established that Black cannot, or explained how in principle we could determine which it is. CLAUS: Well, obviously I can't do any of that. So you won't allow me to claim that it's one or the other? CONNIE: That's right. Of course, I do agree with you that if Black can castle then l . Ph5xPg5 e.p. leads to mate, and also that if Black cannot castle then l . Ke6 leads to mate. 2 But given our present knowledge, I don't think it's correct to go the extra step and conclude, as you do, that either l .Ph5xPg5 e.p. leads to mate or l . Ke6 does. Claus strokes his chin. CONNIE: Let me put it another way. If one were given any legal history of this board position, one could legitimately assert the disjunction: l . Ph5xPg5 e.p. leads to mate or l . Ke6 does. For given any such history, one could determine whether White can capture e.p. and whether Black cannot castle. And on the basis of this determination, one could establish which particular move leads to mate; that is, one could establish one of the two disjuncts, and so the entire disjunction. What I don't think you can do is to assert the disjunction simpliciter. That's like taking an egg without paying for it! CLAUS: Let me try something else. Will you agree that it is not the case that both White cannot capture e.p. and Black can castle? CONNIE: Of course! The assumption that Black can castle forces the conclusion that Black's last move was a double pawn move which pennits White to capture en passant. Thus the assumption that both White cannot capture e.p. and Black can castle leads to a contradiction; and that in­ deed justifies (by reductio ad absurdum) the claim that it cannot be that both conditions hold. CLAUS: Ah, now we're making progress! You've just granted a claim of the form "not-(not-X and Y)," where X is "White can capture e.p. " and Y is "Black can castle." Right? CONNIE: Yes, that's right. CLAUS: But this is just equivalent to the claim "X or not-l"'! So, you should be willing to assert that either White can capture e.p. or Black cannot castle. And from this, you agree, it follows that either l .Ph5xPg5 e.p. leads to mate or l.Ke6 does. CONNIE: I do agree that if we were in a position to assert that either White can capture e. p. or Black cannot castle, then we would indeed be in a position to claim that either l.Ph5xPg5 e.p. leads to mate or l . Ke6 does. But that is be­ cause I think that to be in a position to assert that either White can capture e.p. or Black cannot castle, one would have to be in a position to assert that White can capture e.p. or in a position to assert that Black cannot castle (or in possession of a method that would put us in one of those positions). And the problem is that knowing that these two conditions cannot both hold does not put us in either of those positions (and does not supply us with such a method)!

CLAUS: Are you saying that you deny that it's legitimate to infer from "not-(not-X and Y)" to "X or not-}"'!? CONNIE: That's right! 3 Claus paces in silence around the table. CLAUS: OK. Earlier you agreed that if Black can castle, then White can capture e.p. , right? CONNIE: Absolutely. CLAUS: Good, good. Now, that's a claim of the form "if Y then X." And such a claim is equivalent to the assertion "not- Y or X," which of course is just equivalent to "X or not­ Y." So you see, you must accept that either White can cap­ ture e.p. or Black cannot castle. CONNIE: You find equivalences where I don't! I do not be­ lieve that the conditional claim "if Y then X'' even entails the disjunctive one "not- Y or X. " I accept the conditional claim because I agree that if one could establish that Black can castle, then on that basis one could establish that White can capture e.p. : the castling would force Black's last move to have been the relevant double pawn move. But I don't see how any of that helps one in the least to establish that White can capture e.p. or to establish that Black cannot castle (or to provide a recipe for determining which is the case)-and that's what you'd need to do to earn the right to assert the disjunction. CLAUS: (Sighs.) Well, it seems that I'm prepared to assert baldly that either l.Ph5xPg5 e.p. leads to mate or l.Ke6 does, but you're not. I think we're going to have to agree to disagree. coNNIE: I'm not so sure . . . . CLAUS: Huh!? CONNIE: Well, sometimes, when divergences between peo­ ple get substantial enough, they really aren't disagreeing with one another so much as talking about different mat-

A U THOR

ALEXANDER GEORGE

Department of Philosophy Amherst College Amherst, MA 0 1 002 USA

e-mail: [email protected] Alexander George is a New Yorker, and a graduate of the Ly­

cee FraJ)9ais de New York and of Columbia College. His doc­ torate is in philosophy from Harvard. After

an

appointment at

Oxford, he is now Professor of Philosophy at Amherst College.

ters. The clearest cases of disagreement are those that take place against a background of agreement-and I'm not sure we've located that common ground yet. CLAUS: (Alarmed.) Gosh! We can't even agree on whether we're disagreeing!? Where does that leave us? CONNIE: (She pats hirn reassuringly on the hand. ) It's only chess and logic. That's all. 4

3Later that week, Conn1e broke the news to Claus that this inference also 1nvolves mov1ng from "not -not -X' to

either. Claus was dumbfounded.

"X'.

and that this was something she could not accept

cLAus: But surely 1f you've shown that "not-X'' Isn't correct, then it follows that X must hold'

CONNIE: Why IS that?

cLAus· Because there are only two possibilities: either X holds or not-X holds!

cONNIE: Now there you go again! You're getting that disrunct1on for free by invoking that Law of the Excluded Middle-which I rerect' If you're going to assert that diS­

junction you'll have to earn the nght to 1t: you'll have to give grounds either for X's holding or for not-X's holding (or provide a method that in pnnciple will supply such grounds). And you haven't done that. There are no free lunches in logic! 4Actually, mathematics too. For more Information about the philosophical, logical, and mathematical contrasts between cLAus's classical reason1ng and CONNIE's con­ structiVIsm, see Alexander George and Daniel

J.

Velleman,

Philosophies of MathematiCS,

Blackwell, 2002; especially Chapter 4, "Intuitionism . "

The Chess Amateur i n 1 922. The Chess Mystenes of Sherlock Holmes, Times Book, 1 994.

The retrograde analysis ("retroanalysis") problem considered above was composed b y W . Langstaff and appeared 1n For further examples of retroanalysis problems, see Raymond M . Smullyan,

I do not believe that the idea of using retroanalys1s problems to illustrate the difference between class1cal and constructive reasoning IS original to me, but all my ef­ forts to locate a source have failed. My thanks to Daniel

J.

Velleman for his help.

Mathematician's Dict u m When you come to a fork i n the road-take it! -Folksay, collected by Louis Nirenberg (2002)

VOLUME 25. NUMBER 3 2003

55

I il§'h§'.'fj

O s m o Peko n e n ,

Editor

I

Ou en sont les

mathematiques� Jean-Michel Kantor, Editor PARIS. VUIBERT/SOCIETE MATHEMATIOUE DE FRANCE.

2002. 440 pp .. €60, ISBN 2-7 1 1 7-8994-2

REVIEWED BY VICTOR GUILLEMIN

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.

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56

T

he Gazette des Mathematiciens plays roughly the same role in the francophone world as does the Notices of the AMS in the United States; and like the Notices it aspires (perhaps suc­ ceeding even more than the Notices) to publish lucid, clearly written exposi­ tory articles that are intended for an audience of mathematically sophisti­ cated non-experts. However, because the total number of subscribers to the Gazette here in the States is so small (compared, say, to the number of sub­ scribers to the Notices in France), the readership for these articles includes very few Americans. Fortunately, the Societe Mathematique de France has collected a number of the best of these articles in the present accessible vol­ ume, Ou en sont les mathematiques? [roughly, What Is the CurTent State of Mathematics ?]. The articles they have chosen to in­ clude cover a wide range of topics and provide a panoramic view of activity at the frontiers of mathematics over the last two decades. A small sam­ pling: Marcel Berger on "Encounters with a geometer: Mikhail Gromov," Jacques Dixmier on "Some aspects of the theory of invariants," Pierre Samuel on "The history of Hilbert's fif­ teenth problem," Mark Gotay and James Isenberg on "The symplectifi­ cation of science," Michel Brion on "Integer convex polytopes," and Claude Weber on "Topological ques­ tions in molecular biology. " I would like to single out one of these articles because it deals with an aspect of mathematics which is rarely dealt with in popular expositions and which I wish were more often high-

THE MATHEMATICAL INTELLIGENCER © 2003 SPRINGER-VERLAG NEW YORK

lighted. This is the first article listed above, the one by Marcel Berger. Berger's ostensible reason for writing it is to give a survey of Gromov's achieve­ ments in the many areas of geometry and analysis in which he has worked, some of which Berger feels are under­ appreciated. However, Berger's real aim is much more ambitious: to give one a sense of how Gromov thinks about mathematics. I remember being present once when Gromov was explaining the Gauss-Bonnet theorem to a group of undergraduates. Rather than showing them one of the standard ways of prov­ ing it-triangulations, or the index of the Gauss map, or connection forms on the tangent bundle-Gromov began by drawing a sphere on the b lackboard and then "bashing it in," and showing them that even though the "bashing in" creates a lot of negative curvature, it creates enough positive curvature to keep the "average curvature" un­ changed. In this article Berger shows that this is the paradigmatic Gromov method of attack Berger's comments, in fact, are much more specific: Almost all of MG's big results have three features. At the start is a very simple, even naive, idea, so simple that it does not seem possible to do anything serious with it! The second feature is that the development to the conclusion is very elegant and tech­ nical, sometimes using subtle compu­ tations, but always introducing new tools. The third feature is that MG in the course of his proof introduces one or more new invariants, i.e., new con­ cepts. These are often as naive as the starting idea, but they play a basic role in the proof and most often have the paradoxical property that they are impossible to compute explicitly, even for the simplest spaces. Much of this article consists of illus­ trations of how this three-pronged attack plays out in concrete examples, with particular emphasis on phase one: the "simple naive idea" of which the

"bashing in" of the sphere that I wit­ nessed is a perfect example. It's probably true of most of us when we attempt to describe the results of our peers, or of our peers when they pay us the compliment of describing our results to others, that a more or less factual account suffices: such-and­ such a theorem was proved in article A and then generalized to n dimensions in article B, and has C, D, and E as corollaries. However, for someone like Gromov, this approach misses the boat, because his "theorems" are often merely a hint of a deep underlying geo­ metric reality, which he can see clearly in his own mind but finds hard to con­ vey to us. To quote Berger again, We have seen time and again that MG 's papers are like icebergs: most of the results lie under the surface and are accessible only to exceptional math­ ematicians who are willing to devote their time to them. So why does MG not write his results in detail? We think that the best answer to this and other questions is to let MG speak for himself: "Checking the proof in full detail in my head was already so painful that I was left with no energy for more. " Berger continues, If MG has a muse, it is not the ax­ iomatic one of Euclid. MG is instead guided by concepts such as softness versus rigidity, computability and physical reality of objects. In partic­ ular, when talking about results, he is concerned with the robustness of the invariants used. Gromov is fortunate to have a partisan as perceptive as this writing an in­ depth analysis of this kind, and I wish that more such in-depth analyses were attempted. (I am thinking in particular of another hero of mine of whom it could equally well be said that theo­ rems were merely an attempt to give one a hint of a deep underlying geo­ metric reality: Rene Thorn.) Berger's article, incidentally, is the only one in the volume to have been translated into English; the English version appeared in vol. 37, nos. 2, 3 of the Notices.

The present collection has been beautifully edited by Jean-Michel Kan­ tor, and each article is preceded by a commentary written by an expert re­ porting on developments in the area under discussion since that article's first appearance. In this critic's judg­ ment-highly recommended! Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 021 39 USA e-mail: vwg@math . m it.edu

Visualiser Ia quatrieme dimension Franr;ois Lo Jacomo VUIBERT. PARIS. PRICE €1 5

2002. 1 2 6 PAGES ISBN 2-7 1 1 7-531 5-8

00

REVIEWED BY THOMAS F. BANCHOFF

F

ran<;ois Lo Jacomo is a mathemati­ cian and an artist. For him, mathe­ matics includes both the art of deduc­ tion and the art of construction. The constructions that fascinate him the most involve visualization of objects from the fourth dimension. He describes his twenty-year fascination with these objects in an engaging way, unencum­ bered for the most part by technical lan­ guage. He does introduce notions about groups of symmetries and quaternions when these help to simplify the presen­ tation, and he presents numerous illus­ trations to go along with his discus­ sion. A measure of the success of the book is the degree to which the author engages his readers in following his de­ scriptions by making their own draw­ ings and models. There is a great deal of stimulus for any reader who is willing to make the effort, whether the book is used as an introduction to the geometry of figures in four-dimensional space or as a collection of new viewpoints for those who have been familiar with these notions for a long time. The Problem of the Mirror

The author begins his book by dis­ cussing an old problem that is only par­ tially connected with the main theme

of the work, namely, "Why does an or­ dinary mirror reverse left and right but not top and bottom?" Martin Gardner's answer in "The New Ambidextrous Universe" is that a mirror does not re­ verse left and right but rather front and back The way we talk about what we expect a mirror to do is the source of the difficulties. I tend to accept Gard­ ner's treatment, although I do appreci­ ate the author's comments relating the problem to art, linguistics, physics, and philosophy. Once this discussion is out of the way, the author proceeds in earnest to the main object of his study-visualizing objects in the fourth dimension. Generating Cubes and Hypercubes

The author likes to generalize con­ structions in three-space, some of which are quite familiar and others of which generate old examples in new settings. The chapter on the hypercube begins with a point moving to trace out a seg­ ment, a segment moving to trace out a square, a square tracing out a cube, and, analogously, a cube moving to trace out a hypercube. A segment will have two vertices, a square, four, a cube, eight, doubling at each stage to pro­ duce a hypercube with sixteen ver­ tices. A segment has two bounding ver­ tices, a square, four bounding edges, and a cube, six bounding squares. Anal­ ogously, the hypercube will be a regu­ lar "hyperpolyhedron" with eight cubi­ cal faces. Half-Cubes and Half-Hypercubes

An important theme in this book is the study of "subfigures" of a given figure. One of the first such examples is the "half-cube," a figure formed by color­ ing the vertices of a cube black or white so that the vertices of any edge of the figure have different colors. Such a coloring is completely deter­ mined once one vertex is colored, black for example. The three adjacent vertices will then be colored white, and the three adjacent to them, black, with the final vertex colored white. The four black vertices form the "half-cube," which is seen to be a regular tetrahe­ dron with six edges which are face di­ agonals of the original cube. Dropping

VOLUME 25, NUMBER 3, 2003

57

back one dimension, we see that a "half-square" is just a diagonal, and a "half-segment" is just one of its ver­ tices. What about going up to the fourth dimension? For a hypercube obtained by moving an ordinary three-cube per­ pendicular to itself, the coloring of the vertices on the end cube is exactly op­ posite the coloring on the beginning cube. In each of the eight bounding three-cubes, there will be a regular tetrahedron, and each white vertex is the center of another tetrahedron with all vertices black. In this way, we obtain sixteen congruent regular tetrahedra. This "demi-hypercube" shows up in another context when we look at duals of figures. The dual of a polygon is formed by vertices at the midpoints of the edges of the polygon, so the dual of a square is another square. The dual of a polyhedron has vertices in the cen­ ters of the faces of the original poly­ hedron, so the dual of a cube will have six vertices. The centers of three squares meeting at a vertex form a face of the dual polyhedron, so we obtain eight equilateral triangles forming a regular octahedron, one of the five Platonic fig­ ures. Analogously, the dual of the hy­ percube will have eight vertices, one for each of the eight bounding cubes. The centers of the four cubes that come together at each vertex of the hy­ percube form a regular tetrahedron that is a face of the dual, so we obtain a four-dimensional figure with eight ver­ tices and sixteen tetrahedra. This hy­ percube dual is identical in form to the demi-hypercube constructed above! It is this sort of elegant coincidence that makes the fourth dimension such an intriguing place. As the author points out, this situation does not at all generalize to dimensions higher than four. An n-dimensional cube will have 2n vertices, so the demi-n-cube will have half as many, or 2n- l . The dual of the n-cube, on the other hand, will have 2n vertices, one for each of the bound­ ing (n - 1 )-cubes. Only for n = 4 will 2n = 2" - 1 . Enumerating Regular Polyhedra and Regular Hyperpolyhedra It

would not be so hard to explain to an intelligent Flatlander the proof found in Euclid that there are at most five regu-

58

THE MATHEMATICAL INTELLIGENCER

lar polyhedra in three-dimensional space. Around any vertex of a convex polyhedron, the sum of the angles of the faces has to be less than 27T. For a reg­ ular polyhedron, with all faces congru­ ent regular polygons, this linlits the choice to three, four, or five triangles, three squares, or three pentagons. That proof, although quite convincing, would not necessarily help the Flatland math­ ematician to visualize the five polyhe­ dra that satisfy these specifications. To try to "see" what the figures would look like, Flatlanders would probably study projections of the figures into their space, constructed by using coordinate geometry, and they might also construct nets of polygons in the plane, along with rules for assembly in three-space, even if no such assembly would be possible in a two-dimensional world. In four-dimensional space, there are at most six regular polytopes, the ana­ logues of regular polygons and polyhe­ dra. To prove this, we can use an ana­ logue of Euclid's approach. We look at the numbers of regular polyhedra that can be fitted around an edge, and, by estimating the dihedral angles, we find six possibilities: three, four, or five tetrahedra, three cubes, three octahe­ dra, and three dodecahedra. But do these six possibilities actually exist in some four-dimensional space?

to the hypercube, where we now have four coordinates for each point instead of three. The hypercube has eight cu­ bical faces, and it is often called an 8cell. The dual to the hypercube has ver­ tices in the centers of the cubical faces, with 1 or - 1 in one coordinate and 0 in the others. This figure has 16 tetra­ hedral faces, one for each vertex of the hypercube, and it is known as a 16-cell in nearly every book that treats regu­ lar polytopes, those four-dimensional figures analogous to regular polyhedra in three-space. The author of the book under re­ view chooses to call the 16-cell a "hy­ peroctahedron." Similarly, the four-di­ mensional analogue of a tetrahedron, with five vertices and five tetrahedral faces, is called a "hypertetrahedron" rather than a 5-cell or pentatope, the more common names in the literature. This should not cause much confusion in the case of these three basic figures in four-space. The author points out that the same recipes in n-dimensional space will produce a 2n-cell or n-dimensional cube, a 2n-cell dual to the n-cube, and an (n+ 1)-cell analogous to the tetra­ hedron. The analogue of Euclid's proof shows that these are the only possible regular figures in n dimensions when n is greater than four.

Coordinates for Cubes and

The Other Regular

Hypercubes, Octahedra, and

Hyperpolyhedra

Hyperoctahedra

In four-space, however, there are three more possibilities for regular poly­ topes. We have already considered the hypertetrahedron, with three tetrahe­ dra around each edge, and the hyper­ octahedron, with four tetrahedra around each edge. What kind of figure has five tetrahedra around each edge? The author spends most of his effort in the latter part of the book describ­ ing his efforts to visualize what he calls the "hypericosahedron," conventionally known as the 600-cell with 600 tetra­ hedral faces. The dual to this figure, with 120 dodecahedral faces, is called a 120-cell or "hyperdodecahedron."

For Euclid, the existence of regular polyhedra in three-space was estab­ lished by explicit constructions. Most modern proofs involve the use of co­ ordinates, three numbers for each ver­ tex. For example, the cube can be de­ scribed by specifying eight vertices for which each coordinate is 1 or - 1. This figure, with six square faces, is also called a hexahedron. The polyhedron dual to the cube can then be deter­ mined by the six vertices in the centers of the faces of the cube, namely the points with 1 or - 1 in one coordinate and 0 in the remaining coordinates. This regular polyhedron is the octahe­ dron with eight triangular faces, one for each vertex of the cube. The same specifications of coordi­ nates give the hypercube and the dual

The Granatohedron and Hypergranatohedron

The key to visualizing the hypericosa­ hedron depends on understanding the

sixth regular polytope, with 24 octahe­ dral faces, three around each edge. For this 24-cell, the author uses the name "hypergranatohedron," analogous to the French term for what in English is known as a rhombic dodecahedron. The "granatohedron" in three-space has fourteen vertices given by the eight ver­ tices of a cube and the six centers of the cubes adjacent to the central cube. Each of the twelve faces of this figure is a rhombus with one diagonal given by an edge of the original cube. A typical rhombus in this polyhedron has vertices (2,0,0), ( 1 , 1 , 1), ( 1 , 1 , - 1) and (0,2,0), based on the edge with ver­ tices (1,1,1) and ( 1 , 1 , - 1). The analogue of the rhombic do­ decahedron in four-space will have 24 vertices, 16 from the hypercube with 1 or - 1 in each coordinate, and eight vertices with 2 or - 2 in one coordinate and 0 in the remaining coordinates. A typical polyhedron in the boundary of this figure has vertices (2,0,0,0), (1, 1,1,1), (1,1,- 1,1), (1,1, - 1, - 1), (1,1, 1, - 1) and (0,2,0,0). These six vertices determine a regular octahedron, based on the square with vertices (1,1,1,1), (1, 1, - 1, 1), (1,1, - 1, - 1), and (1,1,1, - 1). Similarly we get a regular octahedral face based on each of the 24 square faces of the hypercube. In contrast with the granatohedron with (non-regular) rhombic faces, the hyper­ granatohedron, or 24-cell, has regular octahedral faces. This is the sixth of the regular polytopes in four-dimensional space. Bicolorings and Tricolorings

In our initial example, we discussed hi­ colorings of the cube and hypercube. The black vertices of a bicolored cube determine a regular tetrahedron, as do the white vertices. The union of these two tetrahedra is known as the "Stella Octangula," and the intersection of the tetrahedra is the octahedron inscribed in the cube. In four-space, when the 16 vertices of a hypercube are bicolored, the eight black vertices determine a "half-hyper­ cube" with 16 tetrahedral faces, a hy­ peroctahedron. The same is true for the white vertices, and the intersection of these two hyperoctahedra is the 24cell inscribed in the hypercube. It is possible to color the six vertices

of an octahedron black, white, and gray such that the vertices of every triangle have different colors. Using this color­ ing, there is a standard way of de­ scribing an icosahedron inside the oc­ tahedron. We start by finding a triangle inscribed in each triangle obtained by moving the black vertex a fixed dis­ tance toward the white one, moving the white vertex the same distance to­ ward the gray one, and the gray one the same distance toward the black This produces a figure with twelve vertices, one for each edge of the octahedron. In addition to the eight equilateral tri­ angles in the faces of the octahedron, there are twelve congruent isosceles triangles. When the twelve points reach the midpoints of their respective edges, the figure that they determine is a cuboctahedron, with eight triangular faces and six square faces (formed by two isosceles right triangles). The cube can be reconstructed by placing on each of the square faces a cap in the form of a half-octahedron. When the twelve points reach a position deter­ mined by the Golden Ratio, the twelve isosceles triangles are equilateral and the figure formed is a regular icosahe­ dron situated within the octahedron. What happens in four-dimensional space? It is possible to color the 24 ver­ tices of a 24-cell or hypergranatohedron black, white, and gray, so that every tri­ angle has vertices of different colors. By carrying out the above procedure for each of the 24 octahedra, we obtain 24 icosahedra inside the octahedra. The points also determine 24 regular tetra­ hedra, one for each vertex of the 24-cell, and on each triangular face of one of these tetrahedra there is a tetrahedron with one equilateral face and three isosceles triangular faces. When the points reach a position on their edges re­ lated to the Golden Ratio, all of these tri­ angles become congruent and the tetra­ hedra are all congruent. We thus obtain a semi-regular polytope with 24 icosa­ hedral faces and 120 tetrahedral faces. The author now completes his con­ struction of the 600-cell by a "capping­ off' process. We can construct an icosahedron by adding to each of the pentagonal faces of a pentagonal an­ tiprism, a pyramidal cap with a pen­ tagonal base and five equilateral trian-

gles. In an analogous way we can form a cap in four-space with an icosahedral base and twenty regular tetrahedra. We replace each of the 24 icosahedral faces in the semi-regular figure described above by one of these caps. The cap­ ping-off procedure produces a regular polytope with 480 new regular tetrahe­ dra in addition to the 120 we already had, for a total of 600 tetrahedral cells. This way of visualizing the structure of the 600-cell can be found in the lit­ erature, for example in H. S. M. Cox­ eter's Regular Polytopes where it is called "Gosset's construction. " The value of the description in this book is not that it represents something en­ tirely new, but rather that it stimulates the imagination in ways that send the reader to sketch pictures and to make models. The whole construction is quite elegant and creative, a fitting climax to this marriage of art and mathematics. There are many other topics treated in the book, including the relationship between quatemions and symmetric objects in four-space. A very good ref­ erence for the entire topic is Kaleido­ scopes, Selected Writings of H. S. M. Coxeter (1995, Wiley-Interscience). As a special treat there is a previously un­ published essay "Two Aspects of the 24-Cell." See also the reviewer's article "Torus Decompositions of Regular Poly­ topes in 4-Space" in Shaping Space (1988, Birkhauser-Verlag). In summary, I recommend the book for anyone who likes geometry. My only disappointment is that the author relies almost entirely on projections and slices of higher-dimensional ob­ jects when he describes visualizations. He downplays one of the techniques that I most appreciate, namely fold­ outs, the three-dimensional analogues of two-dimensional nets of polygons that can be folded up in three-space. The fold-out models suggested by the constructions in this book are truly beautiful, and I encourage any reader to construct them for his or her own pleasure and edification. Department of Mathematics Brown University Providence, Rl

0291 2

USA e-mail: thomas_banchoff@brown .edu

VOLUME 25, NUMBER 3, 2003

59

Alles Mathematik: Von Pythagoras zum CD Player Martin Aigner, Ehrhard Behrends, editors BRAUNSCHWEIGIWIESBADEN 2002: FRIEDR VIEWEG & SOHN VERLAGSGESELLSCHAFT mbH' €24.90; ISBN 3-528- 1 3 1 3 1 -4 (2nd enlarged edrtron) 342 pp

REVIEWED BY MANFRED STERN

T

his volume is a collection of lec­ tures presented in the years after 1990 to the audience of the Urania, a traditional educational institution with a broad range of topics for non­ specialists. Before 1990 the topics of Urania Berlin included arts and sci­ ences, current politics, popular medi­ cine-but no mathematics. This situa­ tion was remedied by the editors, who asked a number of experts to give pop­ ular lectures on mathematics and its applications. The outcome of this en­ terprise are 22 articles, most of them summaries of the lectures, which re­ fute the common prejudice that math­ ematics is a dry subject. The first edi­ tion of the book appeared in 2000; this is the second, enlarged edition, re­ flecting the extremely favorable recep­ tion of the lectures by the audience. The book consists of the following sec­ tions: Prologue; Concrete case studies; Topics of current discussion; The red thread; Mathematics and music; Epi­ logue. The Prologue

In the Prologue Mathematics becomes a cult-description of a hope, Gero von Randow states that victims are necessary in order to popularize math­ ematics-hinting at Jurassic Park, a film in which a mathematician-inci­ dentally a specialist of chaos theory­ is bitten by a dinosaur. There are also other films indicating that the chances are good for mathematics and mathe­ maticians to become more popular than ever. Mathematicians should use their chance-as they did in this book. Concrete Case Studies

Jack H. van Lint expounds The math­ ematics of the compact disc. Why are

60

THE MATHEMATICAL INTELLIGENCER

CDs of better quality than traditional LPs? Because there is mathematics in­ side! More precisely, a branch of dis­ crete mathematics, viz., the theory of er­ ror-correcting codes. This article deals with the application of such codes to the design of the Compact Disc Audio Sys­ tem developed by Philips Electronics and Sony. One will notice that not only bits, but also pits play an important role here. In Theory planning with virtual cancer patients, Peter Deuflhard ex­ plains tools applied in hyperthermia, a cancer therapy. The therapy planning system developed by the author's re­ search group at the Konrad-Zuse-Zen­ trum can predict with a high degree of reliability whether a cancer patient can be treated by hyperthermic therapy. The corresponding medtech devices will be fully efficient only when com­ bined with suitable mathematical soft­ ware. Heinz-Otto Peitgen, Carl Evertsz, Bernhard Preim, Dirk Selle, Thomas Schindewolf, and Wolf Spindler report on Image processing and visualiza­ tion for operation planning-an ex­ ample from liver surgery. This article deals with computer-assisted radiol­ ogy. The authors present methods of extracting and measuring relevant anatomical structures from layer im­ ages. In this way one obtains spatial representations that can be manipu­ lated interactively and that, in turn, sup­ port radiological diagnosis and surgical planning. In particular the model-based computation of individual liver seg­ ments relies heavily on mathematics. Ralf Borndorfer, Martin Grotschel, and Andreas Lobel explain The quick­ est way to get to our destination. The concept of "way" is associated with roads, transport, traffic-and mathe­ matics. The authors discuss Branch & Bound, Branch & Cut, and Branch & Price as the main methods combining enumeration with empirically efficient techniques. Concerning the Shortest Path Problem, they mention Schiller's "Wilhelm Tell" as the possibly oldest source of problems of this kind known from the literature (" . . . a shorter se­ cret way to Arth and Ki.iBnacht, not via Steinen, but via Lowerz . . . "). As early as 1291 Wilhelm Tell was good not only

at shooting with his crossbow, but ob­ viously also solved a graph-theoretical optimization problem in order to liber­ ate Switzerland! In the remaining part the authors discuss questions, methods and possibilities for dealing mathe­ matically with path problems. Basic ex­ amples are taken from passenger in­ formation in public short-distance traffic and from vehicle applications planning. In the course of their presen­ tation they illustrate the results of Dantzig, Gomory, Kantorovich, Dijk­ stra, and others, and suggest that Com­ puter-Aided Scheduling (CAS) will ac­ quire the same importance in logistics as CAD and CAM did in the field of pro­ duction. Bernold Fiedler reports on Romeo and Juliet, spontaneous pattern forma­ tion, and Turing's instability. Crystals, snowflakes, soap-bubbles, water-waves, dunes, mountain-valleys, fir-cones, em­ bryological development, sunflowers, heartbeat, and many other things seem to indicate that ordered structures, pat­ terns, and regularities obviously evolve "by themselves." Turing predicted that under certain circumstances structure may develop "spontaneously"-in a precise mathematical sense. The au­ thor explains the essence of Turing's idea using only the four arithmetical operations, thereby sparing the reader any study of ordinary and partial dif­ ferential equations. Instead the author adapts Shakespeare's Romeo and Juliet and also discusses the love story of Roberto and Julietta, Roberto being an identical mathematical twin of Romeo and Julietta an identical mathe­ matical twin of Juliet. In this scenario the sisters' loquacity ("chat-coefficient") and the brothers' boasting ("brag-coef­ ficient") play a disastrous role. While Romeo and Juliet are on their way to catastrophe, Roberto and Julietta soon get divorced by mutual consent: seem­ ingly because of insuperable aversion and in spite of having precisely the same genetic disposition as Romeo resp. Juliet-the true reason, however, being Turing's instability. Stefan Mi.iller lectures on Mathe­ matics and smart materials, where mathematics plays the key role. Smart materials are materials that react adap­ tively to their surroundings. An exam-

John Milnor commented i n

1967, . . . 3 dimen­

sition as if they had a "memory" for this

sented his doctoral thesis Theorie de la speculation, in which he formulated his famous dictum L 'esperance math­ ematique du speculateur est nulle. The

position. Such metals are used, for ex­

present article starts with an explana­

is a proof. " The history of this conjec­

ample, in stents to stabilize coronary

tion of mortality tables, illustrates the

ture is spell-binding, and the authors

pie are smart metals: at low tempera­ tures

they can

be

bent,

but when

heated they return to their original po­

"

the corresponding problem in

sions remains unsolved. This is a scan­ dalous situation. . . . All that is missing

arteries. The memory of these metals

Bachelier model, and then discusses

sketch the contributions made by Le­

has to do with their microstructure.

the Black-Scholes formula, an extraor­

gendre, Gauss, Thue, Fejes T6th, and

Mathematical models help clarify which

dinary achievement that was honored

others. In the course of the discussion

structures occur and how they affect

by the

Nobel Prize for Economics

the reader gets a feeling of why the prob­

the behavior of the material. But math­

awarded to R. Merton and M. Scholes

lem is that nasty and controversial. Not

ematics also helps design new materi­

in

1997. Like Radon, F. Black was a No­

without reason has the solution led to

als, as shown by the example of mag­

bel Prize candidate who missed out be­

disputes during recent years--disputes

neto-elastic materials.

cause he died earlier.

that are still going on. Nearly every as­

Dis­ crete tomography: from battleships to nanotechnology. Wilhelm Conrad Ront­

money-quite im­ possible or already reality? Means of

computer verifications.

gen was the first to be awarded the No­

payment in general, and coins or paper

J. C. Lagarias, the Hales-Ferguson proof,

Albrecht

Peter Gritzmann writes about

Beutelspacher

asks

the

question Electronic

pect of the proof given by Thomas C. Hales and Samuel Ferguson relies on According

to

tour de force of

( 190 1) for his pi­

money in particular, have undergone a

assumed correct, is a

oneering work on what he called X

process of increasing dematerializa­

nonlinear optimization. The current sta­

rays. Rontgen's rays were later used in

tion and virtualization. A banknote has

tus it that it appears to be sound. The

the diagnostic technique of computer­

almost no material value, but its imma­

proof was examined by a team of re­

ized axial tomography (CAT). In this

terial value is high. Electronic money

viewers, but it is so long and compli­

technique, information obtained from

cated that it seems difficult for any one

X

Tu ri n g p red icted

person to check it. The authors close

bel Prize for Physics

rays taken

by

scanners

rotating

around the patient is combined by a computer to yield a high-resolution im­ age of a slice of the body. The inven­

that under certa i n

tors of this technology, G. N. Hounsfield

c i rcu mstances

and A Cormick, shared the Nobel Prize for Physiology and Medicine awarded in

struct u re

1979. The gist of the problem of these imaging techniques is mathematical, and was basically already solved in

may develop

1917 by

(1887-1956). Were Radon 1979, he might have shared

Johann Radon still alive in

" s pontaneously . "

the Nobel Prize with Hounsfield and

their article with a number of unsolved problems around Kepler's col\iecture. Next comes Ehrhard Behrends with

How do quanta compute? The new world of quantum computers. The physicist R. Feynman observed in 1982 that there are reasons to believe that classical computers cannot simulate quantum processes efficiently, and there­ fore we may need computers based on quantum-mechanical

phenomena.

Deutsch developed in

1985 a physically

D.

of virtualization

(in theory) realizable model of quan­

re­

even further: it is a material nonentity,

tum computers, a quantum analogue of

construction. The author illustrates the

it's only a string of bits, in other words

probabilistic

reconstruction problem with the game

a number, but this number is worth its

next milestone was P. Shor's contribu­

Cormick. The problem at the heart of these techniques is what is called

of battleships

carries the process

Turing

machines.

The

(1994, 1997): he showed that inte­

(reconstruction of the

money. This immateriality implies that

tion

ship position) we played in our youth.

the security of payment systems can­

ger factorization, which is of utmost

He then proceeds to explain the re­

not be hardware-supported, but must

importance for secret-key cryptogra­

construction of crystalline structures,

be associated with numbers. The au­

phy, could be solved in low p olynomial

which is central to the further devel­

thor first gives a definition of money,

time on quantum computers. At the

opment of computer tomography.

and then goes on to illustrate crypto­

ICM Berlin in

graphic mechanisms.

the Nevanlinna Prize, the most impor­

Martin Henk and Giinter Ziegler give

Topics of Current Discussion

1998, Shor was awarded

tant award for achievements in theo­

In

The role of mathematics in finan­ cial markets Walter Schachmeyer pre­

Spheres in the computer­ Kepler's conjecture, according to D. J.

sents an easily accessible survey of

Muder one of those problems that tell us

been constructed in reality-the phys­

stochastic financial mathematics.

In

that we are not as smart as we think we

ical obstacles seem insurmountable.

a report on

retical computer science. In spite of all this, no quantum computer has yet

particular, he points out the advantages

are. Kepler's col\iecture appears as part

The present article is devoted to the

and drawbacks of modeling financial

of Problem

question: What kind of mathematics is

markets by stochastic processes. Math­

1958 C. A Rogers wrote, " . . . many

required to make full use of the new

ematical finance was born on March

In

18 in Hilbert's celebrated list.

mathematicians believe, and all physi­

possibilities? How could a presently

29, 1900, when Louis Bachelier pre-

cists know, that the col\iecture is true. "

la

secure cryptosystem be attacked a

VOLUME 25, NUMBER 3, 2003

61

Shor if physicists succeeded in over­

"difficult" problems, as well as the

everywhere the same mean curvature.

coming the difficulties, viz., how to fac­

sharp contrast between primality tests

A physical soap bubble cannot pene­

tor large numbers by means of a quan­

and the factorization problem.

trate itself and therefore it is a sphere,

tum computer?

Oscillations-from Pythagoras to the CD-player, Ehrhard

so the physicist does not see any prob­ lem at all. However, a mathematical

Behrends talks about the omnipres­

soap bubble is allowed to penetrate it­

gives simple examples showing that a

ence of oscillations, about Fourier and

self. Now the question was whether

stable deterministic system need not

his revolutionary contributions,

and

there exist exotic mathematical soap

be predictable in any practically useful

about Shannon's theorem and the ubiq­

bubbles which are not spheres. The au­

In

Volker EnB 's The pendulum: it's all predetermined and yet not predictable

his

article

sense. The ubiquity of chaotic systems

uitous CD-player. In a short digression

thor explains the Wente-Abresch ex­

reminds us to be modest and to mis­

we learn of the trigonometric back­

ample of such an exotic surface in

trust our own long-term predictions as

ground of Cantor's set theory.

dimensional space.

well as those of "highly reputable in­ stitutes." However, we should not be

Elmar Voigt presents The mathe­ matics of knots in a nutshell, starting

Mathematics and Music

fatalistic either, remembering that the

with

In

frog in the milk-tub was saved by his

Alexander (the Great) and later men­

Musical applications of stochastic and recursive methods, Orm Finnen­

own staying power.

tioning another Alexander (James Wad­

dahl explains how he worked with

Fermat's Last The­ orem-the solution of a 300-year-old problem outlines the sensational de­

dell) and his even greater contributions.

Markov chains when composing his

In between we learn of many other cel­

"Wheel of Fortune," and with recursive

Jtirg Kramer in

velopments concerning Fermat's con­ jecture. In an article of only ten pages the author sketches the "anti-Fermat

shared the

Nash,

who

German mathematician Reinhard Sel­ ten. As the author points out, the Nash

say and he says it-in distinct contrast

it."

i n stitutes . " Martin Aigner is back with

"fractal" music. He has something to

music: "I have nothing to say, but I say

" h i g h ly rep utable

nomics with the Hungarian-American

tions where he developed a kind of

to John Cage who is cited on his own

wel l as those of

1994 Nobel Prize for Eco­

economist John C. Hars:inyi and the

methods in another of his composi­

p red ictions as

not exist.

beautiful mind of John

by

l ong -term

cates why the anti-Fermat world does

mind of John von Neumann to the

given

o u r own

tween these worlds. Finally, he indi­

kar Morgenstern via the transcendent

solution

We m u st m i strust

ular world, " as well as the bridges be­

A short his­ tory of the Nash equilibrium from Os­

robust

ebrated results of knot theory.

world," the "elliptic world," the "mod­

Karl Sigmund presents

the

3-

In Mathematics made sounding­ the dynamic stochastic synthesis of Iannis Xenakis, Peter Hoffmann ana­ lyzes the computer-generated music of the architect and composer Xenakis. We learn that the random walks, not of George P6lya but rather of the drunken sailor, who changes his direction con­ stantly, have lots to do with that mu­

The

sic.

geometry of the world, discussing eu­

In the Epilogue, Philip J. Davis casts

clidean geometry, the role of the par­

A glance into the future: mathematics in a multimedia civilization. He

equilibrium is an aid-not for making

allel

the right decisions but rather for ask­

geometries.

The Poincare model is

makes an attempt to deal with what he

ing the right questions. This reminds us

presented, leading to the surprising

calls the stress of structure, e.g., dis­

of Georg Cantor, another genius next

answer that hyperbolic geometry is

crete vs. continuous, deterministic vs.

door to madness, and his famous dic­

consistent if and only if euclidean

random, thinking vs. clicking, obser­

tum "In re mathematica ars proponendi

geometry is. A side-step to Godel in­

vation vs. proof, platonistic (deistic)

quaestionem pluris facienda est quam

dicates the undecidability of the con­

vs. humanistic mathematics; he con­

solvendi."

sistency problem. The article contin­

cludes quoting Chaitin: The computer

ues with Poincare's question whether

is the reason why mathematicians

The Red Thread

the universe has holes, and finally we

change their habits.

Prime numbers, secret codes, and the limits of computability how the prime num­

learn of the exciting results around Poincare's conjecture.

bers have moved to the center of ap­

Ferns talks about the mathematics of

plications, and what role they play in

soap bubbles (their forms, not their

D-06 1 20 Halle

our "encoded" life. He illustrates "easy"

colors). A mathematical soap bubble is

Germany

problems (polynomial complexity) and

defined as a closed surface having

e-mail: [email protected]

Martin Aigner demonstrates in

62

THE MATHEMATICAL INTELLIGENCER

postulate ,

In his report

and

non-euclidean

This whole book makes very pleas­ ant reading indeed.

On soap bubbles, Dirk Kiefernweg 8

i--j@Wi .MQ•h•J§i

Ro b i n Wi l s o n

I

The Philamath' s C Alphabet-B Robin Wilson

Babbage

harles Babbage (1791-1871) pio­ neered the computer age with his designs for a "difference engine", a com­ plex arrangement of gears and levers for mechanising the calculation of mathe­ matical tables, and an "analytical en­ gine", containing a store (or memory) and programmed by punched cards. Nei­ ther was built during his lifetime. Stefan Banach ( 1892-1945) helped to create modem functional analysis and develop the links between topol­ ogy and algebra; the terms Banach space and Banach algebra are named after him. This stamp was issued to cel­ ebrate the International Congress of Mathematicians in Warsaw. Jakob Bernoulli (1654--1 705) proved the law of large numbers, that if an ex­ periment is performed frequently, the outcome will be as predicted, with high probability. With his brother Johann he applied Leibniz's calculus to polar co­ ordinates and to the catenary and other curves. He invented the term integral. Friedrich Wilhelm Bessel (17841846) made measurements on over 50,000 stars and was the first to use the method of parallax to measure inter­ stellar distances. While investigating a problem of Kepler, he introduced the

Bessel functions Jn(X) which satisfy a certain second-order differential equa­ tion: these have applications through­ out physics, especially in acoustics and electromagnetism. Janos Bolyai (1802-1860) con­ structed one of the earliest non-Eu­ clidean geometries (independently of Lobachevsky), in which, given any line l and any point p not lying on l, there are infinitely many lines parallel to l and passing through p; Euclidean geometry has just one such line. Un­ fortunately, Bolyai's pioneering work made little impact during his lifetime. Bernhard Bolzano ( 1 78 1 - 1 848) helped to formalise the idea of a con­ tinuous function, and proved the inter­ mediate-value theorem. He also de­ scribed a function that is continuous at all points, but differentiable nowhere. Living in Prague, he was isolated from mainstream European mathematics, and his contributions were little known until later.

Bolyai Bernoulli

Banach

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, The Open University, M i lton Keynes. M K7 6AA, England e-mail: [email protected] . uk

64

THE MATHEMATICAL INTELLIGENCER

Bessel

© 2003 SPRINGER-VERLAG NEW YORK

Bolzano