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.
Formulae for sums of consecutive square roots
For a real number x, let [x] denote the largest integer not exceeding x.The fol lowing result might be surprising. Theorem 1.
(i) (ii) (iii) (iv)
The following formulae hold for every positive integer n.
[Vn +vn+l] = [v'4n +1] [Vn +vn+1 +vn+2] [Y9n +8] [Vn +vn+1 + vn+2 +vn+3] = [Y16n +20] [Vn +vn+1 + vn+2 +vn+3 +vn+ 4] = [Y25n +49]. =
Formula (i) is folklore; (ii) is a problem in [1]; (iii) can be found in [2, p.274]. The purpose of this note is to prove (iv) and consider related questions. Proof of (iv). For positive numbers xi= y we have vX + Vy < v'2(x +y). Using this inequality twice we get Vn +vn+1 +vn+2 +vn+3 +vn+4 = cvn + vn+4) +cvn+I +vn+3) +vn+2 < v'4n +8 +v'4n +8 +vn+2 =5vn+2 = v'25n +50. Thus Vn +vn+1 +vn+2 +vn+3 +vn+4 < v'25n +50.
(1)
Usin� the fact that vX > H(x +i)�-(x-i)�} for any real number x?: 1, we obtam Vn +vn+1 + vn+2 +Vn+3 +vn+4 >
{(
2 n +2 9 3
) (n 21 ) } " 2
-
-
" 2
•
(2)
Now we show that when n ?: 12, 2
3
{(n + 29 )" (n 2 )"} -
2
-
1
2
>
Y25n +49.
(3)
Letj(x) =�{(x +t)�- (x-i)�}-Y25x +49.Thenf(l2) > 0, limx-.ooJtx) = O,j(x) is increasing on [12, 14841/400] and decreasing on [14841/400, ) Sof(x) is positive on [12, ) and (3) is proved.Combining (1), (2), and (3), we deduce that when n ?: 12, co .
co
Y25n +49 < Vn +vn+1 +vn+2 +vn+3 +vn+4 < v'25n +50. Since no integer lies strictly between v'25n +49 and v'25n +50, we conclude that (iv) is valid for the case n 2: 12.The cases n 1, 2, ..., 11 are verified by the computer software Matlab.This completes the proof. D =
In view of Theorem 1, it is natural to suspect that for any positive integer k there is a constant c depending on k such that [Vn +vn+1 + 4
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc.
·
·
·
+v'n +k - 1]
=
[Yk2n + c]
(4)
holds for all positive integers n. This is not the case.It is shown in [2, pp. 725-727] that for sufficiently large k no such c exists.Our next result shows that 6 is the first k for which (4) cannot hold for all n. For any real number c, there is a positive integer n such that
Theorem 2.
Proof Let s(n) = ['Vn + v'n+1 + vn+2 + v'n+3 + v'n+4 + \l'n+5]. Using Matlab we find that s(1) 10 < 1 1 s(11) = 22 > 21 =
Therefore, when c 2: 85, [ Y36 X 1 1 + c].
s(l)
0
Prompted by the evidence in Theorems 1 and 2, I pose the following conjec ture. For any positive integer k 2: 6, no constant c depending only on k exists such that (4) is valid for all positive integers n.
Conjecture.
We also have the following related question. Question.
For which positive integers k does there exist an integer c such that
lvn + vn+1 +
·
·
·
+
Vn + k - 1 - Yk2n +
cl < 1
holds for all positive integers n? When such a c exists, determine it. Acknowledgment
The author thanks the referee for valuable comments.
REFERENCES
1 . F. David Hammer, Problem E301 0, Amer. Math. Monthly 95 (1 988), 1 33-1 34. 2. Z. Wang, A City of Nice Mathematics (in Chinese). The Democracy and Construction Press,
Beijing, 2000. Xingzhi Zhan Department of Mathematics East China Normal University Shanghai 200062 China e-mail:
[email protected] © 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
5
The Matrix by Jim Demmel . l.auix!
1attix! r ad or writ ,
by- byt
In the eaehe, or byt
-
What immortal U1 ory uld
Uly f arful yrnm try'?
fram
What lh h) a constant. For the upper side of the square to remain at constant altitude d, the support function of the wheel has to satisfy, for all q, r(q) = d - a(q + 1r) d - (h - a(q)) = (d- h) + a(q). =
It can be easily verified that r" + r 2:: 0, so r(q) is indeed a support function of a convex set. Therefore, the wheel we are looking for is a figure par allel to the axle. For example, Figure 8 shows three stages of the movement of a constant-breadth wheel when the axle is a Reuleaux Triangle. This is the structure we used in our wooden model. Triangular-based Wheels
Now, consider instead the following structure:
for all angles e. [5] Curves with constant breadth h and triangular sets in side a triangle of height h share the property of having perimeter 7Th.
•
Constant-Breadth Wheels
As before, the aim is to find the shape of the wheel such that, when it rotates, the distance from the ground to the upper side of the triangle remains constant. Let h denote the length of the triangle, a(q) the su� rt function of the axle, and d a constant greater than V 3/4h. For the upper side of the square to remain at constant al titude d, the support function of the wheel has to satisfy, for all q, r(q) d - a(q + 7T).
To employ non-circular wheels we need to ensure that when the vehicle moves, it maintains the same distance from the ground. Consider the following structure: • • •
A square attached rigidly to the vehicle (with two of its sides parallel to the ground). A constant-breadth axle that rotates inside the square, maintaining contact with its sides. A wheel that is attached rigidly to the axle.
Let us try to find the shape of the wheel such that, when it rotates, the distance from the ground to the upper side of the square remains constant.
Figure 7. Non-circular triangular set.
10
THE MATHEMATICAL INTELLIGENCER
• •
An equilateral triangle rigidly attached to the vehicle (with its upper side parallel to the ground). A triangular set as the axle that rotates inside the triangle, maintaining contact with its sides. A wheel that is attached rigidly to the axle.
=
And we need to make sure that r is a support function of a convex set. We require that r(q) + r"(q) = d - (a(q
+
1r) + a"(q + 1r)) 2:: 0
'r/q.
It follows that d has to be at least equal to any radius of curvature of the axle. It can be easily verified that r(q) is
Figure 8. The movement of a constant-breadth wheel.
h
r(e)
Figure 1 0. Notation. Figure 9. A triangular wheel.
also a triangular function and the wheel is a triangular set (see Figure
9).
Given a figure that rotates inside, the position of the triangle or parallelogram determines the shape of the wheel. For instance, if the upper side of the triangle or parallelogram is parallel to the ground, the support func
General Non-Circular Wheels Generalizing, we can attach any convex polygon vehicle, have an axle
A
P
to the
that rotates inside (touching at all
times the sides of the polygon), and seek a wheel such that, when it rotates, the polygon remains at constant altitude
a(q d - a(q + 1r)
tion of the wheel is obtained by subtracting from a constant altitude
d,
i.e.,
r(q)
=
+ 1r)
as in
the previous sections. In case the triangle is attached to the vehicle with its lower side parallel to the ground,
from the ground. Since every convex polygon possesses at least one of the following properties: •
•
it is a parallelogram, the extension of three of its sides forms a triangle that contains the polygon,
we can focus on axles that rotate inside parallelograms and triangles.
An axle that rotates inside a parallelogram has constant a(8) satisfies a(8) + a(8 + 1r) = constant.
breadth; therefore its support function
On the other hand, an axle that rotates inside a fixed tri angle (with vertices DEF) has a support function
a(8)
that
satisfies
a(8)
��� + dist(D,EF)
--
a(8
+ (1r- L EDF))
�--�--�----�
dist(E,FD)
+
a(8)
+
L FED)) . d1st(F,ED)
a(8 - (1r-
= 1,
a"(8) :=:::: 0.
The reason is that a point xis contained in the triangle DEF
if and only if
distance(x,EF) distance(D,EF)
+
distance(x,ED) distance(F,ED) +
distance(x,FD) distance(E,FD)
=
1.
For every triangle there exists at least one figure that can rotate inside: the inscribed circle. The main question is whether there exist non-circular figures that can rotate inside as well. We have found such figures for the trian
3'7T/5, 1TI5, 1TI5, and believe that there are case 2'7T/3, 1rl6, 1rl6. The general case is still
gle with angles none for the open.
Figure 1 1. Traces for constant-breadth and triangular-based wheels.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
11
It is asi r to mali ian
uar a ir 1
Ulan to g
t round a math
-Aug ·tus d Morgan
to the wheel rim, or apply rotato:ry force to the axle-just as for familiar vehicles. Non-circular wheels may be a great advance in technol ogy or just a curiosity, who knows? What remains is to study the advantages of these wheels; maybe one day cars will have "triangular" or "squared" wheels. Acknowledgments
Figure 12. Different wheels.
then the wheel has to be parallel to the axle so the tri angle remains at a constant altitude. We can also allow the case where none of the sides of the triangle or par allelogram is parallel to the ground. It can be easily shown that, for the triangle or parallelogram to remain at a constant altitude, the support function of the wheel must be r(8) = p(O) + p( (} - _( 7T - a)) sin + (8) cos + h, f3 f3 p tan a sm a
(
)
where • • •
h is the distance from the triangle or parallelogram to the
ground. a is the angle of the triangle or parallelogram at the lower vertex. f3 is the angle between the ground and one of the lower sides of the triangle or parallelogram (see Figure 10).
"Squared" and "Triangular" Wheels
is clear that at whatever speed a circular wheel moves, it will always look round. For our non-circular wheels, the path that is traced (relative to the vehicle) by a point on the wheel is no longer a circle. Therefore, the wheel while rotating may not look round. For instance, if the axle is a Reuleaux trian gle rotating inside the square, the wheel traces a path that is almost a square; the only deviation is at the comers, where there is a slight rounding. If the vehicle moves fast enough, the wheel will resemble a square. Similarly, if the wheels are triangular sets rotating inside a triangle, then the path traced resembles a triangle with rounded comers (see Figure 1 1). It
Final Remarks
In
mathematics, your preconceptions are not barriers.Now we know a vehicle can have wheels of different shapes and still move at a constant altitude from the ground (Figure 12). To move it, one can push or tow the vehicle, or apply force
12
THE MATHEMATICAL INTELLIGENCER
Chronologically, we would like to thank all the people who collaborated with us during the development of non-circu lar wheels. First of all, we are grateful to Aisha Najera for her participation in the beginning of this project. We appreciate the help offered by Luis Montejano, for mer Director of the Mathematics Institute of the UNAM, and Concepcion Ruiz, former Director of the Mexican Sci ence Museum Universum, for their help and for giving us the opportunity of presenting this in various forums. It is our pleasure to thank Jorge Urrutia for many stim ulating discussions on convexity and his guidance during the completion of the first part of the work; and Margaret Schroeder for her help with the translation of a first ver sion of this paper. We appreciate their patience. We would like to thank Hector Lomeli and Jose Luis Farah for their comments in the final editing of this paper. We would credit the creator of the cartoon in Figure 1, only we don't know who it is. Congratulations to the car toonist on having your work pass into folklore. Finally, we thank COSI Columbus, David Eppstein, and Stan Wagon for providing us the photos in Figure 3. From top to bottom: • • •
Square-wheeled car from Cleveland Science Museum ex hibit.Photo courtesy of COSI Columbus. Photo courtesy of David Eppstein from the Exploratorium at the Palace of Fine Arts in San Francisco, California Stan Wagon on his bicycle at Macalester College. Photo courtesy of S. Wagon.
REFERENCES
[1 ] M. Gardner. Mathematical games: Curves of constant width, one
of which makes it possible to drill square holes. Scientific American 208, no. 2 (1963), 1 48. [2] L. Montejano. Cuerpos de Ancho Constante. Fonda de Cultura
Econ6mica, Mexico, 1998.
[3] H. Rademacher and 0. Toeplitz. The Enjoyment of Mathematics.
Princeton University Press, Princeton, NJ, 1957. [4] L. Santa16. Integral Geometry and Geometric Probability. Addison
Wesley, Reading, 1976.
[5] I. M. Yaglom and V. G. Boltyanskii. Convex Figures. Holt, Rinehart
and Winston, New York, 1961 .
AUTHORS
SEBASTIAN YON WUTHENAU MAYEA
CLAUDIA MASFERAER LEON
!nst1tut0 Techno16gco AutOnomo de M8xico
lnstrtuto Techno16g100 AutOnomo de MexiCO
San
Angel, MeXICO City,
M9XICO
San
Angel, MexiCO C1ty, Mexico
e-mail:
[email protected] e-mail:
[email protected] Wuthenau partiCipated in
Bom tn Mexico 1n 1980. Claudia Masferrer partiCipated tn the Mex
lnst1tute at the Weizmann lnstrtute. This experience led to his
slltute at the Weizmann Institute. This experience led to her deci
tion he worked tor McKinsey & Co. MBXJCO as a bus1ness analyst.
director of her unrversity literary joumal. She Intends to go on to
Bom In M9X1Co 1n 1980. SebastiAn
von
the Mex1can MathematiCS Olympiad, and later 1n the Summer Sci ence
decisiOn to study Apphed MathematiCS at the !TAM. After gradua He
will begin graduate stud1es
next year; hiS ma1n Interests are
geomelly and d1screte mathematiCS. tn partiCUlar cryptography. He
enjoys outdoor sports and travel.
ican Mathematics Olympiad, and later 1n the Summer Science ln
Sion to study Applied MathematiCS at the ITAM. In 2004 She was
graduate school, but at present she
IS
wor1pesetting
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28
THE MATHEMATICAL INTELLIGENCER
lt%Mj.i§j.@iil£11§%§4flhi l§.id ..
Meekness in Ornation: How the Weirdoes Collude
M ichael Kleber and Ravi Vaki l , Editors
T
he answers to the four italicized clues were the eight-letter words THEOREMS, CALLIOPE, SCHOOLED, and couNTESS. Each is the result of a per fect riffle shuffle of two four-letter words-THE oREM s, for example. The constituent four-letter words appeared as the unclued entries, artfully arranged so that each's uncrossed let ter was ambiguous.
The italicized words in the instruc tions, WEIRDOES and COLLUDES, share this property. Thanks to David Miller and Thomas Colthurst for suggestions. The Entertainments editors welcome submissions of crosswords or other puzzles with similar appeal. They should specifically target the mathematically in clined audience of this publication, but otherwise should be broadly accessible.
Michael Kleber
Solution to the Weirdoes Puzzle published in val. 2 7 no. 3
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.
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Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-21 25 , USA
e-mail:
[email protected] © 2005 Springer Science+Business Media, Inc., Volume 27, Number 4, 2005
29
Brouwer's grave and the glass plate with inscriptions.
30
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+Bus1ness Media, Inc,
liiJ$•1,ijj,J§i.£hi.JihtJI i%11
A Blaricum Topology for Brouwer Dirk van Dalen
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made,
D i r k H uylebrouck, Editor
ne hundred years ago, L. E. J. Brouwer settled in one of the most attractive Dutch villages, Blaricum. The village had a reputation for undis turbed landscapes, for artists, and for experiments in social communes. One of these communes, the Christian An archists, was led by the charismatic Professor Van Rees. When this com mune fell apart, Brouwer bought part of the land and asked his friend Rudolph Mauve (son of the famous painter Anton Mauve) to design a small cottage for him. The cottage, called "the hut," was ready in 1904. In that year Brouwer and his bride Lize moved in; they remained faithful to the hut and Blaricum for the rest of their lives. The property contained some rem-
O
I
nants of the old commune (e.g., a ro tating "tuberculosis" hut). In the course of time Brouwer added some small buildings (e.g., the Padox). In the 1920s he bought a neighboring villa, De Pim pernel at the Torenlaan. In 1925-26 the Hut and De Pimpernel were the center of the Dutch topological school, with Alexandrov, Menger, Newman, Vi etoris, and even Emmy N oether as short-term visitors. The village, and the whole area, called Het Gooi, were for a long time the home of a rich variety of artists (e.g., Piet Mondriaan); it also attracted the attention of well-known Dutch ar chitects. Even today, the village offers a panorama of original (small) farm houses, interesting eccentric houses,
the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels?
Huizerweg 526)
If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor,
Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium
Map of the center of Blaricum, showing the cemetery (above) and
e-mail:
[email protected] the Torenlaan (below), where Brouwer lived.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
31
Red Brouwer (t)
Save Brouwer ft ) Translat ion of De Volkskrant arti ·le
90
mb
r 2000
Een doodgewoon graf is het, op de Gemeentelijke Begraalplaats aan de Woensbergweg in het noorden van Blaricwn. L. E. J. Brouwer staat er op de sobere steen, 18811966, emaast het graf van de in 1959 overleden mevrouw Brou wer-de Holl. Oat hier volgens de kenners de grootste Nederlandse wiskundige sinds Christiaan Huy gens rust, blijkt nergens wt Tach, zegt Brouwers biograaf prof. dr. Dirk van Dalen, filosoof te Utrecht, is dit graf een beschei den bedevaartsoord vooi ingewij den, buitenlandse wiskundigen
vooral, die weten hoe de piepjon
ge Nederlander begin twintigste eeuw een born onder de gevestig de wiskunde legde. Sinds vorig jaar op Torenlaan 70 zonder par don Brouwers oude houten woon huis ('de hut') is gesloopt - om plaats te maken voor een forse nieuwbouwvilla - is het graf het enige spoor dat nog van de grote wiskundige re.'t. En dat, meldt de hevig veront ruste Van Dalen, kan binnenkort ook wei eens verleden tijd zijn. Bij navraag ontdekte ,hij onlangs dat beide graven van de Brouwers aan de gemeente Blaricwn zijn verval len omdat de grafrechten al lang niet meer waren voldaan. 'De gra ven kunnen zo geruimd worden. Oat zou na het verlies van de hut verschikkelijk zijn.' Van Dalen heeft de zaak-Brou wer met de burgemeester per soonlijk opgenomen. Die bleek oud-wiskundeleraar en had dus van nature enige voeling met de uak. Van Dalen: 'lruniddels heeft hij me gezegd de zaak wei te wit
len proberen te redden.' Uiteindelijk is het echter de ge-
Blaricum, Waalman,
C met
also
rvic responsibl
ry, witlt unmistakabl
e'\"ant card
ut of the card-ind
·
reads out Th
removal
rv
rtcd to th
nt
no plans of th
il would have to approve That
·
Slt
ably corn Mart11n
32
van
m
due tim
"
CalmthOut
THE MATHEMATICAL INTELLIGENCER
rvi
rL Anyway, th
cultuurhistoriscbe monument voor Nederland te behouden'. Maar hoe dat precies zou moeten, ook linancieel, weet de Utrechtse filosoof nog niet. 'Het gaat er nu om een eventuele ruiming te blok keren', zegt hij. Hoofd Waalman van de Buite!l' dienst van de Gemeente Blari cwn, tevens verantwoordelijk voor de Gemeentelijke Begraaf plaats, haalt met hoorbare tegen zin de betreffende kaart uit de kaartenbak. Brouwer, L. E. J., dat is inderdaad perceel J 32. Het graf is al in 1994 vervallen aan de ge meente, leest hij voor. Er kan dus inde.rdaad geruimd worden. Maar een ruimingsbesluit is er niet en de dienst Buitendienst heeft momen tcel ook geen plannen in die rich1 ting. En dan nog zou de gemeente raad die eerst moeten goedkeu ren. Vandaar, waarschijnlijk, dat hij als verantwoordelijke de burge meester nog niet over de graf kwestie heeft gehoord. 'Oat zal dan nog we! komen.'
1-
L.E.J.,
town
has
coun-
h plaru.
why he, as the person in
heard about the matter of tlt
r
gemeente 'Wegen te vinden dit
ould therefore take
place. But t here · · n o such d(>('ision, and t h at the mom
the
box. Brouw r,
which tS indeed lot J 32. Th grave
in 1994, he
for th
reluctance fish
meentetaad die beslist. En dus heeft Van Dalen deze week een brief naar de Gooise gemeente op de post gedaan waarin hlj nog eens bet cultureel-wetenschappe lijke belang van de in 1966 overle den Blaricummer uiteenzet. 'Bia ricwn', besluit hlj, 'kan terecht trots zijn een geleerde van het for maat van een Newton of Gauss onder haar bewoners te hebben geteld. Het is ondenkbaar dat Cambridge of Gottingen de gra ven van Newton of Gauss zou op geven.' Van Dalen vraagt in de brief de
gr,w
harge, has not y t
.� That
y,;U pr b Copy of the original newspaper article.
Hartijn VIlli Calmthout
A cottage designed in 1904 by Brouwer's friend Rudolph Mauve. It was always referred to as the hut.
Brouwer at work in his Blaricum place (Brouwer Archive).
© 2005 Springer Science+ Business Media, Inc. . Volume 27, Number 4 . 2005
33
"Pimpernel," the villa in Blaricum, adjacent to the hut.
A small structure on a rotating base that could follow the sun. These little houses were often used by tuberculosis patients, hence the name "TBC hut."
34
THE MATHEMATICAL INTELL IGENCER
"The Padox," a prefab house, used for guests.
and opulent villas in the pre-war style. In 2000, Brouwer's hut and the other small buildings fell victim to property developers; fortunately De Pimpernel escaped the demolition crews. The fate of the hut raised fears that Brouwer's grave, for which the lease had run out, could also be cleared out. The national press voiced its concern (see inset), and the town of Blaricum acted with a great sense of responsi bility; it decided to preserve the graves of Lize and Bertus Brouwer and to care for the graves. The Dutch mathematical community (repre sented by the Royal Dutch Mathemat-
ical Society) and the University of Amsterdam acted fittingly by placing a modest memorial-a glass plate etched with the text "Luitzen Egber tus Brouwer, Mathematician-Philoso pher. Father of the New Topology. Founder of Intuitionism," followed by a text in Brouwer's handwriting etched into the glass plate. A bus from the train station in Hil versum takes the visitor to the center of Blaricum, from which it is a 10minute walk to the cemetery. For hikers, there is a path round the IJsselmeer, the Zuiderzeepad, which passes by the cemetery (see map).
More historical information on Brouwer can be found in my biography Mystic, Geometer and Intuitionist: The Life of L.E.J. Brouwer. Vol l. The Dawning Revolution; vol. 2, Hope and Disillusion, Oxford University Press, 1999 and 2005, resp. The unique pho tographs that accompany this contri bution are by Dokie van Dalen.
Department of Philosophy Utrecht University 3508 TC Utrecht The Netherlands e-mail:
[email protected] © 2005 Springer Science +Business Media, Inc., Volume 27, Number 4, 2005
35
la§'jl§l.lfj
Osmo Pekonen , Editor
I
Cogwheels of the M ind. The Story of Venn Diagrams by A. W. F. Edwards with a foreword by Ian Stewart THE JOHNS HOPKINS UNIVERSITY PRESS, BALTIMORE, MARYLAND, USA. 2004, xvi+ 1 1 0 pp. $25.00 ISBN:
0-8018-7434-3.
REVIEWED BY PETER HAMBURGER
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Column Editor: Osmo Pekonen, Agora Center, University of Jyvaskyla, Jyvaskyla,
40351 Finland
e-mail:
[email protected] 36
his book purports to be on recre ational mathematics. It purports to popularize an area of discrete geome try, namely Venn diagrams. Unfortu nately, the text is confusing and fl.lled with historical and mathematical mis takes; many of the figures are useless or meaningless; many statements are pretentious or marred by the author's seeming vendetta against mathemati cians. I will detail some of these com plaints below and also try to set straight a record the author has misreported. The famous three-circle Venn dia gram, which is known to most people, had already been used by Euler. Venn himself calls this diagram "Euler's fa mous circles." So why do we speak of Venn diagrams and not Euler diagrams? I believe there are two reasons. It was John Venn who first gave a rigorous de finition of the notion (though he did not always follow it consistently); and he was the first to prove that the desired diagrams exist for any number of sets. A modem definition is this. A planar Venn diagram is a set of n closed non-self-intersecting continuous planar curves, intersecting each other in iso lated points, and such that the con nected components of the complement (which are bounded by unions of arcs of these curves) are 2n in number. Then these regions can be assigned distinct binary codes, in the following manner. Label the curves 1, 2, . . . , n. If a region is inside the curve i, then write 1 in the ith place in its binary code, otherwise write 0. As the n-digit binary codes are
T
THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc.
exactly 2n in number, the definition of Venn diagram means that they allow all the codes to be assigned to regions. Branko Griinbaum wrote the fol lowing [8]: "Venn diagrams were intro duced by J. Venn in 1880 [ 12] and pop ularized in his book [13]. Venn did consider the question of existence of Venn diagrams for an arbitrary number n of classes, and provided in [12] an in ductive construction of such diagrams. However, in his better known book [13], Venn did not mention the con struction of diagrams with many classes; this was often mistakenly in terpreted as meaning that Venn could not fmd such diagrams, and over the past century many papers were pub lished in which the existence of Venn diagrams for n classes is proved."
H is "cogwheels of the mind" have a mon key wrench in the works .
It is unsettling to find, in this book's introduction by an eminent mathemati cian, the mistake perpetuated: "But what of five, six . . . any number of sets? On this question, the great man was silent" (p. xi). The author, though denying that he means to make priority claims (p. xv), seems to do so when he says (falsely) that the possibility of drawing Venn di agrams on the sphere was ignored "for more than a century until it indepen dently resurfaced and inspired the gen eral solution. It is as though Venn had no geometric insight . . . " (p. 15). Again, in the introduction the author enthusi astically declares that Edwards "has added further conditions to the shapes he seeks-conditions like symmetry. He has become a world expert on Venn di agrams." But no: it was Henderson [9] who posed the problem of symmetric Venn diagrams; he found some with five or seven curves, as did Griinbaum,
Schwenk, and Gointly) Savage and Winkler, all earlier than Edwards. Edwards knows that Griinbaum showed in 1975 [8] the possibility of constructing Venn diagrams with any number of convex curves, and he recognizes that this was a "re markable advance," but he gratuitously disparages others' figures in comparison to his own, and he misrepresents the history. This development began with another remarkable advance by A. Renyi, V. Renyi, and J. Suranyi in 1951 [10]. Edwards must have been aware of the paper [ 10], the start ing point for all modem study of Venn diagrams, for it is given its due in [8]. Griinbaum's result is stronger than Ed wards quotes: not only may all the n curves be chosen so that they are convex, but also so that the 2n - 1 interior in tersection regions, and also their union, are convex. A Venn diagram is called reducible if there is some one of its curves whose deletion results in a Venn diagram with one less curve. It is called simple if at every intersection at most two curves meet. It is known that there are irreducible Venn diagrams, and Edwards refers to this "counterintuitive property" (p. 23)-diagrams with 5 curves can even be sim ple irreducible-but Edwards says falsely (p. 43) that if a Venn diagram can be built up by adjoining n curves one by one, that determines its topological (graph-theoretic) struc ture uniquely. Some of the reducible structures can be re alized by curves all of which are convex, and some can not; even among those which can, there are many graphically different ones. This richness is one of the attractions of the subject to the geometer. In Chapter 6 Edwards discusses the dual graph, stating, "The dual graph of a Venn diagram is a maximal planar sub graph of a Boolean cube" (p. 77). He says he realized this in 1990 but his paper on the subject was rejected. (Well may it have been! The dual graph is always a planar sub graph, but it need not be a maximal one unless the given Venn diagram was simple.) He cites a 1996 paper [2] but says "the proof, though trivially short, assumes a knowl edge of graph theory and is therefore omitted here" (p. 83). Who can be the intended reader? Someone who would be daunted by a trivial proof in graph theory and yet can cope with maximal planar subgraphs of the hypercube ! One of the most disturbing mistakes in the book is when Edwards presents an induction argument to prove a state ment, namely, every Venn diagram can be colored with two colors such that no regions with common arc boundary have the same color, p. 23. This statement does not need inductive proof, and the proof offered is incorrect. It is one of the types of "proof' that college instructors have diffi culty explaining to their students why it is incorrect; Johns Hopkins University Press by publishing this book under mines their teaching efforts. After publication of the fundamental [ 10], there was a pause before the study of Venn diagrams was revived by Griinbaum [6], [7], [8] and Peter Winkler [ 14]. Their deep understanding and challenging conjectures have motivated more recent work. Let me mention two advances here. In [2] the authors show that it is possible to extend any pla nar Venn diagram to a planar Venn diagram with one more
curve. In [5] the authors show that for every prime number p there is a planar Venn diagram with p curves and p-rota tional symmetry. In both problems, it remains unknown whether the Venn diagrams can be chosen simple. The lat ter of these problems is surveyed by Barry Cipra [3]. Read ers may consult an online, regularly updated survey [ 1 1]. An accurate essay by M. E. Baron [1] gives the history of representations of logic diagrams up to the time of Venn. The author's sniping at mathematicians reflects a pro found ambivalence, shown explicitly in this passage (pp. xx-xvi):
Mathematical discovery is perhaps the most delightful ex perience which Academic life has to offer. The pure math ematician G. H. Hardy (1877-1947) wrote in A Mathe matician's Apology, "It will be obvious by now that I am interested in mathematics only as a creative art, " but Hardy was a mathematician's mathematician and most of us cannot appreciate his work. One of thejoys of work ing with Venn diagrams is that there have been simple delights still to be uncovered that can be appreciated by the far wider audience of amateur mathematicians (amongst whom I count myself, for my Cambridge col lege, Trinity Hall, declined to admit me to read the math ematical tripos, for which I am grateful because it meant I became a scientist instead). Hardy created beautiful mathematics, but working with Venn diagrams has been much more of a voyage of discovery. Though Hardy felt his research to be a creative en deavor, he surely regarded it as discovery! One does not begrudge Edwards his post-retirement hobby of venturing in our domains, and if he takes satisfaction in being a sci entist rather than a mere mathematician, let him. But his lofty position as College Dean and status as scientist (not to be undermined, I hope, by any humorless book reviewer) do not entitle him to publish his dabblings without bring ing them up to the standards of our science. Truly his "cog wheels of the mind" have a monkey wrench in the works! REFERENCES
[1 ] M. E. Baron, "A Note on the Historical Development of Logic Di agrams: Leibniz, Euler and Venn, Mathematical Gazette 53 (1 969), 1 1 3-1 25. [2] K. B. Chilakamarri, P. Hamburger, R. E. Pippert, "Hamilton Cycles
in Planar Graphs and Venn Diagrams," Journal of Combinatorial The ory Series B 67 (1 996), 296-303.
[3] B. Cipra, "Venn Meets Boole in Symmetric Proof, " SIAM News 37,
no. 1 (January/February 2004).
[4] L. Euler, Lettres a une Princesse d'AIIemangne, St. Petersburg, 1 768. English translation: H. Hunter, Letters to a German Princess,
London, (1 795). [5] J. Griggs, C. E. Killian, C. D. Savage, "Venn diagrams and sym
metric chain decompositions in the Boolean lattice," The Electronic Journal of Cornbinatorics (2004) [6] B. Grunbaum, "Venn Diagrams and Independent Families of Sets," Mathematics Magazine 48 (1 975), 1 2-22.
[7] B. Grunbaum, "The construction of Venn diagrams, " College Math ematics Journal 1 5 (1 984), 238-247.
© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 4, 2005
37
[8] B. Gri.inbaum, "Venn diagrams 1," Geombinatorics 1 (1 992), 5-1 2 . [9] D. W. Henderson, "Venn Diagrams for More Than Four Classes, " American Mathematical Monthly 7 0 (1 963), 424-426.
[1 O] A. Renyi, V. Renyi, and J. Suranyi, "Sur l'lndependance des Do maines Simples dans I'Espace Euclidien a n-dimensions," Collo quium Mathematicum 2 (1 95 1 ) , 1 30-1 35.
[1 1 ] F. Ruskey, M . Weston, The Electronic Journal of Combinatorics, www.combinatorics.org/SurveysNennEJC.html
[1 2] J . Venn, "On the diagrammatic and mechanical representation of propositions and reasonings," The London, Edinburgh, and Dublin Philos. Mag. and J. Sci. 9 (1 880), 1 -1 8.
[1 3] J. Venn, Symbolic Logic, Macmillan, London, 1 881 , second edi tion 1 894. [1 4] P. Winkler, "Venn diagrams: some observations and an open prob lem," Congressus Numerantiurn 45 (1 984), 267-274. Department of Mathematical Sciences Indiana University- Purdue University Fort Wayne, Fort Wayne, IN 46805 USA e-mail:
[email protected] In the Light of Logic b y Solomon Fejerman OXFORD U NIVERSITY PRESS, 1 998, 352 PP. $ 60.00 US, ISBN 01 95080300
REVIEWED BY ANDREW ARANA
P
oincare famously compared the logician's understand ing of mathematics to the understanding we would have of chess if we were only to know its rules. "To understand the game," Poincare wrote, "is wholly another matter; it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of moves a sort of organized whole." (P, pp. 217-218] The Dutch mathematician L. E. J. Brouwer took a position similar to Poincare's: genuinely mathematical rea soning is not simply a matter of logical inference. It is, as Poincare put it, a matter of mathematical insight. Despite those views concerning logic, Poincare and Brouwer believed in studying the foundations of mathe matics, and indeed they carried out fundamental work in this area. This might strike contemporary minds as a bit odd, but it is a consistent view. Mathematical logic and the foundations of mathematics are frequently lumped to gether, as though they are the same. They are not. Mathe matical logic is a mature mathematical subdiscipline, with its own problems generated by reflecting on what is known from other logic problems and solution attempts. Like any mature mathematical subdiscipline, what counts as a good problem is largely determined by factors "internal" to the subdiscipline, such as how the problem contributes to other work in progress and to what is already known. Foun dations of mathematics, on the other hand, has a different 38
THE MATHEMATICAL INTELLIGENCER
standard. It raises questions about the objects and struc tures of mathematics: what are they, and how do we know anything about them? It raises questions about mathemat ical statements: how should we go about discovering and justifying them? It raises questions about mathematical proofs: what is a proof, what kinds of proofs do we prefer, and for what reasons? Foundations of mathematics is there fore not a mathematical subdiscipline at all, but rather a body of reflections on mathematics itself. A striking insight reached by David Hilbert and others in the early twentieth century was that the foundations of mathematics could be studied by the application of math ematical logic. By taking mathematical objects and struc tures to be described by axioms in formal languages, these axioms and their consequences could be studied using mathematical logic. In this way, contra Poincare and Brouwer, logic could be used to shed light on the founda tions of mathematics, the light of logic to which the title of Feferman's excellent book refers. Ofthose who have shed light on the foundations of math ematics using logic, there is one figure whose influence and views tower over the rest: Kurt Godel. His incompleteness theorems both answered existing questions and raised many new ones, thereby deepening considerably the study of the foundations of mathematics. On account of that, his specter haunts almost every page of Feferman's book Feferman classifies the essays (all previously published) of the book into five parts based on their topics, and for each topic, Godel's work and views are of utmost importance. In Part I, Feferman raises as a problem the role of transfinite set theory in mathematics. Because transfinite sets are sup posed to be infinite objects about which facts are true inde pendent of our abilities to verify them, it seems that these ab stract entities must exist independent of human thoughts or constructions. This family of beliefs about sets is frequently called platonism. Feferman finds platonism philosophically unsatisfying, and thus he presents the three projects aimed at avoiding platonism: L. E. J. Brouwer's "intuitionism," David Hilbert's "finitism," and Hermann Weyl's "predicativism." Feferman characterizes Brouwer's solution as excessively radical, leaving Hilbert's and Weyl's as acceptable options. Feferman believes that Godel's incompleteness theorems cast doubt on the viability of Hilbert's project, as is commonly (but not universally) thought. This leaves Weyl's predica tivism as Feferman's preferred alternative to platonism. I will return to predicativism shortly. The discussion in Part I sets the agenda for the rest of the book Finding an acceptable alternative to platonism emerges as one central theme. Another is the question of whether there is any justification for new axioms for set theory. These two themes are tied together by GOdel's view that platonism could be used to justify new axioms for set theory. These new axioms assert the existence of sets which Godel thought the platonist had every reason to be lieve in, on account of their uniformity with sets already believed to exist, and on account of a sense-perception-like faculty he thought we possess for experiencing mathemat ical objects. In addition, he supported new axioms for set
theory because he thought they would eventually be used
sis, are just sets; but this requires that we justify our use
to solve open mathematical problems, just as they can be
of sets. Feferman is critical of platonist attempts to justify
used to prove the arithmetically unprovable sentences that
set theory, and offers instead a predicativist view.
he had studied in his work on the incompleteness theo
I think there are four main reasons why Feferman thinks
rems. We may justly view Feferman's book as a wrestling
that predicatively definable sets are justifiable, as follows.
match with Godel, the arch-platonist. It is unsurprising,
(1) Consider Grelling's paradox. Suppose we define a word
therefore, that Feferman dedicates one of the book's five
as being heterological if it does not describe itself. The word
parts-Part III-to essays on Godel's life and work
"heterological" is heterological if and only if it is not het
Though these central themes are explored in every part
erological. This definition is infelicitous, since it does not
V to his preferred
determine whether "heterological" is heterological. Pre
alternative to platonism, predicativism. Here Feferman
dicative definitions avoid these vicious circles, as follows.
argues first against attempts to show that transfinite set the
We typically define sets as consisting of all objects satis
of the book, Feferman returns in Part
ory is
necessary
for ordinary finite mathematics. Respond
fying some condition. In predicative definitions, the satis
ing to arguments of Godel and Harvey Friedman, Feferman
faction of this condition for all objects is determined inde
concludes that "the case remains to be established that any
pendently of the set being defined. Hence, there are no
the mathematics of the finite in the everyday sense of the
finable sets entails commitment to whatever is needed for
word" (p. 243). Instead, he supports a much more restricted
Peano Arithmetic, presumably just countably infinite sets.
use of the Cantorian transfinite beyond � 0 is necessary for
vicious circles.
(2) Our commitment to predicatively de
view on the transfinite, maintaining that only predicatively
(3) Predicatively definable sets suffice for doing all scien
definable sets should be
admitted. A set is predicatively de
finable if it is defined by way of the system of natural num
tifically applicable mathematics, so working with just them is adequate for the applicability of mathematics. (4) Pre
bers, or by way of predicatively definable sets that have al
dicatively definable sets suffice for doing all ordinary finite
ready been defined. Sets defined by way of a collection of
mathematics, perhaps the minimum part of mathematics
sets that includes the set to be defined are thereby excluded,
for which any reasonable foundation must account.
such as the "set" of all sets that do not contain themselves,
I will comment briefly on these four reasons.
as used in Russell's paradox. Feferman explains how he used methods from modem logic to develop Weyl's pre
1. The avoidance of vicious circle paradoxes does not en
dicative set theory, yielding a system in which, he argues,
sure the consistency of predicative mathematics. Pre
This system is up to such a task, he argues, because analy
dicative mathematics, but that does not mean that it is
sis, both classical and modem, can be formalized within it.
perfectly secure. Indeed, as Feferman showed in
Yet any (first-order) truth that can be proved in this system
the consistency of predicative analysis cannot be proved
can be proved from the (first-order) Peano Arithmetic ax
predicatively, though it can be proved impredicatively
all "scientifically applicable mathematics" can be proved.
dicative mathematics may be more secure than impre
1964,
1-30] . Furthermore, this characterization of the
ioms, which formalize elementary number theory. Feferman
[F, pp.
argues that this vindicates his view that the predicativist
value of predicativity leaves it open whether predicative
need not admit any transfinite sets beyond the countably in
definitions have any other value. One reason to be wor
finite, because, he maintains, commitment to Peano Arith
ried about this is that there are many sets that can be
metic entails commitment only to the countably infinite.
defined predicatively, but whose impredicative defini
In Parts II and IV of the book, Feferman discusses how
tions mathematicians find more natural. For instance,
logic can be used to shed light on aspects of mathematical
the closure of a set in a topological space is naturally
practice besides that part already formalized within set the
defined as the intersection of all closed sets containing
ory. He critically examines Imre Lakatos's views on math
the set, but this is impredicative. Mathematicians typi
ematical discovery, comparing them with George P6lya's
cally find this definition unproblematic because the ex
views on discovery. He explains how logic can help clarify
istence of the sets involved follows from set-theoretic
vague mathematical concepts such as
construction, infin
axioms such as ZFC. Predicativists reject existence-in
itesimal,
In particular, Fefer
ZFC as sufficient for set existence, demanding instead a
man uses his expertise in proof theory, a branch of math
description (in some weaker axiomatic system, perhaps)
and
natural well-ordering.
ematical logic, to emphasize its utility for understanding
of how a set may be generated from other sets already
As he explains, proof theory can be used to
known to exist. Consider also the following example:
clarify what parts of mathematics can be reduced to other
given a homeomorphism of a compact space, there is al
mathematics.
parts, and in what ways. Feferman's moral is that logic is
ways a "minimal" non-empty closed invariant subset.
useful for more than just the systematic organization of pre
The standard proof uses Zorn's lemma and intersections,
existing, well-understood bodies of mathematics-though
and is thus impredicative. There is a predicative proof,
it is useful for that too.
but it is more involved than the standard proof [BHS, p.
Part of accounting for mathematical practice is saying
152]. (Thanks to Jeremy Avigad for pointing out this ex
how we are justified in admitting the objects we seem to
ample to me.) Predicativity thus exacts a toll, in that it
need to do mathematics in specific areas like analysis. Fre
costs us natural definitions and proofs-leaving what is
quently this is done by saying that the objects of, e.g., analy-
natural unspecified but, I take it, uncontroversial in
© 2005 Springer Sc1ence+Bus1ness Media, Inc., Volume 27. Number 4, 2005
39
these cases. We must weigh the apparent security pur chased by requiring predicative definitions against the burden of having to abandon in many cases what we, as mathematicians, consider natural definitions. 2. It is unclear exactly what objects we are committed to when we are committed to Peano Arithmetic. There are plenty of problems in number theory whose proofs use analytic means, for instance. Does commitment to Peano Arithmetic entail commitment to whatever objects are needed for these proofs? More generally, does commit ment to a mathematical theory mean commitment to any objects needed for solving problems of that theory? If so, then Godel's incompleteness theorems suggest that it is open what objects commitment to Peano Arithmetic entails. 3. As Feferman admits, it is unclear how to account pre dicatively for some mathematics used in currently ac cepted scientific practice, for instance, in quantum me chanics. In addition, I think that Feferman would not want to make the stronger claim that all future scien tifically applicable mathematics will be accountable for by predicative means. However, the claim that currently scientifically applicable mathematics can be accounted for predicatively seems too time-bound to play an im portant role in a foundation of mathematics. Though it is impossible to predict all future scientific advances, it is reasonable to aim at a foundation of mathematics that has the potential to support these advances. Whether or not predicativity is such a foundation should be studied critically. 4. Whether the use of impredicative sets, and the un countable more generally, is needed for ordinary finite mathematics, depends on whether by "ordinary" we mean "current." If so, then this is subject to the same worry I raised for (3). It also depends on where we draw the line on what counts as finite mathematics. If, for in stance, Goldbach's conjecture counts as finite mathe matics, then we have a statement of finite mathematics for which it is completely open whether it can be proved predicatively or not. In emphasizing the degree to which concerns about predicativism shape this book, I should not overempha size it. There is much besides predicativism in this book, as I have tried to indicate. In fact, Feferman advises that we not read his predicativism too strongly. In the pref ace, he describes his interest in predicativity as con cerned with seeing how far in mathematics we can get without resorting to the higher infinite, whose justifica tion he thinks can only be platonic. It may tum out that uncountable sets are needed for doing valuable mathe matics, such as solving currently unsolved problems. In that case, Feferman writes, we "should look to see where it is necessary to use them and what we can say about what it is we know when we do use them" (p. ix). Nevertheless, Feferman's committed anti-platonism is a crucial influence on the book. For mathematics right now, Feferman thinks, "a little bit goes a long way," as one of the essay titles puts it. The full universe of sets
40
THE MATHEMATICAL INTELLIGENCER
admitted by the platonist is unnecessary, he thinks, for doing the mathematics for which we must currently ac count. Time will tell if future developments will support that view, or whether, like Brouwer's view, it will re quire the alteration or outright rejection of too much mathematics to be viable. Feferman's book shows that, far from being over, work on the foundations of mathe matics is vibrant and continuing, perched deliciously but precariously between mathematics and philosophy. REFERENCES
[BHS] A. Blass, J. Hirst, and S. Simpson, "Logical analysis of some theorems of combinatorics and topological dynamics," in Logic and Combinatorics (ed. S. Simpson), AMS Contemporary Mathematics
val. 65, 1 987, pp. 1 25-1 56. [F] S. Feferman, "Systems of Predicative Analysis," Journal of Symbolic Logic 29, no. 1 (1 964), 1 -30.
(P] H. Poincare, The Value of Science (1 905), in The Foundations of Science, ed. and trans. G. Halsted, The Science Press, 1 946.
Department of Philosophy Kansas State University Manhattan, KS 66506 USA e-mail:
[email protected] The SIAM 1 00-Digit Challenge: A Study in High-Accuracy Numerical Computing by Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jorg Waldvogel SIAM. PHILADELPHIA, PA, USA 2004, Xll+306 PP. SOFTCOVER, ISBN 0-8987 1 -561 -X, US$57.00
REVIEWED BY JONATHAN M. BORWEIN
L
ists, challenges, and competitions have a long and pri marily lustrous history in mathematics. This is the story of a recent highly successful challenge. The book under re view makes it clear that with the continued advance of com puting power and accessibility, the view that "real mathe maticians don't compute" has little traction, especially for a newer generation of mathematicians who may readily take advantage of the maturation of computational pack ages such as Maple, Mathematica, and MATLAB. Numerical Analysis Then and Now
George Phillips has accurately called Archimedes the first nu merical analyst [2, pp. 165-169]. In the process of obtaining his famous estimate 3 + 10/71 < TT < 3 + 1n, he had to mas ter notions of recursion without computers, interval analy sis without zero or positional arithmetic, and trigonometry without any of our modem analytic scaffolding. . . . Two millennia later, the same estimate can be obtained by a computer algebra system [3].
Example 1. A modem computer algebra system can tell one that
0
n2> .. .nk� l
THE MATHEMATICAL INTELLIGENCER
M= and other examples. Chapter 5 is devoted to a number of miscellaneous subjects: prime number conjectures, repre sentation of integers by xy + yz + zx, Grobner bases and metric invariants, spherical designs, and many others. Chap ter 6 is a sequel to Chapter 5 of the first volume: its intent is to illustrate that very classical undergraduate theorems of real but especially complex analysis are amenable to prac tical computation, and in fact help to solve computational problems. The final Chapter 7, which is a sequel to Chapter 6 of the first volume, gives a number of other numerical tech niques. The authors have little to say on the Wilf-Zeilberger algorithm of creative telescoping, primality testing, comput ing complex roots of polynomials, or the use of Euler MacLaurin for infinite series summation. Nevertheless, Sec tion 7.4 contains a description of numerical quadrature methods, including the remarkably efficient doubly-expo nential methods such as the tanh-sinh method. Once again I urge the reader to pursue the study of these methods. To conclude, these two books contain a wealth of di verse examples (I did not count, but it may reach 1000), al though the reader must be warned that there are many mathematical misprints. The two volumes are very enjoy able reading and belong on the bookshelves of any math ematician or graduate student who does mathematics for pleasure (which one hopes is the case for most of them!). Laboratoire A2X UFR de Mathematiques et lnformatique Universite Bordeaux I 33405 Talence Cedex France e-mail:
[email protected] .fr
lndra's Pearls. The Vision of Felix Klein b y David Mumford, Caroline Series, and David Wright
1
where the si are positive integers. The latter have been in recent years the object of a vast literature, in number the ory and analysis of course, but also in knot theory, combi natorics, and theoretical physics. These values are espe cially well suited to experimentation, because they can be easily computed to hundreds or thousands of decimals (al though the algorithms for computing them are not com pletely trivial), and constant recognition algorithms such 58
as PSLQ allow the experimenter to find remarkable rela tions between these values. One such experimentally dis covered relation by Zagier gives the value of �(3, 1, 3, 1 . . . , 3, 1) as a rational multiple of a power of 1r, and the authors include a proof of this relation. Chapter 4 is devoted to partition functions, special values of theta functions, Madelung's constant
CAMBRIDGE UNIVERSITY PRESS, CAMBRIDGE, 2002 396 PAGES, £33.00 HARDCOVER ISBN-1 0: 0521 352533 -ISBN-13: 9780521 352536)
REVIEWED BY LINDA KEEN
I
recently received this e-mail message from the distraught mother of a twelve-year-old girl: Dear Mrs. Keen, I left you a message this afternoon requesting your help. I realize this is an inconvenience for you but my 12 year
old daughter, Megan, has been assigned to write about
the text from becoming too cluttered. There are also very
your life and to list one of your theories and to explain
informative boxes giving biographical information on vari
As you can imagine this is difficult to do. I've attempted to read 4 of your papers and was
ous (dead) mathematicians who have made substantial
lost after the intros! I passed calculus with only a B and
projects for readers, involving both computer programs and
this is WAY over my head! Would you be able to pick
colored pencils and paper.
it in her own terms.
contributions. In addition, there are many well-thought-out
one theory and briefly explain it in layman's language.
The book begins with a discussion of symmetry as the
She needs to report what the theory is and how is it used.
basis of geometry as propounded by Felix Klein. In his
I really appreciate your help
view, geometry is the study not only of objects, triangles
I replied that Megan's teacher had given both her and
are just affine motions, but Klein's motions were more
me an impossible task I then suggested that they think
general. The key here is that the motions that preserve
and such, but also of motions. In Euclidean geometry these
about a curve that was so crinkled that no matter how much
symmetrical objects form a group. All the essential infor
it was magnified, it looked as crinkled as ever, and I indi
mation is encoded by the group. Thus, the first chapter is
cated how they might construct it. Megan wrote back
devoted to a discussion of symmetries of the plane, that
thanking me and telling me that the teacher read my e-mail
is, tilings of the plane by regular shapes. A group is de
to the class and they tried to work out the construction and
fined by the properties of the set of transformations
found it interesting. Megan and her mother were going to
needed to move one tile onto another. There are lots of
spend some more time on it over the weekend.
illustrations, both hand-drawn and computer-drawn. The
This incident points out the problem we, as individual
authors talk about their programming techniques and in
mathematicians, and as a community, have. How can we
troduce the first pseudo-code programs to create the tiling
convey to others what it is about mathematics that excites
pictures.
us and makes us get up in the morning? We even have trou ble telling other mathematicians what we do.
The next chapter tackles the next basic concept-com plex numbers. After a bit of history, the authors discuss the
There are some problems in mathematics whose state
arithmetic operations of complex numbers and computer
ments, at least, are accessible to laymen-for example, the
programs to implement them. They quickly progress to the
twin primes conjecture. But, how do we develop the ideas
Riemann sphere and stereographic projection. The impor
and the language that goes with them to tell anyone but a
tant motions here, of course, are inversions. The third chap
few specialists what we are working on?
ter completes the presentation of the basic material by dis
Since the late 1970s, as computers have become more
cussing Mobius transformations of the plane. These are
powerful and their graphics more sophisticated, mathemati
compositions of an even number of inversions. In addition
cians have been able to draw mathematical pictures that ap
to good pictorial descriptions of how these maps act, com
pear beautiful, even to non-specialists. Just as a nonmusi
puter programs are given that find the image of a circle un
cian can listen to a Haydn symphony and enjoy the music
der such a map.
without being able to articulate what it is about the under
Having assembled the mathematical and programming
lying structure that appeals, nonmathematicians can look at
tools, the authors move on to the groups of symmetries that
many of these pictures and fmd them pleasing.
they are really interested in. They start with two pairs of cir
The book under review,
Indra 's Pearls,
is on one level
an attempt to tell a broad audience something about math
cles in the plane, CA, Ca and CB, cb, and two Mobius maps
a
and b, where a maps the exterior of CA to the interior of Ca
b.
ematics. The book grew out of all three authors' enchant
and similarly for
ment with the computer pictures they have been making
pairs determines the symmetrical pattern for the group
The initial arrangement of these circle
a, b
for the last thirty years in their study of discrete groups of
formed by applying
Mobius transformations, otherwise known as Kleinian
binations to the pairs of circles. In the simplest case, the
and their inverses in various com
4 mutually disjoint disks, and the basic tile
groups. The pictures reminded them of the ancient Bud
circles bound
dhist dream of Indra's net. The infinite net, stretching
for the pattern is their common exterior. Tiling, using the
across the heavens, is made from diaphanous threads, and
group elements, covers the whole plane with the exception
at each intersection there is a reflecting pearl. In each pearl,
of a Cantor set-which the authors call "Fractal Dust"
all the other pearls are reflected, and in each reflection
that is invariant under any element of the group.
there are again infinitely many reflected pearls. They
To describe how to plot the fractal dust, the authors take
wanted to share the pictures with a broad audience of math
an excursion into the realm of search algorithms for trees.
ematicians and non-mathematicians alike. Even more, they
I have taught this often from computer science textbooks,
wanted to enable the reader to write programs to recreate
and this is the best description of these algorithms I have
the computer pictures.
seen.
To this end, the book is written in a chatty informal style
In the next two chapters, the initial arrangement of the
and begins at the beginning. The illustrations are both hand
circle patterns is changed. First, the circles are moved in
and computer-drawn. They are well chosen, and the cap
the plane until they just touch and form a "necklace"-CA
tions, set in the margins, give a good description of them.
is tangent to CB, CB is tangent to Ca, Ca is tangent to cb,
There are boxes containing calculations and asides to keep
and cb is tangent to CA. The map
a
sends CA to Ca and
© 2005 Spnnger Science+ Business Media, Inc., Volume 27, Number 4, 2005
59
sends the other three circles, now thought of as beads, to
relations, so each family depends on one complex parame
three smaller beads inside
Ca. The "necklace" is now made cb, CA, CB, and the three inside Ca. The
ter.
up of the six beads
variant sets move, but retain their basic characteristics: for
A, B,
b,
As the parameter varies in the complex plane, the in
act similarly, replacing each
example, the number of components of the complement of
of the original beads with three smaller beads. Iterating this
the fractal invariant. At some points in the plane, these char
other three maps,
and
process indefinitely, the beads of the necklace become
acteristics change: circles may appear in the fractal creating
smaller and smaller and more and more numerous, and
new components, components may disappear, etc. Such
form "Indra's necklace," a fractal continuum invariant un
points form the boundary for the family.
der the group generated by the maps
a
and
b.
In the Riley and Maskit families, there seem to be round
Then the initial arrangement is changed again, so that in addition to the tangencies above, the circles
CA and Ca are
also tangent. The circles no longer form a necklace. Nev
disks, albeit overlapping, in the simply connected invariant component. These disks form circle chains with discernible patterns.
As the parameter is varied appropriately, the
ertheless, the process of forming the necklace, applying the
chains persist until, at points called "accidentally parabolic
group transformations to the circles again and again, re
points, " they become chains of tangent disks. These are
sults in an invariant set for the group that is recognizable
boundary points of the family. The existence of the circle
as the classical Apollonian gasket.
chains reflects the relationship between the elements of the
After this, the going gets tougher and the mathematics
group as words in the generators and continued fractions.
becomes very deep. The groups are divided into families
The parameter spaces for these families of groups can
based on the pattern of the four initial circles. In each fam
also be drawn by computer. It is possible to get the com
ily, the invariant set has certain characteristics. These fam
puter to find the parameter values of enough accidentally
ilies of groups have historically been named after individ
parabolic boundary points to draw the boundary and see
uals-which is unfortunate because it makes it even more
that it is fractal and has an interesting structure of its own.
difficult to keep straight which is which. There are Classi
The book ends with a discussion of the relationship of the
cal Schottky groups whose invariant set is fractal dust,
material to three-dimensional topology and Thurston's work
Fuchsian groups whose invariant set is a circle, and Quasi
So, have the authors succeeded in their attempt to make
fuchsian groups whose invariant set is a closed fractal
this mathematics accessible to a broad audience? Could
curve. There are also Riley groups and Maskit groups,
Megan's mother understand the book? I doubt it, but she
whose fractal invariant sets divide the plane into one sim
might get something from it. The book is written to be read
ply connected invariant component with fractal boundary
on many levels, and a given reader will have to find his own.
and infinitely many components whose boundary is a round
An undergraduate who has taken some algebra and complex
circle.
analysis can certainly get something out of the book, espe
These families of groups depend on parameters, the triple
ta, tb, tab.
cially the material and projects in the first three chapters; a
subject to
sophisticated mathematician can get the flavor of the subject;
certain relations depending on the family. For Fuchsian
and a graduate student can work her way from the beginning
of traces of the Mobius transformations,
As
groups, the parameters are real and there is one relation, so
to interesting research problems in some of the projects.
the family depends on two real parameters. For the Riley and
an "expert," I er\ioyed it very much-and in fact, I learned
Maskit groups the parameters are complex and there are two
some new things.
Grothendieck on Triviality Alexand r Grothendi ck was again ·t "I I told him of a
