error, a number of bibliographical numbers are off by 1 or 2, but this is usually easy to spot. I now summarize the contents with some comments. The first volume, by Jonathan Borwein and David Bailey, is titled "Mathematics by Experiment: Plausible Reasoning in the 21st Century." In Chapter 1 the authors explain their views on experimental mathematics, including a number of examples and many challenge problems, ending with a large list of Internet-based resources. I was surprised to see that the popular Pari/GP software which I developed and is one of the most commonly used in number theory is not even mentioned! In Chapter 2, ten miscellaneous but highly interesting examples of experimental mathematics in action are discussed, and 50 additional examples are given more briefly. Chapter 3 is devoted entirely to ~r and similar constants--how to compute them using AGM-type or BBP-type algorithms, and so on. Chapter 4 is more theoretical and discusses normality of expansions of numbers. As is well known, this is essentially a hopeless subject, but the authors give their views about it, including possible links with BBP algorithms. Chapter 5 also gives a number of miscellaneous results, the philosophy being to concentrate on constructive proofs as opposed to abstract ones. Chapter 6 is the first chapter devoted to the explicit numerical algorithms used in experimental work: Fourier transforms (FFT and DFF), multiprecision arithmetic, including fast high-precision evaluation of exp and log using the AGM and Newton, and constant recognition (PSLQ). Chapter 7 is a concluding chapter containing little information. The second volume, by Jonathan Borwein, David Bailey, and Roland Girgensohn is titled "Experimentation in Mathematics: Computational Paths to Discovery." Chapter 1 deals with miscellaneous results and proofs on sequences, series, products, and integrals. The structure of this chapter is similar to that of Chapter 2 of the first volume: first 10 examples are described in detail, then almost 60 additional examples are offered as problems with indications. This chapter makes for very enjoyable reading, and although most of the problems are classical, there are some real gems. Chapter 2 contains a serious and quite classical exposition of Fourier series, Fourier integrals, and summation kernels. The most amusing part of this chapter is probably Section 2.5, dealing with sinc integrals (sinc(x) = sin(r162 together with several of the additional examples. Chapter 3 is devoted entirely to zeta functions and multizeta values, essentially sums of the form

as PSLQ allow the experimenter to find remarkable relations between these values. One such experimentally discovered relation by Zagier gives the value of ~(3, 1, 3, 1 . . . , 3, 1) as a rational multiple of a power of 7r, and the authors include a proof of this relation. Chapter 4 is devoted to partition functions, special values of theta functions, Madelung's constant M

2_.

(m,n,p)

r (0,0,0)

(-1)m+n+p (m 2 + n 2 + p2)1/2 ,

and other examples. Chapter 5 is devoted to a number of miscellaneous subjects: prime number conjectures, representation of integers by x y + y z + zx, Gr6bner bases and metric invariants, spherical designs, and many others. Chapter 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 practical 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 techniques. The authors have little to say on the Wilf-Zeilberger algorithm of creative telescoping, primality testing, computing complex roots of polynomials, or the use of EulerMacLaurin for infinite series summation. Nevertheless, Section 7.4 contains a description of numerical quadrature methods, including the remarkably efficient doubly-exponential 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 diverse examples (I did not count, but it may reach 1000), although the reader must be warned that there are many mathematical misprints. The two volumes are v e r y enjoyable r e a d i n g and belong on the bookshelves of any mathematician or graduate student who does mathematics for pleasure (which one hopes is the case for most of them!). Laboratoire A2X UFR de Mathematiques et Informatique Universite Bordeaux I 33405 Talence Cedex France e-mail: [email protected] .fr

Indra's Pearls. The Vision of Felix Klein by D a v i d M u m f o r d , Caroline Series, a n d D a v i d Wright

1 ~(81, S 2 , . . . ,

Sk)

~-

~ n l ~n2~...nk~

l

nsl I n~ 2

9 9 9

nSkk '

where the si are positive integers. The latter have been in recent years the object of a vast literature, in number theory and analysis of course, but also in knot theory, combinatorics, and theoretical physics. These values are especially well suited to experimentation, because they can be easily computed to hundreds or thousands of decimals (although the algorithms for computing them are not completely trivial), and constant recognition algorithms such

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THE MATHEMATICALINTELLIGENCER

CAMBRIDGE UNIVERSITYPRESS, CAMBRIDGE, 2002 396 PAGES, s HARDCOVER ISBN-10: 0521352533--1SBN-13:9780521352536)

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 your life and to list one of your theories and to explain it in her own terms. As you can imagine this is difficult to do. I've attempted to read 4 of your papers and was lost after the intros! I passed calculus with only a B and this is WAY over my head! Would you be able to pick one theory and briefly explain it in layman's language. She needs to report what the theory is and how is it used. I really appreciate your help I replied that Megan's teacher had given both her and me an impossible task. I then suggested that they think about a curve that was so crinkled that no matter how much it was magnified, it looked as crinkled as ever, and I indicated how they might construct it. Megan wrote back thanking me and telling me that the teacher read my e-mail to the class and they tried to work out the construction and found it interesting. Megan and her mother were going to spend some more time on it over the weekend. This incident points out the problem we, as individual mathematicians, and as a community, have. How can we convey to others what it is about mathematics that excites us and makes us get up in the morning? We even have trouble telling other mathematicians what we do. There are some problems in mathematics whose statements, at least, are accessible to laymen--for example, the twin primes conjecture. But, how do we develop the ideas and the language that goes with them to tell anyone but a few specialists what we are working on? Since the late 1970s, as computers have become more powerful and their graphics more sophisticated, mathematicians have been able to draw mathematical pictures that appear beautiful, even to non-specialists. Just as a nonmusician can listen to a Haydn symphony and enjoy the music without being able to articulate what it is about the underlying structure that appeals, nonmathematicians can look at many of these pictures and fmd them pleasing. The book under review, Indra's Pearls, is on one level an attempt to tell a broad audience something about mathematics. The book grew out of all three authors' enchantment with the computer pictures they have been making for the last thirty years in their study of discrete groups of MObius transformations, otherwise known as Kleinian groups. The pictures reminded them of the ancient Buddhist dream of Indra's net. The infinite net, stretching across the heavens, is made from diaphanous threads, and at each intersection there is a reflecting pearl. In each pearl, all the other pearls are reflected, and in each reflection there are again infinitely many reflected pearls. They wanted to share the pictures with a broad audience of mathematicians and non-mathematicians alike. Even more, they wanted to enable the reader to write programs to recreate the computer pictures. To this end, the book is written in a chatty informal style and begins at the beginning. The illustrations are both handand computer-drawn. They are well chosen, and the captions, set in the margins, give a good description of them. There are boxes containing calculations and asides to keep

the text from becoming too cluttered. There are also very informative boxes giving biographical information on various (dead) mathematicians who have made substantial contributions. In addition, there are many well-thought-out projects for readers, involving both computer programs and colored pencils and paper. The book begins with a discussion of symmetry as the basis of geometry as propounded by Felix Klein. In his view, geometry is the study not only of objects, triangles and such, but also of motions. In Euclidean geometry these are just affine motions, but Klein's motions were more general. The key here is that the motions that preserve symmetrical objects form a group. All the essential information is encoded by the group. Thus, the first chapter is devoted to a discussion of symmetries of the plane, that is, tilings of the plane by regular shapes. A group is defined by the properties of the set of transformations needed to move one tile onto another. There are lots of illustrations, both hand-drawn and computer-drawn. The authors talk about their programming techniques and introduce the first pseudo-code programs to create the tiling pictures. The next chapter tackles the next basic concept--complex numbers. After a bit of history, the authors discuss the arithmetic operations of complex numbers and computer programs to implement them. They quickly progress to the Riemann sphere and stereographic projection. The important motions here, of course, are inversions. The third chapter completes the presentation of the basic material by discussing MObius transformations of the plane. These are compositions of an even number of inversions. In addition to good pictorial descriptions of how these maps act, computer programs are given that find the image of a circle under such a map. Having assembled the mathematical and programming tools, the authors move on to the groups of symmetries that they are really interested in. They start with two pairs of circles in the plane, CA, Ca and CB, C0, and two MSbius maps a and b, where a maps the exterior of CA to the interior of Ca and similarly for b. The initial arrangement of these circle pairs determines the symmetrical pattern for the group formed by applying a, b and their inverses in various combinations to the pairs of circles. In the simplest case, the circles bound 4 mutually disjoint disks, and the basic tile for the pattern is their common exterior. Tiling, using the group elements, covers the whole plane with the exception of a Cantor set--which the authors call "Fractai D u s t " that is invariant under any element of the group. To describe how to plot the fractal dust, the authors take an excursion into the realm of search algorithms for trees. I have taught this often from computer science textbooks, and this is the best description of these algorithms I have seen. In the next two chapters, the initial arrangement of the circle patterns is changed. First, the circles are moved in the plane until they just touch and form a "necklace"--CA is tangent to CB, CB is tangent to Ca, Ca is tangent to Cb, and Cb is tangent to CA. The map a sends CA to Ca and

9 2005 Springer Science+Business Media, Inc., Volume 27, Number 4, 2005

59

sends the other three circles, now thought of as beads, to three smaller beads inside Ca. The "necklace" is now made up of the six beads Cb, CA, CB, and the three inside Ca. The other three maps, A, B, and b, act similarly, replacing each of the original beads with three smaller beads. Iterating this process indefinitely, the beads of the necklace become smaller and smaller and more and more numerous, and form "Indra's necklace," a fractal continuum invariant under the group generated by the maps a and b. 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. Nevertheless, the process of forming the necklace, applying the group transformations to the circles again and again, results in an invariant set for the group that is recognizable as the classical Apollonian gasket. After this, the going gets tougher and the mathematics becomes very deep. The groups are divided into families based on the pattern of the four initial circles. In each fanlily, the invariant set has certain characteristics. These families of groups have historically been named after individu a l s - w h i c h is unfortunate because it makes it even more difficult to keep straight which is which. There are Classical Schottky groups whose invariant set is fractal dust, Fuchsian groups whose invariant set is a circle, and Quasifuchsian groups whose invariant set is a closed fractal curve. There are also Riley groups and Maskit groups, whose fractal invariant sets divide the plane into one simply connected invariant component with fractal boundary and infinitely many components whose boundary is a round circle. These families of groups depend on parameters, the triple of traces of the MObius transformations, ta, tb, tab, subject to certain relations depending on the family. For Fuchsian groups, the parameters are real and there is one relation, so the family depends on two real parameters. For the Riley and Maskit groups the parameters are complex and there are two

60

THE MATHEMATICALINTELLIGENCER

relations, so each family depends on one complex parameter. As the parameter varies in the complex plane, the invariant sets move, but retain their basic characteristics: for example, the number of components of the complement of the fractal invariant. At some points in the plane, these characteristics change: circles may appear in the fractal creating new components, components may disappear, etc. Such points form the boundary for the family. In the Riley and Maskit families, there seem to be round disks, albeit overlapping, in the simply connected invariant component. These disks form circle chains with discernible patterns. As the parameter is varied appropriately, the chains persist until, at points called "accidentally parabolic points," they become chains of tangent disks. These are boundary points of the family. The existence of the circle chains reflects the relationship between the elements of the group as words in the generators and continued fractions. The parameter spaces for these families of groups can also be drawn by computer. It is possible to get the computer to find the parameter values of enough accidentally parabolic boundary points to draw the boundary and see that it is fractal and has an interesting structure of its own. The book ends with a discussion of the relationship of the material to three-dimensional topology and Thurston's work. So, have the authors succeeded in their attempt to make this mathematics accessible to a broad audience? Could Megan's mother understand the book? I doubt it, but she might get something from it. The book is written to be read on many levels, and a given reader will have to find his own. An undergraduate who has taken some algebra and complex analysis can certainly get something out of the book, especially the material and projects in the first three chapters; a sophisticated mathematician can get the flavor of the subject; and a graduate student can work her way from the beginning to interesting research problems in some of the projects. As an "expert," I enjoyed it very much--and in fact, I learned some new things.