Jet Wimp*

When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias by Arthur T. Winfree Princeton, NJ: Princeton University Press, 1987. 384 pp., Hardcover, US $65.00, softcover, US $19.95 (ISBN 0-691-02402-2)

The Geometry of Biological Time by Arthur T. Winfree N e w York: Springer-Verlag, 1990. 530 pp., softcover, US $32.00 (ISBN 0-387-52528-9)

Reviewed by Leon Glass But living materials are diverse in ways that often defy the mathematics evolved for doing physics and thus in those terms seem imprecise and unanalyzable. Biologists have recently recovered from this illusion in many ways. The way of particular pertinence here is the recognition that there are modes of mathematics--of "'reasoning with symbols"---other than the ones that make living organisms look imprecise. The topological mode offers special promise. It is this mode--indeed one tiny theorem in one part of this mode--that is celebrated here. Take note that this book remains tightly focused on experimental biology and chemistry. There will be no explicit mathematics. There is almost none behind the scenes either; the kinds of topology involved really boil down to little more than geometric intuition applied with patient tenacity. --A. T. Winfree 1987

In the late 1960s Art Winfree, a graduate student in biology at Princeton University, was studying the effects of light on the circadian rhythm in fruit flies, a biological rhythm of about 24 hours. To understand the experimental results, he found it necessary to develop a novel mathematical construct, that of a phase singularity. He realized that phase singularities not only were important in his experiments with fruit flies but also might arise in many other settings. Now, a generation later, hundreds of theoretical and experi-

* Column Editor's address: D e p a r t m e n t U n i v e r s i t y , P h i l a d e l p h i a , P A 19104 USA.

of M a t h e m a t i c s ,

Drexel

mental papers have proved phase singularities to be a perdurable research topic. The Geometry of Biological Time (GBT) and When Time Breaks Down (WTBD) are books of stunning beauty and originality. In these books, Winfree describes h o w phase singularities can be observed either directly or indirectly by a variety of cunning experiments. The modest disclaimer that there is little mathematics here must not be taken literally. These books are gold mines of mathematical ideas. It is true that hard-nosed mathematicians who are comfortable only with lemmatheorem-proof expositions will not be h a p p y with Winfree's chatty style. At the other extreme, biologists with little mathematical inclination will be befuddled by the subtle geometrical notions. Nevertheless, anyone with a curiosity about phenomenology in the natural sciences--and how it can be described with mathematics--will enjoy them greatly. The w a y that I find easiest to explain what phase singularities are is to start with nonlinear oscillations. The phase of an oscillator is a measure of the time since the last occurrence of some marker event of the oscillation. The marker event might be, for example, waking up, the contraction of the heart, or an excitation in a neuron. One usually normalizes the phase to the intrinsic cycle length of the oscillator and often uses a point of a circle to represent it. Differential equations serve as the most common theoretical models of biological oscillators. If one requires n variables to describe the oscillator, the state space then is n-dimensional, and the values of the variables as the oscillation evolves lie on a closed curve. Using the definition of phase above, we consider the phase to vary from 0 to 1 along the closed curve. We associate the values of 0 and 1 with the phase of the marker event. One also can define phase for values of variables that do not lie on the cycle. Assume that the cycle is locally stable and all points in its neighborhood asymptotically approach it. We say that two initial conditions lie on the same isochron if the time evolution with an initial condition starting at either of the points is identical for long times. The isochrons end at a single point or set of points and this is the phase singularity,

THE MATHEMATICALINTELLIGENCERVOL. 15, NO. 1 9 1993 Springer-Verlag New York 67

!

!

Figure 1. Isochrons ending at a phase singularity. (Adapted from The Geometry of Biological Time (GBT).)

see Figure 1. More generally, the term phase singularity refers to a locus of points where the phase is undefined [1]. How can this picture be translated into predictions about the results of experimental manipulations? The simplest concept to explore experimentally is the phase resetting of biological oscillations. One can perturb any biological r h y t h m provided one has an appropriate stimulus. As examples: for the daily cycle of waking and sleeping, an appropriate stimulus is an exposure to bright light, whereas an electrical shock delivered to the heart alters cardiac rhythm. Following any perturbation, it is possible to measure the timing of subsequent biological events and to compare this with what it would have been in the absence of the stimulus. Research workers have used this protocol countless times to analyze activity in diverse organisms. Winfree shows that to interpret the results it is more relevant to have a grasp of the topology of oscillators sketched out above rather than the biochemistry or neurophysiology of whatever biological oscillator is involved. To see the importance of topological considerations, consider stimulation of a neural oscillator by an electric shock. The experimenter can vary the phase and also the amplitude of the stimulus. For every combination of stimulus, amplitude, and phase, the oscillator will find itself sitting on some isochron. Figure 2 shows the plot of the isochrons as a function of phase and amplitude of the stimulus for a mathematical model of a neural oscillator. Each shade corresponds to a unique phase. The phase singularities end up as black holes. For certain combinations of stimulus, amplitude, and phase, the oscillation will be annihilated, corresponding to the values in the black hole. Phase resetting 68

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Figure 2. Isochrons for a model of a neural oscillator. (Adapted from WTBD).

using a low-amplitude stimulus is topologically different from phase resetting using a high-amplitude stimulus. In the first case, isochrons go all around the rainbow; in the latter case, they do not. About one-third of each book under review fleshes out this picture and discusses the experiments researchers have devised to test the theory's predictions. Winfree shows how phase singularities can be captured in physical space. The seminal idea here is to imagine a plane filled with oscillators, all of which are in synchrony. Along one axis of the plane are stimuli with graded amplitude; along the orthogonal axis, the stimuli occur at different phases of the cycle. The net result is that different oscillators will be knocked onto different isochrons. The isochrons tell us how the timing of the oscillators evolves. If the oscillator array traps the phase singularity, spiral waves arise. Winfree argues that the trapping of the phase singularity also should be possible in excitable systems. These systems do not spontaneously oscillate, but can support a large excursion from steady state position before returning to steady state. Winfree documents the case with computer simulations as well as with physiological and chemical experiments that show that excitable systems exhibit spiral waves. A recent triu m p h of the theoretical predictions occurred w h e n Davidenko and his co-workers [2], with the aid of volt-

Figure 3. Spiral waves in cardiac tissues. Reproduced from [2] with permission.

age-sensitive fluorescent dyes, observed spiral waves of electrical activity of a small slice of cardiac tissue using stimulation protocols suggested by Winfree's work, Figure 3. Winfree argues that potentially fatal arrhythmias in the intact heart may be associated with spiral waves of activity. However, in the threedimensional heart, it is also possible to have a whole panoply of other exotic geometries that now exist in the memories and graphics of high-speed computers but have not yet been documented in the biological domain; see Figure 4. Winfree has not written his books in a linear fashion. Much of the text is displayed in boxes, and that material develops peripheral points---historical information, anecdotes, or research suggestions. The boxes often point forward or backward to related topics. This gives the books a lively but somewhat disjointed spirit. The subject matter sketched out above forms the backbone of both books, but there are differences. Winfree uses equations sparingly in GBT and not at all in WTBD. GBT has a "bestiary" of examples, which are described in 13 (out of a total 23) chapters. Each of these chapters describes a specific experimental problem. These problems range from the flowering of morning glories to the dynamics of the female ovulatory cycle (deflowering on glorious mornings?), including along the way patterns in slime molds, insect cuticles, and fungi. Winfree has omitted most of the exotic examples from WTBD. Instead, he focuses on the best-developed examples from neurophysiology, cardiology, and chemistry, organizing the material along theoretical lines. The lack of equations in WTBD may simplify the reading for some, but I like to see equations occasionally to help fix ideas. Winfree emphasizes the nonretraction theorem in WTBD, but only refers to it once in GBT. (A version of the theorem occurs in GBT on p. 28 but is not identified as the nonretraction theorem.) This theorem (the "tiny theorem" in the quote heading this review) states that a compact manifold with a boundary cannot be mapped to its boundary by a continuous map that leaves the

boundary pointwise fixed. A beautiful exposition of this theorem and its application to the matters at hand was given by Steve Strogatz in Mathematical Intelligencer in 1985 [3]. Despite citing this theorem, Winfree tries hard to avoid jargon and high falutin' terms and deftnitions. However, more precise definitions and statement of results would help readers who are not in resonance with Winfree's idiosyncratic style. The Springer Study Edition of GBT under review here is essentially a reprinted version of the 1980 edition, which had been out of print for several years. There have been some minor changes in the text, e.g., on p. 153, "separatrix" becomes "separator." Though this book is now over 10 years old, I find it remarkable that the basic ideas were already in place by 1980. Thus, though WTBD contains some new material concerning the playing out of the ideas in chemistry and

Figure 4. A possible geometry for cardiac activity. From C. Henze, A. T. Winfree, Int. J. Bif. Chaos 1, 891-922, 1991. THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993

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cardiology and the structure of w a v e s in threedimensional excitable media, most of the material will be familiar to those w h o have already read GBT. This is not to say that the material is out of date. It is still fresh. Readers of Physical Review Letters will be familiar with the prevailing interest in the geometry and motion of spiral waves in excitable media [4,5], and the study of spiral waves in cardiology is intense; see Figure 3. Still, many of the problems, particularly those in the bestiary, lie dormant. I expect that researchers interested in biological oscillators will be digging in these books for ideas buried in their pages well into the next century. GBT is indispensable to all researchers with an interest in biological rhythms or the applications of nonlinear dynamics to the natural sciences. Already, research workers often cite it as a "classic." I am sure its stature will continue to grow. WTBD covers narrower territory than GBT. Becal4se it contains no equations, it is more accessible to people with weaker mathematical backgrounds. For biologists and those with a recreational interest in mathematics, it provides a good introduction to Winfree's approach. I close on a personal note. I have known Art Winfree since the fall of 1969 when we both had offices in a building on 57th Street in Chicago that housed the Meat Research Institute and the now defunct Department of Theoretical Biology. We have kept in contact since then. I consider it a privilege to have witnessed the discoveries recounted in these books. It is clear to me that the topological viewpoint that Winfree espouses is just taking hold. Yet there are still great mysteries in understanding the development and dynamics of organisms. The solution of these problems will require novel blends of mathematics, biology, and physics. Winfree's pioneering work has shown us that it can be done.

Apology In the review of J. Stillwell's Mathematics and Its History by J. Fauvel and A. Shenitzer (The Intelligencer 14, no. 3 (1992)), it was the reviewers' intent in the third paragraph of p. 69 to contrast the accounts of pre-1800 mathematics readily findable in popular books with the exceptional book of Stillwell, which also reports more recent mathematics. Unfortunately, the sense of this paragraph was altered in the printed version.

Ramsey Theory by Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer N e w York: Wiley-Interscience, Series in Discrete Mathematics and Optimization, Second Edition, 1990, xi + 196 pp., hardcover, US $49.95 (ISBN 0-471-50046-1). Reviewed by Richard K. Guy Combinatorics is at last seen as a respectable mathematical discipline, in no small part because of the emergence of Ramsey theory as one of its several main streams of thought. It is now a mathematical chestnut to ask to show that if six people are at a party, then three of them are already mutual acquaintances or three of them are strangers to each other. This is the simplest application of Ramsey's theorem, unless you count Kleitman's observation that among three ordinary people, two must have the same gender. Very roughly, Ramsey's theorem states that if a structure is big enough, it must contain a copy of a prescribed substructure. As Motzkin has put it: complete disorder is impossible. It was van der Waerden, in 1927, who proved the first theorem of Ramsey Theory: if the positive integers Notes and References are partitioned into two classes, then at least one of the 1. For a mathematical treatment of phase singularities, see classes must contain arbitrarily long arithmetic proJ. Guckenheimer, Isochrons and phaseless sets, J. Math. gressions. In 1935, Erd~s and Szekeres rediscovered Biology 1 (1975), 259-273. Ramsey's 1930 theorem. Behrend, Dilworth, and espe2. J. M. Davidenko, A. V. Pertsov, R. Salomonsz, W. Bax- cially Rado and Turin along with Erd6s, wrote importer, and J. Jalife, Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature 355 (1992), tant papers in the forties and fifties which, with hindsight, are seen to be central to Ramsey theory. A fairly 349-351. 3. S. Strogatz, Yeast oscillations, Belousov-Zhabotinsky early paper was Greenwood and Gleason (1955). But it waves, and the non-retraction theorem, Mathematical In- was the sixties and seventies which saw the amalgamtelligencer 7 (1985), 9-17. ation into a coherent theory. Of the 132 references in 4. A. Karma, Scaling regime of wave propagation in singlethe book, 85 are from this period. diffusive media, Phys. Rev. Lett. 68 (1992), 397-400. The importance of the subject has demanded a sec5. D. A. Kessler, H. Levine, and W. N. Reynolds, Spiral core in singly diffusive excitable media, Phy. Rev. Lett. 68 ond edition. What has transpired in the eighties? Most (1992), 401-404. momentous, perhaps, is Shelah's discovery of much improved recursive bounds for the Hales-Jewett and Department of Physiology van der Waerden theorems. Indeed, the insertion of a McGill University new w and the related amplification of the old secMontreal, Quebec, H3G 1Y6 tion, EEEEENORMOUS UPPER BOUNDS, on "ackerCanada 70

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manic" functions, are the only significant changes from the 1980 edition. Although there is half a page of "Notation" on p. xi, there is a desperate need for a more whole-hearted glossary. Notation is the aspect of the subject that puts most people off. Suppose that I start to read about the application of Topological Dynamics to Ramsey Theory, and find that it will prove van der Waerden's theorem and Hindman's theorem. The former I know, but I need to be reminded of the latter, which I find on p. 88: "If N is finitely colored, there exists S C NS infinite, such that ~(S) is monochromatic." What are N, NS and ~(S)? N is perhaps the natural numbers, and I confirm this on p. xi, although it seems that zero is not a natural number. However, in Chapter 1, on "Sets," where I might expect to find it, the only place where N occurs is on pp. 25-26, where it denotes a particular number. N next occurs on pp. 34, 39, 40; in each case for a specific integer. Then on p. 41 it has three different uses: as a particular number, then as the set of natural numbers, then as a fixed number of dimensions. We have now reached Gallai's theorem: "Let the vertices of R m be finitely colored. For all finite V C R m there exists a monochromatic W homothetic to V." We can't learn from p. xi (nor from anywhere else?) what R or R m are. Presumably the reals and m-dimensional Euclidean space. Is it usual to refer to its points as vertices? It is more suggestive of a polyhedron. To go back, @(S) is defined on p. 81, but not on p. xi. The mystery of NS was not revealed until I borrowed a colleague's copy of the first edition it should read N, S---a sentence of very different syntax. There are dozens of other misprints (almost none of which occur in the first edition), even among the more well-known names; H i n d m a n , Leeb, Ramsey, and Szekeres appear on pp. 88, 10, and 25 as Hinderman, Leep, Ramsey and Szekers; Tur~in sometimes has his accent, sometimes not; Erd(~s never gets his correct one. There are some atrocious line-breaks in the middles of formulas. My limited experience with TEX shows that line-breaks in the text can be handled very well. Why are many commercial word-processing systems so bad at it? Here we find homoge-neous, independent, appro-priate, elemen-tary, mathemati-cians, terminology. Would the program handle gene, depend, proper, element, mathematics, or term, in the same way? "Proven" is the part principle of preave, an archaic verb meaning "test" and not the modern meaning of mathematical proof. "Denote" is a transitive verb. All these things combine to obliterate the considerable efforts that the authors have made to expound the m a n y combinatorial proofs that most readers find so difficult. But the importance and elegance of the subject shine through, in spite of the blemishes of production.

References F. A. Behrend, "On sets of integers which contain no three in arithemetic progression," Proc. Nat. Acad. Sci., 23 (1946), 331-332. R. P. Dilworth, "A decomposition theorem for partially ordered sets," Ann. Math., 51 (1950), 161-166. P. Erd6s & G. Szekeres, "A combinatorial problem in geometry," Compositio Math., 2 (1935), 464--470. R. E. Greenwood & A. M. Gleason, "Combinatorial relations and chromatic graphs," Canad. J. Math., 7 (1955), 1-7. A. W. Hales & R. I. Jewett, "Regularity and positional games," Trans. Amer. Math. Soc., 106 (1963), 222-229. N. Hindman, "Finite sums from sequences within cells of a partition of N," J. Combin. Theory Ser. A, 17 (1974), 1-11. R. Rado, "Verallgemeinerung eines Satzes von van der Waerden mit Anwendung auf ein Problem der Zahlentheorie," Sonderausg. Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., 17 (1933), 1-10. F. P. Ramsey, "On a problem of formal logic," Proc. London Math. Soc., 30 (1930), 264-286. S. Shelah, "Primitive recursive bounds for van der Waerden numbers," J. Amer. Math. Soc., 1 (1988)~ 683-697. B. L. van der Waerden, "Beweis einer Baudetschen Vermutung," Nieuw Arch. Wisk., 15 (1927), 212-216. Department of Mathematics and Statistics The University of Calgary Calgary, Alberta T2N 1N4 Canada

U n s o l v e d Problems in G e o m e t r y by H. T. Croft, K. J. Falconer, and R. K. Guy New York: Springer-Verlag, 1991, 240 pp. US$39.95 Reviewed by Dennis DeTurck One brisk winter day, the editor of this column called me to ask if I would be willing to review a "little problem book on geometry" that looked interesting to him. Having agreed to do it, and being a close neighbor (the Mathematics Departments of Penn and Drexel are less than two city blocks apart), I walked over to pick up a copy of the book. That night, I settled down to begin looking at the book, and read as far as problem A1 (on page 9)--the equichordal point problem. For the record, an equichordal point of a plane convex curve is one having the property that every chord through it has the same length. The problem (which dates back to 1917 and the likes of Fujiwara, Blaschke, Rothe, and Weitzenb6ck) is to decide whether there is a closed convex plane curve having two distinct equichordal points. Despite warnings in the book about the difficulty of the problem, it was three weeks and many hours of fruitless work later that I read problem A2 and beyond. Well, maybe not completely fruitless one oddball idea I explored a bit was to have my computer start with a pair of points, say (1,0) and ( - 1,0), which were to be the equichordal points, a length L > 6 (the common length of all the chords), and an initial point (x0,Y0), and to generate a sequence of points (xi,Yi) (all THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993 7 1

on the putative curve) by drawing "chords" of length L f r o m (xi.l,yi.1) a l t e r n a t e l y t h r o u g h t h e t w o equichordal points. Each time, the s e q u e n c e of "chords" converged to a horizontal one. It is an amusing exercise to prove that this always happens. Anyhow, to get back to the book at hand, the authors have compiled a remarkable collection of geometry problems, none of which requires a great deal of mathematical sophistication to understand, yet which remain (in many cases after decades or even centuries) unsolved. There are 148 problem sections in all, where a problem section may consist of a single paragraph with the barest statement of a problem together with a reference or two, or else could be several pages long, containing information on published partial solutions, statements of related problems, and interesting anecdotes. The sections are patterned after (and in some instances the problems themselves come from) Victor Klee's "Research Problems" sections of the American Mathematical Monthly, which began running in the late 1960s. The problems are grouped into seven chapters, although many of the problems transcend the authors' classification scheme. Each chapter begins with a concise review of the terminology used therein. The problems in the first chapter (Convexity) have the property that the word "convex" appears in each problem. Here one will find problems ranging from the familiar (reconstruction of convex bodies from shadows and sections, variations on Queen Dido's isoperimetric problem, etc.) to more specialized, less-known problems. An example of the latter is problem A14: What closed convex 3-dimensional surfaces have the property that a regular tetrahedron can rotate to any orientation within them, keeping all four vertices always on the surface? Another intriguing problem in this chapter concerns the existence of convex sets with "universal sections.'" The second chapter contains problems about polytopes--there are real surprises among the first few open questions posed here. One's reaction is inevitably "You mean the answer to that question is unknown?!" This chapter contains the only problem about which I am aware of recent progress: problem B18. One of B18's subproblems asks, "Is every convex polyhedron equivalent to one with all its edges touching a sphere?" Here, equivalence means combinatorial equivalence (i.e., having the same relative arrangement of vertices, edges, and faces, as for instance a cube and a parallelepiped), and the answer to the question is yes. Koebe is responsible for the proof for a certain class of convex polyhedra, with proof for the general case being contained in Chapter 13 of Thurston's Princeton Notes on The Geometry of 3-manifolds. Very recently, Oded Schramm ("How to cage an egg," Inventiones Math. 107 (1992), 543-560) generalized the result: It is still true w h e n the sphere is replaced by an arbitrary smooth convex body. 72

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The other chapters concern tiling and dissection (decompositions of rectangles, polyominoes, rep-tiles), packing and covering (packing pennies, filling containers), combinatorial geometry (problems about lattice points and variations on Minkowski's problem), problems about finite sets of points (number of distinct distances, triangles, etc., determined by such sets; lots of Erd6s problems here), and "general geometric problems." The book is a wonderful source of diversion. Rather than being a book to be read from beginning to end, the book invites the reader to open it almost at random and consider whatever problems are encountered. There is real peril involved, however. Many of the problems are irresistible, and all of them are difficult. The one serious criticism I have of the book concerns not what is between the covers, but what is on the back cover, and what is contained in the promotional material circulated by the publisher. To quote from the back-cover blurb: "The book is an invaluable reference for the research mathematician and also for t h o s e . . . who wish to keep abreast of progress on geometrical problems. For the mathematically minded layman or student, the book provides an insight into current mathematical research." Statements such as these do not accurately represent the nature or the value of the book. Although some of the problems in the later chapters (mostly the combinatorial problems) are au courant in research circles, by and large the problems in the book are not at the focus of late twentieth-century geometry. There are few differential geometry problems in the book, nor are there geometric analysis problems. The author index is not a Who's Who of geometry in the nineties. Rather, the authors have succeeded in producing a charming and valuable collection of accessible unsolved problems, the solution of any of which would attract the attention of some segment of the mathematical community. One can only hope that some day the solutions manual will appear!

Department of Mathematics University of Pennsylvania Philadelphia, PA 19104-6395 USA

Old and N e w Unsolved Problems in Plane Geometry and Number Theory by Victor Klee and Stan Wagon Washington, DC: Mathematical Association of America, 1991. 352 pp. paperback, US$22.00. (ISBN 0-88-385-3159)

Reviewed by Kenneth Falconer Most mathematicians have a favourite problem that they would very much like to solve; a few achieve this aim, but many more enjoy investigating aspects of the

problem, perhaps producing partial solutions or answering simpler related questions, or just convincing themselves that the problem is indeed difficult. Some devote considerable time to an unsolved problem in the hope that a flash of inspiration will lead to their immortalisation in mathematical history. Problems studied by professional mathematicians may be highly technical, but there are many simply stated questions that have occupied innumerable hours of amateurs and professionals alike, Fermat's Last Theorem being, perhaps, the best known example. Appealing problems, problems that anyone can understand but that are hard to solve, are probably most prevalent in the fields of number theory and two- or three-dimensional geometry. We all encounter primes, rationals, convex sets, and plane closed curves at an early age; surely any questions on such familiar topics ought to be tractable! Not surprisingly, m a n y of the unsolved problems in collections and in mathematical journals tend to be on number theory or geometry. This book covers both areas; in a sense it contains two books in one--the halves on geometry and number theory having little in common except for the intuitive nature of the problems. The selection of problems is inevitably somewhat subjective. The authors have dearly decided to include the "big" unsolved problems. Does there exist a plane convex set such that every chord through one of two given points has unit length (the equichordal point problem)? Does there exist a polygon that only tiles the plane aperiodically? (There is a pair of polygons, the famous "Penrose tiles," that tile the plane aperiodically but not periodically.) We also encounter Fermat's Last Theorem and the Riemann Hypothesis. But some problems included are less central, for example, does every simple closed curve contain the vertices of a

Does every simple closed curve in the plane contain all four vertices of some square?

square? Does there exist a box with the lengths of sides, face-diagonals, and main diagonals all integers? The inclusion of these problems reflects the authors" interests. Many readers will have their own favourites that could equally well have formed the basis of sections. The reviewer's own selection might have included: Can a rectangle be dissected into three congruent Jordan regions in a nontrivial way? In the disc of radius 1, what is the subset of largest area for which a distance of 1 between any two of its points is not attained (surely it must be an open disc of diameter 1)? If a convex set C is covered by a collection of parallelsided strips or "planks" of widths wi (1 ~ i ~ m), is it the case that X(wi/Wi) t> 1, where Wi is the width of C in the direction perpendicular to the ith strip? (This is the affine-width version of Bang's plank problem.) Any book of this nature is out of date before publication. The section on "squaring the circle" (can a circle be decomposed into finitely m a n y pieces that can be rearranged to form a square?) has already capitulated to Laczkovitch's brilliant and unexpected proof that it can indeed be done. Still, such advances raise many further problems; for example, in this instance, how few pieces are required? (Somewhere between 3 and 10,000,000 seems all that is known.) How essential is the axiom of choice in any solution? The book is well-written and readable. As well as the main problem stated at the beginning of each section, there is a description of the state-of-the-art related theorems (sometimes accompanied by proofs where these are instructive and fairly elementary) and subsidiary problems. Each part of each section concludes with a nice selection of exercises; working through these (with the odd glance at the Hints and Solutions, if required) will probably not help anyone solve the main problems, but it will increase familiarity with the area and highlight some elegant ideas and slick methods. The authors have appended a useful bibliography to each chapter. Considering the literature that has accumulated on some of the more notorious problems, these bibliographies are by no means complete, but they provide an adequate start for those wishing to study a problem in greater depth. I find the layout of this book highly irritating. Chapter 1 on "Two-dimensional Geometry" has two parts. The first part discusses each of 12 problems at a basic level in half a dozen pages or so. The second part reconsiders each problem, in turn, in greater depth and with historical details and references. Consequently, between Problems 6.2 and 6.3 on the number of connecting lines through points in a finite plane set, one encounters Problem 7.2 on tiling the plane by pentagons, as well as Problem 1.5 on the number of billiard paths on a convex table. The authors have split Chapters 2 and 3 on "Number Theory" and "Interesting Real Numbers" in the same way. Thus, when flicking back and forth between the two Sections 18 on THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993 7 3

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33 The 3n + 1 Problem: Is every positive integer eventually taken to 1 under iteration of the function f(n) = n/2 (if n is even) and fin) = 3n + 1 (if n is odd)?

prime factorization algorithms, one is seduced by an eye-catching problem on approximation of tetrahedra by rational tetrahedra, or by the graph of the Riemann Zeta function, and forgets the original problem. In my frustration, I nearly cut up the book and reassembled the pages to unify the sections! This is an attempt to put two books into one: the first, consisting of the Parts 1, suitable for mathematics undergraduates, the second, consisting of Parts 1 and 2 together, for more advanced s t u d e n t s and researchers. However, it would have been better to keep the material on each problem (including references) together. This book will inevitably be compared with the two Springer-Verlag Unsolved Problems in Intuitive Mathematics rifles, Unsolved Problems in Number Theory by Richard Guy and Unsolved Problems in Geometry by Hallard Croft, Richard Guy, and the reviewer, particularly with the latter book, which appeared at about the same

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THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993

time as the book under review. Though I feel it is inappropriate to praise or scorn m y own book (looking back at some past reviews and refutations, I feel this may differ from Mathematical Intelligencer editorial policy!), some remarks seem in order. Both works have u n d o u b t e d l y benefited from correspondence and manuscripts that have circulated privately for up to 30 years. Many well-known mathematicians have, at least indirectly, influenced both books (this is clear even from comparing the acknowledgements), and there are many common problems. The Springer books contain far more problem sections, but with less discussion and proof in each section. I feel the books complement each other: If you enjoy reading one, then you probably will enjoy meeting new, but not entirely unfamiliar, material in the other. I have to admit (grudgingly, because the matter was outside m y control) that, at $22, this book is more reasonably priced than t h e Springer ones. From m y experience, I know that the book will result in a considerable mailbag for the authors. Some letters will be irritating: "Why haven't you cited my vaguely related paper in the Outer Hebridean Mathematical Society Proceedings?" Others will give purported solutions to the problems. Some will contain genuine misconceptions; others will be just cranky. Some, from both amateurs and professionals, may contain a proof or an idea that will be a genuine contribution to one of the topics. I agree with the publisher's claim that the book will appeal to a wide range of readers, from spare-time amateur mathematicians, to teachers at all levels, students, and university researchers. I personally found many facts and problems to fascinate both on familiar and unfamiliar topics. The sections on prime factorisation, Fermat's Last Theorem, etc., are useful for nonspecialists to update their mathematical general knowledge. Even the sections on areas that I am most conversant with, and to w h i c h I have m a d e minor contributions, contained unfamiliar material. Several people have commented to me that they had "been unable to put the Springer Unsolved Problems in Geometry d o w n . " (I refrain from the Wodehousian crack that this was because they had not picked it up in the first place!) The same is true of Klee and Wagon's book. It is compulsive reading and will fill your mind with problems that will come back to haunt you again and again during idle moments.

School of Mathematics University of Bristol Bristol, BS8 1TW England