\(\pi\): A Biography of the World's Most Mysterious Number

Sporting a bold title that intimates a promise of surprising revelations within for anyone curious about the number π, this fast-paced, eminently readable account does not disappoint. Treating topics such as the history of π, curiosities involving the decimal expansion of π, and the pervasive role of π in geometry, the authors provide brief, engaging anecdotes, derivations, and historical accounts that at once entertain and educate. This enjoyable overview extends well beyond the confines of π, a number that makes surprising appearances in many branches of mathematics. So in following π where it takes them, the authors end up supplying a good dose of mathematical literacy, providing a cursory, yet gentle initiation into not only the mysteries of π, but also the discipline it inhabits.

The adventure is previewed in the preface, where each of the book's seven chapters is outlined in a paragraph. Chapter 1 then lays some groundwork. Surveying various aspects of π, this first chapter sets a tone for the rest of the book with its format of one- to two-page capsules, many of which are self-contained. Found in the bite-sized segments are, among other things, a definition of π, a derivation of the familiar formulas C = 2πr and A = πr², and a discussion of π's status as a transcendental number, a guarantee that its exact value will remain forever a mystery. Some sections act as a warm-up for the second chapter by narrating some history. Other sections, featuring geometry problems, provide a foretaste of the final portion of the book.

Chapter 2 tells the story of π from prehistory through the 21st century, reviewing developments in ancient Egypt, Greece, and China, in Europe from the Renaissance through the 19th century, and worldwide in the 20th and 21st centuries. The main theme is the generally improving approximations to π as time marches on and mathematics develops from one culture to another. Among the ancient Greeks, Archimedes, one of the greatest mathematicians ever, receives special treatment for his intuitive and enduring approximation method. The discussion of the 20th and 21st centuries centers on the computer, a tool that has greatly increased the rate at which we achieve increasingly accurate approximations to π.

Finding these better approximations, that is, increasing the number of known digits in the decimal expansion of π, is the subject of Chapter 3. Archimedes reprises his important role with his method of approximating the circumference of a circle (which effectively approximates π, since C = 2πr) by the perimeter of a circumscribed polygon. By letting the polygon have more and more sides, a better approximation is achieved. For many centuries after him, Archimedes's was the method of choice among digit seekers. But other methods exist, including drawing a circle on graph paper and counting the squares inside the circle, throwing darts at a circular target and counting how many hit versus how many miss (a statistical simulation called the Monte Carlo method), and various infinite series approximations whose partial sums converge to π.

Chapter 4,"π Enthusiasts," is dedicated to the many members of the π fan club. (The biographical blurbs of the enthusiastic authors should be transferred here from the dust jacket!) A short chapter, it documents enthusiasts' fascination with the digits of π. Included here are a description of π Day, March 14 (The first three digits of π are 3.14.), several poems in various languages whose word lengths correspond to the successive digits of π, and even a song chronicling the history of π, including the quest for more and more of its decimal expansion.

Fun with the digits of π continues in Chapter 5,"π Curiosities," with a slightly more serious tone. On display are many "mysterious coincidences" involving the digits of π. Apropos is a statement from the previous chapter: "As with any endless randomly generated list of numbers, you can make just about anything happen within them that you wish." So whereas some of the chapter's showcased surprises mark the spot of bona fide buried mathematics, others, were one to dig under them, would doubtless yield little treasure. Discriminating between the significant curiosities and the mere coincidences the authors leave to the reader. While skeptics unwilling to dig in vain can negotiate it quickly, the chapter provides more leisurely enthusiasts (including, it is hoped, high school students and undergraduates) both diversion and some intriguing projects to pursue.

"Applications of π" departs from the discussion of the digits of π, their calculation, and their various curiosities and instead seeks to observe and engage π in its most natural habitat, geometry. Readers of Chapter 6 should enjoy looking at a diagram and answering such questions as "What is the area of this shape in the diagram compared to the area of that shape?" Indeed, the greater part of the chapter consists in diagram after diagram, each composed of circles and parts of circles, and each distilled into a few simple equations. Sharing the chapter with many anonymous shapes are the racetrack oval, the Reuleaux triangle (a kind of "roundish" triangle that is the basis of some car engines), and the Chinese yin-yang symbol.

The final chapter, "Paradox in π," is to the geometry of π what "π Curiosities" is to its digits. But whereas that chapter exhibits several results of questionable significance, this chapter poses riddles whose answers, though unexpected, all point to deeper truths. For example, assuming the earth is a perfectly smooth sphere, imagine wrapping a rope tautly around the equator. Now lengthen the rope by 1 meter (so that it is no longer taut). Question: Can a mouse crawl easily under the rope? The answer will surprise you. Summoning the reader to know π more intimately, this collection of thought-provoking paradoxes forms a fitting finale to the main part of the biography.

A 29-page epilogue supplies the first one hundred thousand digits of π (!). In an afterword, Nobel laureate Herbert Hauptman, with enthusiasm equal to the authors', recaps and supplements some of the main ideas in the text, elaborating in particular on Archimedes's approximation method and on certain infinite series whose sums involve π. Four short appendices, a list of references, and a twelve-page index round out this very accessible book.

"We hope to present π to you in a very `reader-friendly' way..." The authors have thoroughly succeeded. Anyone with some high school math should be able to read and enjoy this book. The one "advanced" subject drawn upon is, not surprisingly, some basic trigonometry. A knowledge of the sine, cosine, and tangent functions suffices, and they are defined where needed.

Indeed, most terms and concepts that might give trouble to an unseasoned math enthusiast are defined in footnotes. (Examples include parallelogram, irrational, regular polygon, iteration, series, and annulus.) So obliging is the book that some terms are even defined in more than one place. Other footnotes complement the main text by reminding the reader of similar material covered earlier, by taking a bit further the topic under discussion, or by providing historical background.

In the very few places where the book is not fully accommodating, the reader is given ample warning, such as: "We will now take a giant leap to the general case, where we will try to sandwich in the value of π. This may be a bit complicated for the reader no longer familiar with the intricacies of high school mathematics, yet it is the conclusion of this generalization that is of greater importance than the process." Surely few readers will find intimidating a book that quite naturally features a song about the number π and reassuringly acknowledges "the intricacies of high school mathematics."

In addition to being strikingly readable, the book helps shatter the myth, for the general reader, that mathematics is merely tedious computation. "It is our intention to make the general reader aware of the myriad topics surrounding π that contribute to making mathematics beautiful." For the authors, much of this beauty reveals itself in the surprising relationships between different, apparently unrelated, areas of mathematics, with π acting as a lamp to illuminate the connections as it shines a light here, there, and everywhere. In describing both π's seeming ubiquity and the connections it elucidates, "astonishing" is one of the author's favorite adjectives. Illustrating one astonishing connection is Buffon's needle: Throw a needle onto a wood plank floor over and over again, counting the percentage throws for which the needle lands touching two boards (versus landing entirely on one board). These repeated trials are clearly calculating a probability, yet the percentage obtained, when inserted into a certain simple formula, yields an approximation to π, whose native realm is geometry. That probability and geometry are thus connected is indeed astonishing. So is the answer to the rope-around-the-equator riddle mentioned earlier, as well as numerous other results presented in the book.

This is not to say that readers will take in every page with mouth agape. One person's astonishment can be another's expectation. In Chapter 6, for example, the authors repeatedly point out that π continually reappears in the formulas being obtained. But in a chapter filled with diagrams of circles, is it really surprising that π plays a prominent role? At the other end of the spectrum: That the probability a number chosen at random from the set of natural numbers (1, 2, 3, 4,...) has no repeated prime divisors is 6/π² will surely surprise, even astonish, most readers. Between these two extremes is the revelation that there exist two shapes, one bounded by lines and one bounded by curves, that have equal areas. Is this astonishing or not?

The authors certainly think so. They develop a two-dimensional example of such an equivalence in Chapter 2, and in the first appendix, they include the award-winning three-dimensional example developed by Howard Eves. The award citation states that Eves's construction would have had "geometers of ancient times inscribe it on their tombstones" (a la Archimedes), so evidently mathematicians both ancient and modern do consider it exciting that the area of a shape with "straight sides" can equal the area of a shape with "round sides." Here, then, is a glimpse into not only some interesting mathematics but also into the way mathematicians perceive it.

Though ostensibly a biography of π, the book's greatest strength is this gentle way of introducing some deep mathematics, from mathematical content, such as the nature of different kinds of numbers, to mathematical perspective, such as what mathematicians find astonishing, to mathematical method, such as approximating numbers like π with a computer or proving mathematical facts with a picture and a short derivation. The cumulative effect is to issue readers an invitation to join the mathematics community by absorbing some of the beauty of the discipline, reveling in its surprising connections, and trying their hands at developing and proving some simple results.

Indeed, for those with little experience constructing proofs of mathematical facts, the many derivations throughout the book afford a fruitful training ground. Woven into the discussion, these are not boredom-inducing formal proofs. Instead, they usually involve inspecting a labeled diagram and drawing some conclusions, the conclusions almost always involving π, of course. Along the way, some important and illustrative proof techniques are put to use, including examining extreme cases and analyzing dependence among variables. Here is a great opportunity for aspiring math fans to read the discussion and then close the book and see if they can replicate the results. The parade of diagrams that is Chapter 6 supplies almost endless practice. Enterprising readers will want to create some diagrams of their own.

Much of the mystery in some of the results obtained in Chapters 6 and 7 involves proportional relationships, especially among the radius, circumference, and area of a circle. The authors seem to hope that readers will piece together the ideas for themselves over the course of the last two chapters. For readers who do not "see it", the paradoxes are still fun, but they will have missed a significant learning opportunity. Perhaps the authors could have been more explicit. When discussing the curious "Lost Circle Area," for example, a question such as the following could have been asked: "How much larger is a 14-inch pizza than a 10-inch pizza?" The solution recalls the fact that a circle's area is proportional to the square of its radius, not to the radius itself. This, along with the fact that the circumference, in contrast, is proportional to the radius itself, are at the heart of many of the surprises and paradoxes toward the end of the book. To be fair, the authors are challenging the reader in the "Lost Circle Area" section to explain what is going on, a pedagogical approach that generally deserves applause. But surely it would not ruin the fun to spell out these proportional relationships here, and it would augment nicely the excellent prior discussion of the nature of π. (It's a ratio, but it's not rational!)

Despite being lively in its parts, the text as a whole fails to flow. It is less narrative than it is a series of historical anecdotes, insightful geometric derivations, and surprising revelations. Those expecting an integrated plot to unfold in this biography will be disappointed. On the other hand, those interested in only certain aspects of π will find the piecemeal arrangement advantageous, as they can effortlessly join the friendly discussion midstream, handling references to earlier material quickly and painlessly. The writing is so inviting and enthusiastic that it can be a joy to pick up the book and read a few sections. And faulting a lack of cohesion is a small and perhaps unfair complaint about a popular book that touches meaningfully on so many aspects of mathematics.

Indeed, because of its broad coverage, many general readers will find this book a fit introduction to the world of mathematics. Teachers of grades 11-14 could make it suggested reading for their students, for whom await within ample leads for mathematical projects. The many diagrams and their accompanying derivations suit the book as a supplementary resource in university geometry courses serving future high school teachers. And certainly anyone curious to know more about the number π should enjoy this entertaining account.

David A. Huckaby (david.huckaby@angelo.edu) is assistant professor of mathematics at Angelo State University.