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Strange Curves, Counting Rabbits, and Other Mathematical Explorations

Keith Ball
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Stacy G. Langton
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Keith Ball's book Strange Curves, Counting Rabbits, and other Mathematical Explorations is a gem. Give a copy to any bright high-school student you know. Recommend it to your own students for additional reading, or just for fun. And enjoy it yourself.

The book originated in a series of talks which Ball gave to math clubs at secondary schools. There are ten chapters, which deal with error-correcting codes and information theory; Pick's Theorem (a formula for the area of a lattice polygon in terms of the number of lattice points in the interior and on the boundary); decimal expansions of the reciprocals of primes; space-filling curves; the birthday problem (given n people in a room, what is the chance that two of them will have the same birthday?); Stirling's formula; the coin-weighing problem (given 9 coins, 8 of the same weight and one heavier, how many weighings are required to find the heavy one? — this gets applied to testing samples of blood for abnormalities); Fibonacci and Lucas numbers; Padé approximations and continued fractions; and irrational numbers, especially the irrationality of e and π.

Each chapter preserves, pretty much, the format of a math club talk. Each topic is taken up in a setting which immediately generates interest. There is a development section which introduces some appropriate tools and shows how to use them to solve the initial problem. And then the talk is over. It is followed by a section which suggests directions for further reading. Half a dozen or so exercises, some routine and some more difficult, are scattered through each chapter. Complete solutions are provided.

A book like this must be hard to write. You need to find interesting problems which can be tackled with a minimum of mathematical preparation. (Secondary students in London — where the author is on the faculty of University College — must be better prepared than secondary students in the United States. Ball takes for granted that the students he is writing for are comfortable applying elementary calculus.)

Of course, we could all probably think of some well-worn topics that would do. Ball's achievement is to have come up with a selection of topics which are fresh and unusual for this level. Several of the topics — such as Stirling's formula and Padé approximations — are probably unfamiliar even to many college mathematics majors. But Ball has treated even the better-known topics in a non-standard way. For example, in the chapter on Fibonacci and Lucas numbers, Ball works out the connection between these numbers and continued fractions. (The treatment of continued fractions here using 2x2 matrices is similar to the one in Harold Stark's book on number theory.) In addition, Ball shows that the approximation to the golden ratio coming from the Fibonacci numbers, via continued fractions, is related to Newton's method for solving equations. And he ends with a result, which appears to be new, about Lucas numbers which are prime.

Similarly, he shows how the birthday problem leads to the Central Limit Theorem of probability theory.

Often, ideas which are developed in one chapter reappear in later chapters, thus illustrating the unity of mathematics. For example, Ball applies Pick's Theorem to prove the basic arithmetic fact that if p and q are relatively prime numbers, then there are integers a and b such that ap - bq = 1. This of course is the essential step in proving the fundamental theorem of arithmetic, which is needed in the chapter on decimal expansions.

Similarly, he applies the Central Limit Theorem, developed in the chapter on the birthday problem, to evaluate the numerical coefficient in Stirling's formula. Continued fractions, first mentioned in connection with Fibonacci numbers, appear again in the chapter on Padé approximations; and the continued fraction expansions of ex and arctan x are applied in the final chapter to get the irrationality of e and π.

Other beautiful touches appear throughout the book. Thus, the integral of the Gaussian probability distribution is evaluated by a very elementary geometric method which does not require knowledge of the formula for transforming to polar coordinates in a double integral.

The tone of the book is that of a relaxed conversation about why things are the way they are. In some cases (for example, in the discussion of the Central Limit Theorem), Ball does not give a complete, rigorous proof, but instead treats an illuminating special case, from which the reader can grasp the main ideas of the more general argument. And he often makes comments, not strictly required by the logical argument, to explain to the reader why that argument is proceeding in a certain direction. Thus, on p. 89, he points out that expectations are easier to handle than probabilities.

It is a pleasure to report that the book is written in limpid, graceful, elegant English prose — nowadays a nearly vanished species.

The book is not without flaws, however. Some of Ball's historical references are unclear or wrong.

On p. 101, he attributes to Pólya the saying that a mathematician is someone to whom it is obvious that

This saying is found, attributed to Kelvin, in the biography of Kelvin by Silvanus P. Thompson, published in 1910, when Pólya was 13 years old.

On p. 220, Ball attributes to the Pythagoreans the standard modern proof of the irrationality of the square root of 2. (Write p2/q2 = 2, where p and q are relatively prime, and derive that both must be even, a contradiction.) This proof occurs (expressed in the language of incommensurability) in the manuscripts of Euclid's Elements (at the end of Book X), but is now considered to be an interpolation into Euclid's text. A remark which Aristotle makes in his Prior Analytics may refer to this proof. We simply don't know, however, whether this proof is due to the Pythagoreans, or whether they even had a proof.

On p. 155, referring to the exact formula for the nth Fibonacci number in terms of the square root of 5, Ball writes, "Such a formula is usually credited to the French mathematician Binet, although it was certainly well understood a century earlier (in particular by Bernoulli)." He gives no reference, nor does he say which Bernoulli he has in mind. Presumably the reference is to Daniel Bernoulli's work on "recurrent series", which led him to "Bernoulli's method" for solving polynomial equations. Indeed, the explicit formula for the Fibonacci numbers can be found in section 7 of Bernoulli's 1728 paper "Observationes de seriebus [recurrentibus]" (Werke, Band 2, p. 53).

In sum, Ball's book is a very good book. But more attention to historical accuracy would have made it even better.


Harold M. Stark, An Introduction to Number Theory, MIT Press, 1978, ISBN: 0-262-69060-8. (Originally Markham Publishing Company, 1970.)

Silvanus P. Thompson, The Life of Lord Kelvin, Chelsea Publishing Company, 1976, ISBN: 0-8284-0292-3. (Originally published as The Life of William Thomson, Baron Kelvin of Largs, Macmillan, London, 1910.) The anecdote about the normal probability integral is on p. 1139 of volume II.

Aristotle's reference to the irrationality of the square root of 2 is found in Prior Analytics, I.23, 41a 23-30. For the Euclidean interpolation, see Thomas L. Heath, The Thirteen Books of Euclid's Elements, 2d edition, vol. III (Cambridge University Press, 1926; reprinted by Dover Publications, 1956, ISBN: 0-486-60090-4), p. 2. For what we know of the Pythagoreans and the incommensurability of the diagonal, see David H. Fowler, The Mathematics of Plato's Academy, Oxford University Press, 1987, ISBN: 0-19-853912-6; 2d edition, 1999, ISBN: 0-19-850258-3; especially chapters 8 and 10 (2d edition).

L. P. Bouckaert and B. L. van der Waerden, eds., Die Werke von Daniel Bernoulli, Band 2, Birkhäuser, Basel, 1982, ISBN: 3-7643-1084-7. For Bernoulli's method for solving equations, see Leonhard Euler, Introduction to Analysis of the Infinite, Book I, Chapter XVII (translated by John D. Blanton, Springer-Verlag, 1988, ISBN: 0-387-96824-5; pp. 283--302.). For a modern treatment, see Peter Henrici, Essentials of Numerical Analysis (Wiley, 1982, ISBN: 0-471-05904-8), pp. 136--151.

Stacy G. Langton ( is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.

Preface xi
Acknowledgements xiii
Chapter One
Shannon's Free Lunch 1
1.1 The ISBN Code 1
1.2 Binary Channels 5
1.3 The Hunt for Good Codes 7
1.4 Parity-Check Construction 11
1.5 Decoding a Hamming Code 13
1.6 The Free Lunch Made Precise 19
1.7 Further Reading 21
1.8 Solutions 22
Chapter Two
Counting Dots 25
2.1 Introduction 25
2.2 Why Is Pick's Theorem True?27
2.3 An Interpretation 31
2.4 Pick's Theorem and Arithmetic 32
2.5 Further Reading 34
2.6 Solutions 35
Chapter Three
Fermat's Little Theorem and Infinite Decimals 41
3.1 Introduction 41
3.2 The Prime Numbers 43
3.3 Decimal Expansions of Reciprocals of Primes 46
3.4 An Algebraic Description of the Period 48
3.5 The Period Is a Factor of p 150
3.6 Fermat's Little Theorem 55
3.7 Further Reading 56
3.8 Solutions 58
Chapter Four
Strange Curves 63
4.1 Introduction 63
4.2 A Curve Constructed Using Tiles 65
4.3 Is the Curve Continuous? 70
4.4 Does the Curve Cover the Square? 71
4.5 Hilbert's Construction and Peano's Original 73
4.6 A Computer Program 75
4.7 A Gothic Frieze 76
4.8 Further Reading 79
4.9 Solutions 80
Chapter Five
Shared Birthdays, Normal Bells 83
5.1 Introduction 83
5.2 What Chance of a Match? 84
5.3 How Many Matches? 89
5.4 How Many People Share? 91
5.5 The Bell-Shaped Curve 93
5.6 The Area under a Normal Curve 100
5.7 Further Reading 105
5.8 Solutions 106
Chapter Six
Stirling Works 109
6.1 Introduction 109
6.2 A First Estimate for n 110
6.3 A Second Estimate for n 114
6.4 A Limiting Ratio 117
6.5 Stirling's Formula 122
6.6 Further Reading 124
6.7 Solutions 125
Chapter Seven
Spare Change, Pools of Blood 127
7.1 Introduction 127
7.2 The Coin-Weighing Problem 128
7.3 Back to Blood 131
7.4 The Binary Protocol for a Rare Abnormality 134
7.5 A Refined Binary Protocol 139
7.6 An Eficiency Estimate Using Telephones 141
7.7 An Eficiency Estimate for Blood Pooling 144
7.8 A Precise Formula for the Binary Protocol 147
7.9 Further Reading 149
7.10 Solutions 151
Chapter Eight
Fibonacci's Rabbits Revisited 153
8.1 Introduction 153
8.2 Fibonacci and the Golden Ratio 154
8.3 The Continued Fraction for the Golden Ratio 158
8.4 Best Approximations and the Fibonacci Hyperbola 161
8.5 Continued Fractions and Matrices 165
8.6 Skipping down the Fibonacci Numbers 169
8.7 The Prime Lucas Numbers 174
8.8 The Trace Problem 178
8.9 Further Reading 181
8.10 Solutions 182
Chapter Nine
Chasing the Curve 189
9.1 Introduction 189
9.2 Approximation by Rational Functions 193
9.3 The Tangent 202
9.4 An Integral Formula 207
9.5 The Exponential 210
9.6 The Inverse Tangent 213
9.7 Further Reading 214
9.8 Solutions 215
Chapter Ten
Rational and Irrational 219
10.1 Introduction 219
10.2 Fibonacci Revisited 220
10.3 The Square Root of d 223
10.4 The Box Principle 225
10.5 The Numbers e and p 230
10.6 The Irrationality of e 233
10.7 Euler's Argument 236
10.8 The Irrationality of p 238
10.9 Further Reading 242
10.10 Solutions 243
Index 247