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The BEST WRITING on MATHEMATICS 2011 Pitici.indb 1 8/20/2011 2:22:41 PM Pitici.indb 2 8/20/2011 2:22:42 PM The BEST WRITING on MATHEMATICS 2011 Mircea Pitici, Editor princeton university press princeton and oxford Pitici.indb 3 8/20/2011 2:22:43 PM Copyright © 2012 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved ISBN 978-0-691-15315-5 This book has been composed in Perpetua Std Printed on acid-free paper. ∞ Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 Pitici.indb 4 8/20/2011 2:22:43 PM To my teacher, Ioan Candrea In memoriam Pitici.indb 5 8/20/2011 2:22:44 PM Pitici.indb 6 8/20/2011 2:22:44 PM Contents Foreword: Recreational Mathematics Freeman Dyson xi Introduction Mircea Pitici xvii What Is Mathematics For? Underwood Dudley 1 A Tisket, a Tasket, an Apollonian Gasket Dana Mackenzie 13 The Quest for God’s Number Rik van Grol 27 Meta-morphism: From Graduate Student to Networked Mathematician Andrew Schultz 35 One, Two, Many: Individuality and Collectivity in Mathematics Melvyn B. Nathanson 43 Reflections on the Decline of Mathematical Tables Martin Campbell-Kelly 51 Under-Represented Then Over-Represented: A Memoir of Jews in American Mathematics Reuben Hersh 55 Did Over-Reliance on Mathematical Models for Risk Assessment Create the Financial Crisis? David J. Hand 67 Pitici.indb 7 8/20/2011 2:22:45 PM viii Contents Fill in the Blanks: Using Math to Turn Lo-Res Datasets into Hi-Res Samples Jordan Ellenberg 75 The Great Principles of Computing Peter J. Denning 82 Computer Generation of Ribbed Sculptures James Hamlin and Carlo H. Séquin 93 Lorenz System Offers Manifold Possibilities for Art Barry A. Cipra 115 The Mathematical Side of M. C. Escher Doris Schattschneider 121 Celebrating Mathematics in Stone and Bronze Helaman Ferguson and Claire Ferguson 150 Mathematics Education: Theory, Practice, and Memories over 50 Years John Mason 169 Thinking and Comprehending in the Mathematics Classroom Douglas Fisher, Nancy Frey, and Heather Anderson 188 Teaching Research: Encouraging Discoveries Francis Edward Su 203 Reflections of an Accidental Theorist Alan H. Schoenfeld 219 The Conjoint Origin of Proof and Theoretical Physics Hans Niels Jahnke 236 What Makes Mathematics Mathematics? Ian Hacking 257 What Anti-realism in Philosophy of Mathematics Must Offer Feng Ye 286 Seeing Numbers Ivan M. Havel 312 Pitici.indb 8 8/20/2011 2:22:45 PM Contents ix Autism and Mathematical Talent Ioan James 330 How Much Math is Too Much Math? Chris J. Budd and Rob Eastaway 336 Hidden Dimensions Marianne Freiberger 347 Playing with Matches Erica Klarreich 356 Notable Texts 367 Contributors 371 Acknowledgments 379 Credits 381 Pitici.indb 9 8/20/2011 2:22:46 PM Pitici.indb 10 8/20/2011 2:22:46 PM Foreword: Recreational Mathematics Freeman Dyson Hobbies are the spice of life. Recreational mathematics is a splendid hobby which young and old can equally enjoy. The popularity of Sudoku shows that an aptitude for recreational mathematics is widespread in the population. From Sudoku it is easy to ascend to mathematical pursuits that offer more scope for imagination and originality. To enjoy recre- ational mathematics you do not need to be an expert. You do not need to know the modern abstract mathematical jargon. You do not need to know the difference between homology and homotopy. You need only the good old nineteenth-century mathematics that is taught in high schools, arithmetic and algebra and a little geometry. Luckily for me, the same nineteenth-century mathematics was all that I needed to do useful calculations in theoretical physics. So, when I decided to become a professional physicist, I remained a recreational mathematician. This foreword gives me a chance to share a few of my adventures in recre- ational mathematics. The articles in this collection, The Best Writing on Mathematics 2011, do not say much about recreational mathematics. Many of them describe the interactions of mathematics with the serious worlds of education and finance and politics and history and philosophy. They are mostly looking at mathematics from the outside rather than from the inside. Three of the articles, Doris Schattschneider’s piece about Maurits Escher, the Fergusons’ piece on mathematical sculpture, and Dana Mackenzie’s piece about packing a circle with circles, come closest to being recre- ational. I particularly enjoyed those pieces, but I recommend the others too, whether they are recreational or not. I hope they will get you inter- ested and excited about mathematics. I hope they will tempt a few of you to take up recreational mathematics as a hobby. I began my addiction to recreational mathematics in high school with the fifty-nine icosahedra. The Fifty-Nine Icosahedra is a little book Pitici.indb 11 8/20/2011 2:22:47 PM xii Foreword published in 1938 by the University of Toronto Press with four authors, H.S.M. Coxeter, P. DuVal, H. T. Flather, and J. F. Petrie. I saw the title in a catalog and ordered the book from a local bookstore. Coxeter was the world expert on polyhedra, and Flather was the amateur who made models of them. The book contains a complete description of the fifty- nine stellations of the icosahedron. The icosahedron is the Platonic solid with twenty equilateral triangular faces. A stellation is a symmetrical solid figure obtained by continuing the planes of the twenty faces out- side the original triangles. I joined my school friends Christopher and Michael Longuet-Higgins in a campaign to build as many as we could of the fifty-nine icosahedra out of cardboard and glue, with brightly colored coats of enamel paint to enhance their beauty. Christopher and Michael both went on later to become distinguished scientists. Christopher, now deceased, was a theoretical chemist. Michael is an oceanographer. In 1952, Michael took a holiday from oceanography and wrote a paper with Coxeter giving a complete enumeration of higher-dimensional poly- topes. Today, if you visit the senior mathematics classroom at our old school in England, you will see the fruits of our teenage labors grandly displayed in a glass case, looking as bright and new as they did seventy years ago. My favorites among the stellations are the twin figures consisting of five regular tetrahedra with the twenty vertices of a regular dodecahe- dron. The twins are mirror images of each other, one right-handed and the other left-handed. These models give to anyone who looks at them a vivid introduction to symmetry groups. They show in a dramatic fash- ion how the symmetry group of the icosahedron is the same as the group of 120 permutations of the five tetrahedra, and the subgroup of rotations without reflections is the same as the subgroup of 60 even permutations of the tetrahedra. Each of the twins has the symmetry of the even permutation subgroup, and any odd permutation changes one twin into the other. Another book which I acquired in high-school was An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright, a wonderful cornucopia of recreational mathematics published in 1938. Chapter 2 2n contains the history of the Fermat numbers, Fn 5 2 1 1, which Fer- mat conjectured to be all prime. Fermat was famously wrong. The first four Fermat numbers are prime, but Euler discovered in 1732 that 32 F5 5 2 1 1 is divisible by 641, and Landry discovered in 1880 that Pitici.indb 12 8/20/2011 2:22:51 PM Foreword xiii 64 F6 5 2 1 1 is divisible by 274,177. Since fast computers became avail- able, many larger Fermat numbers have been tested for primality, and not one has been found to be prime. Hardy and Wright provide a simple argument to explain why F5 is divisible by 641. Since 641 5 1 1 5a 5 24 1 54 with a 5 27, we have 4 4 4 4 4 4 F5 ϭ2 a ϩ 1ϭ(1 ϩ 5aϪ 5 ) a ϩ 1ϭ(1 ϩ 5a ) a ϩ(1 Ϫ (5a)), which is obviously divisible by 1 1 5a. I was always intrigued by the question of whether a similarly elementary argument could be found to explain the factorization of F6. Sixty years later, I found the answer. This was another joyful piece of recreational mathematics. The answer turned out to depend on a theorem concerning palindromic continued frac- tions. If a and q are positive integers with a , q, the fraction a/q can be expressed in two ways as a continued fraction: a/ q ϭ 1/(p1 ϩ1/( p 2 ϩ 1/...(pnϪ1 ϩ1/ pn )...)), where the partial quotients pj are positive integers. The fraction is palin- dromic if pj 5 pn112j for each j. The theorem says that the fraction is palindromic if and only if a2 1 (21)n is divisible by q. Landry’s factor of F6 has the structure q ϭ274177 ϭ1ϩ 28 f ,(f ϭ 26 Ϫ 1)(24 ϩ1), where f is a factor of 224 2 1, so that 224 Ϫ 1 ϭ fg,(g ϭ26 ϩ 1)(2 8 Ϫ24 ϩ 1), and 232 ϭ 28(1 ϩ fg),ϭgq Ϫ a a ϭg Ϫ 28 ϭ15409. The partial quotients of the fraction a/q are (17, 1, 3, 1, 5, 5, 1, 3, 1, 17).
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