**The Best Writing on Mathematics 2011** Author Mircea Pitici Isbn 9780691153155 File size 4 05MB Year 2011 Pages 416 Language English File format PDF Category Mathematics This anthology brings together the year s finest mathematics writing from around the world Featuring promising new voices alongside some of the foremost names in the field The Best Writing on Mathematics 2011 makes available to a wide audience many articles not easily found anywhere else and you don t need to be a mathematician to enjoy them These writings offe

Publisher :

Author : Mircea Pitici

ISBN : 9780691153155

Year : 2011

Language: English

File Size : 4.05MB

Category : Mathematics

The BEST

WRITING on

MATHEMATICS

2011

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The BEST

WRITING on

MATHEMATICS

2011

Mircea Pitici, Editor

p r i nc e ton u n i v e r s i t y p r e s s

p r i nc e ton a n d ox f or d

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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

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To my teacher, Ioan Candrea

In memoriam

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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

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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

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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

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356

Notable Texts

367

Contributors

371

Acknowledgments

379

Credits

381

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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 recreational 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 recreational 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 recreational. I particularly enjoyed those pieces, but I recommend the others

too, whether they are recreational or not. I hope they will get you interested 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

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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 fiftynine 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 outside 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 polytopes. 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 dodecahedron. 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 fashion 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 inn 1938. Chapter 2

contains the history of the Fermat numbers, Fn 5 22 1 1, which Fermat conjectured to be all prime. Fermat was famously wrong. The first

four Fermat numbers are prime, but Euler discovered in 1732 that

F5 5 232 1 1 is divisible by 641, and Landry discovered in 1880 that

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Foreword

F6 5 264 1 1 is divisible by 274,177. Since fast computers became available, 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

F5 24 a 4 1 (1 5a 54 )a 4 1 (1 5a)a 4 (1 (5a)4 ),

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 fractions. 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 1/(p1 1/(p2 1/...(pn1 1/pn )...)),

where the partial quotients pj are positive integers. The fraction is palindromic 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 274177 1 28 f , f (26 1)(24 1),

where f is a factor of 224 2 1, so that

224 1 fg, g (26 1)(28 24 1),

and

232 28 (1 fg) gq a, a g 28 15409.

The partial quotients of the fraction a/q are (17, 1, 3, 1, 5, 5, 1, 3, 1,

17). A beautiful palindrome, and the palindrome theorem tells us that

a2 1 1 5 qu with u integer. Therefore F6 5 1 1 (gq 2 a)2 5 q(g2q 2

2ga 1 u) is divisible by q. I was particularly proud of having discovered

the palindrome theorem, until I learned that I had been scooped by the

French mathematician Joseph Alfred Serret, who published it in 1848.

To be scooped by one of the lesser-known luminaries of the nineteenth

century is part of the game of recreational mathematics. On another

occasion I was scooped by Riemann, but that is a long story and I do not

have space to tell it here.

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xiv

Foreword

When I was an undergraduate at Cambridge University, I was intrigued by a famous discovery of the Indian prodigy Ramanujan concerning the arithmetical properties of the partition function p(n). This

discovery was lovingly described in Hardy and Wright’s An Introduction

to the Theory of Numbers. For any positive integer n, p(n) is the number of

ways of expressing n as a sum of positive integer parts. Ramanujan discovered that p(5k 1 4) is always divisible by 5, p(7k 1 5) is divisible

by 7, and p(11k 1 6) is divisible by 11. I wanted to find a way of actually

dividing the partitions of 5k 1 4 into 5 equal classes, and similarly for 7

and 11. I found a simple way to do the equal division. The “rank” of any

partition is defined as the biggest part minus the number of parts. The

partitions of any n can be divided into 5 rank classes, putting into class

m the partitions that have rank of the form 5j 1 m for m 5 1, 2, 3, 4, 5.

I found to my delight that the 5 rank classes of partitions of 5k 1 4 are

exactly equal. The same trick works for 7 but not for 11. It was easy to

check numerically that the rank classes were equal for the partitions of

5k 1 4 and 7k 1 5 all the way up to 100, but I failed to find a proof.

I also conjectured the existence of another property of a partition that

would do the same job for the partitions of 11k 1 6, and I called that

hypothetical property the “crank.”

Ten years later, Oliver Atkin and Peter Swinnerton-Dyer succeeded

in proving the equality of the rank-classes for 5k 1 4 and 7k 1 5, and

45 years later, Frank Garvan and George Andrews identified the crank.

Garvan and Andrews not only found the crank, but also proved that it

provides an equal division of the crank classes for all three cases, 5, 7,

and 11. More recently, in 2008, Andrews made another discovery as

beautiful as Ramanujan’s original discovery. Andrews was looking at another function S(n) enumerating the smallest parts of partitions of n.

S(n) is defined as the sum, over all partitions of n, of the number of

smallest parts in that partition. Andrews discovered and also proved that

S(5k 1 4) is divisible by 5, S(7k 1 5) is divisible by 7, and S(13k 1 6) is

divisible by 13. The appearance of 13 instead of 11 in this statement is

not a typographical error. It is a big surprise and adds a new mystery to

the mysteries discovered by Ramanujan.

The question now arises whether there exists another property of

partitions with their smallest parts, like the rank and the crank, allowing

us to divide the partitions with their smallest parts into 5 or 7 or 13

equal classes. I conjecture that such a property exists, and I offer a chal-

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Foreword

xv

lenge to readers of this volume to find it. To find it requires no expert

knowledge. All that you have to do is to study the partitions and smallest

parts for a few small values of n, and make an inspired guess at the property that divides them equally. A second challenge is to prove that the

guess actually works. To succeed with the second challenge probably

requires some expert knowledge, since I am asking you to beat George

Andrews at his own game.

My most recent adventure in recreational mathematics is concerned

with the hypothesis of Decadactylic Divinity. Decadactylic is Greek for

ten-fingered. In days gone by, serious mathematicians were seriously

concerned with theology. Famous examples were Pythagoras and Descartes. Each of them applied his analytical abilities to the elucidation of

the attributes of God. I recently found myself unexpectedly following in

their footsteps, applying elementary number theory to answer a theological question. The question is whether God has ten fingers. The evidence in favor of a ten-fingered God was brought to my attention by

Norman Frankel and Lawrence Glasser. I hasten to add that Frankel and

Glasser were only concerned with the mathematics, and I am solely responsible for the theological interpretation. Frankel and Glasser were

studying a sequence of rational approximations to p discovered by

Derek Lehmer. For each integer k, there is a rational approximation

[R1(k)/R2(k)] to p, with numerator and denominator defined by the

identity

∞

R1 (k ) R2 (k )π ∑ [(m!)2/(2m)!]2m mk .

m1

The right-hand side of this identity has interesting analytic properties

which Frankel and Glasser explored. The approximations to p that it

generates are remarkably accurate, beginning with 3, 22/7, 22/7,

335/113, for k 5 1, 2, 3, 4. Frankel and Glasser calculated the first

hundred approximations to high accuracy, and found to their astonishment that the kth approximation agrees with the exact value of p to

roughly k places of decimals. I deduced from this discovery that God

must calculate as we do, using arithmetic to base ten, and it was then

easy to conclude that He has ten fingers. It seemed obvious that no other

theological hypothesis could account for the appearance of powers of

ten in the approximations to such a transcendental quantity as p.

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Foreword

Unfortunately, I soon found out that my theological breakthrough

was illusory. I calculated precisely the magnitude of the error of the kth

Lehmer approximation for large k, and it turned out that the error does

not go like 102k but like Q2k, where Q 5 9.1197 . . . is a little smaller

than ten. For large k, the approximation is in fact only accurate to 0.96k

places of decimals, where 0.96 is the logarithm of Q to base ten. Q is

defined as the absolute value of the complex number q 5 1 1 (2pi/

ln(2)). When we are dealing with complex numbers, the logarithm is a

many-valued function. The logarithm of 2 to the base 2 has many values,

beginning with the trivial value 1. The first nontrivial value of log2(2)

is q. This is the reason why q determines the accuracy of the Lehmer

approximations. This calculation demonstrates that God does not use

arithmetic to base ten. He uses only fundamental units such as p and

ln(2) in the design of His mathematical sensorium. The number of His

fingers remains an open question.

Two of these recreational adventures were from my boyhood and two

from my old age. In between, I was doing mathematics in a more professional style, finding problems in the understanding of nature where elegant nineteenth-century mathematics could be usefully applied. Mathematics can be highly enjoyable even when it is not recreational. I hope

that the articles in this volume will spark readers’ interest in digging

deeper into some aspect of mathematics, whether it is puzzles and games,

history of mathematics, mathematics education, or perhaps studying for

a professional degree in mathematics. The joys of mathematics are to be

found at all levels of the game.

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Introduction

Mircea Pitici

This new volume in the series of The Best Writing on Mathematics brings

together a collection of remarkable texts, originally printed during 2010

in publications from several countries. A few exceptions from the strict

timeframe are inevitable, due to the time required for the distribution,

surveying, reading, and selection of a vast literature, part of it coming

from afar.

Over the past decade or so, writing about mathematics has become a

genre, with its own professional practitioners—some highly talented,

some struggling to be relevant, some well established, some newcomers.

Every year these authors, considered together, publish many books. This

abundance is welcome, since writing on mathematics realizes the semantic component of a mental activity too often identified to its syntacticprocedural mode of operation.The appropriation to the natural language

of meaningful intricacies latent in symbolic formulas opens up paths toward comprehending the abstraction that characterizes mathematical

thinking and some mathematical notions; it also offers unlimited expressive, imaginative, and cognitive possibilities. In the second part of the

introduction, I mention the books on mathematics that came to my attention last year; the selection in this volume concerns mostly pieces that

are not yet available in book form—either articles from academic journals or good writing in the media that goes unobserved or is forgotten

after a little while. The Best Writing on Mathematics reflects the literature

on mathematics available out there in myriad publications, some difficult

to consult even for people who have access to exceptional academic resources. In editing this series I see my task as restitution to the public, in

convenient form, of excellent writing on mathematics that deserves enhanced reception beyond the initial publication. By editing this series I

also want to make widely available good texts about mathematics that

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Introduction

otherwise would be lost in the deluge of information that surrounds us.

The content of each volume builds itself up to a point; I only give it a

coherent structure and present it to the reader. That means that every

year some prevailing themes will be new, others will reappear.

Most readers of this book are likely to be engaged with mathematics

in some way, at least by being curious about it. But most of them are inevitably engaged with only a (small) part of mathematics. That is true

even for professional mathematicians, with rare exceptions. Mathematics has far-reaching tentacles, in pure research branches as well as in

mundane applications and in instructional contexts. No wonder the

stakeholders in the metamorphosis of mathematics as a social phenomenon can hardly be well informed about the main ideas and developments

in all the different aspects connected to mathematics. Solipsism among

mathematicians is surely not as common as the general public assumes it

is; yet specialization is widespread, with many professionals finding it

difficult to keep abreast of developments beyond their strict areas of interest. By making this volume intentionally eclectic, I aim to break some

of the barriers laid by intense specialization. I hope that the enterprise

makes it easier for readers, insiders and outsiders, to identify the main

trends in thinking about mathematics in areas unfamiliar to them.

Anthologies of writings on mathematics have a long—if sparse and

irregular—history. Countless volumes of contributed collections in particular fields of mathematics exist but, to my knowledge, only a handful

of anthologies that include panoramic selections across multiple fields.

Soon after the Second World War,William Schaaf edited Mathematics, Our

Great Heritage, which included writings by G. H. Hardy, George Sarton,

D. J. Struik, Carl G. Hempel, and others. A few years later James Newman edited The World of Mathematics in four massive tomes, a collection

widely read for decades by many mathematicians still active today. In

parallel, in francophone countries circulated Le Lionnaise’s Les Grand

Courants de la Pensée Mathématique, translated into English only several

years ago. During the 1950s and 1960s a synthesis in three volumes

edited by A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrentiev,

including contributions by Soviet authors, was translated and widely

circulated. Very few similar books appeared during the last three decades of the twentieth century; notable was Mathematics Today, edited by

Lynn Arthur Steen in the late 1970s. In the present century the pace

quickened; several excellent volumes were published, starting with

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Introduction

xix

Mathematics: Frontiers and Perspectives edited in 2000 by Vladimir Arnold,

Michael Atiyah, Peter Lax, and Barry Mazur, followed by the voluminous tomes edited by Björn Engquist and Wilfried Schmid in 2001,

Timothy Gowers in 2008, as well as the smaller collections edited by

Raymond Ayoub in 2004 and Reuben Hersh in 2006 (for complete references, see the list of works mentioned at the end of this introduction).

The BestWriting on Mathematics builds on this illustrious tradition, capitalizing on an ever more interconnected world of ideas and benefiting

from the regularity of yearly serialization.

Overview of theVolume

The texts included in this volume touch on many topics related to mathematics. I gave up the thematic organization adopted in the first volume,

since some of the texts are not easy to categorize and some themes

would have been represented by only one or very few pieces.

Underwood Dudley argues that mathematics beyond the elementary notions is the best preparation for reasoning in general and that

most people value it primarily for that purpose, not for its immediate

practicality.

Dana Mackenzie describes the overt and the hidden properties of the

Apollonian gasket, a configuration of infinitely nested tangent circles

akin to a fractal.

Rik van Grol tells the story of finding the optimal number of steps

that solve scrambled Rubik’s cubes of different sizes—starting with the

easy cases and going to the still unsolved ones.

Andrew Schultz writes on the friendly professional interactions that

shape the career of a mathematician, from learning the ropes as a graduate student to becoming an accomplished academic.

In a polemical reply to a text we selected in last year’s volume of this

series (Gowers and Nielsen), Melvyn Nathanson argues that the most

original mathematical achievements are distinctively individual, rather

than results of collaboration.

Martin Campbell-Kelly meditates on the long flourishing popularity

and recent demise of mathematical tables.

Reuben Hersh ponders on the post–World War II abundance of Jewish mathematicians at American universities, in contrast to the pale prewar representation of Jews among American mathematicians.

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