Game, Set and Math: Enigmas and Conundrums

Game, Set, and Math

Gatne, Set, and Math Enigmas and Conundrums Ian Stewart

Basil Blackwell

Copyright © Ian Stewart 1989 First published 1989 Basil Blackwell Ltd 108 Cowley Road, Oxford, OX4 1JF, UK Basil Blackwell, Inc. 3 Cambridge Center Cambridge, Massachusetts 02142, USA All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.

British Library Cataloguing in Publication Data A CIP catalogue record for this book is available from the British Library.

Library of Congress Cataloging in Publication Data

Stewart, Ian. Game, set, and math: enigmas and conundrums/Ian Stewart. p. em. ISBN 0-631-17114-2 1. Mathematical recreations. 1. Title. QA95.S725 1989 793./'4--dc20 89-17652 CIP

Typeset in 10 on 12 pt Palatino by OPUS, Oxford Printed in Great Britain by T.J. Press, Padstow, Cornwall

Contents

1 2 3 4 5 6

Preface

Mother Worm's Blanket

Build your own Virus

1 15 31 41 55 71

Parity Piece

Close Encounters of the Fermat Kind

Pascal's Fractals The Worm Returns

10 11 12

The Drunken Tennis-Player The lnfinormatics Laboratory The Autovoracious Ourotorus Fallacy or Ycalla£?

All Parallels Lead to Rome The Twelve Games of Christmas

107 125 141 155 171

Preface

A few years ago, Philippe Boulanger asked me to suggest someone to write a "Mathematical Visions" column in Pour Ia Science. That's the French translation of Scientific American; Philippe is the editor. I first came across that magazine in my teens, and for me the high point was Martin Gardner's "Mathematical Games" column. When Gardner ceased writing it, the column eventually metamorphosed into A. K. Dewdney's admirable "Computer Recreations ". The change is perhaps symbolic of our times. But in France, the idea that computers are here to replace mathematics was resisted, and "Mathematical Games" lived on, in tandem with "Computer Recreations", under the name "Visions mathematiques". That fits my world-view: computing and mathematics have a symbiotic relationship, each needing the other. Anyway, the anchorman of the column had departed for pastures new, and Philippe was looking for a replacement. Did I know of anyone suitable? Of course I did, and modestly offered my advice. " Me. " He took it - with, I suspect, a few qualms. Two years later, the column has found its own identity and settled into its own style. I write it in English, and Philippe translates it (with considerable skill and also considerable licence) into French. I try to come up with puns that will work in French: for instance, "the twelve games of Christmas". That translates as "les douze jeux de Noel", whereas "the twelve days of Christmas" is "les douze jours de Noel ". And nowadays, whenever I encounter an interesting piece of mathematics, one part of my mind is thinking "I wonder if I could explain that in Pour Ia Science . . . ? " It offers a very different perspective; and on at least one occasion an idea that I had when thinking about "Visions Mathematiques" turned out to be useful in serious research. Anyway, here it is: Game, Set, and Math, a selection of twelve articles that present serious mathematics in less than serious fashion. I've edited

Preface vii them, updated them, and put the English puns back. People sometimes try to sell the idea that "mathematics can be fun". I think that gets the emphasis wrong. To me, mathematics is fun, and this book is a natural consequence of the way I approach the subject. Mind you, I can understand why most people find that statement baffling. To see why mathematics is fun, you have to find the right perspective. You have to stop being overawed by symbols and jargon, and concentrate on ideas; you have to think of mathematics as a friend, not as an enemy. I'm not saying that mathematics is always a joyous romp; but you should be able to enjoy it, at whatever level you operate. Do you enjoy crossword or jigsaw puzzles? Do you like playing draughts, or chess? Are you fascinated by patterns? Do you like working out what "makes things tick"? Then you have the capacity to enjoy mathematical ideas. And, perhaps, if you do enjoy them, you might even become a mathematician. We could do with more mathematicians. Mathematics is fundamental to our lifestyle. How many people, watching a television programme, realize that without mathematics there would be nothing to watch? Mathematics was a crucial ingredient in the discovery of radio waves. It controls the design of the electronic circuits that process the signals. When the picture on the screen rolls up into a tube and spins off to reveal another picture, the quantity of mathematics that has come to life as computer graphics is staggering. But that's mathematics at work. What this book is about is the flip side: mathematics at play. The two are not that far apart. Mathematics is a remarkable sprawling riot of imagination, ranging from pure intellectual curiosity to nuts-and bolts utility; and it is all one thing. The last few years have witnessed a remarkable re-unification of pure and applied mathematics. Topology is opening up entire new areas of dynamics; the geometry of multi dimensional ellipsoids is currently minting money for AT&T; obscure items such as p-adic groups turn up in the design of efficient telephone networks; and the Cantor set describes how your heart works. Yesterday's intellectual game has become today's corporate cash-flow. However, what you'll find here is the playful side of mathematics, not the breadwinning one. Some items are old favourites, some are hot off the press. Most chapters include problems for you to solve, with answers at the end; there are things to make and games to play. But there's a more serious intention, too. I'm hoping that at least some of you might be inspired to find out more about the remarkable mental world that lies behind the jokey presentation. The ideas you will encounter all have connections with real mathematics - though you might be forgiven if you didn't see through the heavy disguise. "Mother Worm's

viii

Preface

Blanket" is a problem in geometric measure theory, and "The Drunken Tennis-Player'' is about stochastic processes and Markov chains. "Parity Piece" introduces algebraic topology; "The Autovoracious Ourotorus" leads to coding theory and telecommunications. On the other hand, I can assure you that "Close Encounters of the Fermat Kind" has nothing whatsoever to do with space travel or the motion picture industry. Or does it? Wait a minute . . . Ian Stewart

Coventry

Mother Worm's Blanket

"Bother!" said Mother Worm. "Something the matter, dear?" "It's our sweet little Wermentrude. I know I shouldn't criticize the child, but sometimes - well - her blanket's come off again! She'll be chilled to the bone!" "Anne-Lida, worms don't have bones." "Well, chilled to her endodermic lining, then, Henry! The problem is that when she goes to sleep, she wriggles around and curls up into almost any position, and the blanket falls off."

2 Game, Set, and Math "Does she move once she's asleep?" "No, Henry, she sleeps like a log. " She even looks like a log, thought Henry Worm, but did not voice the thought. "Then wait until she's asleep before you cover her up, dearest." "Yes, Henry, I've thought of that. But there is another problem." "Tell me, my pet. " "What shape should the blanket be?" It took a while for Henry to sort that one out. It turned out that Mother Worm wanted to make a blanket which would completely cover her worm-child, no matter how she curled up. Just the worm, you understand: not the area she surrounds. The blanket can have holes. But, being thrifty, Mother Worm wished the blanket to have as small an area as possible. "Ah," said Father Worm, who -as you will have noticed -is something of a pedant. "We may choose units so that the length of the little horr- . . . dear little Wermentrude is 1 unit. You're asking what shape is the plane set of minimal area that will cover any plane curve of length 1. And no doubt you also wish to know what this minimal area is." "Precisely, Henry. " "Hmmmmmmmm. Tri-cky . . . " When you start thinking about Mother Worm's blanket, the greatest difficulty is to get any kind of grip. The problem tends to squirm away from you. But as Henry explained to his wife - in order to distract her attention from his inability to answer the question - there are some general principles that can form the basis of an attack. Suppose that we know where some points of the worm are: what can we say about the rest? He pointed outtwo such principles (box 1.1): they depend upon the fact that the shortest distance between two points is the straight line joining them. "Excellent," said Father Worm. "Now, Anne-Lida my dear, we can make some progress. An application of the Circle Principle shows that a circle of diameter 2 will certainly keep Wermentrude warm. Lay the centre of the blanket over Baby's tail, my dear: the rest of her cannot be more than her total length away! How big is the blanket? Well, a circle of diameter 2 has an area of 1t, which you'll recall is approximately 3.14159... . " "That's enough, Henry! I've already thought of something much better. Suppose that you (mentally!) chop Wermentrude into two at her mid-point. Each half lies inside a circle of radius ! centred on her mid point. If I place a circular blanket of radius ! - that is, diameter 1- so that its centre lies over Baby's mid-point, it will cover the dear little thing."

What's the area now? Remember pi-r-squared? In fact this is the smallest circle that will always cover Baby, because if she stretches out straight she can poke out of any circle of diameter less

Mother Worm's Blanket 3 Box 1.1

Blanket Regulations

The Circle Principle Suppose we have a portion of worm, of lengthL, and we know that one end of it is at a point P. Then that portion lies inside a circle of radiusL, centreP. The reason: every point on the portion is distance L or less away from P, measured along the worm. The straight line distance is therefore also L or less. But such points lie inside the circle of radius L. The Ellipse Principle Suppose we have a portion of worm, of length L, and we know where both ends are. Let the ends be at pointsP and Q in the plane. Form a curve as follows. Tie a string of length L between P and Q, insert the point of a pencil, and stretch the line taut. As the pencil moves, it describes an ellipse whose foci areP and Q. The points inside this ellipse are those points X for which P X+ XQ is less than or equal to L. Therefore every point on the portion of worm concerned lies inside this ellipse (figure 1 .1).

Circle

Ellipse

1.1 A portion of worm of length L, one point P of which is known, lies inside a circle of radius L, centre P. Iftwo points P and Qare known then the portion lies inside an ellipse with P and Qas foci, consisting ofall points X such that PX + XQ

=L.

than 1. But might a shape different from a circle be more economical? "It had better be, " groaned Father Worm, who would have to pay for the blanket, as he retired to his study. Two hours later he emerged with several sheets of paper and announced that Anne-Lida's proposal, a circle of diameter 1, is at least twice as large as is necessary.

Game, Set, and Math

"Good news, my dear. A semicircle of diameter 1 is big enough to cover Baby no matter how much the little pest - er, pet - squirms before snoozing. "

That cuts the area down even more: to what? As I said, Henry Worm is a pedant. He won't say anything like that unless he's absolutely certain it's true. So he hasn't just spent his time doing experiments with semicircles: he has a proofthat the unit semicircle (a semicircle of diameter 1) always works.lt isn't an easy proof, and if you want to skip it I wouldn't blame you. But proof is the essence of mathematics, and you may be interested to see Father Worm's line of reasoning. If so, it's in box 1.2. "Very clever, Henry, " sniffed Anne-Lida. "But I think the same idea shows that you can cut some extra pieces off the semicircle. You see, when P and Q are closer together than b, the distance between X and Y is less than 1. That must leave room for improvement, surely?" "Hrrrumph. You may well be right, my dear. But it gets very complicated to work out what happens next. " And Henry rapidly changed the topic of conversation. My more persistent readers may wish to pursue the matter, because Anne-Lida is right: the unit semicircle is not the best possible shape. Indeed, nobody knows what shape Baby Worm's blanket should be. The problem is wide open. Remember, it must cover her no matter what shape she squirms into;and you mustgive aproofthat this is the case! If you can improve on�, let me know. Later that evening, Henry suddenly threw down his newspaper, knocking over a glass of Pupa-Cola and soaking the full-size picture of Maggot Thatcher on the front page. "Anne-Lida! We've forgotten to ask whether a solution exists at all!" You can't keep a good pedant down. But he has a point. Plane sets can be a lot more complicated than traditional things like circles and polygons. The blanket may not be convex: in fact it might have holes! For that matter, what do we mean by the "area" of a complicated plane set? " My God," said Henry. "Perhaps the minimal area is zero!" "Don't be silly, dear. Then there would be no blanket at all! " Henry poured a replacement and sipped at it with a superior smirk. "Anne-Lida, it is obviously time I told you about the Cantor set. " "What have those horrible horsey snobs got to do with . . . " "Cantor, my dear, not canter. Georg Cantor was a German mathematician who invented a very curious set in about 1883. Actually, it was known to the Englishman Henry Smith in 1875 - but 'Smith set' wouldn't sound very impressive, would it? To get a Cantor set you start with a line segment of length 1, and remove its middle third. Now

Mother Worm's Blanket 5 Box 1. 2

Father Worm's Proof.

A line that meets Wermentrude at some point or points, but such that she lies entirelyonone side of it,is called a support line(figure 1 .2). Support lines exist in any direction. Just start with a line pointing in that direction and slide it until it first hits the worm. Notice that support lines may meet the worm in more than one point.

1.2

Support lines.

First, suppose that every support line meets Wermentrude in exactly one point. Then she must be curled up in a closed convex loop, perhaps with other bits of her inside (figure 1 .3). Suppose she touches a support line at a point P. Then all points on the loop are at a distance � or less from P, measured along the worm; hence also measured in a straight line. So are the other points inside the loop. Therefore Wermentrude lies inside the circle of radius � centre P. But she also lies on one side of the diameter of this circle formed by the support line. Thus she lies inside a unit semicircle. Alternatively, some support line meets Wermentrude in at least two points P and Q. These points divide her into three segments A, B, C of lengthsa,b,c, wherea+b+c 1 (figure 1 .4). The distance betweenP and Q is at most b because segment B joinsPto Q. By the Circle Principle,segment A lies inside a circle centre P radius a; but it also lies on one side of the =

6 Game, Set, and Math

1.3 If every support line meets the worm in a single point, then the worm determines a convex loop ofperimeter less than or equal to 1 and hence lies inside a unit semicircle. support line, so it actually lies inside a semicircle of radius segment C lies inside a semicircle of radius c.

Similarly

1.4 If a support line meets the worm in two points, then the worm lies inside a figure obtained by overlapping two semicircles and a semiellipse.

Mother Worm's Blanket 7 (a)

(b)

-----

(c)

1 .5 There are three possible arrangements of the semicircles and the semiellipse. In all three cases the distance XY is at most 1 . Therefore the semicircle on XY as diameter (dotted) fits inside the outer unit semicircle. So does the worm .

Game, Set, and Math What about segment B? By the Ellipse Principle, it lies within an ellipse whose foci are at P and Q,traced by stretching a string of length b. Because

of the support line, B actually lies inside a semiellipse (half an ellipse).

Thus the entire worm lies inside a rather complicated figure formed by overlapping two semicircles and a semiellipse. Let X andY be the extreme points of this figure on the support line. There is a minor complication now. The point X may be either on the semicircle centre P or the semiellipse; similarly Y may be either on the semicircle centre Q or the semiellipse. However, in each case it is not hard to show that the distance between X andY is 1 or less (figure 1 .5). Now, the ellipse is "flatter'' than a circle; so both semicircles and the semiellipse fit inside a semicircle whose diameter is XY. Since XY is 1 or less, Wermentrude fits inside a unit semicircle.

remove the middle third of each remaining piece.Repeat, forever.What is left is the Cantor set." (Figure 1.6) "I don't see how there can be anything left, Henry." "Oh, but there is.All the end-points of all the smaller segments are left, for a start.And many others.But you are right in one way, my dear.What is the length of the Cantor set? " "Its ends are distance 1 apart, Henry." "No, I meant the length not counting the gaps." ._______________________________________________�StageO

.__________ Stage

•

••--_..stage 2

Stage 3

......_.

.,......

��

-�Stage 4

__ __

_ __

Cantor set

1.6 Construction of the Cantor set by removing middle thirds. Its length is zero, but it contains infinitely many points.

Mother Worm's Blanket

"I have no idea, Henry.But it looks very small to me.The set is mostly holes." "Yes, like Wermentrude's sock." "Are you criticizing me? I'm going to dam her sock tomorrow!Of all the ..." "No, no, my dear; nothing was further from my mind. Hrrumph. The length reduces to � the size at each stage, so the total length after the nth stage is = T

= 0.8660 . . . .

5. Henry's cube-halving method fails in four dimensions, because the diagonal of a half-size hypercube is ...J ( ( !)2 + ( !)2 + ( !)2 + ( !)2 ) = 1 . 6. Cut the hypercube in half in three directions but into three equal parts along the fourth. This yields twenty-four smaller hypercuboids, each of diameter ..J( (!)2 + (!)2 + (!)2 + (�)2 ) =..J ( �) = 0.9279 . . . But perhaps you can improve on that? .

FURTHER READING

V. Boltjansky and I. Gohberg, Results and Problems in Combinatorial Geometry (Cambridge: Cambridge University Press, 1985) K. Borsuk, "Drei 5atze iiber die n-dimensionale Sphlire" ,Fundamenta mathematicae, 20 (1933), pp. 177-90 H. G. Eggleston, Convexity (Cambridge: Cambridge University Press, 1955) BrankoGriinbaum, "A Simple Proof of Borsuk' sConjecture in Three Dimensions", Proceedings of the Cambridge Philosophical Society, 53 (1957), pp. 776-8. I. Yaglom and V. Boltjansky, Convex Sets (New York: Holt, Rinehart and Winston, 1961)

All Parallels Lead to Rome

The city that invented apartment blocks has an insoluble housing shortage. The city that invented the public sewer has no adequate sewerage system. The city which in 45 BC banned wagons from its centre during daylight hours has an average traffic speed of 6 km/hr. There are three cars for every metre of road.

156

Game, Set, and Math

The city is noisy, filthy, and heavily in debt - and one of the most beautiful in the world. A living paradox. No wonder that a one-sided surface is named after it. No, there's no city named "Mobius" . This is Rome. And when in Rome . . . We sat at a table in the Via Vittorio Veneto, which winds downhill from the gardens of the Villa Borghese until it runs into the Piazza Barberini. An empty chianti bottle lay among the remains of a pasta lunch. A second, half full, stood beside it. "Good job chianti doesn't come in Klein bottles," I said. "I know that klein is German for 'small'," said Enrico, "and I agree that it's a good job chianti doesn't come in small bottles - but why have you lapsed into German all of a . . . ?" "No," I said. "I didn't mean that. It was a mathematician's joke. A Klein bottle is one whose inside and outside are the same." "It would save on corks," said Elena. "No, it would always leak," said Enrico. Enrico and Elena Macaroni: Henry and Helen. But it sounds so much more elegant in Italian. He runs an art gallery, and she runs him. "How can a bottle have its inside the same as its outside?" Elena asked, serious now. "It's a complicated story," I said. "The truthis thatit doesn't really have an inside or an outside . . . And it isn't really a bottle." "That explains a great deal." "Who was Klein?" asked Enrico. "Felix Klein was one of the greatest mathematicians that Germany ever produced," I said. "He was the second person to invent a surface with only one side. The first was August Mobius." I took a paper napkin, tore off a narrow strip, and joined its ends with a half-twist. "See: a Mobius band (figure1 1 .1 ) . But the Mobius band has an edge. Klein's bottle, invented in 1882, has no edges, it's a closed surface." (Figure 1 1 .2(a )) "And it has only one side?" "Imagine trying to paint the surface. You start on what looks like the outside, and carry on painting the tube. But it bends round, passes through itself, and then kind of turns inside-out. At that point you find that you're painting what you originally thought was the inside. There's only one side to the surface: it all joins up." "But that's because it passes through itself," said Elena. "No, it's because it turns inside-out and then joins up. I admit that it has to pass through itself if you want to make a model in three dimensional space. In four-dimensional space it doesn't cross itself, but it

All Parallels Lead to Rome

1 1 .1

157

The Mobius band.

(a)

(b)

(c)

1 1 .2 Three views of the Klein bottle. (a) Embedded in 3-space. (b) Embedded in 4space, thefourth dimension being illustrated by the depth ofshading. The self-intersection in 3-space does not occur in 4-space (despite the way the picture looks when projected into 3-space as here): the positions in the fourth dimension of the two sheets of surface (that is, their shades) are different at the apparent intersection. (c) A less familiarform of the Klein bottle obtained by joining a figure 8 to itself with a half-twist. The shades distinguish the two lobes of the figure 8.

158

Game, Set, and Math

still has only one side. Of course you have to learn how to think in four dimensions to see that." (Figure 1 1.2(b)) "Oh." "Another way to obtain a Klein bottle is to take a figure 8, move it round a circle, and give it a half-twist as you do so (figure 1 1.2(c) ). Butthat doesn't look very bottle-shaped. Actually," I went on, " I have a private theory about the name Klein bottle. I think it was originally Klein's surface. You see, in Klein's day there was quite an industry involving German mathematicians inventing new surfaces and getting them named after themselves. Kummer's surface and Steiner's surface, for instance, originally Kummersche Flache and Steinersche Flache, the '-sche' being a possessive ending and 'Flache' being German for 'surface'. So it probably started out as Kleinsche Flache, 'Klein's surface'. But it looks like a bottle, and the German for bottle is Flasche, so . . . "Some graduate student called it the Kleinsche Flasche!" said Elena. "'Klein's bottle'! A German pun!" "Exactly. Or maybe it was mistranslated. I do know that in Hilbert and Cohn-Vossen's famous book Anschauliche Geometrie they refer to 'Klein's surface, also known as the Klein bottle'. Maybe Hilbert invented the pun." "Fascinating," said Enrico. "Not very relevant to the real world, though." "Don't be so sure," I said. "You're an art-dealer, right?" "You know that." "Italy is famous for beautiful paintings. Masaccio, Canaletto, Gozzoli, Veneziano, della Francesca. Wonderful perspective, right?" "Perspective was invented in Italy." "Perspective drawing was invented in Italy. The basic idea was discovered by Brunelleschi,inabout 1420. And the geometry ofperspective was published by another Italian, Alberti, in 1436, in his book Della pittura. It's called projectivegeometry, and it describes the way in which the eye sees the world. The basic surface is known as the projective plane. In the projective plane there are no parallel lines: any two lines meet at a single point." (Figure 1 1 .3) "Crazy." "Furthermore, as Klein showed in 1874, the projective plane has only one side." "Not so good for painting, then," said Enrico. "No, you're wrong: you can get twice the size of painting on the same size of canvas!" Elena pointed out. Curiously, the projective plane was invented long before the Klein bottle; but it's virtually unknown outside mathematical circles, whereas the Klein bottle is famous. Below, we'll examine some of the possible "

All Parallels Lead to Rome

159

1 1 .3 The Annunciation of Domenico Veneziano (fl. 1438-61). The edges of the walls, in reality parallel, appear to the eye to meet "at infinity".

reasons for this; but first we need to become familiar with the projective plane. In ordinary geometry, there is a unique line joining any two distinct points. Most pairs of lines intersect in a unique point, but some - parallel lines - do not. But, from the right viewpoint . . . I led Enrico and Elena from the Via Vittorio Veneto to the nearby Via XX Settembre, part of a long, straight stretch running for almost 4 km from the middle of Rome towards the suburbs. "What do you see?" I asked them. "Traffic. Jammed solid, as usual." "No, I mean, something geometric." "Nothing special." "The two edges ofthe road, they're a pair of parallellines. Parallel lines don't meet. Look at them: do they look as if they don't meet?" Enrico and Elena humoured me by staring down the long, straight road. "They do seem to meet," said Elena. "On the horizon," said Enrico. "Precisely," I said. "When the eye looks at parallel lines, they appear to meet. In the geometry of the visual system, parallel lines do not exist. So we need a new kind of geometry, in which any two lines meet.

160 Game, Set, and Math "How far away is the point on the horizon where the two sides of the road would meet - if they were extended far enough?" "Ooh, about 50 kilometres," said Elena. "On a spherical Earth, yes. But on a plane?" "Well . . . At the edge." "It's a long way to the edge of a plane," said Enrico. "Infinitely long," I said. "The place where parallel lines appear to meet is at infinity. In the usual Euclidean plane, infinity doesn't exist. You can go as far as you like, but you can't actually get to infinity. But in projective geometry, you can. To achieve this you have to add extra 'ideal' points 'at infinity' to the plane (figure 1 1.4). The points 'at infinity' form an extra line, so you have to add that too. What you get then is a slightly larger plane, so to speak, in which any two points are joined by a unique line and any two lines meet in a unique point."

(a)

(b)

(c)

1 1 .4 The Euclidean plane (a) plus a line at infinity (b) forms the projective plane, provided we agree (c) that opposite pairs of points on the boundary such as AA or BB represent the same point of the projective plane.

"But parallel lines meet in two points," said Elena. "One at one end, one at the other." "Mmmm," I said. "But it would be nice if they met in only one, right? Prettier. More symmetric and elegant. More like actual lines." "Yes," she said doubtfully. "So we have to pretend that the two points at opposite ends of a pair of parallel lines are the same," I said. "That's silly." "Not as silly as it sounds. Have you ever been to infinity to see for yourself?"

All Parallels Lead to Rome 1 61 "No." "Mathematically, infinity is just an abstract construct, so we can endow it with any properties we want. I happen to want lines to meet in only one point. So I insist that the 'two' points at infinity, at either end of a pair of parallels, are to be considered as the same. It may sound odd, but it works. It's sort of like bending the lines round into a circle - except that they stay straight." "Clear as mud." "Good. So we get our first model of the projective plane: it's the usual plane, plus a 'line' at infinity, plus the rule that the opposite ends of pairs of parallels meet the 'line' at infinity in the same point." (Figure 1 1 .5)

In projective geometry,

looking south

looking north

1 1 .5 Looking south along a straight railway line we see two parallels meeting at infinity. Looking north, we see them meet again. Because two lines should meet in a unique point, we must identify these two "opposite" points at infinity.

"I'm having trouble visualizing it." "On the contrary, Elena, it's how your visual system actually works." "Well, I'm having trouble getting it into my head in one piece. And it's not at all clear to me why the projective plane has only one side, as you say. The ordinary plane has two sides: top and bottom." "Yes, but the top surface and the bottom surface get joined together at infinity because of the rule about the end-points of parallels being the same," I said. There are several different ways to "see" the shape of the projective plane, and some of them make it clearer than others that it has only one side. Probably the simplest is to take a topologist's viewpoint. As far as

162

Game, Set, and Math

a topologist is concerned, the whole infinite plane can be squashed up inside a circular disc (figure 1 1 .4(a)) - minus its boundary, of course. Then the extra "points at infinity" can be added in by gluing on the boundary as well (figure 1 1 .4(b)). It looks circular, but that's not a problem to a topologist. To accommodate the rule about the opposite ends being the same, we have to (mentally) "glue" opposite points of the boundary circle together (figure 1 1 .4(c)). If you try to bend the disc in three-dimensional space, so that this happens, then you have to pass it through itself (figure 1 1 .6). The top half of the picture is called a cross-cap.

11.6 If we attempt to identify opposite points in Figure 1 1 .4(c) by physically bending the plane it is necessary for the resulting surface to pass through itself, forming a cross cap. Along the self-intersection, the two "sides" of the plane join together to create a single-sided surface. The point at the top is singular: the surface near it cannot be continuously deformed into one or more separate discs.

The cross-cap cuts through itself along a line. Just as for the Klein bottle, this line of self-intersection is an artefact caused by the way we draw the surface in three-dimensional space. Mathematically, it isn't "really" there. But it helps us to visualize it. To get rid of it, we should think of a disc, whose opposite boundary points are identified mentally, rather than by actually bending the disc around to bring them together. You can see that this version of the projective plane only has one side. If you start painting the "outside" and cross the line of self-intersection, you end up on the "inside". You can see it in another way. If we cut out a strip that crosses the disc (figure 1 1 .7(a)) then we really can glue the ends together - and we get a Mobius band (figure 1 1 .7(b)). So the inside and outside already join up in this part of the projective plane. In fact we can

All Parallels Lead to Rome

(a)

(b)

163

(c)

1 1 .7 A strip (shaded) that runs across the projective plane (a) forms a Mobius band (b) because opposite points on the boundary are identified. Suitably deformed, the remaining two pieces join to form a disc (c). Abstractly, we can form a projective plane by sewing a Mobius band and a disc together along their edges.

see that a projective plane is just a Mobius band with a disc sewn on along the edge (figure 1 1 .7(c)). "It's all a bit unnatural," said Enrico. "I agree," I told him. "Your sense of artistic elegance is functioning very well. But there's another model for the projective plane that is both geometric and natural. Of course, it has its own peculiarities." "Of course." ''The idea is to increase the dimension of everything by one. When I say 'point' you must think 'line through the origin'. In ordinary three dimensional space. When I say 'line' you must think 'plane through the origin'. When I say 'two points lie on a line' you must think 'two lines lie on a plane'. OK?" "If it keeps you amused." 'Well, the abstract essence of geometry is the way that 'lines' and 'points' relate to each other, it's not their actual shape. The names are just useful labels. Here, it's useful to modify the labels to make the relationship clearer. In this 'beefed-up' version of geometry, any two 'points' lie on a unique 'line'. That is, any two lines through the origin lie in a unique plane (figure 1 1 .8(a)). You agree?"

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"Of course." "But in addition, any two 'lines' meet in a unique 'point'. That is, any two planes through the origin meet in a line (figure 1 1 .8 (b)) . So we have exactly the properties required in projective geometry. The projective plane is just three-dimensional space, but with a new meaning attached to 'point' and 'line'. Geometric, and natural."

(a)

(b)

1 1.8 In ordinary 3-space, two lines through the origin determinea unique plane (a) and two planes through the origin determine a unique line (b). Each line cuts the sphere (shaded) in a pairofopposite points; each plane cuts it in a great circle. The projective plane can thus be interpreted either as the geometry of lines and planes through the origin in 3-space, or as that of point-pairs and great circles on a sphere.

''Natural?" "Natural enough for a mathematician." "But how can you call three-dimensional space a plane?" asked Elena. "Because we've increased all the dimensions by one." I reminded her. "If a 'line' is a plane through the origin, then a 'plane' has to correspond to a three-dimensional object - which must be the whole space." "Not only that: you can show that this new version of the projective plane is just the original one in disguise." "How? It doesn't look like it to me!"

All Parallels Lead to Rome 165 "It's a rather heavy disguise. Imagine a sphere centred at the origin. lt cuts every 'point' of the projective plane - that is, every line through its centre - in a pair of opposite points. It cuts every 'line' - plane through its centre - in a great circle. So the geometry of the projective plane is just the geometry of the sphere, with 'point' interpreted as a pair of antipodal points, and 'line' interpreted as great circle." "Fine. But we've got pairs of points, not individual ones." ''That doesn't really matter," I said. "Not in the abstract. But we can overcome that by thinking just of a hemisphere. That cuts most pairs down to single points." "Except points on the boundary of the hemisphere." "Precisely, Enrico! Well done! So we haveto identify opposite pointson the boundary of the hemisphere (figure 1 1 . 9). Just as our first model of the projective plane identified opposite points on the boundary of a disc.

11.9 To obtain single points rather than point pairs we can restrict attention to a hemisphere, obtaining a geometry ofpoints and great semicircles. But opposite points on the boundary must still be identified. This version of the projective plane is therefore a topological distortion offigure 1 1 .4. "And, topologically speaking, a hemisphere is a disc. It's just got a bit bent. So the new model is just the old one in disguise." "Bravo!" They applauded . I suspected irony, but I playeu along and gave a low bow.

166 Game, Set, and Math "Bis!" yelled Elena, meaning "encore", getting carried away by the spirit of things. Enrico tried to shut her up but the damage was done. I began to perform my encore. "There are lots of different ways to visualize the projective plane," I said. "Dozens." (Enrico groaned.) "One of them was discovered by Jacob Steiner. Well, sort of. The year was 1844, and by coincidence he was visiting Rome, so he called it the Roman surface (figure 1 1 .10). It's one of the few mathematical objects named after a place. In actual fact, he constructed it in a highly complicated fashion using pure geometry. Now, using coordinates, every surface is determined by some equation. For instance, a sphere of radius 1 centred at the origin has equation 2 / + y 2 + z = 1 in coordinates (x,y,z). Steiner was a wonderful geometer but hopeless at algebra, and he couldn't work out the equation for his surface. A year before Steiner died he asked Karl Weierstrass to work the equation out. Weierstrass, a much more versatile mathematician than Steiner, found the equation with no trouble at all·. 2 2

2 2

x y + y z + z x + xyz = 0. It's beautifully symmetric, just like the surface." Enrico and Elena admired the elegant symmetry of their native city's surface.

1 1 .10 Steiner 's Roman surface: six cross-capsjoined together. It has the same symmetry as a tetrahedron.

All Parallels Lead to Rome 167 "The Klein bottle has an equation, too," I said. "It's formed by the points (x,y,z) such that 2

(

(x + y + z + 2y - 1) (x + y + z - 2y - 1) - 8z 2 2 2 16xz (x + y + z - 2y - 1) = 0.

It's not as symmetric; but then neither is the surface. "The Roman surface is just a projective plane in yet another disguise. But it has a flaw." They shook their heads in horror at this news. "Like the cross-cap, it has (several) singular points. Those are places where it doesn't just cross through itself in two or more separate sheets, but the sheets get all tangled up and merge together. Like the top of the cross-cap. The Klein bottle, on the other hand, has no singular points. It crosses itself, but in clearly defined separate sheets. Maybe that's why most people don't realize that the projective plane is really a simpler example of a one-sided surface. It's a lot easier to draw a convincing Klein bottle." "It's a snappier name, too. Someone did a better public relations job." "Could be. For a long time it was actually an unsolved question whether the projective plane can be arranged in three-dimensional space so that it has no singular points - only self-intersections. In fact David Hilbert, one of the greatest mathematicians who ever lived, conjectured that it can't be done - and told his student Werner Boy to prove it. Boy, like any good research student, followed his own nose and disproved Hilbert's conjecture instead, producing what is now known as Boy's surface (figure 1 1 .1 1 ) . That's yet another incarnation of the projective plane." "It looks sort of funny," said Elena. ''Yes. It's a bit like three doughnuts stuck together, but the dough of each doughnut runs into the hole in the next. There's a polyhedral model for the Boy surface, which you can make from cardboard (figure 1 1 .12). That may give you a better idea of the shape." "Does the Boy surface have a pretty equation, like Steiner's?" asked Elena. I thought it was a highly intelligent question, and told her so. "That's perhaps the greatest curiosity of them all," I said. "Until very recently, nobody knew the answer. They could draw the surface, they could study its topology, but they couldn't decide whether or not it had a polynomial equation, pretty or not. In 1978 Bernard Morin, a French geometer who, incidentally, is blind, found equations for a projective plane without singularities but nobody could prove it was the same as the Boy surface. In 1985 J. F. Hughes found an empirical formula using polynomials of degree 8. But both formulas are parametrizations; that is, instead of an equation 'something in x,y,z = 0' they take the form 'x, y, and z = certain expressions in some other variables'. In principle you can

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1 1 .11 Boy 's surface, topologically equivalent to the projective plane and having no singular points. Hilbert conjectured that no such surface exists. The self-intersections (solid lines) form a "bouquet" of three loops joined at a common point. These two very different views are topologically equivalent. In each, sections of the surface have been cut away to reveal the interior.

All Parallels Lead to Rome

1 69

1 1 .1 2 To make a polyhedral model ofBoy 's surface, cut this shapefrom thin card and join the edges with the same numbers.

eliminate the new variables and get some hugely complicated equation in x, y, and z, but I don't think anybody has done it. "In 1986 Fran

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