Q.E.D.: Beauty in Mathematical Proof (Wooden Books)

QED

Copyright © 2004 by Burkard Polster All rights reserved. No part of this book may be reproduced or transmitted in any form Jr by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the Publisher. First published in the United States of America in 2004 by Walker Publishing Company, Inc. Published simultaneously in Canada by Fitzhenry and Whiteside, Markham, Ontario LBR 4T8 For information about pemiission to reproduce selections from this book, write to _Permissions, Walker & Company, 104 Fifth Avenue, New York, New York 10011. Library of Congress Cataloging-in-Publication Data Polster, Burkard. Q.E.D.: beauty in mathematical proof / written and illustrated by Burkard Polster. p. cm. ISBN 0-8027-1431-5 (alk. paper) 1. Proof theory. 2. Logic, Symbolic and mathematical. I. Title: QED: beauty in mathematical proof II. Title. QA9.54.P65 2004 51 1 .3'6——dc22 2004041905 Visit Walker & Company’s Web site at www.walkerbooks.com Printed in the United States of America ‘ 4 6 ,8 10 9 7 5

QED Beauty in Mathematical Proof written and iilustrated by Burkard Polter Wooden Books Walker and Company New York

With love to Anu, who understands it all . . . I am indebted to the many giathematicians ofthe past and present fom whose ideas this book has been distilled. I am grateful to Marty Ross and john Stillwellfw their criticism and insightful comments. Finally, many, many thanks to _]ohn Martineau and Daud Sutton for being patient guides and accomplices in opening this visual vortex into the beautjul world cyr mathematical procys. E 1 3 *1 ‘ I E E‘Er‘ J V ; k` 1/4 % l I ` ? I is ; 2 Q , 2 Q 1/3 L 1 1/% F ;l!i`.l\.['1l.lH` l'_` 'l ‘ NN ' ly lt" , ' r` Y _` .i’(`.? ~,;7}¤.?'.- I Z'} ·) ‘? "" X" Two infinite sums—boxed and ready to be served.

Comrms Introduction 1 Treacherous Truth 2 Pythagoras’s Theorem 4 Plane and Simple 6 From Pie to Pi 8 Cavalieri’s Principle 10 Cavalier Cone Carving 12 A Frustrating Frustum 14 Archimedes’ Theorem 16 Inside Out 18 Mathematical Dominoes 20 The Inhnite Staircase 22 Circling the Cycloid 24 Slicing Cones 26 Folding Conics 28 Knotting Polygons 30 Cutting Squares 32 Power Sums 34 Never—ending Primes 36 The Nature of Numbers 38 The Golden Ratio 40 The Numbers of Nature 42 Euler’s Formula 44 Possible Impossibilities 46 Appendix I: One Theorem, Many Proots 48 Appendix II: All for One and One for All 50 Appendix III: Looks Can Be Deceiving 52 Appendix IV: Triangles of Generality 54 Appendix V: Polytopes of Analogy 56

v Squdré I lpfllrégflll -· `j g { h€Xq·90Tl h€Pl*q~g0Tl U0(.;l'qgl7Tl _; regular polpgon is a convexfigure withequal sides and angles.- (lhete are itfnitely many regular pohgons. _ _ _ _ _ tetrahedron octahedronl 3 icosahedron _ _ cube dodecahedron ,2 regular polyhedron is a convex body with identical regular polygons as faces. and the same numherncgc Z.. faces meeting at corner. Shown at the top are the dyjlerent ways of joining three or more identical · regularapolxgons to a corner with space left to fold up into three dimensions. flhese possibilities cflruilding __ i spatial corners then can beshown to extend in a unique way to the famous flwregularpolyhedra. __ "; `__` _ . _ _