Fractal Music, ~~~ercardsand More . Mathematical Recreations from SCIENTIFICAMERICAN Magazine Fractal Music, Mathematical Recreations from SCIENTIFICAMERICAN Magazine Martin Gardner W. H. Freeman and Company New York Library of Congress Cataloging-in-Publication Data Gardner, Martin, 1914- Fractal music, hypercards and more : mathematical recreations from Scientific: American I by Martin Gardner. p. cn1. Includes index. ISBN 0-7 167-2 188-0. -ISBN 0-7 167-2 189-9(pbk.) 1. Mat hematical recreations. I. Scientific American. 11. Title. QA95.Gi16 1991 793.8-dc20 91-17066 CIP Copyright O 1992 by W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted or otherwise copied for public or private use, without written permission from the publisher. Printed in the United States of America To Douglas Hofstadter For opening our eyes to the "strange loops" inside our heads, to the deep mysteries of memory, intelligence, and self-awareness, and for the incomparable insights, humor, and wordplay in his writ- ings. Preface ix Chapter One White, Brown, and Fractal Music 1 Chapter Two The Tinkly Temple Bells 24 Chapter Three Mathematical Zoo 39 Chapter Four Charles Sanders Peirce 61 Chapter Five Twisted Prismatic Rings 76 Chapter Six The Thirty Color Cubes 88 Chapter Seven Egyptian Fractions 100 Chapter Eight Minimal Sculpture 1 10 Chapter Nine Minimal Sculpture I1 133 Chapter Ten Tangent Circles 149 Chapter Eleven The Rotating Table and Other Problems 167 Chapter Twelve Does Time Ever Stop? Can the Past B;e Altered? 191 Chapter Thirteen Generalized Ticktacktoe 202 Chapter Fourteen Psychic Wonders and Probability 2 I14 Chapter Fifteen Mathematical Chess Problems 228 Chapter Sixteen Douglas Hofstadter's adel, Escher, Bach 243 Chapter Senenteen Imaginary Numbers 257 Chapter Eighteen Pi and Poetry: Some Accidental Patterns 271 Chapter Nineteen More on Poetry 281 Chapter Twenty Packing Squares 289 Chapter Twenty-one Chaitin's Omega 307 Name Index 3 2 1 his book reprints my Mathematical Games columns from the 1978 and 1979 issues of Scientific American Magazine. It is theT fourteenth such collection, and I have one more to go be- fore running out of columns. As in previous anthologies, ad- denda to the chapters update the material with information supplied by faithful readers, and by papers published after the columns were written. In two cases-a column c)n pi and poetry, and one on minimal sculpture-there was so much to add that I have written new chapters which appear here for the first time. The book is dedicated to my friend Douglas Hofstadter, whom I first met when he was seeking a publisher for his classic Godel, Escher, Bach, and who later became my suc- cessor at Scientific American. I had the privilege of r~eviewing GEB in a column that is reprinted here. ~kte,Brown, and Fracta 1 Music "For when there are no words [accompanying music] it is very difficult to recognize the meaning of the harmony and rhythm, or to see that any wor- thy object is imitated by them." -PLATO, Laws, Book I1 lato and Aristotle agreed that in some fashion all the fine arts, P including music, "imitate" nature, and from their day until the -late 18th century "imitation" was a central concept in west- ern aesthetics. It is obvious how representational painting and sculpture "represent," and how fiction and the stage copy life, but in what sense does music imitate? By the mid-18th century philosophers and critics were still arguing over exactly how the arts imitate and .whether the term is relevant to music. The rhythms of music may be said to imitate such natural rhythms as heartbeats, walking, running, flapping wings, waving fins, water waves, the peri- odic motions of heavenly bodies and so on, but this does not explain why we enjoy music more than, say, the souind of ci- cadas or the ticking of clocks. Musical pleasure derives mainly from tone patterns, and nature, though noisy, is singularly de- void of tones. Occasionally wind blows over some object to produce a tone, cats howl, birds warble, bowstrings twang. A Greek legend tells how Hermes invented the lyre: he found a turtle shell with tendons attached to it that produced musical tones wlhen they were plucked. Above all, human beings sing. Musical instruments may be said to imitate song, but what does singing imitate? A sad, happy, angry or serene song somehow resembles sadness, joy, anger or serenity, but if a melody has no words and invokes no speci~almood, what does it copy? It is easy to understand Plato's mystification. There is one exception: the kind of imitation that plays a role in "program music." A lyre is severely limited in the nat- ural sounds it can copy, but such limitations do not apply to sympho~~icor electronic music. Program music has no diffi- culty featuring the sounds of thunder, wind, rain, fire, ocean waves and brook murmurings; bird calls (cuckoos and crow- ing coclrs have been particularly popular), frog croaks, the gaits of animals (the thundering hoofbeats in Wagner's Ride of the Valkyries), the flights of bumblebees; the rolling of trains, the clang of hammers; the battle sounds of marching soldiers, clashing: armies, roaring cannons and exploding bombs. S1aughtc.r on Tenth Avenue includes a pistol shot and the wail of a police-car siren. In Bach's Saint Matthew Passion we hear the earthquake and the ripping of the temple veil. In the Al- pine Syvnphony by Richard Strauss, cowbells are imitated by the shaking of cowbells. Strauss insisted he could tell that a certain female character in Felix Mottl's Don Juan had red hair, and he once said that someday music would be able to distinguish the clattering of spoons from that of forks. Such imitative noises are surely a trivial aspect of music even when it accompanies opera, ballet or the cinema; be- sides, siuch sounds play no role whatsoever in "absolute mu- sic," music not intended to "mean" anything. A Platonist might argue th~atabstract music imitates emotions, or beauty, or the divine harmony of God or the gods, but on more mundane lev- els musilc is the least imitative of the arts. Even nonobjective paintings resemble certain patterns of nature, but nonobjec- tive music resembles nothing except itself. Since the turn of the century most critics have agreed that "imitation" has been given so many meanings (almost all White, Brown, and Fractal Music 3 are found in Plato) that it has become a useless synonym for "resemblance." When it is made precise with referenlce to lit- erature or the visual arts, its meaning is obvious and trivial. When it is applied to music, its meaning is too fuzzy to be helpful. In this chapter we take a look at a surprising discov- ery by Richard F. Voss, a physicist from Minnesota who joined the Thomas J. Watson Research Center of the International Business Machines Corporation after obtaining his 1Ph.D. at the University of California at Berkeley under the guidance of John Clarke. This work is not likely to restore "innitation" to the lexicon of musical criticism, but it does suggest a cu- rious way in which good music may mirror a subtle statistical property of the world. The key concepts behind Voss's discovery are what math- ematicians and physicists call the spectral density (or power spectrum) of a fluctuating quantity, and its "autocorrelation." These deep notions are technical and hard to understand. Be- noit Mandelbrot, who is also at the Watson Research Center, and whose work makes extensive use of spectral densities and autocorrelation functions, has suggested a way of avoiding them here. Let the tape of a sound be played faster or slower than normal. One expects the character of the sound to change considerably. A violin, for example, no longer sounds like a violin. There is a special class of sounds, however, that be- have quite differently. If you play a recording of such a sound at a different speed, you have only to adjust the volume to make it sound exactly as before. Mandelbrot calls sucl~sounds "scaling noises." By far the simplest example of a scaling noise is what in electronics and information theory is called white noise (or "Johnson noise"). To be white is to be colorless. White noise is a colorless hiss that is just as dull whether you play it faster or slower. Its autocorrelation function, which measures how its fluctuations at any moment are related to previous fluctua- tions, is zero except at the origin, where of course it must be 1. The most commonly encountered white noise is the ther- mal noise produced by the random motions of electrons through an electrical resistance. It causes most of the static in a radio or amplifier and the "snow" on radar and television screens when there is no input. With randomizers such as dice or spinners it is easy to generate white noise that can then be used for composing a random "white tune," one with no correlation between any two notes. Our scale will be one octave of seven white keys on a piano: do, re, me, fa, so, la, ti. Fa is our middle fre- quency. Now construct a spinner such as the one shown at the left in Figure 1. Divide the circle into seven sectors and label them with the notes. It matters not at all what arc lengths are assigned to these sectors; they can be completely arbi- trary.