Fragments of Infinity

Ivars Peterson

Description

A visual journey to the intersection of math and imagination, guided by an award-winning author
Mathematics is right brain work, art left brain, right? Not so. This intriguing book shows how intertwined the disciplines are. Portraying the work of many contemporary artists in media from metals to glass to snow, Fragments of Infinity draws us into the mysteries of one-sided surfaces, four-dimensional spaces, self-similar structures, and other bizarre or seemingly impossible features of modern mathematics as they are given visible expression. Featuring more than 250 beautiful illustrations and photographs of artworks ranging from sculptures both massive and minute to elaborate geometric tapestries and mosaics of startling complexity, this is an enthralling exploration of abstract shapes, space, and time made tangible.
Ivars Peterson (Washington, DC) is the mathematics writer and online editor of Science News and the author of The Jungles of Randomness (Wiley: 0-471-16449-6), as well as four previous trade books.
Preface.

1. Gallery Visits.

2. Theorems in Stone.

3. A Place in Space.

4. Plane Folds.

5. Grid Fields.

6. Crystal Visions.

7. Strange Sides.

8. Minimal Snow.

9. Points of View.

10. Fragments.

Further Readings.

Credits.

Index.

Sujets

• Sciences formelles
• Mathématiques générales
• Mathématiques générales
• Ivars Peterson
• Criticism
• Theory
• Science News
• Art

Informations

• Publié par: Turner Publishing Company
• Date de parution: 02 mai 2008
• Nombre de lectures: 0
• EAN13: 9780470341124
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Fragments of Infinity
A Kaleidoscope of Math and Art
I VARS P ETERSON

John Wiley & Sons, Inc.
This book is printed on acid-free paper.
Copyright © 2001 by Ivars Peterson. All rights reserved
Published by John Wiley & Sons, Inc.
Published simultaneously in Canada
Design and production by Navta Associates, Inc.
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, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone]. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, [phone], fax [phone], email: [e-mail].
This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional person should be sought.
Library of Congress Cataloging-in-Publication Data
Peterson, Ivars
Fragments of infinity : a kaleidoscope of math and art / Ivars Peterson.
p. cm.
Includes bibliographical references and index.
ISBN 0-471-16558-1 (cloth : alk. paper)
1. Mathematics in art. 2. Visual perception. I. Title.
N72.M3 P48 2001
701'.05—dc21
2001024026
10 9 8 7 6 5 4 3 2 1
Contents
Preface

1. Gallery Visits

2. Theorems in Stone

3. A Place in Space

4. Plane Folds

5. Grid Fields

6. Crystal Visions

7. Strange Sides

8. Minimal Snow

9. Points of View

10. Fragments
Further Readings
Credits
Index
Preface
This book is about creativity and imagination at the intersection of mathematics and art. It portrays the work of several contemporary mathematicians who are also artists or whose mathematical thoughts have inspired others to create. It provides glimpses of artists enthralled by the unlimited possibilities offered by mathematically guided explorations of space and time. It delves into the endlessly fascinating mysteries of one-sided surfaces, four-dimensional spaces, self-similar structures, and other seemingly bizarre features of modern mathematics.
In 1992 I was invited to present the opening address at a remarkable meeting devoted to mathematics and art, organized by mathematician and sculptor Nat Friedman of the State University of New York at Albany. My invitation to the pathbreaking meeting came about because of articles I had written for Science News highlighting the increasing use of visualization in mathematics, particularly the burgeoning role of computer graphics in illuminating and exploring mathematical ideas, from soap-film surfaces, fractals, and knots to chaos, hyperbolic space, and topological transformations. One of my articles had focused on Helaman Ferguson, a sculptor and mathematician who not only works with computers but also carves marble and molds bronze into graceful, sensuous, mathematically inspired artworks.
Friedman’s lively gathering introduced me to many more people who are fascinated by interactions between art and mathematics, and with them, I have attended and participated in subsequent meetings. Many of the artists and mathematicians mentioned in this book belong to this peripatetic tribe of math and art enthusiasts. The tribe’s diversity of thought and custom, however, also brings to mind difficult questions of what constitutes mathematical art, what beauty means in that context, and what explicit role, if any, mathematics ought to play in the visual arts.
The following chapters offer a glimpse of the ways in which we can stretch our minds to imagine and explore exotic geometric realms. They highlight the processes of creativity, invention, and discovery intrinsic to mathematical research and to artistic endeavor.
The book’s title echoes thoughts of the Dutch graphic artist M. C. Escher, who sought to capture the notion of infinity in visual images. In 1959, in his essay “Approaches to Infinity,” Escher described the reasoning behind one of his intricately repeating designs, which featured a parade of reptiles, as follows: “Not yet true infinity but nevertheless a fragment of it; a piece of ‘the universe of reptiles.’
“If only the plane on which [the tiles] fit into one another were infinitely large, then it would be possible to represent an infinite number of them,” he continued. “However, we aren’t playing an intellectual game here; we are aware that we live in a material, three-dimensional reality, and we cannot manufacture a plane that extends infinitely in all directions.”
Escher’s solution to his immediate artistic dilemma was to “bend the piece of paper on which this world of reptiles is represented fragmentarily and make a paper cylinder in such a way that the animal figures on its surface continue to fit together without interruptions while the tube revolves around its lengthwise axis.” It was just one of many highly original schemes that Escher devised in his attempts to capture infinity visually. Other artists share this passion (or perhaps obsession) for visualizing creations of the mind, whether theorem or dream, and rendering them in concrete form, and they, too, must overcome limitations of their tools and their place in the natural world to present glimpses or fragments of these conceivable yet elusive realms.
Special thanks go to Helaman Ferguson and Nat Friedman, who introduced me to many of the people who and the ideas that inspired and encouraged my travels in the surprisingly wide and diverse world of mathematical art.
I also wish to thank the following persons for their help in explaining ideas, providing illustrations, or supplying other material for this book: Don Albers, Tom Banchoff, Bob Brill, Harriet Brisson, John Bruning, Donald Caspar, Davide Cervone, Benigna Chilla, Barry Cipra, Brent Collins, John Conway, H. S. M. Coxeter, Erik Demaine, Ben Dickins, Stewart Dickson, Doug Dunham, Claire Ferguson, Mike Field, Eric Fontano, George Francis, Martin Gardner, Bathsheba Grossman, George Hart, Linda Henderson, Paul Hildebrandt, Tom Hull, Robert Krawczyk, Robert Lang, Howard Levine, Cliff Long, Robert Longhurst, Shiela Morgan, Eleni Mylonas, Chris K. Palmer, Doug Peden, Roger Penrose, Charles Perry, Cliff Pickover, Tony Robbin, John Robinson, Carlo Roselli, John Safer, Reza Sarhangi, Doris Schattschneider, Dan Schwalbe, Marjorie Senechal, Carlo Séquin, John Sharp, Rhonda Roland Shearer, Arthur Silverman, John Sims, Clifford Singer, Arlene Stamp, Paul Steinhardt, John Sullivan, Keizo Ushio, Helena Verrill, Stan Wagon, William Webber, Jeff Weeks, and Elizabeth Whiteley. My apologies to anyone I have inadvertently failed to include in the list.
I am grateful to my editors at Science News, Joel Greenberg, Pat Young, and Julie Ann Miller, for allowing me to venture occasionally into topics that didn’t always fit comfortably within the purview of newsworthy scientific and mathematical research advances. Some of the material in this book has appeared in a somewhat different form in Science News .
I wish to thank my wife, Nancy, for many helpful suggestions while reviewing the original manuscript. I greatly appreciate the efforts of everyone at Wiley who worked so hard to transform an unwieldy stack of manuscript pages and numerous illustrations in a wide variety of formats into the finished book.
1
Gallery Visits

In Washington, D.C, the National Gallery of Art’s East Building, which opened to the public in 1978, features a facade that teases the eye-where wails unexpectedly meet at acute and obtuse angles rather than commonplace right angles.
Few of us expect to encounter mathematics on a visit to an art gallery. At first (or even second) glance, art and mathematics appear to have very little in common, although both are products of the human intellect.
Walking up to the buildings of the National Gallery of Art in Washington, D.C., and strolling through its rooms, however, can be immensely illuminating when the viewing is done with a mathematical eye. The gallery’s East Building, designed by the architect I. M. Pei, is an eye-teasing festival of vast walls, sharp edges, odd angles, and unexpected shapes. A playful visual illusion writ large, it is itself a work of mathematical art.

To fit the National Gallery’s East Building on a trapezoid-shaped site, architect I. M. Pei based his design on a division of the trapezoid into an isosceles triangle and a smaller right triangle.
Such a close connection between architecture and mathematics shouldn’t be surprising. Geometry has long occupied a prime position in the architect’s and builder’s toolbox. In commenting on the inspiration for his East Building design, Pei noted in a 1978 article in National Geographic, “ I sketched a trapezoid on the back of an envelope. I drew a diagonal line across the trapezoid and produced two triangles. That was the beginning.”
Conceived as explorations of form, space, light, and color, sculptures and paintings can themselves embody a variety of mathematical principles, expressed not only in such obviously geometric objects as triangles, spheres, and cones, but also through depictions of motion and metamorphosis. Renaissance painting of the fifteenth century celebrated the precisely mathematical use of proportion and perspective to achieve startlingly natural images of the visual world. Centuries later, artists such as Pablo Picasso (1881–1973), Salvador Dali (1904–1989), and René Magritte (1898–1967) could play with those conventions and illusio