The Visual Mind II edited by Michele Emmer The Visual Mind II leonardo Roger F. Malina, series editor The Visual Mind, edited by Michele Emmer, 1993 Leonardo Almanac, edited by Craig Harris, 1994 Designing Information Technology, Richard Coyne, 1995 Immersed in Technology: Art and Virtual Environments, edited by Mary Anne Moser with Douglas MacLeod, 1996 Technoromanticism: Digital Narrative, Holism, and the Romance of the Real, Richard Coyne, 1999 Art and Innovation: The Xerox PARC Artist-in-Residence Program, edited by Craig Harris, 1999 The Digital Dialectic: New Essays on New Media, edited by Peter Lunenfeld, 1999 The Robot in the Garden: Telerobotics and Telepistemology in the Age of the Internet, edited by Ken Goldberg, 2000 The Language of New Media, Lev Manovich, 2000 Metal and Flesh: The Evolution of Man: Technology Takes Over, Ollivier Dyens, 2001 Uncanny Networks: Dialogues with the Virtual Intelligentsia, Geert Lovink, 2002 Information Arts: Intersections of Art, Science, and Technology, Stephen Wilson, 2002 Virtual Art: From Illusion to Immersion, Oliver Grau, 2003 Women, Art, and Technology, edited by Judy Malloy, 2003 Protocol: How Control Exists after Decentralization, Alexander R. Galloway, 2004 At a Distance: Precursors to Art and Activism on the Internet, edited by Annmarie Chandler and Norie Neumark, 2005 The Visual Mind II, edited by Michele Emmer, 2005 CODE: Collaborative Ownership and the Digital Economy, edited by Rishab Aiyer Ghosh, 2005 From Technological to Virtual Art, Frank Popper, 2005 The Visual Mind II edited by Michele Emmer The MIT Press Cambridge, Massachusetts London, England © 2005 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or informa- tion storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email special_sales@mitpress. mit.edu or write to Special Sales Department, The MIT Press, 5 Cambridge Center, Cambridge, MA 02142. This book was set in Bell Gothic and Garamond by SNP Best-set Typesetter Ltd., Hong Kong and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data The visual mind II / edited by Michele Emmer. p. cm. — (Leonardo) Includes bibliographical references and index. ISBN 0-262-05076-5 (hc : alk. paper) 1. Art—Mathematics. 2. Geometry. 3. Aesthetics. I. Title: Visual mind 2. II. Title: Visual mind two. III. Emmer, Michele. IV. Leonardo (Series) (Cambridge, Mass.) N72.M3V58 2005 701¢.5—dc22 2004057850 To Valeria Max Bill Fred Almgren H. S. M. Coxeter Contents introduction Michele Emmer xi Section 1 Mathematics and Aesthetics 1 1 the phenomenology of mathematical beauty Gian-Carlo Rota 3 2 mathematical beauty and the evolution of the standards of mathematical proof James W. McAllister 15 3 aesthetics for computers, or how to measure harmony Jaroslav Nesˇetrˇil 35 4 visual mathematics: mathematics and art Michele Emmer 59 Section 2 Geometry and Art 91 5 life through art Carmen Bonell 95 6 john robinson’s symbolic sculptures: knots and mathematics Ronald Brown 125 7 geometries of curvature and their aesthetics Brent Collins 141 8 poetry in curves: the guggenheim museum in bilbao Giuseppa Di Cristina 159 9 eightfold way: the sculpture Helaman Ferguson with Claire Ferguson 187 10 the geometric aesthetic George W. Hart 215 11 art and the age of the sciences Charles Perry 235 12 some aspects of the use of geometry in my artistic work Sylvie Pic 253 Section 3 Mathematics and Art 269 13 local/global in mathematics and painting Capi Corrales Rodrigáñez with an Appendix by Laura Tedeschini-Lalli 273 14 visual knots: concerning geometry and visuality in the work of marcel duchamp Manuel Corrada 309 15 lunda symmetry: where geometry meets art Paulus Gerdes 335 16 four-dimensional space or space-time? the emergence of the cubism-relativity myth in new york in the 1940s Linda Dalrymple Henderson 349 17 “reverse perspective”: historical fallacies and an alternative view Clemena Antonova and Martin Kemp 399 18 four-dimensional projection: art and reality Tony Robbin 433 19 rational design versus artistic intuition in stained- glass art Tomás García Salgado 449 Section 4 Geometry, Computer Graphics, and Art 469 20 dynamics, chaos, and design Michael Field 473 21 paul klee on computer: biomathematical models help us understand his work Roberto Giunti 495 22 parameterized sculpture families Carlo H. Séquin 527 23 the aesthetic value of optimal geometry John M. Sullivan 547 Contents viii Section 5 Mathematics, Visualization, and Cinema 565 24 mathematics and cinema Michele Emmer 569 25 some organizing principles Peter Greenaway 601 26 figures and characters in the great book of nature Jean-Marc Lévy-Leblond 623 27 circle packings and the sacred lotus Tibor Tarnai and Koji Miyazaki 647 28 meander mazes on polysphericons Anthony Phillips 667 contributors 685 name index 689 subject index 697 Contents ix Introduction Michele Emmer The project “Mathematics and Art” started in 1976. Perhaps better said, I started to think about the project that year. I started thinking about it for essentially two, or perhaps three, reasons. First, in 1976 I was at the University of Trento, in the north of Italy, working in the area called the calculus of variations, in particular, minimal surfaces and capillarity problems. I had received my degree from the Uni- versity of Rome in 1970 and had started my career at the University of Ferrara, where I was very lucky to work with Mario Miranda, favorite graduate student of Ennio De Giorgi. Then I met Enrico Giusti and Enrico Bombieri. It was the period in which, in the investigations of partial dif- ferential equations, of the calculus of variations, and the perimeter theory— introduced by Renato Caccioppoli and established by De Giorgi and Miranda—the Italian School of the Scuola Normale Superiore of Pisa was one of the best in the world. Just in the year 1976 Enrico Bombieri received the Fields medal. I was, by chance, in the right place at the right moment. All the mathematicians in the world working in these areas of research had to be up-to-date on what was happening in Italy. Also in 1976 Jean Taylor proved a famous conjecture posed experimen- tally by the Belgian physicist Joseph Plateau, over a hundred years earlier, having to do with the types of singularities, i.e., of edges, that soap films generate when they meet. Plateau had experimentally observed that the angles generated by soap films are of only two kinds. Jean Taylor, using the theory of integral currents introduced by Federer and researched by Allard and Almgren, was able to prove that the result was true. A few years before, Ennio De Giorgi proved in his generality the existence of the solution of the Plateau problem, that for any chosen boundary it is possi- ble to find a minimal surface that has this boundary. He was also able to prove the isoperimetric property of the sphere in every dimension n. In three dimensions this is the case with soap bubbles: there is a surface with assigned mean curvature that must contain a fixed volume of air.1 In 1976 Scientific American asked Jean Taylor and Fred Almgren (they were married a few months before) to write a paper on the more recent results on the topic of minimal surfaces and soap bubbles. A professional photographer was asked to realize the pictures for the paper. The same year they were invited to the University of Trento as visiting professors, and during the summer they gave a course in Cortona, near Arezzo. When Almgren and Taylor came to Trento in 1976, their paper in Scientific American had just been published. The photographs accompanying the article and the cover were quite beautiful and interesting. Looking at the pictures inspired me to make a movie about soap film in order to show very well, close-up and in slow motion, their shapes and geometries. I dis- cussed the project with my wife Valeria, who lived in Rome with our two sons, and she was very pleased and interested by the idea. For me, thinking about making a film was quite natural. My father, Luciano Emmer, is a film director. (Marcello Mastroianni made his first film with him, Domenica d’agosto, in 1949.) When I was a child, I was always involved in filmmaking—as collaborator, organizer, even as an actor in several of my father’s movies. Both Almgren and Jean Taylor were very interested in my project. In any case, my idea was not to make a small sci- entific film, a sort of scientific spot just to show some little experiments with soap bubbles and soap films. I was not at all interested in filming a lesson by Almgren and Taylor, with them explaining their results and here and there inserting some images of soap bubbles and soap films. Almgren and Taylor concurred. Now the second reason. I was working at the University of Trento while my family lived in Rome. Every Friday I left Trento for Rome (seven hours by train), and every Monday I returned to Trento. I have always been a lover of art—of any kind, of any culture and period. Of course I have my own favorite artists. In Trento I learned of an exhibition in Parma dedi- cated to one of the most important artists of the last century: Max Bill. I was already familiar with some of the sculptures of the Swiss artist, but I had not had the occasion to visit a large exhibition like the one in Parma. Michele Emmer xii As Parma was more or less on my way from Trento to Rome, I decided to stop on my way to see the exhibition.