Recent progress in research, teaching and communication has arisen from the use of new tools in visualization. To be fruitful, visualization needs precision and beauty. This book is a source of mathematical illustrations by mathematicians as well as artists. It offers examples in many basic mathematical fields including polyhedra theory, group theory, solving polynomial equations, dynamical systems and differential topology. For a long time, arts, architecture, music and painting have been the source of new developments in mathematics. And vice versa, artists have often found new techniques, themes and inspiration within mathematics. Here, while mathematicians provide mathematical tools for the analysis of musical creations, the contributions from sculptors emphasize the role of mathematics in their work. This book emphasizes and renews the deep relation between Mathematics and Art. The Forum Discussion suggests to develop a deeper interpenetration between these two cultural fields, notably in the teaching of both Mathematics and Art.
Mathematics and Visualization
Series Editors Gerald Farin Hans-Christian Hege David Hoffman Christopher R. Johnson Konrad Polthier
Springer-Verlag Berlin Heidelberg GmbH
Claude P. Bruter
Editor
Mathematics and Art Mathematical Visualization in Art and Education
With 284 Figures, 127 in Color
Springer
Editor Claude P. Bruter Universite Paris XlI Mathematiques UER Sciences 61 Avenue du General de Gaulle 94010 Creteil Cedex e-mail:
[email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Mathematics and art: mathematical visualization in art and education I Claude P. Bruter ed .. (Mathematics and visualization)
The cover figure reproduces a classical Kleinian tessellation of the hyperbolic plane by triangles (Klein, 1878-1879). In the present case, the angles of each triangle a.:re (11' /2, 7r /3, Jr,/7). All the triangles have the same area, 11' / 42 : they are the smallest triangles with which the hyperbolic plane can be tiled.
Mathematics Subject Classification (2000): 97D20, 97CSO, 97D30, 97U99, ooBIO ISBN 978-3-642-07782-1 ISBN 978-3-662-04909-9 (eBook) DOI 10.1007/978-3-662-04909-9 This work is subject to copyright. All rights arc reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002
Originally published by Springer-Verlag Berlin Heidelberg New York in 2002 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production GmbH, Heidelberg 46/3111LK - 5 43
2 - Printed
on acid-free paper
Born in 1910, Alexandre VITKINE,
[email protected], became a photographer and a graphic artist after a career in the industry. He is now a sculptor. His drawing