Mathematics and Art: Mathematical Visualization in Art and Education

337 Pages·2002·25.78 MB·English

by  Claude-Paul Bruter (auth.)

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 trian
gles (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
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© 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 pro
tective 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 photogra
pher and a graphic artist after a career in the industry. He is now a sculptor.
His drawings, photos (obtained through electronic equipments made by
himself), and his sculptures produced by computer-controlled machines (info
sculptures), are based on mathematical forms, mainly those of Lissajous
curves.
The drawing above was chosen as the basis for the poster announcing the
Colloquium.
Preface
I am convinced that the work of the artists
is to create order from chaos
Fred Uhlman 1
A Colloquium on Mathematics and Art was hold in the French city of Mau
beuge in September 2000. The scientific committee included Jacek Bochnak
(Amsterdam), Ronald Brown (Bangor), Claude-Paul Bruter (Paris 12), Ma
nuel Chaves (Porto), Michele Emmer (Roma), Tzee-Char Kuo (Sydney),
Richard Palais (Brandeis) and Valentin Poenaru (Paris 11). We would like
to warmly thank Francis Trincaretto and his team who arranged to have the
meeting in such an agreable venue: the "Theatre du Manege".
Placed at the transition of the second and the third millennium, this Col
loquium presented original ideas related to the development of new forms
of civilization based on the many recent and rapid technological advances
in communication and computation. With the strong encouragement of the
local organizer, Francis Trincaretto, the Colloquium was - unlike more for
mal mathematical conferences - videoed and could be attended on the web.
The speakers were true artists and mathematicians of rather unusual stan
dard: while the artists were partly inspired by advanced mathematics, or even
were sometimes ahead of mathematics, the mathematicians intended to show
the beauty of their work and to share their feeling with the greater part of
the population. They used all the old and new means of static and dynamic
visualizations. Their works may be understood as symbolic and iconic repre
sentations of our environment and as essential tools for the understanding of
our world, and the development of mankind.
Indeed, this Colloquium can be related to a renewal in the ways of dif
fusion and of teaching of mathematics. While schools of plastic or musical
art are beginning to ask for some mathematics, mathematicians are seriously
thinking of setting forth the artistic qualities of their work to attract the
mind, and to support and facilitate the learning of their discipline.
We hope that readers may find in these proceedings ideas, projects and
realizations which can contribute towards the inspiration and promotion of
new cultural developments in Society.
The order of the articles follows that of the talks. Pictures and images
appear in grey in the articles. Images appear also in colour in the Appendix.
Such an image is labelled [XY]k where XY are the main initials of the author,
while k numbers the image.
1 (1901-1985) German advocate, then British painter and writer: his booklet Re
union is a true masterpiece.
VIII Preface
To conclude, we would like to thank Mike Field who accepted to help the
translations into English of most French written texts, Bill Mac Callum who
did that work for the second part of my first contribution, and the Springer
team who accepted to publish these Proceedings, and provided their help to
the editor.
Claude-Paul Bruter
Table of Contents
Presentation of the Colloquium. The ARPAM Project. . . . . . . . . . . . . . . 1
Claude-Paul Bruter
Solid-Segment Sculptures ........ .... .. .. ... .... ... ... .. .... ..... 17
George W. Hart
Visualizing Mathematics - Online. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29
Konrad Polthier
The Design of 2-Colour Wallpaper Patterns Using Methods Based on
Chaotic Dynamics and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43
Michael Field
Machines for Building Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61
Maria Dedo
The Mathematics of Tuning Musical Instruments - a Simple Toolkit
for Experiments ....... ..... .... ...... .... .......... .. .. .... ... . 79
Erich Neuwirth
The Garden of Eden. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 89
Charles O. Perry
Visualization and Dynamical Systems ... .... ......... ............. 91
John Hubbard
Solving Polynomials by Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
Scott Crass
Mathematical Aspects in the Second Viennese School of Music ... .. ... 105
Carlota Simoes
Mathematics and Art: The Film Series ... .. ..... ...... ...... ..... . 119
Michele Emmer
Guided Tours of Buried Galleries (Inside a Computer) ............... 135
Jean-Franr;ois Colonna
A Mathematical Interpretation of Expressive Intonation ............. 141
Yves Hellegouarch
Symbolic Sculptures ........ .... ...... ..... .. ...... ...... .. ... ... 149
John Robinson
X Table of Contents
FORUM: How Art Can Help the Teaching of Mathematics? .. ..... ... 153
Claude-Paul Bruter
Forum Discussion .......... .... .. .... .. .......... ... ... .. ..... .. 155
Ronnie Brown
Forum Discussion: Presentation of the Atractor .............. ... ... . 160
Manuel Arala Chaves
Forum Discussion .... .... ..... ...... .................. .... ...... 166
Michele Emmer
Forum Discussion .... ..... .. .... ..... .................. ..... ... . 168
Michael Field
Getting Out of the Box and Into the Sphere. . . . . . . . . . . . . . . . . . . . . . . . 173
Dick Termes
Constructing Wire Models .... ..... ........ ................ ..... . 179
Franr;ois Apery
Sphere Eversions: from Smale through "The Optiverse" ... ....... ... . 201
John M. Sullivan
Tactile Mathematics .. ........ ...... ..... .. .. .... ... ......... .... 213
Stewart Dickson
Hyperseeing, Knots, and Minimal Surfaces ..... .. .. ...... .... .. .. .. 223
Nathaniel A. Friedman
Ruled Sculptures .......... .... .. ..... ..... ........... ...... ..... 233
Philippe Charbonneau
A Gallery of Algebraic Surfaces .... .. .. ... .... .... .......... .. ... . 237
Bruce Hunt
The Mathematical Exploratorium .... ... ...... ... .... .... .... ..... 267
Richard S. Palais
Copper Engravings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Patrice Jeener
Appendix: Color Plates ....... ..... ... ...... ........... .. ........ 275
Index ..... .... ........ .. ........... ... ......... .... .... ....... . 335
Presentation of the Colloquium.
The ARPAM Project
Claude-Paul Bruter
Mathematiques, Universite Paris 12, Av. du General de Gaulle, 94010 Creteil,
France <bruter@univ-paris12>
1 The Colloquium
1.1 Introduction
My intervention has two parts. The first one is devoted to a general presenta
tion of the Colloquium, through an evocation of the works of the artists who
are present among us. Thus that presentation does not address mathemati
cians in a particular way. It reveals some of the reasons which have directed
the scientific organisation of the Colloquium. As its architecture shows, it
turns over the art of visualisation of mathematics, either for the general pub
lic, or for the one of mathematicians. The second part is devoted to a succinct
description of the ARPAM project.
1.2 Presentation of the Colloquium
On the foundations of the relations between Mathematics and Arts
As a preliminary comment, it is fitting to say a few words on the relations
which tie Mathematics and the Arts: they are so tight that sometimes Math
ematics is compared with one of the Fine Arts.
One of the reasons, the main one to my eyes, which solders the arts to
mathematics is probably the following: the tangible object, the living being,
are not only present in space, and are evolving in space, but are moreover
highly elaborated constructions, obtained from the unfolding of the properties
of the primordial space.
In other respects, the existence of the object, that is its inward properties
of stability, are themselves dependent on the stability of its constituents, of
their internal arrangement according the various levels of integration. This
existence also depends upon the capabilities of the object to resist against
shocks of any kind, of internal or external origin, created by all that makes its
environment, close or distant, into space and time. Thus, knowledge of this
environment, in all its modalities, is the essential means whereby the being
can guarantee its spatio-temporal stability. So we are always brought back
to the fundamental problem of the knowledge of the space, of all the richness
of its manifestations.
C. P. Bruter (ed.), Mathematics and Art
© Springer-Verlag Berlin Heidelberg 2002