Mathematics and Art: Mathematical Visualization in Art and Education

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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.

Author(s): Claude-Paul Bruter (auth.), Claude P. Bruter (eds.)
Series: Mathematics and Visualization
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2002

Language: English
Pages: 337

Cover
Title Page
Copyright Page
Author Introduction Page
Preface
Table of Contents
1 Presentation of the Colloquium. The ARPAM Project
1 The Colloquium
1.1 Introduction
1.2 Presentation of the Colloquium
2 The ARPAM project
2.1 The pedagogical data
2.2 General characteristic of the project
2.3 A word on the gardens
2.4 Succinct presentation of some folies
2.5 The future of the project
References
2 Solid-Segment Sculptures
1 Sculpture by 3D Printing
2 Visual presentation of segment structure
3 Statement of problem
4 New approach
5 Examples
5.1 Deep Structure
5.2 Five-Legged-Bee Hive
5.3 120-cell projected as "Schlegel Diagram"
5.4 Truncated 120-cell
6 Conclusions
References
3 Visualizing Mathematics - Online
1 Multimedia of the Old Masters
2 Gaining Mathematics from Visualization
3 Discrete Minimal Surfaces - Online
4 Mathematical Videos
5 EG-Models Journal: An Archive for Electronic Geometry Models
6 Dissertation Online
7 JavaView
8 Hardware for 3D Visualization
9 Conclusion
References
4 The Design of 2-Colour Wallpaper Patterns Using Methods Based on Chaotic Dynamics and Symmetry
1 Introduction
2 The Theory of Designer Chaos
2.1 Planar symmetry groups
2.2 Attractors
2.3 Symmetric Attractors
2.4 Numerical implementation
2.5 Colouring
3 Wallpaper Patterns
3.1 Geometry of wallpaper patterns
3.2 Dynamics
3.3 Wallpaper patterns through dynamics
4 Two-colour Patterns
4.1 Two-colour wallpaper patterns
4.2 Two-colour quilts from dynamics
4.3 Numerical Algorithms
4.4 Examples of two-colour designs
References
5 Machines for Building Symmetry
1 Introduction
2 2The mathematics underlying the "machines for building symmetry"
2.1 Coxeter groups
2.2 Two-dimensional machines and plane crystallographic groups
2.3 Three-dimensional machines and polyhedra
2.4 Elliptic, euclidean (and hyperbolic) geometry
3 What can be done with the "machines for building symmetry"
3.1 Classification with respect to symmetry, for different visitors
3.2 The role of interactivity
3.3 The role of mathematical proofs
3.4 The role of beauty
References
6 The Mathematics of Tuning Musical Instruments - a Simple Toolkit for Experiments
7 The Garden of Eden
8 Visualization and Dynamical Systems
1 Introduction
2 Complex dynamics and visualization
3 The dynamics of the pendulum
4 Some conclusions
5 References
9 Solving Polynomials by Iteration
1 Introduction
2 Preliminary Background
2.1 Polynomials
2.2 Symmetric and Alternating Groups
2.3 Group Actions
2.4 Maps
3 Polynomials, Symmetry, and Dynamics
4 The Quintic - S_5 Acts in Three Dimensions
4.1 Invariant Polynomials
4.2 A Surface Generated by Lines
4.3 Special Orbits
4.4 Maps with Symmetry
4.5 Quadric-preserving Maps
4.6 A Special Map in Degree Six
5 The Sextic - A_6 Acts in Two Dimensions
5.1 Basics of A_6 and Valentiner's Group
5.2 Valentiner Geometry
5.3 Polynomials and Maps with Valentiner Symmetry
5.4 A Case of Inelegant Dynamics
5.5 A Special Icosahedral Map of Degree 19
6 Gallery of Basin Portraits
References
10 Mathematical Aspects in the Second Viennese School of Music
1 The Second Viennese School of Music
2 The Twelve-Tone Method
3 Mathematical Formalization of Twelve-tone Music
4 Series related to a given series
5 The Matrix of Series
6 Hexachords
7 Creating 12-tone music
8 Final Comments
References
11 Mathematics and Art: The Film Series
1 The Mathematics and Art Project
2 Moebius Band
3 Labyrinths
4 Dimensions
5 Final comments
References
12 Guided Tours of Buried Galleries (Inside a Computer)
1 Introduction
2 Art Galleries
3 Mine Galleries
4 Follow the Guide
5 Conclusion
13 A Mathematical Interpretation of Expressive Intonation
1 Introduction
2 "Petite £leur"
3 Scale Constructions
4 Playing "Petite Fleur" in the Pythagorean Scale
5 Mathematical Interpretation of Expressive Intonation
6 Pure Intonation
7 Conclusion
References
14 Symbolic Sculptures
15 FORUM: How Art Can Help the Teaching of Mathematics (Claude-Paul Bruter)?
16 Forum Discussion(Ronnie Brown)
17 Forum Discussion: Presentation of the Atractor(Manuel Arala Chaves)
18 Forum Discussion(Michele Emmer)
19 Forum Discussion(Michael Field)
20 Getting Out of the Box and Into the Sphere
1 What Is a Termesphere
2 How this Concept Was Found from Inside a Box
3 The Geometry
4 Inside or Outside the Spherecube
5 In the sphere
21 Constructing Wire Models
1 Introduction
2 The variational problem
2.1 The functional space
2.2 The potential energy of a springy inextensible wire
2.3 The energy of the derived path
2.4 Computing the derivative of v
2.5 Constraint of the half tangents at the extremities
2.6 Isoperimetric constraint
2.7 Euler-Lagrange equation
2.8 The example of the circular helix
3 The case of plane curves
3.1 Reduction of the Euler-Lagrange equation
3.2 Case μ = 1
3.3 Case μ > 1
3.4 Case 0<μ<1
3.5 Summing up
4 Wire models
4.1 The halfway model
4.2 The gastrulation
5 Conclusion
References
22 Sphere Eversions: from Smale through "The Optiverse"
1 A Short History of Sphere Eversions
2 Bending Energy and the Minimax Eversions
3 Topological Stages in the Eversion
4 References
23 Tactile Mathematics
1 Background: Visual Computing/Visual Mathematics
2 Computer 3-D Printing - Concrete Mathematics
3 4 th, 5th, . .. Dimensions
4 Tactile Fonts, 3-D Textures
5 Conclusion / Future Work
6 References
24 Hyperseeing, Knots, and Minimal Surfaces
1 Hyperseeing
2 Knots
3 Kenneth Snelson Sculptures
4 Soapfilm Minimal Surfaces of Knots
5 Framed Knots
6 Multiple Mobius Bands
References
25 Ruled Sculptures
1 3rd degree ruled surfaces
2 Drawings through pendular plotting
26 A Gallery of Algebraic Surfaces
1 Introduction
2 Algebraic surfaces
3 Symmetry
4 Singularities
4.1 Maximal numbers of singularities
4.2 Deformations
4.3 Degenerations
5 Enumerative geometry
6 Conclusion
27 The Mathematical Exploratorium
1 Introduction
2 Rationale
3 What is it?
4 Who will the audience be?
5 What are the Goals ?
6 Principle
7 Logical Organization
8 Administration
9 Financing
28 Copper Engravings
Appendix: Color Plates
Index