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Mathematics and Modern Art

Proceedings of the First ESMA Conference, held in Paris, July 19-22, 2010

herausgegeben von: Claude Bruter

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Proceedings in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The link between mathematics and art remains as strong today as it was in the earliest instances of decorative and ritual art. Arts, architecture, music and painting have for a long time been sources of new developments in mathematics, and vice versa. Many great painters have seen no contradiction between artistic and mathematical endeavors, contributing to the progress of both, using mathematical principles to guide their visual creativity, enriching their visual environment with the new objects created by the mathematical science.

Owing to the recent development of the so nice techniques for visualization, while mathematicians can better explore these new mathematical objects, artists can use them to emphasize their intrinsic beauty, and create quite new sceneries. This volume, the content of the first conference of the European Society for Mathematics and the Arts (ESMA), held in Paris in 2010, gives an overview on some significant and beautiful recent works where maths and art, including architecture and music, are interwoven.

The book includes a wealth of mathematical illustrations from several basic mathematical fields including classical geometry, topology, differential geometry, dynamical systems. Here, artists and mathematicians alike elucidate the thought processes and the tools used to create their work

Inhaltsverzeichnis

Frontmatter

A Mathematician and an Artist. The Story of a Collaboration

Abstract

In recent years it has become de rigueur for an invited speaker at a conference to “Thank the organizers for inviting me.” Today I can say this with more than the usual sincerity. Paris is my favorite city in the world, I have many fond memories of the Institut Henri Poincaré, and the subject of this conference is very close to my heart. So to Claude, and all the organizers who have worked hard to make this conference a success. Merci bien.

Richard S. Palais

Dimensions, a Math Movie

Abstract

Dimensions is a 2-h animated movie, aimed at a broad audience, produced by Jos Leys, Étienne Ghys and Aurélien Alvarez. The notion of “dimension” in the mathematical sense is explained in nine chapters: “Dimension 2” talks about location on a sphere and stereographic projection. “Dimension 3” explains how 2-dimensional creatures can imagine 3-dimensional objects, which is an introduction to “Dimension 4” where we show how we, as 3-dimensional creatures, can imagine 4-dimensional objects. Next is a visual introduction to complex numbers, leading in to the Hopf fibration as an example of 4-dimensional math. As an epilogue we show a formal proof of a geometric theorem related to stereographic projection. Through this film, the authors wanted to show that math does not need to be “dry”, but that math can produce beautiful imagery. In order to reach as wide an audience as possible, the film is a non-profit project. The DVD has a low price, and the films can be downloaded free of charge from an internet site featuring additional information on the subjects of the film. Furthermore, the film has a “Creative Commons” license, which allows copying of the film (provided there is no commercial gain). The film is available in 8 commentary languages and 20 languages for the subtitles.

Aurélien Alvarez, Jos Leys

Old and New Mathematical Models: Saving the Heritage of the Institut Henri Poincaré

Abstract

During the inaugural conference of the very new european society dedicated to the interaction between mathematics and arts (ESMA), which was held at Intitut Henri Poincaré (IHP) in Paris in July 2010, I was given the opportunity to tell about the state of the substantial collection of mathematical objects, which are displayed in the IHP library or are stored in the underground reserve, because of poor condition or which are in process of being catalogued.

François Apéry

An Introduction to the Construction of Some Mathematical Objects

Abstract

In order to understand and to reconstruct the shape of many objects of the geometric world, mathematicians have focused their attention on singularities and deformations. The purpose of this article is to present these usual topological concepts and tools to artists being a priori unfamiliar with mathematics, with the hope that new beautiful creations will appear in the artistic world.

Claude Paul Bruter

Computer, Mathematics and Art

Abstract

Mathematics could be seen as a “simple” mind game hardly more useful in everyday life than the chess. But their “formidable efficiency” as the language with which are written the laws of Nature could be an evidence they are the Reality. Thus, Mathematics would contain all works of art past, present and future, but also their creators.

Jean-François Colonna

Structure of Visualization and Symmetry in Iterated Function Systems

Abstract

Principles and practices of visualization have always been valuable tools in all fields of research. The formation of representations plays a key role in all aspect of science. Prior research in the area of visualization demonstrates that representation of data hold great potential for enhancing comprehension of abstract concepts and greatly benefit collaborative decision-making and project performance. The purpose of this study is to examine the effectiveness of integrating various methodologies in the production of a visually coherent proposition. The author explores the dynamic of visualization in the digital environment and brings together elements of fractal topology, optical distortions and color theory components in an esthetic statement. The background and information selected for this purpose is based on specific cognitive processes, neurological and topological researches developed at the turn of the twentieth century by various experts in the field of investigative science: mathematicians Cantor and Sierpinski, neuroscientists Hermann, Hering, Wundt and other analytical minds such as Itten and Kandinsky that, at a particular time in history, demonstrated the similarities and overall cohesion of an intellectual and scientific discourse that helped articulate the technological and esthetic world we know today.

Jean Constant

M.C. Escher’s Use of the Poincaré Models of Hyperbolic Geometry

Abstract

The artist M.C. Escher was the first artist to create patterns in the hyperbolic plane. He used both the Poincaré disk model and the Poincaré half-plane model of hyperbolic geometry. We discuss some of the theory of hyperbolic patterns and show Escher-inspired designs in both of these models.

Douglas Dunham

Mathematics and Music Boxes

Abstract

Music boxes which play a paper tape are fantastic tools for visually demonstrating some of the mathematical concepts in musical structure. The literal written notes in a piece can be transformed physically through reflections and rotations, and then easily played on the music box. Principles of topology can be demonstrated by playing loops and Möbius strips. Written music can also be transformed into different types of canons by sending it through multiple music boxes.

Vi Hart

My Mathematical Engravings

Abstract

The work of an engraver is shown through the presentation of three types of engravings concerning minimal surfaces, closed surfaces without singularities, and bi-periodic functions.

Patrice Jeener

Knots and Links As Form-Generating Structures

Abstract

Practical modeling of spatial surfaces is more convenient by means of transformation of their flat developments made as topologically connected kinetic structures. Any surface in 3D space topologically consists of three types of elements: planar facets (F), linear edges (E) and point vertexes (V). It is possible to identify the first two types of these elements with structural units of two common types of transformable systems: folding structures and kinematic nets respectively.

In the paper a third possible type of flat transformable structures with vertexes as form-generative units is considered. In this case flat developments of surfaces are formed by arranged point sets given by contacting crossing points of some classes of periodic knots and links made of elastic-flexible material, so that their crossing points have real physical contacts. A fragment of plane point surface can be reversibly converted into a fragment of a spatial surface with positive, negative or combined Gaussian curvature by means of transformation which saves connectivity between the points, but not the distances and angles between them. It was proved experimentally that this new form-generative method can be applied to modeling of both oriented and non-oriented differentiable topological 2D manifolds. The method of form-generation based upon the developing properties of periodic structures of knots and links may be applied to many practical fields including art, design and architecture.

Dmitri Kozlov

Geometry and Art from the Cordovan Proportion

Abstract

The Cordovan proportion, c = (2-√2)^–1/2, is the ratio between the radius of the regular octagon and its side length. This proportion was introduced by R. de la Hoz in 1973. Recently, the authors have found geometric properties linked with that proportion, related with a family of shapes named by them, Cordovan polygons. These results are summarized and are extended through the works of art of Hashim Cabrera and Luis Calvo, two Cordovan painters who have consciously considered the Cordovan proportion in their recent compositions. In fact, we have checked this ratio in several dissections of the canvases of Cabrera, and looking at the picture of Calvo, we have recognized many of our Cordovan polygons and some new polygons which we have added to our previous collection. We have also discovered some new cordovan dissections of a square, a √2 rectangle and a Silver rectangle.

Antonia Redondo Buitrago, Encarnación Reyes Iglesias

Dynamic Surfaces

Abstract

After discussing some well-known examples in geometry and function theory, we study surfaces in space that are defined by the vanishing of the torsion of integral curves of a given vector field.

Simon Salamon

Pleasing Shapes for Topological Objects

Abstract

Topology is the study of deformable shapes; to draw a picture of a topological object one must choose a particular geometric shape. One strategy is to minimize a geometric energy, of the type that also arises in many physical situations. The energy minimizers or optimal shapes are also often aesthetically pleasing. This article first appeared in an Italian translation [Sullivan, Affascinanti forme per oggetti topologici, 145–156 (2011)].

John M. Sullivan

Rhombopolyclonic Polygonal Rosettes Theory

Abstract

The division of a regular polygon with an even numbers of vertices into a whole number of “isoperimetric” rhombuses (equal sides and different angles) is possible. Elementary reasoning leads to a general theory, which offers many possibilities of plastic applications.

François Tard

Backmatter

Titel: Mathematics and Modern Art
herausgegeben von: Claude Bruter
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-24497-1
Print ISBN: 978-3-642-24496-4
DOI: https://doi.org/10.1007/978-3-642-24497-1

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

A Mathematician and an Artist. The Story of a Collaboration

Dimensions, a Math Movie

Old and New Mathematical Models: Saving the Heritage of the Institut Henri Poincaré

An Introduction to the Construction of Some Mathematical Objects

Computer, Mathematics and Art

Structure of Visualization and Symmetry in Iterated Function Systems

M.C. Escher’s Use of the Poincaré Models of Hyperbolic Geometry

Mathematics and Music Boxes

My Mathematical Engravings

Knots and Links As Form-Generating Structures

Geometry and Art from the Cordovan Proportion

Dynamic Surfaces

Pleasing Shapes for Topological Objects

Rhombopolyclonic Polygonal Rosettes Theory

Backmatter

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