Mathematics and art - Definition and Overview

Mathematics and art have a long historical relationship. The ancient ancient Egyptians and ancient Greeks knew about the golden ratio, regarded as an aesthetically pleasing ratio, and incorporated it into the design of monuments including the Great Pyramid, the Parthenon, the Colloseum. The golden ratio is used in the design and layout of paintings such as The Roses of Heliogabalus. Recent studies show that the golden ratio also plays a role in the human perception of beauty in body shapes and faces.

Melancholia I

The Platonic solids and other polyhedra are a recurring theme in Western art. Examples are:

• A marble mosaic featuring the lesser stellated dodecahedron, attributed to Paolo Uccello, in the floor of the San Marco Basilica in Venice.
• Leonardo da Vinci's outstanding diagrams of regular polyhedra drawn as illustrations for Luca Pacioli's book The Divine Proportion.
• A glass rhombicuboctahedron in Jacopo de Barbari's portrait of Pacioli, painted in 1495.
• A truncated polyhedron (and various other mathematical objects) which feature in Albrecht Dürer's engraving Melancholia I.
• Salvador Dali's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.

Many of the works of artist M. C. Escher contain impossible constructions, made using geometrical objects that cannot exist but are pleasant to the human sight. Some of Escher's tessellation drawings were inspired by conversations with the mathematician H. S. M. Coxeter concerning hyperbolic geometry. Relationships between the works of mathematician Kurt Gödel, artist M.C. Escher and composer Johann Sebastian Bach are explored in Goedel, Escher, Bach, a Pulitzer Prize-winning book.

The processing power of modern computers allows mathematicians and non-mathematicians to visualise complex mathematical objects such as the Mandelbrot set. In the modern industry of computer animation, fractals play a key role in modelling mountains, fire, trees and other natural objects. See fractal art for examples of the use of these mathematical objects with only aesthetic motivations. See low-complexity art for Juergen Schmidhuber's minimal art form explicitly based on short computer programs.

Sculptor Helaman Ferguson has made sculptures in various materials of a wide range of complex surfaces and other topological objects. His work is motivated specifically by the desire to create visual representations of mathematical objects.

A recent study published in Scientific American (December 2002) shows that an interesting property in Jackson Pollock's art is that his works have a fractal dimension, which make them different from purely random strokes.

Literature and film

Mathematical themes and mathematicians have been featured in novels (A Beautiful Mind), plays (Copenhagen), motion pictures (Pi) and even an opera (Fermat's Last Tango).

See also

• Mathematics and architecture
• Mathematics of musical scales
• Musical set theory - the application of group theory and combinatorics to certain aspects of music theory

External links

• Order in Pollock's Chaos (Scientific American article, requires subscription) (http://www.sciam.com/issue.cfm?issueDate=Dec-02)
• Fractal expressionism - the mathematics of Pollock's art (free article) (http://plus.maths.org/issue11/features/physics_world/)
• Polyhedra in Art (http://www.georgehart.com/virtual-polyhedra/art.html)
• Connections in Space - topology in art (http://www.btinternet.com/~connectionsinspace/index.html)
• Helaman Ferguson web site (http://www.helasculpt.com/index.html)
• National University of Singapore's course on Mathematics in Art and Architecture (http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml)
• Mathematics in the movies site (http://mathinthemovies.com)