Mathematics and art

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Mathematics and art have a long historical relationship. The 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 Colosseum. 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 small 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 Dalí's painting The Last Supper in which Christ and his disciples are pictured inside a giant dodecahedron.

$The Mandelbrot set, a common example of a fractal art.$

The Mandelbrot set, a common example of a fractal art.

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 Gödel, 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.

Ferguson has created a sculpture (pdf) called The Eightfold Way at the Berkeley, California, Mathematical Sciences Research Institute. A publication of the institute by the same title is a tribute to that sculpture, and the underlying mathematics that involves the projective special linear group PSL(2,7), a finite group of 168 elements.

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.

[edit] 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).