The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.
Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.
"Dodecafoam I," by Chaim Goodman-Strauss, University of Arkansas (http://mathbun.com/main.php)
Unlike all of the other images in this collection, the symmetry here is not governed by a group action, but rather by a substitution system--a set of replacement rules, based on the stellations of the dodecahedron. Several oddly shaped three-dimensional cells based on the stellations of the dodecahedron are used; a rule then gives a method for dividing each cell into small copies of the others. Such techniques are commonly used to produce highly ordered non-periodic structures; though it may look as if such a structure repeats, in fact it cannot repeat periodically.