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.
"Prime colorings of the sphere, Euclidean plane and the hyperbolic plane," by Chaim Goodman-Strauss, University of Arkansas (http://mathbun.com/main.php)
Tilings of the sphere, the Euclidean plane and the hyperbolic plane are shown. In each case, we have triangular faces, but on the sphere, the triangles meet in fives; in the Euclidean plane, the triangles meet in sixes, and in the hyperbolic plane, they meet in sevens. To a great degree, this is forced. It is impossible, for example, to have a tiling of the sphere with triangles meeting only in sevens (try it!).
In each case, a prime-fold coloring of the pattern is shown. It is helpful to realize that there are more similarities than differences among the three geometries.
The symmetry of the hyperbolic plane pictured above was known to Felix Klein by 1878, and has a tremendous number of interesting topological, geometric and algebraic properties.
This image is from "The Symmetries of Things"� by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss (AK Peters, 2008).