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.
"Hyperbolic Tiling I," by Vladimir Bulatov (Corvallis, OR)
Rapid prototyping sculpture, 200mm x 60mm x 60mm, 2010
This is a visualization of a tiling of the hyperbolic space. The tiling is generated by reflections in the faces of Lambert cube (Coxeter polyhedron), which becomes the fundamental polyhedron of the symmetry group of the tiling. Only 4 out of 6 sides are used, which results in sub-tiling (subgroup) filling only part of the space. It let us see the internal structure of the tiling. We use a cylinder model of the hyperbolic space--a 3D generalization of 2D band model. In this model the Poincare ball is stretched into infinite cylinder. Cylinder's axis becomes one of hyperbolic geodesics. The tiling is oriented to make one it's plane to be orthogonal to the cylinder's axis to have a feet to stand on. The cylinder's axis is close to the axis of a loxodromic transformation of the group, which gives the pieces its spiral twist. The sharp boundary of the piece corresponds to the limit set of the group. The limit set is fractal Jordan curve at the infinity of the hyperbolic space. --- Vladimir Bulatov (http://bulatov.org)