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Home > 2011 Mathematical Art Exhibition
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"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 (

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American Mathematical Society