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Home > 2013 Mathematical Art Exhibition
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"Bended Circle Limit III," by Vladimir Bulatov (Corvallis, OR)

24" x 24", Digital print, 2012
Best photograph, painting, or print, 2013 Mathematical Art Exhibition

M.C. Escher's hyperbolic tessellations Circle Limit III is based on a tiling of the hyperbolic plane by identical triangles. The tiling is rigid because hyperbolic triangles are unambiguously defined by their vertex angles. However, if we reduce the symmetry of the tiling by joining several triangles into a single polygonal tile, such tiling can be deformed. Hyperbolic geometry allows a type of deformation of tiling called bending. Let's extend the tiling of the hyperbolic plane by identical polygons into tiling of hyperbolic space by identical infinite prisms. The prism's cross section is the original polygon. The shape of these 3D prisms can be carefully changed by rotating some of its sides in space and preserving all dihedral angles. Such operation is only possible in hyperbolic geometry. The resulting tiling of 3D hyperbolic space creates 2D tiling on the infinity of hyperbolic 3D space, which is a Riemann sphere. The sphere is stereographically projected to the plane. -- Vladimir Bulatov

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American Mathematical Society