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Surfaces and tiled surfaces

genus two surface genus2.gif genus2.eps
icosahedral tiling of sphere icos.gif icos.eps
(2,4,4) tiling on a torus torus244.gif torus244.eps
(3,3,3) tiling on a torus torus333.gif torus333.eps

Hyperbolic tilings

(2,4,5)- tiling of the hyperbolic plane T245.gif
(2,5,5)- tiling of the hyperbolic plane T255.gif
(3,5,5)- tiling of the hyperbolic plane T355.gif
(4,3,3)- tiling of the hyperbolic plane T433.gif

Divisible tilings

A tiling is divisible if it can be divided into a finer kaleidoscopic tiling. An example is the tiling of the torus by rectangles which is refined by the tiling by (2,4,4) triangles, pictured below. Each divisible tiling of a surface comes from a divisible tiling of the hyperbolic plane. All such tilings have been classified . There are four types of tilings. The description and links to the tables and pictures are below:

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Last modified April 4, 2001