Tiling the Hyperbolic Plane with Regular Polygons

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This Demonstration shows the beginning of the tiling of the hyperbolic plane (in the Poincaré disk model) with regular -gons, of them touching at each vertex inside the tiling. (There may be fewer than polygons touching a vertex near the boundary of the current tiling.) Click the plane to change where the center appears.

Contributed by: Gašper Zadnik (March 2013)
Open content licensed under CC BY-NC-SA



For every pair of positive integers satisfying , there is a tiling of the hyperbolic plane with regular -gons such that -gons touch at each vertex. Since isometries of the hyperbolic plane act transitively on it, the center of the "initial" -gon (from which we start the tiling) can be chosen anywhere in the hyperbolic plane.


[1] Wikipedia. "Uniform Tilings in Hyperbolic Plane." (Mar 1, 2013) en.wikipedia.org/wiki/Uniform_tilings_in _hyperbolic _plane.

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