We have all seen many of the beautiful patterns obtained by tiling the hyperbolic plane H by repeated reflection in the sides of a "kaleidoscopic" polygon. Though there are such patterns on the sphere and the euclidean plane, these positively curved and fiat geometries lack the richness we see in the hyperbolic plane. Many of these patterns have been popularized by the beautiful art of M.C. Escher. For a list of references and a more complete discussion on the construction of artistic tilings see .
Broughton, Sean A., "Constructing Kaleidscopic Tiling Polygons in the Hyperbolic Plane" (1999). Mathematical Sciences Technical Reports (MSTR). 112.