Planar and Spherical Tessellations
By a tessellation T we mean a division of a geometric surface into nonoverlapping congruent polygons, called tiles. T is symmetric if it “looks the same from the viewpoint of any tile”, i.e., if any tile II1 can be mapped onto any tile II2 by an isometry which maps the whole of T onto itself (faces onto faces and edges onto edges). The isometries of T onto itself are called symmetries of T, and they form a group called the symmetry group of T. Thus, we are defining T to be symmetric if its symmetry group contains enough elements to map any tile onto any other tile.
KeywordsSymmetry Group Equilateral Triangle Cayley Graph Hyperbolic Surface Angle Condition
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