Uniform Tilings In Hyperbolic Plane
In geometry, a uniform (regular, quasiregular or semiregular) hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of reflectional, rotational, and translational symmetry.
Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given Schläfli symbol {7,3}.
Uniform tilings may be regular (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive). For right triangles (p q 2), there are are two regular tilings, represented by Schläfli symbol {p,q} and {q,p}.
Read more about Uniform Tilings In Hyperbolic Plane: Wythoff Construction, Right Triangle Domains, General Triangle Domains, Summary of Tilings With Finite Triangular Fundamental Domains, Ideal Triangle Domains, Summary of Tilings With Infinite Triangular Fundamental Domains
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