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
Famous quotes containing the words uniform and/or plane:
“The sugar maple is remarkable for its clean ankle. The groves of these trees looked like vast forest sheds, their branches stopping short at a uniform height, four or five feet from the ground, like eaves, as if they had been trimmed by art, so that you could look under and through the whole grove with its leafy canopy, as under a tent whose curtain is raised.”
—Henry David Thoreau (1817–1862)
“At the moment when a man openly makes known his difference of opinion from a well-known party leader, the whole world thinks that he must be angry with the latter. Sometimes, however, he is just on the point of ceasing to be angry with him. He ventures to put himself on the same plane as his opponent, and is free from the tortures of suppressed envy.”
—Friedrich Nietzsche (1844–1900)