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 maples
Stood uniform in buckets, and the steam
Of sap and snow rolled off the sugarhouse.”
—Robert Frost (1874–1963)
“We’ve got to figure these things a little bit different than most people. Y’know, there’s something about going out in a plane that beats any other way.... A guy that washes out at the controls of his own ship, well, he goes down doing the thing that he loved the best. It seems to me that that’s a very special way to die.”
—Dalton Trumbo (1905–1976)