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:
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—Thomas Jefferson (1743–1826)
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)