Uniform tilings in hyperbolic planeIn hyperbolic geometry, a uniform hyperbolic tiling (or 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 rotational and translational symmetry.
Triheptagonal tilingIn geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}. Compare to trihexagonal tiling with vertex configuration 3.6.3.6. In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.
Order-7 triangular tilingIn geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}. The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces.
Heptagonal tilingIn geometry, a heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex. This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Pavage du planthumb|Pavage constitué de triangles équilatéraux et d'hexagones, dit pavage trihexagonal. thumb|Pavage hexagonal de tomettes provençales en terre cuite. Un pavage du plan est un ensemble de portions du plan, par exemple des polygones, dont l'union est le plan tout entier, sans recouvrement. Plus précisément, c'est une partition du plan euclidien par des éléments d'un ensemble fini, appelés « carreaux » (plus précisément, ce sont des compacts d’intérieur non vide).
