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.
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.
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.
Symbole de WythoffEn géométrie, un symbole de Wythoff est une notation courte, créée par le mathématicien Willem Abraham Wythoff, pour nommer les polyèdres réguliers et semi-réguliers utilisant une construction kaléidoscopique, en les représentant comme des pavages sur la surface d'une sphère, sur un plan euclidien ou un plan hyperbolique. Le symbole de Wythoff donne 3 nombres p,q,r et une barre verticale positionnelle (|) qui sépare les nombres avant et après elle. Chaque nombre représente l'ordre des miroirs à un sommet du triangle fondamental.
List of Euclidean uniform tilingsThis table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.
Configuration de sommetEn géométrie, une configuration de sommet est une notation abrégée pour représenter la figure de sommet d'un polyèdre ou d'un pavage comme la séquence de faces autour d'un sommet. Pour les polyèdres uniformes, il n'y a qu'un seul type de sommet et, par conséquent, la configuration des sommets définit entièrement le polyèdre. (Les polyèdres chiraux existent dans des paires d'images miroir avec la même configuration de sommet). Une configuration de sommet est donnée sous la forme d'une suite de nombres représentant le nombre de côtés des faces faisant le tour du sommet.
