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In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}. Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille). There are 3 regular and 8 semiregular tilings in the plane. There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.) The snub square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number). The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling. An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square. If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces. Example: File:Snub snub square tiling.svg|A [[Conway snub operator|snub operator]] applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons. File:Isogonal snub square tiling-8x8.svg|A related [[isogonal tiling]] that combines pairs of triangles into rhombi File:Triangular heptagonal tiling.svg|A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons. File:P2_dual.png|The [[Cairo pentagonal tiling]] is dual to the snub square tiling.

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Concepts associés (13)

Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain.

Diagramme de Coxeter-Dynkin

En géométrie, un diagramme de Coxeter-Dynkin est un graphe représentant un ensemble relationnel de miroirs (ou d'hyperplans de réflexion) dans l'espace pour une construction kaléidoscopique. En tant que graphe lui-même, le diagramme représente les groupes de Coxeter, chaque nœud du graphe représente un miroir (facette du domaine) et chaque branche du graphe représente l'ordre de l'angle diédral entre deux miroirs (sur une arête du domaine). En plus, les graphes ont des anneaux (cercles) autour des nœuds pour les miroirs actifs représentant un polytope précis.

List of Euclidean uniform tilings

This 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.