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.
Pavage carré adouciIn 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.
Construction de WythoffEn géométrie, une construction de Wythoff, nommée en l'honneur du mathématicien Willem Abraham Wythoff, est une méthode pour construire un polyèdre uniforme ou un pavage plan. On l'appelle souvent construction kaléidoscopique de Wythoff. Elle repose sur le pavage d'une sphère, avec des triangles sphériques. Si trois miroirs sont placés de telle manière que leurs plans se coupent en un point unique, alors les miroirs entourent un triangle sphérique sur la surface d'une sphère quelconque centrée en ce point et par réflexions répétées, on obtient une multitude de copies du triangle.
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.
Polyèdre uniformeUn polyèdre uniforme est un polyèdre dont les faces sont des polygones réguliers et qui est isogonal, c'est-à-dire que pour tout couple de sommets, il existe une isométrie qui applique un sommet sur l'autre. Il en découle que tous les sommets sont congruents et que le polyèdre possède un haut degré de symétrie par réflexion et rotation. La notion de polyèdre uniforme est généralisée, pour un nombre de dimensions quelconque, par celle de . Les polyèdres uniformes peuvent être réguliers, quasi réguliers ou semi-réguliers.
Coxeter elementIn mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there are multiple conjugacy classes of Coxeter elements, and they have infinite order. There are many different ways to define the Coxeter number h of an irreducible root system. A Coxeter element is a product of all simple reflections.
Icosahedral symmetryIn mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120.
Coxeter notationIn geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson. For Coxeter groups, defined by pure reflections, there is a direct correspondence between the bracket notation and Coxeter-Dynkin diagram.
Uniform polytopeIn geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vertex-transitive even-sided polygons that alternate two different lengths of edges). This is a generalization of the older category of semiregular polytopes, but also includes the regular polytopes. Further, star regular faces and vertex figures (star polygons) are allowed, which greatly expand the possible solutions.
E8 latticeIn mathematics, the E_8 lattice is a special lattice in R^8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E_8 root system. The norm of the E_8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8. The existence of such a form was first shown by H.
