11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli type {3,5,3}, with 3 hemi-icosahedra (Schläfli type {3,5}) around each edge. It has symmetry order 660, computed as the product of the number of cells (11) and the symmetry of each cell (60). The symmetry structure is the abstract group projective special linear group L2(11).
Icosahedral honeycombIn geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure. The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.
Configuration (polytope)In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration. Other configurations in geometry are something different. These polytope configurations may be more accurately called incidence matrices, where like elements are collected together in rows and columns. Regular polytopes will have one row and column per k-face element, while other polytopes will have one row and column for each k-face type by their symmetry classes.
Grand hécatonicosachore étoilévignette|243x243px| Projection orthogonale En géométrie, le grand hécatonicosachore étoilé, ou hécatonicosachore 5,5/2,5, est un 4-polytope régulier étoilé ayant pour symbole de Schläfli {5,5/2,5}. C'est l'un des 10 polychores de Schläfli-Hess. C'est l'un des deux polytopes qui est son propre dual. Il a la même que l'hexacosichore et l'hécatonicosachore icosaédral, ainsi que la même disposition de faces que l'hécatonicosachore 5,3,5/2.
Hécatonicosachore 5/2,5,5/2En géométrie, l'hécatonicosachore 5/2,5,5/2 est un 4-polytope régulier étoilé ayant pour symbole de Schläfli {5/2,5,5/2}. C'est l'un des 10 polychores de Schläfli-Hess. Il est l'un des deux polytopes à être son propre dual. Il a la même que le grand hexacosichore et l'hécatonicosachore icosaédral, ainsi que la même disposition de faces que l'hécatonicosachore 5/2,3,5.
24-cell honeycombIn four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}. The dual tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space. The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D4 or F4 root lattice.
Tesseractic honeycombIn four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets. Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex. It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space.
Orthantthumb|Dans un espace de dimension 2, il y a 4 orthants (appelés quadrants) En géométrie, un orthant est la généralisation dans un espace euclidien de dimension quelconque n du quadrant d'un plan ou de l'octant en dimension 3. Un orthant en dimension n peut être considéré comme l'Intersection de n demi-espaces orthogonaux. Par permutation, il y a 2n orthants dans un espace de dimension n. De façon spécifique, un orthant fermé dans est le sous-ensemble défini par une contrainte de signe sur chaque coordonnée cartésienne.
Schläfli orthoschemeIn geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli.
BitruncationIn geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves. Bitruncated regular polytopes can be represented by an extended Schläfli symbol notation t_1,2{p,q,...} or 2t{p,q,...}. For regular polyhedra (i.e. regular 3-polytopes), a bitruncated form is the truncated dual. For example, a bitruncated cube is a truncated octahedron.
