51 relations: Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, E6 polytope, E8 polytope, En (Lie algebra), Face (geometry), Facet, Facet (geometry), Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Hyperplane, Rectified 8-simplexes, Rectified 9-simplexes, Regular polytope, Schläfli symbol, Semiregular polytope, Simplex, Tetrahedron, Thorold Gosset, Triangle, Uniform 1 k2 polytope, Uniform 2 k1 polytope, Uniform k 21 polytope, Uniform polytope, Vertex figure, Wythoff construction, 1 22 polytope, 1 32 polytope, 1 42 polytope, 1 52 honeycomb, 16-cell, 2 31 polytope, 2 41 polytope, 2 51 honeycomb, 3 21 polytope, 5 21 honeycomb, 5-cell, 5-demicube, 5-orthoplex, 5-simplex, 6-demicube, 6-simplex, 7-demicube, 7-simplex, 8-demicube, 8-simplex, 9-demicube, 9-orthoplex, ..., 9-simplex. Expand index (1 more) » « Shrink index
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions.
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry.
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry.
In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k.
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.
Facets are flat faces on geometric shapes.
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles.
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes.
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.
A triangle is a polygon with three edges and three vertices.
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.
In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets.
A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.
In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.
In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.
In geometry, a 6-simplex is a self-dual regular 6-polytope.
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed.
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed.
In geometry, an 8-simplex is a self-dual regular 8-polytope.
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed.
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.
In geometry, a 9-simplex is a self-dual regular 9-polytope.