Table of Contents
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, ... Expand index (1 more) »
- 10-polytopes
Coxeter group
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).
See E9 honeycomb and Coxeter group
Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.
See E9 honeycomb and Coxeter–Dynkin diagram
Cross-polytope
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space.
See E9 honeycomb and Cross-polytope
E6 polytope
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry.
See E9 honeycomb and E6 polytope
E8 polytope
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry.
See E9 honeycomb and E8 polytope
En (Lie algebra)
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.
See E9 honeycomb and En (Lie algebra)
Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.
See E9 honeycomb and Face (geometry)
Facet
Facets are flat faces on geometric shapes.
Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.
See E9 honeycomb and Facet (geometry)
Geometry
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
Gosset–Elte figures
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.
See E9 honeycomb and Gosset–Elte figures
Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician.
See E9 honeycomb and Harold Scott MacDonald Coxeter
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.
See E9 honeycomb and Hyperplane
Rectified 8-simplexes
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
See E9 honeycomb and Rectified 8-simplexes
Rectified 9-simplexes
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.
See E9 honeycomb and Rectified 9-simplexes
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.
See E9 honeycomb and Regular polytope
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
See E9 honeycomb and Schläfli symbol
Semiregular polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes.
See E9 honeycomb and Semiregular polytope
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.
Tetrahedron
In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.
See E9 honeycomb and Tetrahedron
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.
See E9 honeycomb and Thorold Gosset
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.
Uniform 1 k2 polytope
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.
See E9 honeycomb and Uniform 1 k2 polytope
Uniform 2 k1 polytope
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n.
See E9 honeycomb and Uniform 2 k1 polytope
Uniform k 21 polytope
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.
See E9 honeycomb and Uniform k 21 polytope
Uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
See E9 honeycomb and Uniform polytope
Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
See E9 honeycomb and Vertex figure
Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
See E9 honeycomb and Wythoff construction
1 22 polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
See E9 honeycomb and 1 22 polytope
1 32 polytope
In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.
See E9 honeycomb and 1 32 polytope
1 42 polytope
In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
See E9 honeycomb and 1 42 polytope
1 52 honeycomb
In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.
See E9 honeycomb and 1 52 honeycomb
16-cell
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
See E9 honeycomb and 2 31 polytope
2 41 polytope
In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.
See E9 honeycomb and 2 41 polytope
2 51 honeycomb
In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation.
See E9 honeycomb and 2 51 honeycomb
3 21 polytope
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group.
See E9 honeycomb and 3 21 polytope
5 21 honeycomb
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.
See E9 honeycomb and 5 21 honeycomb
5-cell
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol.
5-demicube
In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.
See E9 honeycomb and 5-demicube
5-orthoplex
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.
See E9 honeycomb and 5-orthoplex
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.
See E9 honeycomb and 5-simplex
6-demicube
In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed.
See E9 honeycomb and 6-demicube
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope.
See E9 honeycomb and 6-simplex
7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed.
See E9 honeycomb and 7-demicube
7-simplex
In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.
See E9 honeycomb and 7-simplex
8-demicube
In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed.
See E9 honeycomb and 8-demicube
8-simplex
In geometry, an 8-simplex is a self-dual regular 8-polytope.
See E9 honeycomb and 8-simplex
9-demicube
In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed.
See E9 honeycomb and 9-demicube
9-orthoplex
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.
See E9 honeycomb and 9-orthoplex
9-simplex
In geometry, a 9-simplex is a self-dual regular 9-polytope.
See E9 honeycomb and 9-simplex
See also
10-polytopes
- 10-cube
- 10-demicube
- 10-orthoplex
- 10-simplex
- E9 honeycomb
- Rectified 10-cubes
- Rectified 10-orthoplexes
- Rectified 10-simplexes
- Uniform 10-polytope
References
Also known as 1 62 honeycomb, 2 61 honeycomb, 6 21 honeycomb, 7 21 honeycomb, E10 honeycomb.