43 relations: Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, E7 (mathematics), Edmund Hess, Emanuel Lodewijk Elte, Geometry, Gosset graph, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Honeycomb (geometry), Hosohedron, List of E7 polytopes, N-skeleton, Octadecagon, Petrie polygon, Projection (linear algebra), Rectified 5-cell, Rectified 6-orthoplexes, Rectified 6-simplexes, Regular polytope, Schläfli symbol, Semiregular polytope, Simplex, Tetrahedron, Thorold Gosset, Triangle, Uniform 6-polytope, Uniform 7-polytope, Uniform k 21 polytope, Uniform polytope, Vertex figure, 1 32 polytope, 2 21 polytope, 2 31 polytope, 5-cell, 5-demicube, 5-simplex, 6-orthoplex, 6-simplex, 7-simplex.

## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

New!!: 3 21 polytope and Convex polytope ·

## Coxeter element

In mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group.

New!!: 3 21 polytope and Coxeter element ·

## 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).

New!!: 3 21 polytope 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 the spatial relations between a collection of mirrors (or reflecting hyperplanes).

New!!: 3 21 polytope and Coxeter–Dynkin diagram ·

## Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions.

New!!: 3 21 polytope and Cross-polytope ·

## E7 (mathematics)

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.

New!!: 3 21 polytope and E7 (mathematics) ·

## Edmund Hess

Edmund Hess (1843 – 1903) was a German mathematician who discovered several regular polytopes.

New!!: 3 21 polytope and Edmund Hess ·

## Emanuel Lodewijk Elte

Emanuel Lodewijk Elte (16 March 1881, Amsterdam – 9 April 1943, Sobibór) at joodsmonument.nl was a Dutch mathematician.

New!!: 3 21 polytope and Emanuel Lodewijk Elte ·

## Geometry

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.

New!!: 3 21 polytope and Geometry ·

## Gosset graph

The Gosset graph, named after Thorold Gosset, is a specific regular graph (1-skeleton of the 7-dimensional 321 polytope) with 56 vertices and valency 27.

New!!: 3 21 polytope and Gosset graph ·

## 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.

New!!: 3 21 polytope and Gosset–Elte figures ·

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

New!!: 3 21 polytope and Harold Scott MacDonald Coxeter ·

## Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.

New!!: 3 21 polytope and Honeycomb (geometry) ·

## Hosohedron

In geometry, an ''n''-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

New!!: 3 21 polytope and Hosohedron ·

## List of E7 polytopes

In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry.

New!!: 3 21 polytope and List of E7 polytopes ·

## N-skeleton

In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.

New!!: 3 21 polytope and N-skeleton ·

## Octadecagon

An octadecagon (or octakaidecagon) is a polygon with 18 sides and 18 vertices.

New!!: 3 21 polytope and Octadecagon ·

## Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every (n-1) consecutive sides (but no n) belong to one of the facets.

New!!: 3 21 polytope and Petrie polygon ·

## Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

New!!: 3 21 polytope and Projection (linear algebra) ·

## Rectified 5-cell

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells.

New!!: 3 21 polytope and Rectified 5-cell ·

## Rectified 6-orthoplexes

In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

New!!: 3 21 polytope and Rectified 6-orthoplexes ·

## Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

New!!: 3 21 polytope and Rectified 6-simplexes ·

## Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry.

New!!: 3 21 polytope and Regular polytope ·

## Schläfli symbol

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

New!!: 3 21 polytope 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-uniform and has all its facets being regular polytopes.

New!!: 3 21 polytope 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.

New!!: 3 21 polytope and Simplex ·

## Tetrahedron

In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons) is a polyhedron composed of four triangular faces, three of which meet at each corner or vertex.

New!!: 3 21 polytope and Tetrahedron ·

## Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.

New!!: 3 21 polytope and Thorold Gosset ·

## Triangle

A triangle is a polygon with three edges and three vertices.

New!!: 3 21 polytope and Triangle ·

## Uniform 6-polytope

In six-dimensional geometry, a uniform polypeton (or uniform 6-polytope) is a six-dimensional uniform polytope.

New!!: 3 21 polytope and Uniform 6-polytope ·

## Uniform 7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets.

New!!: 3 21 polytope and Uniform 7-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.

New!!: 3 21 polytope and Uniform k 21 polytope ·

## Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.

New!!: 3 21 polytope and Uniform polytope ·

## Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.

New!!: 3 21 polytope and Vertex figure ·

## 1 32 polytope

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

New!!: 3 21 polytope and 1 32 polytope ·

## 2 21 polytope

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group.

New!!: 3 21 polytope and 2 21 polytope ·

## 2 31 polytope

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

New!!: 3 21 polytope and 2 31 polytope ·

## 5-cell

In geometry, the 5-cell is a four-dimensional object bounded by 5 tetrahedral cells.

New!!: 3 21 polytope and 5-cell ·

## 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 truncated.

New!!: 3 21 polytope and 5-demicube ·

## 5-simplex

In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope.

New!!: 3 21 polytope and 5-simplex ·

## 6-orthoplex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

New!!: 3 21 polytope and 6-orthoplex ·

## 6-simplex

In geometry, a 6-simplex is a self-dual regular 6-polytope.

New!!: 3 21 polytope and 6-simplex ·

## 7-simplex

In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope.

New!!: 3 21 polytope and 7-simplex ·