## Table of Contents

50 relations: Configuration (polytope), Convex polytope, Coxeter element, Coxeter group, Coxeter–Dynkin diagram, Disphenoid, E8 (mathematics), E8 polytope, Emanuel Lodewijk Elte, Equilateral triangle, Geometry, Gosset–Elte figures, Harold Scott MacDonald Coxeter, Honeycomb (geometry), Hyperplane, Isosceles triangle, Octahedron, Orthographic projection, Petrie polygon, Projection (linear algebra), Rectification (geometry), Rectified 5-cell, Rectified 5-cubes, Rectified 5-simplexes, Rectified 6-cubes, Rectified 6-simplexes, Rectified 7-cubes, Rectified 7-simplexes, Rectified 8-cubes, Regular polygon, Schläfli symbol, Tetrahedron, Triangle, Uniform 1 k2 polytope, Uniform 8-polytope, Uniform polytope, Vertex figure, Wythoff construction, 1 22 polytope, 1 32 polytope, 16-cell, 2 41 polytope, 4 21 polytope, 5-cell, 5-demicube, 5-simplex, 6-demicube, 6-simplex, 7-demicube, 8-demicube.

- 8-polytopes
- E8 (mathematics)

## Configuration (polytope)

In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.

See 1 42 polytope and Configuration (polytope)

## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n.

See 1 42 polytope and Convex polytope

## Coxeter element

In mathematics, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections.

See 1 42 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).

See 1 42 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 a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

See 1 42 polytope and Coxeter–Dynkin diagram

## Disphenoid

In geometry, a disphenoid is a tetrahedron whose four faces are congruent acute-angled triangles.

See 1 42 polytope and Disphenoid

## E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.

See 1 42 polytope and E8 (mathematics)

## E8 polytope

In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. 1 42 polytope and E8 polytope are 8-polytopes and e8 (mathematics).

See 1 42 polytope and E8 polytope

## Emanuel Lodewijk Elte

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

See 1 42 polytope and Emanuel Lodewijk Elte

## Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have the same length.

See 1 42 polytope and Equilateral triangle

## Geometry

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.

See 1 42 polytope and Geometry

## 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 1 42 polytope 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 1 42 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.

See 1 42 polytope and Honeycomb (geometry)

## Hyperplane

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension.

See 1 42 polytope and Hyperplane

## Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length.

See 1 42 polytope and Isosceles triangle

## Octahedron

In geometry, an octahedron (octahedra or octahedrons) is a polyhedron with eight faces.

See 1 42 polytope and Octahedron

## Orthographic projection

Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions.

See 1 42 polytope and Orthographic projection

## Petrie polygon

In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no) belongs to one of the facets.

See 1 42 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 (an endomorphism) such that P\circ P.

See 1 42 polytope and Projection (linear algebra)

## Rectification (geometry)

In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.

See 1 42 polytope and Rectification (geometry)

## 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.

See 1 42 polytope and Rectified 5-cell

## Rectified 5-cubes

In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

See 1 42 polytope and Rectified 5-cubes

## Rectified 5-simplexes

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

See 1 42 polytope and Rectified 5-simplexes

## Rectified 6-cubes

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

See 1 42 polytope and Rectified 6-cubes

## 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.

See 1 42 polytope and Rectified 6-simplexes

## Rectified 7-cubes

In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.

See 1 42 polytope and Rectified 7-cubes

## Rectified 7-simplexes

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

See 1 42 polytope and Rectified 7-simplexes

## Rectified 8-cubes

In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube. 1 42 polytope and rectified 8-cubes are 8-polytopes.

See 1 42 polytope and Rectified 8-cubes

## Regular polygon

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

See 1 42 polytope and Regular polygon

## Schläfli symbol

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

See 1 42 polytope and Schläfli symbol

## 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 1 42 polytope and Tetrahedron

## Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

See 1 42 polytope and Triangle

## Uniform 1 k2 polytope

In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n.

See 1 42 polytope and Uniform 1 k2 polytope

## Uniform 8-polytope

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. 1 42 polytope and Uniform 8-polytope are 8-polytopes.

See 1 42 polytope and Uniform 8-polytope

## Uniform polytope

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

See 1 42 polytope 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 1 42 polytope 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 1 42 polytope and Wythoff construction

## 1 22 polytope

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.

See 1 42 polytope and 1 22 polytope

## 1 32 polytope

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

See 1 42 polytope and 1 32 polytope

## 16-cell

In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.

## 2 41 polytope

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group. 1 42 polytope and 2 41 polytope are 8-polytopes and e8 (mathematics).

See 1 42 polytope and 2 41 polytope

## 4 21 polytope

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. 1 42 polytope and 4 21 polytope are 8-polytopes and e8 (mathematics).

See 1 42 polytope and 4 21 polytope

## 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 1 42 polytope and 5-demicube

## 5-simplex

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

See 1 42 polytope 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 1 42 polytope and 6-demicube

## 6-simplex

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

See 1 42 polytope 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 1 42 polytope and 7-demicube

## 8-demicube

In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. 1 42 polytope and 8-demicube are 8-polytopes.

See 1 42 polytope and 8-demicube

## See also

### 8-polytopes

- 1 33 honeycomb
- 1 42 polytope
- 2 41 polytope
- 3 31 honeycomb
- 4 21 polytope
- 7-cubic honeycomb
- 7-demicubic honeycomb
- 7-simplex honeycomb
- 8-cube
- 8-demicube
- 8-orthoplex
- 8-simplex
- A8 polytope
- B8 polytope
- Cantellated 8-simplexes
- Cantic 8-cube
- Cyclotruncated 7-simplex honeycomb
- D8 polytope
- E8 polytope
- Heptellated 8-simplexes
- Hexicated 8-simplexes
- Omnitruncated 7-simplex honeycomb
- Pentellated 8-simplexes
- Quarter 7-cubic honeycomb
- Rectified 8-cubes
- Rectified 8-orthoplexes
- Rectified 8-simplexes
- Runcinated 8-simplexes
- Stericated 8-simplexes
- Truncated 8-cubes
- Truncated 8-orthoplexes
- Truncated 8-simplexes
- Uniform 8-polytope

### E8 (mathematics)

- 1 42 polytope
- 1 52 honeycomb
- 2 41 polytope
- 2 51 honeycomb
- 4 21 polytope
- 5 21 honeycomb
- E8 (mathematics)
- E8 lattice
- E8 manifold
- E8 polytope
- Heterotic string theory

## References

Also known as 0 421 polytope, Birectified 2 41 polytope, Gosset 1 42 polytope, Quadrirectified 4 21 polytope, Rectified 1 42 polytope.