# 6-cube

6-cube
Hexeract

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and the center yellow has 4 vertices
Type Regular 6-polytope
Family hypercube
Schläfli symbol {4,34}
Coxeter diagram
5-faces 12 {4,3,3,3}
4-faces 60 {4,3,3}
Cells 160 {4,3}
Faces 240 {4}
Edges 192
Vertices 64
Vertex figure 5-simplex
Petrie polygon dodecagon
Coxeter group B6, [34,4]
Dual 6-orthoplex
Properties convex

In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.

It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.

## Related polytopes[]

It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes.

Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.

## As a configuration[]

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[1][2]

${\displaystyle {\begin{bmatrix}{\begin{matrix}64&6&15&20&15&6\\2&192&5&10&10&5\\4&4&240&4&6&4\\8&12&6&160&3&3\\16&32&24&8&60&2\\32&80&80&40&10&12\end{matrix}}\end{bmatrix}}}$

## Cartesian coordinates[]

Cartesian coordinates for the vertices of a 6-cube centered at the origin and edge length 2 are

(±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with −1 < xi < 1.

## Construction[]

There are three Coxeter groups associated with the 6-cube, one regular, with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry (D6) or [33,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.

Name Coxeter Schläfli Symmetry Order
Regular 6-cube
{4,3,3,3,3} [4,3,3,3,3] 46080
Quasiregular 6-cube [3,3,3,31,1] 23040
hyperrectangle {4,3,3,3}×{} [4,3,3,3,2] 7680
{4,3,3}×{4} [4,3,3,2,4] 3072
{4,3}2 [4,3,2,4,3] 2304
{4,3,3}×{}2 [4,3,3,2,2] 1536
{4,3}×{4}×{} [4,3,2,4,2] 768
{4}3 [4,2,4,2,4] 512
{4,3}×{}3 [4,3,2,2,2] 384
{4}2×{}2 [4,2,4,2,2] 256
{4}×{}4 [4,2,2,2,2] 128
{}6 [2,2,2,2,2] 64

## Images[]

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane Other B3 B2
Graph
Dihedral symmetry [2] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
 3D Projections 6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. 6-cube quasicrystal structure orthographically projected to 3D using the golden ratio.

## Related polytopes[]

This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

## References[]

1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
• Coxeter, H.S.M. Regular Polytopes, (3rd ion, 1973), Dover ion, ISBN 0-486-61480-8 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
• Klitzing, Richard. "6D uniform polytopes (polypeta) o3o3o3o3o4x - ax".