# 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

This configuration matrix represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[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 edition, 1973), Dover edition, 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".