# 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 | {4,3$^4$} |
| 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 | B$_6$, [3$^4$,4] |
| Dual | 6-orthoplex |
| Properties | convex, Hanner polytope |
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 {4,3$^4$}, 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. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u).
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.
${\begin{bmatrix}{\begin{matrix}64&6&15&20&15&6\\2&192&5&10&10&5\\4&4&240&4&6&4\\8&12&6&160&3&3\\16&32&24&8&60&2\\32&80&80&40&10&12\end{matrix}}\end{bmatrix}}$

## 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 (x$_0$, x$_1$, x$_2$, x$_3$, x$_4$, x$_5$) with −1 < x$_i$ < 1.

## Construction

There are three Coxeter groups associated with the 6-cube, one regular, with the C$_6$ or [4,3,3,3,3] Coxeter group, and a half symmetry (D$_6$) or [3$^{3,1,1}$] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.
| Name | Schläfli | Symmetry | Order |
| Regular 6-cube | {4,3,3,3,3} | [4,3,3,3,3] | 46080 |
| Quasiregular 6-cube | | [3,3,3,3$^{1,1}$] | 23040 |
| hyperrectangle | {4,3,3,3}× | [4,3,3,3,2] | 7680 |
| hyperrectangle | {4,3,3}×{4} | [4,3,3,2,4] | 3072 |
| hyperrectangle | {4,3}$^2$ | [4,3,2,4,3] | 2304 |
| hyperrectangle | {4,3,3}×$^2$ | [4,3,3,2,2] | 1536 |
| hyperrectangle | {4,3}×{4}× | [4,3,2,4,2] | 768 |
| hyperrectangle | {4}$^3$ | [4,2,4,2,4] | 512 |
| hyperrectangle | {4,3}×$^3$ | [4,3,2,2,2] | 384 |
| hyperrectangle | {4}$^2$×$^2$ | [4,2,4,2,2] | 256 |
| hyperrectangle | {4}×$^4$ | [4,2,2,2,2] | 128 |
| hyperrectangle |$^6$ | [2,2,2,2,2] | 64 |

## Projections

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | Other | B$_3$ | B$_2$ |
| Graph | | | |
| Dihedral symmetry | [2] | [6] | [4] |
| Coxeter plane | | A$_5$ | A$_3$ |
| 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. |
| A 3D perspective projection of a hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes. |

## Related polytopes

The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.
The 6-cube is 6th in a series of hypercube:
Petrie polygon orthographic projections :
| Line segment | Square | Cube | 4-cube | 5-cube | 6-cube | 7-cube | 8-cube | 9-cube | 10-cube |
This polytope is one of 63 uniform 6-polytopes generated from the B$_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes :
| β$_6$ | t$_1$β$_6$ | t$_2$β$_6$ | t$_2$γ$_6$ | t$_1$γ$_6$ | γ$_6$ | t$_{0,1}$β$_6$ | t$_{0,2}$β$_6$ |
| t$_{1,2}$β$_6$ | t$_{0,3}$β$_6$ | t$_{1,3}$β$_6$ | t$_{2,3}$γ$_6$ | t$_{0,4}$β$_6$ | t$_{1,4}$γ$_6$ | t$_{1,3}$γ$_6$ | t$_{1,2}$γ$_6$ |
| t$_{0,5}$γ$_6$ | t$_{0,4}$γ$_6$ | t$_{0,3}$γ$_6$ | t$_{0,2}$γ$_6$ | t$_{0,1}$γ$_6$ | t$_{0,1,2}$β$_6$ | t$_{0,1,3}$β$_6$ | t$_{0,2,3}$β$_6$ |
| t$_{1,2,3}$β$_6$ | t$_{0,1,4}$β$_6$ | t$_{0,2,4}$β$_6$ | t$_{1,2,4}$β$_6$ | t$_{0,3,4}$β$_6$ | t$_{1,2,4}$γ$_6$ | t$_{1,2,3}$γ$_6$ | t$_{0,1,5}$β$_6$ |
| t$_{0,2,5}$β$_6$ | t$_{0,3,4}$γ$_6$ | t$_{0,2,5}$γ$_6$ | t$_{0,2,4}$γ$_6$ | t$_{0,2,3}$γ$_6$ | t$_{0,1,5}$γ$_6$ | t$_{0,1,4}$γ$_6$ | t$_{0,1,3}$γ$_6$ |
| t$_{0,1,2}$γ$_6$ | t$_{0,1,2,3}$β$_6$ | t$_{0,1,2,4}$β$_6$ | t$_{0,1,3,4}$β$_6$ | t$_{0,2,3,4}$β$_6$ | t$_{1,2,3,4}$γ$_6$ | t$_{0,1,2,5}$β$_6$ | t$_{0,1,3,5}$β$_6$ |
| t$_{0,2,3,5}$γ$_6$ | t$_{0,2,3,4}$γ$_6$ | t$_{0,1,4,5}$γ$_6$ | t$_{0,1,3,5}$γ$_6$ | t$_{0,1,3,4}$γ$_6$ | t$_{0,1,2,5}$γ$_6$ | t$_{0,1,2,4}$γ$_6$ | t$_{0,1,2,3}$γ$_6$ |
| t$_{0,1,2,3,4}$β$_6$ | t$_{0,1,2,3,5}$β$_6$ | t$_{0,1,2,4,5}$β$_6$ | t$_{0,1,2,4,5}$γ$_6$ | t$_{0,1,2,3,5}$γ$_6$ | t$_{0,1,2,3,4}$γ$_6$ | t$_{0,1,2,3,4,5}$γ$_6$ |