6-cube

Short description: 6-dimensional hypercube

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, 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 {4,3⁴}, 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.

[math]\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} }[/math]

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 < x_i < 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 [3^3,1,1] 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

Projections

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
File:Hexeract.ogv 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 an 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:

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

References

• 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". https://bendwavy.org/klitzing/dimensions/polypeta.htm. 

External links

• Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html. 
• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Measure. 
• Multi-dimensional Glossary: hypercube Garrett Jones