6-cube

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.

6-cube
Hexeract

Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and the center yellow has 4 vertices
TypeRegular 6-polytope
Familyhypercube
Schläfli symbol{4,34}
Coxeter diagram
5-faces12 {4,3,3,3}
4-faces60 {4,3,3}
Cells160 {4,3}
Faces240 {4}
Edges192
Vertices64
Vertex figure5-simplex
Petrie polygondodecagon
Coxeter groupB6, [34,4]
Dual6-orthoplex
Propertiesconvex

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.

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]

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

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

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.

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".
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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