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6-cube graph.svg
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 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-faces 12 {4,3,3,3} 5-cube graph.svg
4-faces 60 {4,3,3} 4-cube graph.svg
Cells 160 {4,3} 3-cube graph.svg
Faces 240 {4} 2-cube.svg
Edges 192
Vertices 64
Vertex figure 5-simplex
Petrie polygon dodecagon
Coxeter group B6, [34,4]
Dual 6-orthoplex 6-orthoplex.svg
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 polytopesEdit

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 configurationEdit

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]


Cartesian coordinatesEdit

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


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


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


orthographic projections
Coxeter plane B6 B5 B4
Dihedral symmetry [12] [10] [8]
Coxeter plane Other B3 B2
Dihedral symmetry [2] [6] [4]
Coxeter plane A5 A3
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 polytopesEdit


  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".

External linksEdit