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6-cube
Hexeract
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.

Contents

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.

Cartesian coordinatesEdit

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.

ConstructionEdit

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

ImagesEdit

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 polytopesEdit

ReferencesEdit

External linksEdit