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6-cube

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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 {4,34}
Coxeter-Dynkin 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

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 polytopes 1
  • Cartesian coordinates 2
  • Images 3
  • Related polytopes 4
  • References 5
  • External links 6

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.

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.

Images

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.

Hexeract Quasicrystal structure orthographically projected
to 3D using the Golden Ratio.

Related polytopes

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


β6

t1β6

t2β6

t2γ6

t1γ6

γ6

t0,1β6

t0,2β6

t1,2β6

t0,3β6

t1,3β6

t2,3γ6

t0,4β6

t1,4γ6

t1,3γ6

t1,2γ6

t0,5γ6

t0,4γ6

t0,3γ6

t0,2γ6

t0,1γ6

t0,1,2β6

t0,1,3β6

t0,2,3β6

t1,2,3β6

t0,1,4β6

t0,2,4β6

t1,2,4β6

t0,3,4β6

t1,2,4γ6

t1,2,3γ6

t0,1,5β6

t0,2,5β6

t0,3,4γ6

t0,2,5γ6

t0,2,4γ6

t0,2,3γ6

t0,1,5γ6

t0,1,4γ6

t0,1,3γ6

t0,1,2γ6

t0,1,2,3β6

t0,1,2,4β6

t0,1,3,4β6

t0,2,3,4β6

t1,2,3,4γ6

t0,1,2,5β6

t0,1,3,5β6

t0,2,3,5γ6

t0,2,3,4γ6

t0,1,4,5γ6

t0,1,3,5γ6

t0,1,3,4γ6

t0,1,2,5γ6

t0,1,2,4γ6

t0,1,2,3γ6

t0,1,2,3,4β6

t0,1,2,3,5β6

t0,1,2,4,5β6

t0,1,2,4,5γ6

t0,1,2,3,5γ6

t0,1,2,3,4γ6

t0,1,2,3,4,5γ6

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)
  • Richard Klitzing, 6D uniform polytopes (polypeta), o3o3o3o3o4x - ax

External links

  • Weisstein, Eric W., "Hypercube", MathWorld.
  • Olshevsky, George, Measure polytope at Glossary for Hyperspace.
  • Multi-dimensional Glossary: hypercube Garrett Jones
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