6-cube

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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-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.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 polytopes[edit]

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[edit]

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[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t0.svg 6-cube t0 B5.svg 4-cube t0.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane Other B3 B2
Graph 6-cube column graph.svg 6-cube t0 B3.svg 6-cube t0 B2.svg
Dihedral symmetry [2] [6] [4]
Coxeter plane A5 A3
Graph 6-cube t0 A5.svg 6-cube t0 A3.svg
Dihedral symmetry [6] [4]
3D Projections

6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.
6Cube-QuasiCrystal.jpg
Hexeract Quasicrystal structure orthographically projected
to 3D using the Golden Ratio.

Related polytopes[edit]

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

References[edit]

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds