6-cube

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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 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 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 6-cube t0 B4.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 polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

6-cube t5.svg
β6
6-cube t4.svg
t1β6
6-cube t3.svg
t2β6
6-cube t2.svg
t2γ6
6-cube t1.svg
t1γ6
6-cube t0.svg
γ6
6-cube t45.svg
t0,1β6
6-cube t35.svg
t0,2β6
6-cube t34.svg
t1,2β6
6-cube t25.svg
t0,3β6
6-cube t24.svg
t1,3β6
6-cube t23.svg
t2,3γ6
6-cube t15.svg
t0,4β6
6-cube t14.svg
t1,4γ6
6-cube t13.svg
t1,3γ6
6-cube t12.svg
t1,2γ6
6-cube t05.svg
t0,5γ6
6-cube t04.svg
t0,4γ6
6-cube t03.svg
t0,3γ6
6-cube t02.svg
t0,2γ6
6-cube t01.svg
t0,1γ6
6-cube t345.svg
t0,1,2β6
6-cube t245.svg
t0,1,3β6
6-cube t235.svg
t0,2,3β6
6-cube t234.svg
t1,2,3β6
6-cube t145.svg
t0,1,4β6
6-cube t135.svg
t0,2,4β6
6-cube t134.svg
t1,2,4β6
6-cube t125.svg
t0,3,4β6
6-cube t124.svg
t1,2,4γ6
6-cube t123.svg
t1,2,3γ6
6-cube t045.svg
t0,1,5β6
6-cube t035.svg
t0,2,5β6
6-cube t034.svg
t0,3,4γ6
6-cube t025.svg
t0,2,5γ6
6-cube t024.svg
t0,2,4γ6
6-cube t023.svg
t0,2,3γ6
6-cube t015.svg
t0,1,5γ6
6-cube t014.svg
t0,1,4γ6
6-cube t013.svg
t0,1,3γ6
6-cube t012.svg
t0,1,2γ6
6-cube t2345.svg
t0,1,2,3β6
6-cube t1345.svg
t0,1,2,4β6
6-cube t1245.svg
t0,1,3,4β6
6-cube t1235.svg
t0,2,3,4β6
6-cube t1234.svg
t1,2,3,4γ6
6-cube t0345.svg
t0,1,2,5β6
6-cube t0245.svg
t0,1,3,5β6
6-cube t0235.svg
t0,2,3,5γ6
6-cube t0234.svg
t0,2,3,4γ6
6-cube t0145.svg
t0,1,4,5γ6
6-cube t0135.svg
t0,1,3,5γ6
6-cube t0134.svg
t0,1,3,4γ6
6-cube t0125.svg
t0,1,2,5γ6
6-cube t0124.svg
t0,1,2,4γ6
6-cube t0123.svg
t0,1,2,3γ6
6-cube t12345.svg
t0,1,2,3,4β6
6-cube t02345.svg
t0,1,2,3,5β6
6-cube t01345.svg
t0,1,2,4,5β6
6-cube t01245.svg
t0,1,2,4,5γ6
6-cube t01235.svg
t0,1,2,3,5γ6
6-cube t01234.svg
t0,1,2,3,4γ6
6-cube t012345.svg
t0,1,2,3,4,5γ6

References[edit]

External links[edit]