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
TypeRegular 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-faces12 {4,3,3,3} 5-cube graph.svg
4-faces60 {4,3,3} 4-cube graph.svg
Cells160 {4,3} 3-cube graph.svg
Faces240 {4} 2-cube.svg
Edges192
Vertices64
Vertex figure 5-simplex
Petrie polygon dodecagon
Coxeter group B6, [34,4]
Dual 6-orthoplex 6-orthoplex.svg
Properties convex, Hanner polytope

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.

Contents

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.

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 configuration

This configuration matrix represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element. [1] [2]

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.

Construction

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-cubeCDel 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 f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{4,3,3,3,3}[4,3,3,3,3]46080
Quasiregular 6-cubeCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png[3,3,3,31,1]23040
hyperrectangle CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png{4,3,3,3}×{}[4,3,3,3,2]7680
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png{4,3,3}×{4}[4,3,3,2,4]3072
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png{4,3}2[4,3,2,4,3]2304
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png{4,3,3}×{}2[4,3,3,2,2]1536
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png{4,3}×{4}×{}[4,3,2,4,2]768
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.png{4}3[4,2,4,2,4]512
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png{4,3}×{}3[4,3,2,2,2]384
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png{4}2×{}2[4,2,4,2,2]256
CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png{4}×{}4[4,2,2,2,2]128
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png{}6[2,2,2,2,2]64

Projections

orthographic projections
Coxeter plane B6B5B4
Graph 6-cube t0.svg 6-cube t0 B5.svg 4-cube t0.svg
Dihedral symmetry [12][10][8]
Coxeter planeOtherB3B2
Graph 6-cube column graph.svg 6-cube t0 B3.svg 6-cube t0 B2.svg
Dihedral symmetry[2][6][4]
Coxeter planeA5A3
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.png
6-cube quasicrystal structure orthographically projected
to 3D using the golden ratio.

The 6-cube is 6th in a series of hypercube:

Petrie polygon orthographic projections
1-simplex t0.svg 2-cube.svg 3-cube graph.svg 4-cube graph.svg 5-cube graph.svg 6-cube graph.svg 7-cube graph.svg 8-cube.svg
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube

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

B6 polytopes
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

Related Research Articles

<span class="mw-page-title-main">Tesseract</span> Four-dimensional analogue of the cube

In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

<span class="mw-page-title-main">5-demicube</span>

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

<span class="mw-page-title-main">5-orthoplex</span>

In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.

<span class="mw-page-title-main">6-orthoplex</span>

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

<span class="mw-page-title-main">6-demicube</span>

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

<span class="mw-page-title-main">7-cube</span> 7-dimensional hypercube

In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.

<span class="mw-page-title-main">7-demicube</span>

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

<span class="mw-page-title-main">8-cube</span> 8-dimensional hypercube

In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

<span class="mw-page-title-main">9-cube</span> 9-dimensional hypercube

In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.

<span class="mw-page-title-main">7-orthoplex</span>

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.

<span class="mw-page-title-main">8-orthoplex</span>

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

<span class="mw-page-title-main">10-cube</span> 10-dimensional hypercube

In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

<span class="mw-page-title-main">Rectified 5-orthoplexes</span>

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

<span class="mw-page-title-main">Rectified 6-cubes</span>

In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.

<span class="mw-page-title-main">Rectified 5-cubes</span>

In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

<span class="mw-page-title-main">Truncated 5-orthoplexes</span>

In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

<span class="mw-page-title-main">Runcinated 5-orthoplexes</span>

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

<span class="mw-page-title-main">Pentic 6-cubes</span>

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

<span class="mw-page-title-main">Runcic 7-cubes</span>

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

References

  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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