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. It is composed of various 5-cubes, at perpendicular angles on the u-axis, forming coordinates (x,y,z,w,v,u). [1] [2]

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. [3] [4]

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.
Hexeract-q1q4-q2q5-q3q6.gif
A 3D perspective projection of a hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes.

The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.

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 9-cube.svg 10-cube.svg
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube 9-cube 10-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

References

  1. Mehdi, Sadiq A.; Ali, Zaydon L. (2019). "A New Six-Dimensional Hyper-Chaotic System". 2019 International Engineering Conference (IEC). pp. 211–215. doi:10.1109/IEC47844.2019.8950634. ISBN   978-1-7281-4377-4.
  2. McCallum, Scott (February 1988). "An improved projection operation for cylindrical algebraic decomposition of three-dimensional space". Journal of Symbolic Computation. 5 (1–2): 141–161. doi:10.1016/S0747-7171(88)80010-5.
  3. Coxeter, Regular Polytopes, sec 1.8 Configurations
  4. 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 compoundsPolytope operations