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 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, 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.
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
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-cube | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3,3,3,3} | [4,3,3,3,3] | 46080 |
Quasiregular 6-cube | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | [3,3,3,31,1] | 23040 | |
hyperrectangle | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3,3,3}×{} | [4,3,3,3,2] | 7680 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3,3}×{4} | [4,3,3,2,4] | 3072 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3}2 | [4,3,2,4,3] | 2304 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3,3}×{}2 | [4,3,3,2,2] | 1536 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3}×{4}×{} | [4,3,2,4,2] | 768 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4}3 | [4,2,4,2,4] | 512 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4,3}×{}3 | [4,3,2,2,2] | 384 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4}2×{}2 | [4,2,4,2,2] | 256 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {4}×{}4 | [4,2,2,2,2] | 128 | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | {}6 | [2,2,2,2,2] | 64 |
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. | ![]() 6-cube quasicrystal structure orthographically projected to 3D using the golden ratio. |
![]() A 3D perspective projection of a hexeract undergoing a triple rotation about the X-W1, Y-W2 and Z-W3 orthogonal planes. |
Related polytopes
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:
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
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 | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
![]() 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)
- Klitzing, Richard. "6D uniform polytopes (polypeta) o3o3o3o3o4x - ax".
External links
- Weisstein, Eric W. "Hypercube". MathWorld.
- Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Multi-dimensional Glossary: hypercube Garrett Jones
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |