6-cube
6-dimensional hypercube
title: "6-cube" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["6-polytopes", "articles-containing-video-clips"] description: "6-dimensional hypercube" topic_path: "general/6-polytopes" source: "https://en.wikipedia.org/wiki/6-cube" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary 6-dimensional hypercube ::
::data[format=table]
| 6-cubeHexeract |
|---|
| [[File:6-cube graph.svg |
| Type |
| Family |
| Schläfli symbol |
| Coxeter diagram |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Petrie polygon |
| Coxeter group |
| Dual |
| Properties |
| :: |
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).
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.
\begin{bmatrix}\begin{matrix}64 & 6 & 15 & 20 & 15 & 6 \ 2 & 192 & 5 & 10 & 10 & 5 \ 4 & 4 & 240 & 4 & 6 & 4 \ 8 & 12 & 6 & 160 & 3 & 3 \ 16 & 32 & 24 & 8 & 60 & 2 \ 32 & 80 & 80 & 40 & 10 & 12 \end{matrix}\end{bmatrix}
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 i
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.
::data[format=table]
| Name | Coxeter | Schläfli | Symmetry | Order | Regular 6-cube | Quasiregular 6-cube | hyperrectangle |
|---|---|---|---|---|---|---|---|
| {4,3,3,3,3} | [4,3,3,3,3] | 46080 | |||||
| [3,3,3,31,1] | 23040 | ||||||
| {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
::data[format=table title="[[orthographic projection]]s"]
| Coxeter plane | B6 | B5 | B4 | Graph | Dihedral symmetry | Coxeter plane | Other | B3 | B2 | Graph | Dihedral symmetry | Coxeter plane | A5 | A3 | Graph | Dihedral symmetry |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| [[File:6-cube t0.svg | 150px]] | [[File:6-cube t0 B5.svg | 150px]] | [[File:4-cube t0.svg | 150px]] | |||||||||||
| [12] | [10] | [8] | ||||||||||||||
| [[Image:6-cube column graph.svg | 150px]] | [[File:6-cube t0 B3.svg | 150px]] | [[File:6-cube t0 B2.svg | 150px]] | |||||||||||
| [2] | [6] | [4] | ||||||||||||||
| [[File:6-cube t0 A5.svg | 150px]] | [[File:6-cube t0 A3.svg | 150px]] | |||||||||||||
| [6] | [4] | |||||||||||||||
| :: |
::data[format=table] | [[File:Hexeract-q1q4-q2q5-q3q6.gif|280px]] 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:
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
References
- H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5),
References
- (2019). "2019 International Engineering Conference (IEC)".
- (February 1988). "An improved projection operation for cylindrical algebraic decomposition of three-dimensional space". Journal of Symbolic Computation.
- Coxeter, Regular Polytopes, sec 1.8 Configurations
- Coxeter, Complex Regular Polytopes, p.117
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