| 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-Dynkin 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 |
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, derived from combining the name 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.
Contents |
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 for the vertices of a 6-cube centered at the origin and edge length 2 are
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with -1 < xi < 1.
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
This hypercube graph is an orthogonal projection. This oriention shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:6:15:20:15:6:1. |
Petrie polygon, Skew orthographic projection |
| 3D Projections | |
6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D. |
Hexeract Quasicrystal structure orthographically projected to 3D using the Golden Ratio. |
This polytope is one of 63 uniform polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.