7-cube
7-dimensional hypercube
From Wikipedia, the free encyclopedia
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.
| 7-cube Hepteract | |
|---|---|
 Orthogonal projection inside Petrie polygon The central orange vertex is doubled | |
| Type | Regular 7-polytope |
| Family | hypercube |
| Schläfli symbol | {4,35} |
| Coxeter-Dynkin diagrams |    
|
| 6-faces | 14 {4,34}  |
| 5-faces | 84 {4,33}  |
| 4-faces | 280 {4,3,3}  |
| Cells | 560 {4,3}  |
| Faces | 672 {4}  |
| Edges | 448 |
| Vertices | 128 |
| Vertex figure | 6-simplex  |
| Petrie polygon | tetradecagon |
| Coxeter group | C7, [35,4] |
| Dual | 7-orthoplex |
| Properties | convex, Hanner polytope |
It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.
As a configuration
This configuration matrix represents the 7-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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 hepteract centered at the origin and edge length 2 are
- (±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.
Projections
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph | |
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| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
Related polytopes
The 7-cube is 7th in a series of hypercube:
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| Line segment | Square | Cube | 4-cube | 5-cube | 6-cube | 7-cube | 8-cube | 9-cube | 10-cube |
The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes.
Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces.