| bgcolor=#e7dcc3 colspan=2 | 7-cube Hepteract | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Orthogonal projection inside Petrie polygon The central orange vertex is doubled | |
| Type | Regular 7-polytope | |
| Family | hypercube | |
| Schläfli symbol | ||
| Coxeter-Dynkin diagrams | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 448 | |
| Vertices | 128 | |
| Vertex figure | ||
| Petrie polygon | tetradecagon | |
| Coxeter group | C7, [3<sup>5</sup>,4] | |
| Dual | 7-orthoplex | |
| Properties | convex, Hanner polytope |
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.
It can be named by its Schläfli symbol, 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.
The 7-cube is 7th in a series of hypercube:
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.
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]
\begin{bmatrix}\begin{matrix} 128&7&21&35&35&21&7\ 2&448&6&15&20&15&6\ 4&4&672&5&10&10&5\ 8&12&6&560&4&6&4\ 16&32&24&8&280&3&3\ 32&80&80&40&10&84&2\ 64&192&240&160&60&12&14\end{matrix}\end{bmatrix}
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.