7-cube

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 {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.

Related polytopes
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.

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.

$$\begin{bmatrix}\begin{matrix} 128 & 7 & 21 & 35 & 35 & 21 & 7 \\ 2 & 448 & 6 & 15 & 20 & 15 & 6 \\ 4 & 4 & 672 & 5 & 10 & 10 & 5 \\ 8 & 12 & 6 & 560 & 4 & 6 & 4 \\ 16 & 32 & 24 & 8 & 280 & 3 & 3 \\ 32 & 80 & 80 & 40 & 10 & 84 & 2 \\ 64 & 192 & 240 & 160 & 60 & 12 & 14 \end{matrix}\end{bmatrix}$$

Cartesian coordinates
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.