8-cube

In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.

It is represented by Schläfli symbol {4,36}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the 4-cube) and oct for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes.

Cartesian coordinates
Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are
 * (±1,±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7) with -1 < xi < 1.

As a configuration
This configuration matrix represents the 8-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces, and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

$$\begin{bmatrix}\begin{matrix} 256 & 8 & 28 & 56 & 70 & 56 & 28 & 8 \\ 2 & 1024 & 7 & 21 & 35 & 35 & 21 & 7 \\ 4 & 4 & 1792 & 6 & 15 & 20 & 15 & 6 \\ 8 & 12 & 6 &  1792 & 5 & 10 & 10 & 5 \\ 16 & 32 & 24 & 8 & 1120 & 4 & 6 & 4 \\ 32 & 80 & 80 & 40 & 10 & 448 & 3 & 3 \\ 64 & 192 & 240 & 160 & 60 & 12 &  112 & 2 \\ 128 & 448 & 672 & 560 & 280 & 84 & 14 & 16 \end{matrix}\end{bmatrix}$$

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.

Derived polytopes
Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.

Related polytopes
The 8-cube is 8th in an infinite series of hypercube: