| bgcolor=#e7dcc3 colspan=2 | 8-cube Octeract | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Orthogonal projection inside Petrie polygon | |
| Type | Regular 8-polytope | |
| Family | hypercube | |
| Schläfli symbol | ||
| Coxeter-Dynkin diagrams | ||
| 7-faces | 16 | |
| 6-faces | 112 | |
| 5-faces | 448 | |
| 4-faces | 1120 | |
| Cells | 1792 | |
| Faces | 1792 | |
| Edges | 1024 | |
| Vertices | 256 | |
| Vertex figure | ||
| Petrie polygon | hexadecagon | |
| Coxeter group | C8, [3<sup>6</sup>,4] | |
| Dual | ||
| Properties | convex, Hanner polytope |
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, 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 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.
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.[1] [2]
\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.
| B8 | k-face | fk | f0 | f1 | f2 | f3 | f4 | f5 | f6 | f7 | k-figure | notes | |||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| A7 | f0 | 256 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | B8/A7 = 2^8*8!/8 | = 256 | ||||
| A6A1 | f1 | 2 | 1024 | 7 | 21 | 35 | 35 | 21 | 7 | B8/A6A1 = 2^8*8!/7 | /2 = 1024 | ||||
| A5B2 | f2 | 4 | 4 | 1792 | 6 | 15 | 20 | 15 | 6 | B8/A5B2 = 2^8*8!/6 | /4/2 = 1792 | ||||
| A4B3 | f3 | 8 | 12 | 6 | 1792 | 5 | 10 | 10 | 5 | B8/A4B3 = 2^8*8!/5 | /8/3! = 1792 | ||||
| A3B4 | f4 | 16 | 32 | 24 | 8 | 1120 | 4 | 6 | 4 | B8/A3B4 = 2^8*8!/4 | /2^4/4! = 1120 | ||||
| A2B5 | f5 | 32 | 80 | 80 | 40 | 10 | 448 | 3 | 3 | B8/A2B5 = 2^8*8!/3 | /2^5/5! = 448 | ||||
| A1B6 | f6 | 64 | 192 | 240 | 160 | 60 | 12 | 112 | 2 | B8/A1B6 = 2^8*8!/2/2^6/6 | = 112 | ||||
| B7 | f7 | 128 | 448 | 672 | 560 | 280 | 84 | 14 | 16 | B8/B7 = 2^8*8!/2^7/7 | = 16 |
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.
The 8-cube is 8th in an infinite series of hypercube: