| 8-cube Octeract |
|
|---|---|
Orthogonal projection inside Petrie polygon |
|
| Type | Regular 8-polytope |
| Family | hypercube |
| Schläfli symbol | {4,36} |
| Coxeter-Dynkin diagram | |
| 7-faces | 16 {4,35} |
| 6-faces | 112 {4,34} |
| 5-faces | 448 {4,33} |
| 4-faces | 1120 {4,32} |
| Cells | 1792 {4,3} |
| Faces | 1792 {4} |
| Edges | 1024 |
| Vertices | 256 |
| Vertex figure | 7-simplex |
| Petrie polygon | hexadecagon |
| Coxeter group | C8, [36,4] |
| Dual | 8-orthoplex |
| Properties | convex |
In geometry, an 8-cube is an eight-dimensional hypercube (8-cube). 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, derived from combining the name tesseract (the 4-cube) with oct for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being a 8 dimensional polytope constructed from 16 regular facets.
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It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called a 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
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7) with -1 < xi < 1.
| B8 | B7 | ||||
|---|---|---|---|---|---|
| [16] | [14] | ||||
| B6 | B5 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
| [8] | [6] | [4] | |||
This 8-cube graph is an orthogonal projection. This oriention shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:8:28:56:70:56:28:8:1. |
Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.