| Octeract (8-cube) |
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|---|---|
 Orthogonal projection inside Petrie polygon |
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| 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 | Octacross |
| Properties | convex |
In geometry, an octeract is an eight-dimensional hypercube (8-cube). The name octeract is 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 made of 16 regular facets.
It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 penteract 5-faces, 112 hexeract 6-faces, and 16 hepteract 7-faces.
It is a part of an infinite family of polytopes, called hypercubes. The dual of an octeract can be called a octacross, and is a part of the infinite family of cross-polytopes.
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Cartesian coordinates
Cartesian coordinates for the vertices of a octeract 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.
Projections
 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. |
Derived polytopes
Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demiocteract, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.
See also
- Hypercubes family
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
External links
- Weisstein, Eric W., "Hypercube" from MathWorld.
- Olshevsky, George, Measure polytope at Glossary for Hyperspace.
- Multi-dimensional Glossary: hypercube Garrett Jones
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