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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.
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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.
Related polytopes
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
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.
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.