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.

## Contents

It is represented by Schläfli symbol {4,3^{6}}, 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 (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) with -1 < x_{i} < 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.