In geometry, a **10-cube** is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces.

## Contents

It can be named by its Schläfli symbol {4,3^{8}}, being composed of 3 9-cubes around each 8-face. It is sometimes called a **dekeract**, a portmanteau of tesseract (the *4-cube*) and *deka-* for ten (dimensions) in Greek, It can also be called an **icosaxennon** or **icosa-10-tope** as a 10 dimensional polytope, constructed from 20 regular facets.

It is a part of an infinite family of polytopes, called hypercubes. The dual of a dekeract can be called a 10-orthoplex or decacross, and is a part of the infinite family of cross-polytopes.

## Cartesian coordinates

Cartesian coordinates for the vertices of a dekeract centered at the origin and edge length 2 are

(±1,±1,±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}, *x*_{8}, *x*_{9}) with −1 < *x _{i}* < 1.

## Derived polytopes

Applying an *alternation* operation, deleting alternating vertices of the *dekeract*, creates another uniform polytope, called a 10-demicube*, (part of an infinite family called demihypercubes), which has 20 demienneractic and 512 enneazettonic facets.*