In geometry, a **10-orthoplex** or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells *4-faces*, 13440 *5-faces*, 15360 *6-faces*, 11520 *7-faces*, 5120 *8-faces*, and 1024 *9-faces*.

## Contents

It has two constructed forms, the first being regular with Schläfli symbol {3^{8},4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {3^{7},3^{1,1}} or Coxeter symbol **7 _{11}**.

It is one of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is the 10-hypercube or 10-cube.

## Alternate names

**Decacross**is derived from combining the family name

*cross polytope*with

*deca*for ten (dimensions) in Greek

**Chilliaicositetraxennon**as a 1024-facetted 10-polytope (polyxennon).

## Construction

There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C_{10} or [4,3^{8}] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D_{10} or [3^{7,1,1}] symmetry group.

## Cartesian coordinates

Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are

(±1,0,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0,0), (0,0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,0,±1)Every vertex pair is connected by an edge, except opposites.