# 8-orthoplex

In geometry, an **8-orthoplex** or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells *4-faces*, 1792 *5-faces*, 1024 *6-faces*, and 256 *7-faces*.

It is a part of an infinite family of polytopes, called cross-polytopes or *orthoplexes*. The dual polytope is an 8-hypercube, or octeract.

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors. ^{[3]}

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C_{8} or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D_{8} or [3^{5,1,1}] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an **8-fusil**.

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are

It is used in its alternated form **5 _{11}** with the 8-simplex to form the

**5**honeycomb.

_{21}