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 has two constructive forms, the first being regular with Schläfli symbol {36,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,31,1} or Coxeter symbol 511.

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

Alternate names

 * Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
 * Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton)

As a configuration
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.

$$\begin{bmatrix}\begin{matrix} 16 & 14 & 84 & 280 & 560 & 672 & 448 & 128 \\ 2 & 112 & 12 & 60 & 160 & 240 & 192 & 64 \\ 3 & 3 & 448 & 10 & 40 & 80 & 80 & 32 \\ 4 & 6 & 4 & 1120 & 8 & 24 & 32 & 16 \\ 5 & 10 & 10 & 5 & 1792 & 6 & 12 & 8 \\ 6 & 15 & 20 & 15 & 6 & 1792 & 4 & 4 \\ 7 & 21 & 35 & 35 & 21 & 7 & 1024 & 2 \\ 8 & 28 & 56 & 70 & 56 & 28 & 8 & 256 \end{matrix}\end{bmatrix}$$

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

Construction
There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C8 or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or [35,1,1] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.

Cartesian coordinates
Cartesian coordinates for the vertices of an 8-cube, centered at the origin are
 * (±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0),
 * (0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

Images
It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb.