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-cell 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.
It has two constructive forms, the first being regular with Schläfli symbol {3<sup>6</sup>,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3<sup>1,1</sup>} or Coxeter symbol 5<sub>11</sub>.
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.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.
There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C<sub>8</sub> 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<sub>8</sub> or [3<sup>5,1,1</sup>] 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
Every vertex pair is connected by an edge, except opposites.
It is used in its alternated form 5<sub>11</sub> with the 8-simplex to form the 5<sub>21</sub> honeycomb.