7-orthoplex

In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448  5-faces, and 128 6-faces.

It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.

Alternate names

 * Heptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek.
 * Hecatonicosoctaexon as a 128-facetted 7-polytope (polyexon).

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

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

Construction
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.

Cartesian coordinates
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
 * (±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (0,0,0,0,0,±1,0), (0,0,0,0,0,0,±1)

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