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-cell 4-faces, 448 5-faces, and 128 6-faces.
It has two constructed forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol 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 in Greek. Hecatonicosaoctaexon as a 128-facetted 7-polytope. Acronym: zee

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.

Construction

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

Name	Coxeter diagram	Schläfli symbol	Symmetry	Order	Vertex figure
regular 7-orthoplex				645120
Quasiregular 7-orthoplex				322560
7-fusil		7		128

Cartesian coordinates

Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
Every vertex pair is connected by an edge, except opposites.