10-orthoplex

In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 tetrahedron cells, 8064 5-cell 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-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 7₁₁.
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.

Alternate names

Decacross is derived from combining the family name cross polytope with deca for ten in Greek. Acronym: kaChilliaicositetraronnon as a 1024-facetted 10-polytope.

Construction

There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C₁₀ or symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D₁₀ or symmetry group.

Cartesian coordinates

Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are
Every vertex pair is connected by an edge, except opposites.