5-orthoplex


In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-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 211.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.

Alternate names

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

Cartesian coordinates

for the vertices of a 5-orthoplex, centered at the origin are

Construction

There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
NameCoxeter diagramSchläfli symbolSymmetryOrderVertex figure
regular 5-orthoplex3840
Quasiregular 5-orthoplex1920
5-fusil-----
5-fusil3840
5-fusil+768
5-fusil+384
5-fusil+2192
5-fusil2+128
5-fusil+364
5-fusil532

Other images


The perspective projection of a stereographic projection of the Schlegel diagram of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.

Related polytopes and honeycombs

This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.