Rectified 5-orthoplexes


In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.

Rectified 5-orthoplex

Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups.
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5-dimensional cross polytope.

Alternate names

  • rectified pentacross
  • rectified triacontiditeron

    Construction

There are two Coxeter groups associated with the rectified pentacross, one with the C5 or Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or Coxeter group.

Cartesian coordinates

for the vertices of a rectified pentacross, centered at the origin, edge length are all permutations of:

Images

Related polytopes

The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb:
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.