Rectified 10-orthoplexes

In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex.

There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex.

These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry.

Rectified 10-orthoplex
In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex.

Rectified 10-orthoplex
The rectified 10-orthoplex is the vertex figure for the demidekeractic honeycomb.
 * or

Alternate names

 * rectified decacross (Acronym rake) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or [4,38] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or [37,1,1] Coxeter group.

Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,0,0,0,0,0,0,0,0)

Root vectors
Its 180 vertices represent the root vectors of the simple Lie group D10. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B10.

Alternate names

 * Birectified decacross

Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,±1,0,0,0,0,0,0,0)

Alternate names

 * Trirectified decacross (Acronym trake) (Jonathan Bowers)

Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,±1,±1,0,0,0,0,0,0)

Alternate names

 * Quadrirectified decacross (Acronym brake) (Jonthan Bowers)

Cartesian coordinates
Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,±1,±1,±1,0,0,0,0,0)