Rectified 10 orthoplexes

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 [3^7,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length 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 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 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 2 are all permutations of:

(±1,±1,±1,±1,±1,0,0,0,0,0)