Trirectified_8-orthoplex

Rectified 8-orthoplexes

Add article description

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

More information Orthogonal projections in A8 Coxeter plane ...

There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.

Rectified 8-orthoplex

Rectified 8-orthoplex
Type	uniform 8-polytope
Schläfli symbol	t₁{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces	272
6-faces	3072
5-faces	8960
4-faces	12544
Cells	10080
Faces	4928
Edges	1344
Vertices	112
Vertex figure	6-orthoplex prism
Petrie polygon	hexakaidecagon
Coxeter groups	C₈, [4,3⁶] D₈, [3^5,1,1]
Properties	convex

The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D₈. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B₈ and C₈ simple Lie groups.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:

(±1,±1,0,0,0,0,0,0)

Birectified 8-orthoplex

More information Birectified 8-orthoplex ...

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:

(±1,±1,±1,0,0,0,0,0)