Truncated 7-orthoplexes

In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.

There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.

Alternate names

 * Truncated heptacross
 * Truncated hecatonicosoctaexon (Jonathan Bowers)

Coordinates
Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of
 * (±2,±1,0,0,0,0,0)

Construction
There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.

Alternate names

 * Bitruncated heptacross
 * Bitruncated hecatonicosoctaexon (Jonathan Bowers)

Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
 * (±2,±2,±1,0,0,0,0)

Tritruncated 7-orthoplex
The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.

Alternate names

 * Tritruncated heptacross
 * Tritruncated hecatonicosoctaexon (Jonathan Bowers)

Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
 * (±2,±2,±2,±1,0,0,0)