Rectified 7-orthoplexes

• rectified heptacross
• rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon[1]

There are two Coxeter groups associated with the rectified heptacross, one with the C₇ or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D₇ or [3^4,1,1] Coxeter group.

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

(±1,±1,0,0,0,0,0)

Its 84 vertices represent the root vectors of the simple Lie group D₇. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B₇ and C₇ simple Lie groups.

• Birectified heptacross
• Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexon[2]

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}\ }$ are all permutations of:

(±1,±1,±1,0,0,0,0)