Cantellated 6-orthoplexes

In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.

There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 5-cube

Alternate names

 * Cantellated hexacross
 * Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates
Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
 * (2,1,1,0,0,0)

Alternate names

 * Bicantellated hexacross, bicantellated hexacontatetrapeton
 * Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates
Cartesian coordinates for the 1440 vertices of a bicantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
 * (2,2,1,1,0,0)

Alternate names

 * Cantitruncated hexacross, cantitruncated hexacontatetrapeton
 * Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates
Cartesian coordinates for the 960 vertices of a cantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
 * (3,2,1,0,0,0)

Alternate names

 * Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton
 * Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.

Coordinates
Cartesian coordinates for the 2880 vertices of a bicantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
 * (3,3,2,1,0,0)

Related polytopes
These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.