Rectified 6-orthoplexes

• rectified hexacross
• rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

There are two Coxeter groups associated with the rectified hexacross, one with the C₆ or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D₆ or [3^3,1,1] Coxeter group.

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

(±1,±1,0,0,0,0)

The 60 vertices represent the root vectors of the simple Lie group D₆. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B₆ and C₆ simple Lie groups.

The 60 roots of D₆ can be geometrically folded into H₃ (Icosahedral symmetry), as to , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:[1]

Rectified 6-orthoplex		2 icosidodecahedra
3D (H3 projection)	A4/B5/D6 Coxeter plane	H2 Coxeter plane

• birectified hexacross
• birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

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

(±1,±1,±1,0,0,0)

orthographic projections

Coxeter plane	B6	B5	B4
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B3	B2
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

It can also be projected into 3D-dimensions as --> , a dodecahedron envelope.