Rectified 6-orthoplexes#Rectified 6-orthoplex

The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.

:{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}}

• rectified hexacross
• rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/rag.htm (o3x3o3o3o4o - rag)]}}

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 $\backslash sqrt\{2\}\backslash$ are all permutations of:

: (±1,±1,0,0,0,0)

{{6-cube Coxeter plane graphs|t4|150}}

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 {{CDD|nodes|3ab|nodes_10lr|split5c|nodes}} to {{CDD|node|3|node_1|5|node}}, creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:[https://blogs.ams.org/visualinsight/2015/01/01/icosidodecahedron-from-projected-d6-root-polytope Icosidodecahedron from D6] John Baez, January 1, 2015

The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.

• birectified hexacross
• birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/brag.htm (o3o3x3o3o4o - brag)]}}

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

: (±1,±1,±1,0,0,0)

{{6-cube Coxeter plane graphs|t3|150}}

It can also be projected into 3D-dimensions as {{CDD|nodes|3ab|nodes|split5c|nodes_01l}} → {{CDD|node|3|node|5|node_1}}, a dodecahedron envelope.

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

{{Hexeract family}}

{{reflist}}

• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• {{KlitzingPolytopes|polypeta.htm|6D uniform polytopes (polypeta) with acronyms}} o3x3o3o3o4o - rag, o3o3x3o3o4o - brag {{sfn whitelist| CITEREFKlitzing}}

• [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
• [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{polytopes}}

Category:6-polytopes