Rectified 6-orthoplexes

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

6-orthoplex	Rectified 6-orthoplex	Birectified 6-orthoplex
Birectified 6-cube	Rectified 6-cube	6-cube
Orthogonal projections in B6 Coxeter plane

There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.

Rectified 6-orthoplex

Rectified hexacross
Type	uniform 6-polytope
Schläfli symbols	t1{34,4} or r{34,4} ${\displaystyle \left\{{\begin{array}{l}3,3,3,4\\3\end{array}}\right\}}$ r{3,3,3,31,1}
Coxeter-Dynkin diagrams	= =
5-faces	76 total: 64 rectified 5-simplex 12 5-orthoplex
4-faces	576 total: 192 rectified 5-cell 384 5-cell
Cells	1200 total: 240 octahedron 960 tetrahedron
Faces	1120 total: 160 and 960 triangles
Edges	480
Vertices	60
Vertex figure	16-cell prism
Petrie polygon	Dodecagon
Coxeter groups	B6, [3,3,3,3,4] D6, [33,1,1]
Properties	convex

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

or

Alternate names

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

Construction

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

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)

Images

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]

Root vectors

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 6-orthoplex

Birectified 6-orthoplex
Type	uniform 6-polytope
Schläfli symbols	t2{34,4} or 2r{34,4} ${\displaystyle \left\{{\begin{array}{l}3,3,4\\3,3\end{array}}\right\}}$ t2{3,3,3,31,1}
Coxeter-Dynkin diagrams	= =
5-faces	76
4-faces	636
Cells	2160
Faces	2880
Edges	1440
Vertices	160
Vertex figure	{3}×{3,4} duoprism
Petrie polygon	Dodecagon
Coxeter groups	B6, [3,3,3,3,4] D6, [33,1,1]
Properties	convex

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

Alternate names

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

Cartesian coordinates

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)

Images

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.

Related polytopes

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.

B6 polytopes
β6	t1β6	t2β6	t2γ6	t1γ6	γ6	t0,1β6	t0,2β6
t1,2β6	t0,3β6	t1,3β6	t2,3γ6	t0,4β6	t1,4γ6	t1,3γ6	t1,2γ6
t0,5γ6	t0,4γ6	t0,3γ6	t0,2γ6	t0,1γ6	t0,1,2β6	t0,1,3β6	t0,2,3β6
t1,2,3β6	t0,1,4β6	t0,2,4β6	t1,2,4β6	t0,3,4β6	t1,2,4γ6	t1,2,3γ6	t0,1,5β6
t0,2,5β6	t0,3,4γ6	t0,2,5γ6	t0,2,4γ6	t0,2,3γ6	t0,1,5γ6	t0,1,4γ6	t0,1,3γ6
t0,1,2γ6	t0,1,2,3β6	t0,1,2,4β6	t0,1,3,4β6	t0,2,3,4β6	t1,2,3,4γ6	t0,1,2,5β6	t0,1,3,5β6
t0,2,3,5γ6	t0,2,3,4γ6	t0,1,4,5γ6	t0,1,3,5γ6	t0,1,3,4γ6	t0,1,2,5γ6	t0,1,2,4γ6	t0,1,2,3γ6
t0,1,2,3,4β6	t0,1,2,3,5β6	t0,1,2,4,5β6	t0,1,2,4,5γ6	t0,1,2,3,5γ6	t0,1,2,3,4γ6	t0,1,2,3,4,5γ6

Notes

1. Icosidodecahedron from D6 John Baez, January 1, 2015

References

• 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, 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.
• Klitzing, Richard. "6D uniform polytopes (polypeta)". o3x3o3o3o4o - rag, o3o3x3o3o4o - brag

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds