Rectified 6-orthoplexes

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

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

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} $\left\{{\begin{array}{l}3,3,3,4\\3\end{array}}\right\}$ r{3,3,3,3^1,1}
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, [3^3,1,1]
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 ${\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} $\left\{{\begin{array}{l}3,3,4\\3,3\end{array}}\right\}$ t₂{3,3,3,3^1,1}
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, [3^3,1,1]
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 ${\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.

Notes

↑ 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

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[1] Icosidodecahedron from D6 John Baez, January 1, 2015

[1]

B6 polytopes
β₆	t₁β₆	t₂β₆	t₂γ₆	t₁γ₆	γ₆	t_0,1β₆	t_0,2β₆
t_1,2β₆	t_0,3β₆	t_1,3β₆	t_2,3γ₆	t_0,4β₆	t_1,4γ₆	t_1,3γ₆	t_1,2γ₆
t_0,5γ₆	t_0,4γ₆	t_0,3γ₆	t_0,2γ₆	t_0,1γ₆	t_0,1,2β₆	t_0,1,3β₆	t_0,2,3β₆
t_1,2,3β₆	t_0,1,4β₆	t_0,2,4β₆	t_1,2,4β₆	t_0,3,4β₆	t_1,2,4γ₆	t_1,2,3γ₆	t_0,1,5β₆
t_0,2,5β₆	t_0,3,4γ₆	t_0,2,5γ₆	t_0,2,4γ₆	t_0,2,3γ₆	t_0,1,5γ₆	t_0,1,4γ₆	t_0,1,3γ₆
t_0,1,2γ₆	t_0,1,2,3β₆	t_0,1,2,4β₆	t_0,1,3,4β₆	t_0,2,3,4β₆	t_1,2,3,4γ₆	t_0,1,2,5β₆	t_0,1,3,5β₆
t_0,2,3,5γ₆	t_0,2,3,4γ₆	t_0,1,4,5γ₆	t_0,1,3,5γ₆	t_0,1,3,4γ₆	t_0,1,2,5γ₆	t_0,1,2,4γ₆	t_0,1,2,3γ₆
t_0,1,2,3,4β₆	t_0,1,2,3,5β₆	t_0,1,2,4,5β₆	t_0,1,2,4,5γ₆	t_0,1,2,3,5γ₆	t_0,1,2,3,4γ₆	t_0,1,2,3,4,5γ₆