Stericated 6-orthoplexes

In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex.

Orthogonal projections in B₆ Coxeter plane
6-orthoplex	Stericated 6-orthoplex	Steritruncated 6-orthoplex
Stericantellated 6-orthoplex	Stericantitruncated 6-orthoplex	Steriruncinated 6-orthoplex
Steriruncitruncated 6-orthoplex	Steriruncicantellated 6-orthoplex	Steriruncicantitruncated 6-orthoplex

There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the stericated 6-cube.

Stericated 6-orthoplex

Stericated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	2r2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	5760
Vertices	960
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Small cellated hexacontatetrapeton (Acronym: scag) (Jonathan Bowers)[1]

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]

Steritruncated 6-orthoplex

Steritruncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	t_0,1,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	19200
Vertices	3840
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Cellitruncated hexacontatetrapeton (Acronym: catog) (Jonathan Bowers)[2]

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]

Stericantellated 6-orthoplex

Stericantellated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbols	t_0,2,4{3⁴,4} rr2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	28800
Vertices	5760
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Cellirhombated hexacontatetrapeton (Acronym: crag) (Jonathan Bowers)[3]

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]

Stericantitruncated 6-orthoplex

Stericantitruncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	t_0,1,2,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	46080
Vertices	11520
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (Jonathan Bowers)[4]

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]

Steriruncinated 6-orthoplex

Steriruncinated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	t_0,3,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	15360
Vertices	3840
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Celliprismated hexacontatetrapeton (Acronym: copog) (Jonathan Bowers)[5]

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]

Steriruncitruncated 6-orthoplex

Steriruncitruncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	2t2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	11520
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Celliprismatotruncated hexacontatetrapeton (Acronym: captog) (Jonathan Bowers)[6]

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]

Steriruncicantellated 6-orthoplex

Steriruncicantellated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbol	t_0,2,3,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	40320
Vertices	11520
Vertex figure
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Celliprismatorhombated hexacontatetrapeton (Acronym: coprag) (Jonathan Bowers)[7]

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]

Steriruncicantitruncated 6-orthoplex

Steriuncicantitruncated 6-orthoplex
Type	uniform 6-polytope
Schläfli symbols	t_0,1,2,3,4{3⁴,4} tr2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces	536: 12 t_0,1,2,3{3,3,3,4} 60 {}×t_0,1,2{3,3,4} × 160 {6}×t_0,1,2{3,3} × 240 {4}×t_0,1,2{3,3} × 64 t_0,1,2,3,4{3⁴}
4-faces	8216
Cells	38400
Faces	76800
Edges	69120
Vertices	23040
Vertex figure	irregular 5-simplex
Coxeter groups	B₆, [4,3,3,3,3]
Properties	convex

Alternate names

Great cellated hexacontatetrapeton (Acronym: gocog) (Jonathan Bowers)[8]

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]

Snub 6-demicube

The snub 6-demicube defined as an alternation of the omnitruncated 6-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [3^2,1,1,1]⁺ or [4,(3,3,3,3)⁺], and constructed from 12 snub 5-demicubes, 64 snub 5-simplexes, 60 snub 24-cell antiprisms, 160 3-s{3,4} duoantiprisms, 240 2-sr{3,3} duoantiprisms, and 11520 irregular 5-simplexes filling the gaps at the deleted vertices.

Related polytopes

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

Notes

Klitzing, (x3o3o3o3x4o - scag)
Klitzing, (x3x3o3o3x4o - catog)
Klitzing, (x3o3x3o3x4o - crag)
Klitzing, (x3x3x3o3x4o - cagorg)
Klitzing, (x3o3o3x3x4o - copog)
Klitzing, (x3x3o3x3x4o - captog)
Klitzing, (x3o3x3x3x4o - coprag)
Klitzing, (x3x3x3x3x4o - gocog)

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)".

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] Klitzing, (x3o3o3o3x4o - scag)

[2] Klitzing, (x3x3o3o3x4o - catog)

[3] Klitzing, (x3o3x3o3x4o - crag)

[4] Klitzing, (x3x3x3o3x4o - cagorg)

[5] Klitzing, (x3o3o3x3x4o - copog)

[6] Klitzing, (x3x3o3x3x4o - captog)

[7] Klitzing, (x3o3x3x3x4o - coprag)

[8] Klitzing, (x3x3x3x3x4o - gocog)

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