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.

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
Orthogonal projections in B6 Coxeter plane

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	t0,1,4{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	t0,2,4{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	t0,1,2,4{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	t0,3,4{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	t0,2,3,4{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	t0,1,2,3,4{34,4} tr2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces	536: 12 t0,1,2,3{3,3,3,4} 60 {}×t0,1,2{3,3,4} × 160 {6}×t0,1,2{3,3} × 240 {4}×t0,1,2{3,3} × 64 t0,1,2,3,4{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.

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. 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)

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

• 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