Stericated 6-orthoplexes

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

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.

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)

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)

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)

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)

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)

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)

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)

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} 40px×40px 160 {6}×t0,1,2{3,3} 40px×40px 240 {4}×t0,1,2{3,3} 40px×40px 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)

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.

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 [1]
    • (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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm. 

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary