Pentellated 6-cubes

6-cube	6-orthoplex	Pentellated 6-cube
Pentitruncated 6-cube	Penticantellated 6-cube	Penticantitruncated 6-cube
Pentiruncitruncated 6-cube	Pentiruncicantellated 6-cube	Pentiruncicantitruncated 6-cube
Pentisteritruncated 6-cube	Pentistericantitruncated 6-cube	Omnitruncated 6-cube
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.

There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex.

Pentellated 6-cube

Pentellated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,5{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	1920
Vertices	384
Vertex figure	5-cell antiprism
Coxeter group	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Pentellated 6-orthoplex
• Expanded 6-cube, expanded 6-orthoplex
• Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)

Pentitruncated 6-cube

Pentitruncated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	8640
Vertices	1920
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Teritruncated hexeract (Acronym: tacog) (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]

Penticantellated 6-cube

Penticantellated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	21120
Vertices	3840
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Terirhombated hexeract (Acronym: topag) (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]

Penticantitruncated 6-cube

Penticantitruncated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	30720
Vertices	7680
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Terigreatorhombated hexeract (Acronym: togrix) (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]

Pentiruncitruncated 6-cube

Pentiruncitruncated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	151840
Vertices	11520
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (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]

Pentiruncicantellated 6-cube

Pentiruncicantellated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,2,3,5{4,3,3,3,3}
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

• Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)

Pentiruncicantitruncated 6-cube

Pentiruncicantitruncated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	80640
Vertices	23040
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Terigreatoprismated hexeract (Acronym: tagpox) (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]

Pentisteritruncated 6-cube

Pentisteritruncated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	30720
Vertices	7680
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)

Pentistericantitruncated 6-cube

Pentistericantitruncated 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	80640
Vertices	23040
Vertex figure
Coxeter groups	B6, [4,3,3,3,3]
Properties	convex

Alternate names

• Tericelligreatorhombated hexeract (Acronym: tocagrax) (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]

Omnitruncated 6-cube

Omnitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams
5-faces	728: 12 t0,1,2,3,4{3,3,3,4} 60 {}×t0,1,2,3{3,3,4} 40px×40px 160 {6}×t0,1,2{3,4} 40px×40px 240 {8}×t0,1,2{3,3} 40px×40px 192 {}×t0,1,2,3{33} 40px×40px 64 t0,1,2,3,4{34}
4-faces	14168
Cells	72960
Faces	151680
Edges	138240
Vertices	46080
Vertex figure	irregular 5-simplex
Coxeter group	B6, [4,3,3,3,3]
Properties	convex, isogonal

The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.

Alternate names

• Pentisteriruncicantituncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
• Omnitruncated hexeract
• Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)

Full snub 6-cube

The full snub 6-cube or omnisnub 6-cube, defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3,3]⁺, and constructed from 12 snub 5-cubes, 64 snub 5-simplexes, 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr{4,3} duoantiprisms, 240 4-s{3,4} duoantiprisms, and 23040 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-cube 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.  x4o3o3o3o3x - stoxog, x4x3o3o3o3x - tacog, x4o3x3o3o3x - topag, x4x3x3o3o3x - togrix, x4x3o3x3o3x - tocrag, x4o3x3x3o3x - tiprixog, x4x3x3o3x3x - tagpox, x4x3o3o3x3x - tactaxog, x4x3x3o3x3x - tocagrax, x4x3x3x3x3x - gotaxog

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary