Pentellated 6-cubes

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

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

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)^[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]

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)^[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]

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)^[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]

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)^[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]

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)^[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]

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)^[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]

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)^[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]

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)^[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]

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)^[9]

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} × 160 {6}×t0,1,2{3,4} × 240 {8}×t0,1,2{3,3} × 192 {}×t0,1,2,3{33} × 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

• Pentisteriruncicantitruncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
• Omnitruncated hexeract
• Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)^[10]

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]

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.

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, (x4o3o3o3o3x - stoxog)
2. Klitzing, (x4x3o3o3o3x - tacog)
3. Klitzing, (x4o3x3o3o3x - topag)
4. Klitzing, (x4x3x3o3o3x - togrix)
5. Klitzing, (x4x3o3x3o3x - tocrag)
6. Klitzing, (x4o3x3x3o3x - tiprixog)
7. Klitzing, (x4x3x3o3x3x - tagpox)
8. Klitzing, (x4x3o3o3x3x - tactaxog)
9. Klitzing, (x4x3x3o3x3x - tocagrax)
10. Klitzing, (x4x3x3x3x3x - gotaxog)

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)". 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

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