Pentic 6-cubes

Orthogonal projections in D₆ Coxeter plane
6-demicube (half 6-cube) =	Pentic 6-cube =	Penticantic 6-cube =
Pentiruncic 6-cube =	Pentiruncicantic 6-cube =	Pentisteric 6-cube =
Pentistericantic 6-cube =	Pentisteriruncic 6-cube =	Pentisteriruncicantic 6-cube =

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

There are 8 pentic forms of the 6-cube.

Pentic 6-cube

Pentic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,4{3,3^4,1} h₅{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	1440
Vertices	192
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .

Alternate names

Stericated 6-demicube/demihexeract
Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)^[1]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±1,±3)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Penticantic 6-cube

Penticantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,1,4{3,3^4,1} h_2,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	9600
Vertices	1920
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .

Alternate names

Steritruncated 6-demicube/demihexeract
cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)^[2]

Cartesian coordinates

The Cartesian coordinates for the vertices of a stericantitruncated demihexeract centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Pentiruncic 6-cube

Pentiruncic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,2,4{3,3^4,1} h_3,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	10560
Vertices	1920
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .

Alternate names

Stericantellated 6-demicube/demihexeract
cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)^[3]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentiruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,1,2,4{3,3^2,1} h_2,3,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	20160
Vertices	5760
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),

Alternate names

Stericantitruncated demihexeract, stericantitruncated 7-demicube
Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)^[4]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentiruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Pentisteric 6-cube

Pentisteric 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,3,4{3,3^4,1} h_4,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	5280
Vertices	960
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),

Alternate names

Steriruncinated 6-demicube/demihexeract
Small cellipriamated hemihexeract (Acronym: cophix) (Jonathan Bowers)^[5]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteric 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±1,±3,±5)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Pentistericantic 6-cube

Pentistericantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,1,3,4{3,3^4,1} h_2,4,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .

Alternate names

Steriruncicantitruncated demihexeract/7-demicube
cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)^[6]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentistericantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Pentisteriruncic 6-cube

Pentisteriruncic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,2,3,4{3,3^4,1} h_3,4,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	15360
Vertices	3840
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .

Alternate names

Steriruncicantellated 6-demicube/demihexeract
Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)^[7]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteriruncic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t_0,1,2,3,4{3,3^2,1} h_2,3,4,5{4,3⁴}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	34560
Vertices	11520
Vertex figure
Coxeter groups	D₆, [3^3,1,1]
Properties	convex

The pentisteriruncicantic 6-cube, , has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), .

Alternate names

Steriruncicantitruncated 6-demicube/demihexeract
Great cellated hemihexeract (Acronym: gochax) ((Jonathan Bowers)^[8]

Cartesian coordinates

The Cartesian coordinates for the vertices of a pentisteriruncicantic 6-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5,±7)

with an odd number of plus signs.

Images

orthographic projections
Coxeter plane	B₆
Graph
Dihedral symmetry	[12/2]
Coxeter plane	D₆	D₅
Graph
Dihedral symmetry	[10]	[8]
Coxeter plane	D₄	D₃
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Related polytopes

There are 47 uniform polytopes with D₆ symmetry, 31 are shared by the BC₆ symmetry, and 16 are unique:

D6 polytopes

h{4,3⁴}	h₂{4,3⁴}	h₃{4,3⁴}	h₄{4,3⁴}	h₅{4,3⁴}	h_2,3{4,3⁴}	h_2,4{4,3⁴}	h_2,5{4,3⁴}
h_3,4{4,3⁴}	h_3,5{4,3⁴}	h_4,5{4,3⁴}	h_2,3,4{4,3⁴}	h_2,3,5{4,3⁴}	h_2,4,5{4,3⁴}	h_3,4,5{4,3⁴}	h_2,3,4,5{4,3⁴}

Notes

↑ Klitzing, (x3o3o *b3o3x3o3o - sochax)
↑ Klitzing, (x3x3o *b3o3x3o3o - cathix)
↑ Klitzing, (x3o3o *b3x3x3o3o - crohax)
↑ Klitzing, (x3x3o *b3x3x3o3o - cagrohax)
↑ Klitzing, (x3o3o *b3o3x3x3x - cophix)
↑ Klitzing, (x3x3o *b3o3x3x3x - capthix)
↑ Klitzing, (x3o3o *b3x3x3x3x - caprohax)
↑ Klitzing, (x3x3o *b3x3x3x3o - gochax)

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.
Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o *b3o3x3o3o - sochax, x3x3o *b3o3x3o3o - cathix, x3o3o *b3x3x3o3o - crohax, x3x3o *b3x3x3o3o - cagrohax, x3o3o *b3o3x3x3x - cophix, x3x3o *b3o3x3x3x - capthix, x3o3o *b3x3x3x3x - caprohax, x3x3o *b3x3x3x3o - gochax

External links

Weisstein, Eric W., "Hypercube", MathWorld.
Olshevsky, George, Measure polytope at Glossary for Hyperspace.
Polytopes of Various Dimensions
Multi-dimensional Glossary

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 4-polytope	5-cell	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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