Pentic 6-cubes

6-demicube (half 6-cube) =	160px Pentic 6-cube =	160px Penticantic 6-cube =
160px Pentiruncic 6-cube =	160px Pentiruncicantic 6-cube =	160px Pentisteric 6-cube =
160px Pentistericantic 6-cube =	160px Pentisteriruncic 6-cube =	160px Pentisteriruncicantic 6-cube =
Orthogonal projections in D6 Coxeter plane

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	t0,4{3,34,1} h5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	1440
Vertices	192
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Penticantic 6-cube

Penticantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,4{3,34,1} h2,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	9600
Vertices	1920
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentiruncic 6-cube

Pentiruncic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,2,4{3,34,1} h3,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	10560
Vertices	1920
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentiruncicantic 6-cube

Pentiruncicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,4{3,32,1} h2,3,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	20160
Vertices	5760
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteric 6-cube

Pentisteric 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,3,4{3,34,1} h4,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	5280
Vertices	960
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentistericantic 6-cube

Pentistericantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,3,4{3,34,1} h2,4,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteriruncic 6-cube

Pentisteriruncic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,2,3,4{3,34,1} h3,4,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	15360
Vertices	3840
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
Dihedral symmetry	[6]	[4]

Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,3,4{3,32,1} h2,3,4,5{4,34}
Coxeter-Dynkin diagram	=
5-faces
4-faces
Cells
Faces
Edges	34560
Vertices	11520
Vertex figure
Coxeter groups	D6, [33,1,1]
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)

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	150px
Dihedral symmetry	[12/2]
Coxeter plane	D6	D5
Graph		150px
Dihedral symmetry	[10]	[8]
Coxeter plane	D4	D3
Graph	150px	150px
Dihedral symmetry	[6]	[4]
Coxeter plane	A5	A3
Graph	150px	150px
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}	h2,3{4,34}	h2,4{4,34}	h2,5{4,34}
h3,4{4,34}	h3,5{4,34}	h4,5{4,34}	h2,3,4{4,34}	h2,3,5{4,34}	h2,4,5{4,34}	h3,4,5{4,34}	h2,3,4,5{4,34}

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.
• 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