6-demicube
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
The pentic 6-cube , 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, centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±1,±3) with an odd number of plus signs.
Images
Penticantic 6-cube
The penticantic 6-cube , penticantellated 6-cube ,
Alternate names
Steritruncated 6-demicube/demihexeract
cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)[ 2]
Cartesian coordinates
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
(±1,±1,±3,±3,±3,±5) with an odd number of plus signs.
Images
Pentiruncic 6-cube
The pentiruncic 6-cube , 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, centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±3,±5) with an odd number of plus signs.
Images
Pentiruncicantic 6-cube
Pentiruncicantic 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,2,4 {3,32,1 }2,3,5 {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 , 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, centered at the origin are coordinate permutations:
(±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
Images
Pentisteric 6-cube
The pentisteric 6-cube , 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, centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±3,±5) with an odd number of plus signs.
Images
Pentistericantic 6-cube
Pentistericantic 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,3,4 {3,34,1 }2,4,5 {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 , pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex),
Alternate names
Steriruncitruncated demihexeract/7-demicube
cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)[ 6]
Cartesian coordinates
The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:
(±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
Images
Pentisteriruncic 6-cube
Pentisteriruncic 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,2,3,4 {3,34,1 }3,4,5 {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 , 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, centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±5,±7) with an odd number of plus signs.
Images
Pentisteriruncicantic 6-cube
Pentisteriruncicantic 6-cube
Type
uniform 6-polytope
Schläfli symbol t0,1,2,3,4 {3,32,1 }2,3,4,5 {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 , 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, centered at the origin are coordinate permutations:
(±1,±1,±3,±3,±5,±7) with an odd number of plus signs.
Images
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 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 }
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 [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)" .
External links
