Steric 6-cubes
In six-dimensional geometry , a steric 6-cube is a convex uniform 6-polytope . There are unique 4 steric forms of the 6-cube.
Steric 6-cube
Alternate names
Runcinated demihexeract/6-demicube
Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)[ 1]
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±1,±3)
with an odd number of plus signs.
Images
Dimensional family of steric n-cubes
n
5
6
7
8
[1+ ,4,3n-2 ] = [3,3n-3,1 ]
[1+ ,4,33 ] = [3,32,1 ]
[1+ ,4,34 ] = [3,33,1 ]
[1+ ,4,35 ] = [3,34,1 ]
[1+ ,4,36 ] = [3,35,1 ]
Steric figure
Coxeter
=
=
=
=
Schläfli
h4 {4,33 }
h4 {4,34 }
h4 {4,35 }
h4 {4,36 }
Stericantic 6-cube
Alternate names
Runcitruncated demihexeract/6-demicube
Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)[ 2]
Cartesian coordinates
The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±3,±5)
with an odd number of plus signs.
Images
Steriruncic 6-cube
Alternate names
Runcicantellated demihexeract/6-demicube
Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)[ 3]
Cartesian coordinates
The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±3,±5)
with an odd number of plus signs.
Images
Steriruncicantic 6-cube
Steriruncicantic 6-cube
Type
uniform 6-polytope
Schläfli symbol
t0,1,2,3 {3,32,1 } h2,3,4 {4,34 }
Coxeter-Dynkin diagram
=
5-faces
4-faces
Cells
Faces
Edges
17280
Vertices
5760
Vertex figure
Coxeter groups
D6 , [33,1,1 ]
Properties
convex
Alternate names
Runcicantitruncated demihexeract/6-demicube
Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)[ 4]
Cartesian coordinates
The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±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 *b3o3x3o - sophax)
^ Klitzing, (x3x3o *b3o3x3o - pithax)
^ Klitzing, (x3o3o *b3x3x3o - prohax)
^ Klitzing, (x3x3o *b3x3x3o - gophax)
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)" . x3o3o *b3o3x3o - sophax, x3x3o *b3o3x3o - pithax, x3o3o *b3x3x3o - prohax, x3x3o *b3x3x3o - gophax
External links