Runcic 6-cubes
In six-dimensional geometry , a runcic 6-cube is a convex uniform 6-polytope . There are 2 unique runcic for the 6-cube.
Runcic 6-cube
Alternate names
Cantellated 6-demicube/demihexeract
Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)[ 1]
Cartesian coordinates
The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±3,±3) with an odd number of plus signs.
Images
Runcic n -cubes
n
4
5
6
7
8
[1+ ,4,3n-2 ]n-3,1 ]
[1+ ,4,32 ]1,1 ]
[1+ ,4,33 ]2,1 ]
[1+ ,4,34 ]3,1 ]
[1+ ,4,35 ]4,1 ]
[1+ ,4,36 ]5,1 ]
Runcic
Coxeter
Schläfli
h3 {4,32 }
h3 {4,33 }
h3 {4,34 }
h3 {4,35 }
h3 {4,36 }
Runcicantic 6-cube
Alternate names
Cantitruncated 6-demicube/demihexeract
Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)[ 2]
Cartesian coordinates
The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations:
(±1,±1,±3,±5,±5,±5) with an odd number of plus signs.
Images
This polytope is based on the 6-demicube , a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
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 *b3x3o3o - sirhax)
^ Klitzing, (x3x3o *b3x3o3o - girhax)
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
