Steric 5-cubes
In five-dimensional geometry , a steric 5-cube or (steric 5-demicube or sterihalf 5-cube ) is a convex uniform 5-polytope . There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes .
Steric 5-cube
Alternate names
Steric penteract, runcinated demipenteract
Small prismated hemipenteract (siphin) (Jonathan Bowers)[ 1] : (x3o3o *b3o3x - siphin)
Cartesian coordinates
The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of
(±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 ]n-3,1 ]
[1+ ,4,33 ]2,1 ]
[1+ ,4,34 ]3,1 ]
[1+ ,4,35 ]4,1 ]
[1+ ,4,36 ]5,1 ]
Steric
Coxeter
Schläfli
h4 {4,33 }
h4 {4,34 }
h4 {4,35 }
h4 {4,36 }
Stericantic 5-cube
Stericantic 5-cube
Type
uniform polyteron
Schläfli symbol t0,1,3 {3,32,1 } h2,4 {4,3,3,3
Coxeter-Dynkin diagram
4-faces
82
Cells
720
Faces
1840
Edges
1680
Vertices
480
Vertex figure
Coxeter groups D5 , [32,1,1 ]
Properties
convex
Alternate names
Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)[ 1] : (x3x3o *b3o3x - pithin)
Cartesian coordinates
The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:
(±1,±1,±3,±3,±5) with an odd number of plus signs.
Images
Steriruncic 5-cube
Steriruncic 5-cube
Type
uniform polyteron
Schläfli symbol t0,2,3 {3,32,1 } h3,4 {4,3,3,3
Coxeter-Dynkin diagram
4-faces
82
Cells
560
Faces
1280
Edges
1120
Vertices
320
Vertex figure
Coxeter groups D5 , [32,1,1 ]
Properties
convex
Alternate names
Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)[ 1] : (x3o3o *b3x3x - pirhin)
Cartesian coordinates
The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±5) with an odd number of plus signs.
Images
Steriruncicantic 5-cube
Steriruncicantic 5-cube
Type
uniform polyteron
Schläfli symbol t0,1,2,3 {3,32,1 } h2,3,4 {4,3,3,3
Coxeter-Dynkin diagram
4-faces
82
Cells
720
Faces
2080
Edges
2400
Vertices
960
Vertex figure
Coxeter groups D5 , [32,1,1 ]
Properties
convex
Alternate names
Great prismated hemipenteract (giphin) (Jonathan Bowers)[ 1] : (x3x3o *b3x3x - giphin)
Cartesian coordinates
The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:
(±1,±1,±3,±5,±7) with an odd number of plus signs.
Images
This polytope is based on the 5-demicube , a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
D5 polytopes
h{4,3,3,3}
h2 {4,3,3,3}
h3 {4,3,3,3}
h4 {4,3,3,3}
h2,3 {4,3,3,3}
h2,4 {4,3,3,3}
h3,4 {4,3,3,3}
h2,3,4 {4,3,3,3}
References
Further reading
Coxeter, H. S. M. (1973). Regular Polytopes New York City : Dover. Retrieved 2022-05-19 .Coxeter, H. S. M. (1995-05-17). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivić (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons . ISBN 978-0-471-01003-6 LCCN 94047368 . OCLC 632987525 . OL 7598569M . Retrieved 2022-05-19 .Coxeter, H. S. M. (1940-12-01). "Regular and Semi Regular Polytopes I" Mathematische Zeitschrift . 46 . Springer Nature : 380– 407. doi :10.1007/BF01181449 . ISSN 1432-1823 . S2CID 186237114 . Retrieved 2022-05-19 .Coxeter, H. S. M. (1985-12-01). "Regular and Semi-Regular Polytopes II" Mathematische Zeitschrift . 188 (4). Springer Nature : 559– 591. doi :10.1007/BF01161657 . ISSN 1432-1823 . S2CID 120429557 . Retrieved 2022-05-19 .Coxeter, H. S. M. (1988-03-01). "Regular and Semi-Regular Polytopes III" Mathematische Zeitschrift . 200 (1). Springer Nature : 3– 45. doi :10.1007/BF01161745 . ISSN 1432-1823 . S2CID 186237142 . Retrieved 2022-05-19 .Johnson, Norman W. (1991). Uniform Polytopes (Unfinished manuscript thesis).Johnson, Norman W. (1966). The Theory of Uniform Polytopes and Honeycombs University of Toronto . Retrieved 2022-05-19 .
External links
