In seven-dimensional geometry , a cantellated 7-cube is a convex uniform 7-polytope , being a cantellation of the regular 7-cube .
There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex .
Cantellated 7-cube
Cantellated 7-cube
Type
uniform 7-polytope
Schläfli symbol rr{4,3,3,3,3,3}
Coxeter diagram File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
16128
Vertices
2688
Vertex figure
Coxeter groups B7 , [4,3,3,3,3,3]
Properties
convex
Alternate names Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)[ 1] Images Bicantellated 7-cube
Bicantellated 7-cube
Type
uniform 7-polytope
Schläfli symbol r2r{4,3,3,3,3,3}
Coxeter diagrams File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
40320
Vertices
6720
Vertex figure
Coxeter groups B7 , [4,3,3,3,3,3]
Properties
convex
Alternate names Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)[ 2] Images Tricantellated 7-cube
Tricantellated 7-cube
Type
uniform 7-polytope
Schläfli symbol r3r{4,3,3,3,3,3}
Coxeter diagrams File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
47040
Vertices
6720
Vertex figure
Coxeter groups B7 , [4,3,3,3,3,3]
Properties
convex
Alternate names Small trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (Jonathan Bowers)[ 3] Images Cantitruncated 7-cube
Cantitruncated 7-cube
Type
uniform 7-polytope
Schläfli symbol tr{4,3,3,3,3,3}
Coxeter diagrams File:CDel node 1.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
18816
Vertices
5376
Vertex figure
Coxeter groups B7 , [4,3,3,3,3,3]
Properties
convex
Alternate names Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)[ 4] Images It is fifth in a series of cantitruncated hypercubes:
Bicantitruncated 7-cube
Bicantitruncated 7-cube
Type
uniform 7-polytope
Schläfli symbol r2r{4,3,3,3,3,3}
Coxeter diagrams File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes 11.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
47040
Vertices
13440
Vertex figure
Coxeter groups B7 , [4,3,3,3,3,3]
Properties
convex
Alternate names Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)[ 5] Images Tricantitruncated 7-cube
Tricantitruncated 7-cube
Type
uniform 7-polytope
Schläfli symbol t3r{4,3,3,3,3,3}
Coxeter diagrams File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel nodes.png File:CDel split2.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
53760
Vertices
13440
Vertex figure
Coxeter groups B7 , [4,3,3,3,3,3]
Properties
convex
Alternate names Great trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)[ 6] Images Related polytopes These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.
See also Notes
↑ Klitizing, (x3o3x3o3o3o4o - sersa)
↑ Klitizing, (o3x3o3x3o3o4o - sibrosa)
↑ Klitizing, (o3o3x3o3x3o4o - strasaz)
↑ Klitizing, (x3x3x3o3o3o4o - gersa)
↑ Klitizing, (o3x3x3x3o3o4o - gibrosa)
↑ Klitizing, (o3o3x3x3x3o4o - gotrasaz)
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. "7D uniform polytopes (polyexa)" . External links
