In six-dimensional geometry , a cantellated 5-cube is a convex uniform 5-polytope , being a cantellation of the regular 5-cube .
There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex
Cantellated 5-cube Alternate names Small rhombated penteract (Acronym: sirn) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:
( ± 1 , ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) ) Images Bicantellated 5-cube
Bicantellated 5-cube
Type
Uniform 5-polytope
Schläfli symbols
2rr{4,3,3,3} = r { 3 , 4 3 , 3 } 2,1,1 } = r { 3 , 3 3 3 }
Coxeter-Dynkin 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 node.png File:CDel split1.png File:CDel nodes 11.png File:CDel 4a3b.png File:CDel nodes.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
4-faces
122
10 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:Schlegel half-solid cantellated 16-cell.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:4-3 duoprism.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:Schlegel half-solid cantellated 5-cell.png
Cells
840
40 File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Uniform polyhedron-43-t1.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:Tetragonal prism.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:Uniform polyhedron-33-t02.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Triangular prism wedge.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Uniform polyhedron-33-t1.svg
Faces
2160
240 File:CDel node.png File:CDel 4.png File:CDel node 1.png File:2-cube.svg File:CDel node 1.png File:CDel 3.png File:CDel node.png File:2-simplex t0.svg File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:2-cube.svg File:CDel node.png File:CDel 3.png File:CDel node 1.png File:2-simplex t0.svg File:CDel node 1.png File:CDel 3.png File:CDel node.png File:2-simplex t0.svg
Edges
1920
960+960
Vertices
480
Vertex figure
File:Bicantellated penteract verf.png
Coxeter groups
B5 , [3,3,3,4]5 , [32,1,1 ]
Properties
convex , uniform
In five-dimensional geometry , a bicantellated 5-cube is a uniform 5-polytope .
Alternate names Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:
(0,1,1,2,2) Images
Cantitruncated 5-cube Alternate names Tricantitruncated 5-orthoplex / tricantitruncated pentacross
Great rhombated penteract (girn) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
( 1 , 1 + 2 , 1 + 2 2 , 1 + 2 2 , 1 + 2 2 ) Images Related polytopes It is third in a series of cantitruncated hypercubes:
Bicantitruncated 5-cube
Bicantitruncated 5-cube
Type
uniform 5-polytope
Schläfli symbol
2tr{3,3,3,4} = t { 3 , 4 3 , 3 } 2,1,1 } = t { 3 , 3 3 3 }
Coxeter-Dynkin 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 node 1.png File:CDel split1.png File:CDel nodes 11.png File:CDel 4a3b.png File:CDel nodes.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
4-faces
122
10 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:Schlegel half-solid cantitruncated 16-cell.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:4-3 duoprism.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:Schlegel half-solid cantitruncated 5-cell.png
Cells
840
40 File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:Uniform polyhedron-43-t12.png File:CDel node.png File:CDel 4.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:Tetragonal prism.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:Uniform polyhedron-33-t012.png File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Triangular prism wedge.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:Uniform polyhedron-33-t01.png
Faces
2160
240 File:CDel node.png File:CDel 4.png File:CDel node 1.png File:2-cube.svg File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:2-simplex t01.svg File:CDel node 1.png File:CDel 2.png File:CDel node 1.png File:2-cube.svg File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:2-simplex t01.svg File:CDel node 1.png File:CDel 3.png File:CDel node.png File:2-simplex t0.svg
Edges
2400
960+480+960
Vertices
960
Vertex figure
File:Bicanitruncated 5-cube verf.png
Coxeter groups
B5 , [3,3,3,4]5 , [32,1,1 ]
Properties
convex , uniform
Alternate names Bicantitruncated penteract
Bicantitruncated pentacross
Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers) Coordinates Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of
(±3,±3,±2,±1,0) Images Related polytopes These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex .
B5 polytopes
File:5-cube t4.svg β5
File:5-cube t3.svg t1 β5
File:5-cube t2.svg t2 γ5
File:5-cube t1.svg t1 γ5
File:5-cube t0.svg γ5
File:5-cube t34.svg t0,1 β5
File:5-cube t24.svg t0,2 β5
File:5-cube t23.svg t1,2 β5
File:5-cube t14.svg t0,3 β5
File:5-cube t13.svg t1,3 γ5
File:5-cube t12.svg t1,2 γ5
File:5-cube t04.svg t0,4 γ5
File:5-cube t03.svg t0,3 γ5
File:5-cube t02.svg t0,2 γ5
File:5-cube t01.svg t0,1 γ5
File:5-cube t234.svg t0,1,2 β5
File:5-cube t134.svg t0,1,3 β5
File:5-cube t124.svg t0,2,3 β5
File:5-cube t123.svg t1,2,3 γ5
File:5-cube t034.svg t0,1,4 β5
File:5-cube t024.svg t0,2,4 γ5
File:5-cube t023.svg t0,2,3 γ5
File:5-cube t014.svg t0,1,4 γ5
File:5-cube t013.svg t0,1,3 γ5
File:5-cube t012.svg t0,1,2 γ5
File:5-cube t1234.svg t0,1,2,3 β5
File:5-cube t0234.svg t0,1,2,4 β5
File:5-cube t0134.svg t0,1,3,4 γ5
File:5-cube t0124.svg t0,1,2,4 γ5
File:5-cube t0123.svg t0,1,2,3 γ5
File:5-cube t01234.svg t0,1,2,3,4 γ5
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 , editied 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. "5D uniform polytopes (polytera)" . External links
