Cantellated 5-orthoplexes
In five-dimensional geometry , a cantellated 5-orthoplex is a convex uniform 5-polytope , being a cantellation of the regular 5-orthoplex .
There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube .
Cantellated 5-orthoplex
Cantellated 5-orthoplex
Type
Uniform 5-polytope
Schläfli symbol
rr{3,3,3,4} rr{3,3,31,1 }
Coxeter-Dynkin diagrams
4-faces
82
10 40 32
Cells
640
80 160 320 80
Faces
1520
640 320 480 80
Edges
1200
960 240
Vertices
240
Vertex figure
Square pyramidal prism
Coxeter group
B5 , [4,3,3,3], order 3840 D5 , [32,1,1 ], order 1920
Properties
convex
Alternate names
Cantellated 5-orthoplex
Bicantellated 5-demicube
Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)[1]
Coordinates
The vertices of the can be made in 5-space, as permutations and sign combinations of:
(0,0,1,1,2)
Images
The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.
Cantitruncated 5-orthoplex
Cantitruncated 5-orthoplex
Type
uniform 5-polytope
Schläfli symbol
tr{3,3,3,4} tr{3,31,1 }
Coxeter-Dynkin diagrams
4-faces
82
10 40 32
Cells
640
80 160 320 80
Faces
1520
640 320 480 80
Edges
1440
960 240 240
Vertices
480
Vertex figure
Square pyramidal pyramid
Coxeter groups
B5 , [3,3,3,4], order 3840 D5 , [32,1,1 ], order 1920
Properties
convex
Alternate names
Cantitruncated pentacross
Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±3,±2,±1,0,0)
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
β5
t1 β5
t2 γ5
t1 γ5
γ5
t0,1 β5
t0,2 β5
t1,2 β5
t0,3 β5
t1,3 γ5
t1,2 γ5
t0,4 γ5
t0,3 γ5
t0,2 γ5
t0,1 γ5
t0,1,2 β5
t0,1,3 β5
t0,2,3 β5
t1,2,3 γ5
t0,1,4 β5
t0,2,4 γ5
t0,2,3 γ5
t0,1,4 γ5
t0,1,3 γ5
t0,1,2 γ5
t0,1,2,3 β5
t0,1,2,4 β5
t0,1,3,4 γ5
t0,1,2,4 γ5
t0,1,2,3 γ5
t0,1,2,3,4 γ5
Notes
^ Klitizing, (x3o3x3o4o - sart)
^ Klitizing, (x3x3x3o4o - gart)
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. "5D uniform polytopes (polytera)" . x3o3x3o4o - sart, x3x3x3o4o - gart
External links