In five-dimensional geometry , a truncated 5-orthoplex is a convex uniform 5-polytope , being a truncation of the regular 5-orthoplex .
There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube .
Truncated 5-orthoplex
Truncated 5-orthoplex
Type
uniform 5-polytope
Schläfli symbol t{3,3,3,4}1,1 }
Coxeter-Dynkin diagrams File:CDel node.png File:CDel 4.png File:CDel node.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 1.png File:CDel nodes.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 1.png
4-faces
42
10 File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:4-cube t3.svg File:CDel node.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 1.png File:4-simplex t01.svg
Cells
240
160 File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:3-simplex t0.svg File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:3-simplex t01.svg
Faces
400
320 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 1.png File:2-simplex t01.svg
Edges
280
240 File:CDel node 1.png File:CDel node 1.png
Vertices
80
Vertex figure File:Truncated pentacross.png ( )v{3,4}
Coxeter groups B5 , [3,3,3,4], order 38405 , [32,1,1 ], order 1920
Properties
convex
Alternate names Truncated pentacross
Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)[ 1] Coordinates Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±2,±1,0,0,0) Images The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex . All edges are shortened, and two new vertices are added on each original edge.
Bitruncated 5-orthoplex
Bitruncated 5-orthoplex
Type
uniform 5-polytope
Schläfli symbol 2t{3,3,3,4}1,1 }
Coxeter-Dynkin 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.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.png
4-faces
42
10 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:4-cube t23.svg 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.png File:4-simplex t12.svg
Cells
280
40 File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node 1.png File:3-cube t2.svg File:CDel node.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:3-simplex t01.svg File:CDel node 1.png File:CDel 3.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:3-simplex t01.svg
Faces
720
320 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 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
720
480 File:CDel node 1.png File:CDel node 1.png
Vertices
240
Vertex figure File:Bitruncated pentacross verf.png { }v{4}
Coxeter groups B5 , [3,3,3,4], order 38405 , [32,1,1 ], order 1920
Properties
convex
The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb .
Alternate names Bitruncated pentacross
Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)[ 2] Coordinates Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of
(±2,±2,±1,0,0) Images The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex .
Related polytopes This polytope is one 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
Notes
↑ Klitzing, (x3x3o3o4o - tot)
↑ Klitzing, (o3x3x3o4o - bittit)
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)" . External links
