Cantellated 5-orthoplexes

  (Redirected from Cantitruncated 5-orthoplex)
5-cube t4.svg
5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t24.svg
Cantellated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t13.svg
Bicantellated 5-cube
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t13.svg
Cantellated 5-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
5-cube t0.svg
5-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
5-cube t012.svg
Cantitruncated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
5-cube t123.svg
Bicantitruncated 5-cube
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t234.svg
Cantitruncated 5-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Orthogonal projections in B5 Coxeter plane

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-orthoplexEdit

Cantellated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol rr{3,3,3,4}
rr{3,3,31,1}
Coxeter-Dynkin diagram          
       
4-faces 82
Cells 640
Faces 1520
Edges 1200
Vertices 240
Vertex figure  
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex

Alternate namesEdit

  • Cantellated 5-orthoplex
  • Bicantellated 5-demicube
  • Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)[1]

CoordinatesEdit

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,0,1,1,2)

ImagesEdit

The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph      
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph    
Dihedral symmetry [4] [4]

Cantitruncated 5-orthoplexEdit

Cantitruncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol tr{3,3,3,4}
tr{3,31,1}
Coxeter-Dynkin diagrams          
       
4-faces 82
Cells 640
Faces 1520
Edges 1440
Vertices 480
Vertex figure  
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex

Alternate namesEdit

  • Cantitruncated pentacross
  • Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)[2]

CoordinatesEdit

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)

ImagesEdit

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph      
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph    
Dihedral symmetry [4] [4]

Related polytopesEdit

These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

NotesEdit

  1. ^ Klitizing, (x3o3x3o4o - sart)
  2. ^ Klitizing, (x3x3x3o4o - gart)

ReferencesEdit

  • 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 linksEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds