Runcinated 5-orthoplexes

  (Redirected from Runcitruncated 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 t14.svg
Runcinated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t03.svg
Runcinated 5-cube
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
5-cube t124.svg
Runcitruncated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t134.svg
Runcicantellated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
5-cube t1234.svg
Runcicantitruncated 5-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t013.svg
Runcitruncated 5-cube
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
5-cube t023.svg
Runcicantellated 5-cube
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
5-cube t0123.svg
Runcicantitruncated 5-cube
CDel node.pngCDel 3.pngCDel node 1.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 runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.

Runcinated 5-orthoplexEdit

Runcinated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,3{3,3,3,4}
Coxeter-Dynkin diagram          
       
4-faces 162
Cells 1200
Faces 2160
Edges 1440
Vertices 320
Vertex figure  
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex

Alternate namesEdit

  • Runcinated pentacross
  • Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)[1]

CoordinatesEdit

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

(0,1,1,1,2)

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]

Runcitruncated 5-orthoplexEdit

Runcitruncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol t0,1,3{3,3,3,4}
t0,1,3{3,31,1}
Coxeter-Dynkin diagrams          
       
4-faces 162
Cells 1440
Faces 3680
Edges 3360
Vertices 960
Vertex figure  
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex

Alternate namesEdit

  • Runcitruncated pentacross
  • Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)[2]

CoordinatesEdit

Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of

(±3,±2,±1,±1,0)

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]

Runcicantellated 5-orthoplexEdit

Runcicantellated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,4}
t0,2,3{3,3,31,1}
Coxeter-Dynkin diagram          
       
4-faces 162
Cells 1200
Faces 2960
Edges 2880
Vertices 960
Vertex figure  
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex

Alternate namesEdit

  • Runcicantellated pentacross
  • Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)[3]

CoordinatesEdit

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

(0,1,2,2,3)

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]

Runcicantitruncated 5-orthoplexEdit

Runcicantitruncated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{3,3,3,4}
Coxeter-Dynkin
diagram
         
       
4-faces 162
Cells 1440
Faces 4160
Edges 4800
Vertices 1920
Vertex figure  
Irregular 5-cell
Coxeter groups B5 [4,3,3,3]
D5 [32,1,1]
Properties convex, isogonal

Alternate namesEdit

  • Runcicantitruncated pentacross
  • Great prismated triacontiditeron (gippit) (Jonathan Bowers)[4]

CoordinatesEdit

The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of 2 are given by all permutations of coordinates and sign of:

 

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]

Snub 5-demicubeEdit

The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram         or           and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.

Related polytopesEdit

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

NotesEdit

  1. ^ Klitzing, (x3o3o3x4o - spat)
  2. ^ Klitzing, (x3x3o3x4o - pattit)
  3. ^ Klitzing, (x3o3x3x4o - pirt)
  4. ^ Klitzing, (x3x3x3x4o - gippit)

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)". x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit

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