Runcinated 5-simplexes

  (Redirected from Runcicantitruncated 5-simplex)
5-simplex t0.svg
5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t03.svg
Runcinated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t013.svg
Runcitruncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t2.svg
Birectified 5-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t023.svg
Runcicantellated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t0123.svg
Runcicantitruncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Orthogonal projections in A5 Coxeter plane

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.

Runcinated 5-simplexEdit

Runcinated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,3{3,3,3,3}
Coxeter-Dynkin diagram          
4-faces 47 6 t0,3{3,3,3}  
20 {3}×{3}
15 { }×r{3,3}
6 r{3,3,3}  
Cells 255 45 {3,3}  
180 { }×{3}
30 r{3,3}  
Faces 420 240 {3}  
180 {4}
Edges 270
Vertices 60
Vertex figure  
Coxeter group A5 [3,3,3,3], order 720
Properties convex

Alternate namesEdit

  • Runcinated hexateron
  • Small prismated hexateron (Acronym: spix) (Jonathan Bowers)[1]

CoordinatesEdit

The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.

ImagesEdit

orthographic projections
Ak
Coxeter plane
A5 A4
Graph    
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph    
Dihedral symmetry [4] [3]

Runcitruncated 5-simplexEdit

Runcitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,3{3,3,3,3}
Coxeter-Dynkin diagram          
4-faces 47 6 t0,1,3{3,3,3}
20 {3}×{6}
15 { }×r{3,3}
6 rr{3,3,3}
Cells 315
Faces 720
Edges 630
Vertices 180
Vertex figure  
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate namesEdit

  • Runcitruncated hexateron
  • Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)[2]

CoordinatesEdit

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,1,2,3)

This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.

ImagesEdit

orthographic projections
Ak
Coxeter plane
A5 A4
Graph    
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph    
Dihedral symmetry [4] [3]

Runcicantellated 5-simplexEdit

Runcicantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,3}
Coxeter-Dynkin diagram          
4-faces 47
Cells 255
Faces 570
Edges 540
Vertices 180
Vertex figure  
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate namesEdit

  • Runcicantellated hexateron
  • Biruncitruncated 5-simplex/hexateron
  • Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)[3]

CoordinatesEdit

The coordinates can be made in 6-space, as 180 permutations of:

(0,0,1,2,2,3)

This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.

ImagesEdit

orthographic projections
Ak
Coxeter plane
A5 A4
Graph    
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph    
Dihedral symmetry [4] [3]

Runcicantitruncated 5-simplexEdit

Runcicantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{3,3,3,3}
Coxeter-Dynkin diagram          
4-faces 47 6 t0,1,2,3{3,3,3}
20 {3}×{6}
15 {}×t{3,3}
6 tr{3,3,3}
Cells 315 45 t0,1,2{3,3}
120 { }×{3}
120 { }×{6}
30 t{3,3}
Faces 810 120 {3}
450 {4}
240 {6}
Edges 900
Vertices 360
Vertex figure  
Irregular 5-cell
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate namesEdit

  • Runcicantitruncated hexateron
  • Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)[4]

CoordinatesEdit

The coordinates can be made in 6-space, as 360 permutations of:

(0,0,1,2,3,4)

This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.

ImagesEdit

orthographic projections
Ak
Coxeter plane
A5 A4
Graph    
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph    
Dihedral symmetry [4] [3]

Related uniform 5-polytopesEdit

These polytopes are in a set of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

NotesEdit

  1. ^ Klitizing, (x3o3o3x3o - spidtix)
  2. ^ Klitizing, (x3x3o3x3o - pattix)
  3. ^ Klitizing, (x3o3x3x3o - pirx)
  4. ^ Klitizing, (x3x3x3x3o - gippix)

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)". x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix

External linksEdit

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