Cantellated 5-simplexes

  (Redirected from Bicantellated 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 t02.svg
Cantellated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t13.svg
Bicantellated 5-simplex
CDel node.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 t012.svg
Cantitruncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t123.svg
Bicantitruncated 5-simplex
CDel node.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 five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.

There are unique 4 degrees of cantellation for the 5-simplex, including truncations.

Cantellated 5-simplexEdit

Cantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol rr{3,3,3,3} =  
Coxeter-Dynkin diagram          
or        
4-faces 27 6 r{3,3,3} 
6 rr{3,3,3} 
15 {}x{3,3} 
Cells 135 30 {3,3} 
30 r{3,3} 
15 rr{3,3} 
60 {}x{3} 
Faces 290 200 {3}
90 {4}
Edges 240
Vertices 60
Vertex figure  
Tetrahedral prism
Coxeter group A5 [3,3,3,3], order 720
Properties convex

The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).

Alternate namesEdit

  • Cantellated hexateron
  • Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)[1]

CoordinatesEdit

The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.

ImagesEdit

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

Bicantellated 5-simplexEdit

Bicantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2rr{3,3,3,3} =  
Coxeter-Dynkin diagram          
or      
4-faces 32 12 t02{3,3,3}
20 {3}x{3}
Cells 180 30 t1{3,3}
120 {}x{3}
30 t02{3,3}
Faces 420 240 {3}
180 {4}
Edges 360
Vertices 90
Vertex figure  
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate namesEdit

  • Bicantellated hexateron
  • Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)[2]

CoordinatesEdit

The coordinates can be made in 6-space, as 90 permutations of:

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

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

ImagesEdit

orthographic projections
Ak
Coxeter plane
A5 A4
Graph    
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph    
Dihedral symmetry [4] [[3]]=[6]

Cantitruncated 5-simplexEdit

cantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol tr{3,3,3,3} =  
Coxeter-Dynkin diagram          
or        
4-faces 27 6 t012{3,3,3} 
6 t{3,3,3} 
15 {}x{3,3}
Cells 135 15 t012{3,3}  
30 t{3,3} 
60 {}x{3}
30 {3,3} 
Faces 290 120 {3} 
80 {6} 
90 {}x{} 
Edges 300
Vertices 120
Vertex figure  
Irr. 5-cell
Coxeter group A5 [3,3,3,3], order 720
Properties convex

Alternate namesEdit

  • Cantitruncated hexateron
  • Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)[3]

CoordinatesEdit

The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 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]

Bicantitruncated 5-simplexEdit

Bicantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2tr{3,3,3,3} =  
Coxeter-Dynkin diagram          
or      
4-faces 32 12 tr{3,3,3}
20 {3}x{3}
Cells 180 30 t{3,3}
120 {}x{3}
30 t{3,4}
Faces 420 240 {3}
180 {4}
Edges 450
Vertices 180
Vertex figure  
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate namesEdit

  • Bicantitruncated hexateron
  • Great birhombated dodecateron (Acronym: gibrid) (Jonathan Bowers)[4]

CoordinatesEdit

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

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

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

ImagesEdit

orthographic projections
Ak
Coxeter plane
A5 A4
Graph    
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph    
Dihedral symmetry [4] [[3]]=[6]

Related uniform 5-polytopesEdit

The cantellated 5-simplex is one 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, (x3o3x3o3o - sarx)
  2. ^ Klitizing, (o3x3o3x3o - sibrid)
  3. ^ Klitizing, (x3x3x3o3o - garx)
  4. ^ Klitizing, (o3x3x3x3o - gibrid)

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)". x3o3x3o3o - sarx, o3x3o3x3o - sibrid, x3x3x3o3o - garx, o3x3x3x3o - gibrid

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