Stericated 5-simplexes

  (Redirected from Expanded 5-simplex)
5-simplex t0.svg 5-simplex t0 A4.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 t04.svg 5-simplex t04 A4.svg
Stericated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t014.svg 5-simplex t014 A4.svg
Steritruncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t024.svg 5-simplex t024 A4.svg
Stericantellated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t0124.svg 5-simplex t0124 A4.svg
Stericantitruncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-simplex t0134.svg 5-simplex t0134 A4.svg
Steriruncitruncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-simplex t01234.svg 5-simplex t01234 A4.svg
Steriruncicantitruncated 5-simplex
(Omnitruncated 5-simplex)
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Orthogonal projections in A5 and A4 Coxeter planes

In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the steriruncicantitruncated 5-simplex is more simply called an omnitruncated 5-simplex with all of the nodes ringed.

Stericated 5-simplexEdit

Stericated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2r2r{3,3,3,3}
2r{32,2} =  
Coxeter-Dynkin diagram          
or      
4-faces 62 6+6 {3,3,3} 
15+15 {}×{3,3} 
20 {3}×{3} 
Cells 180 60 {3,3} 
120 {}×{3} 
Faces 210 120 {3}
90 {4}
Edges 120
Vertices 30
Vertex figure  
Tetrahedral antiprism
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal, isotoxal

A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).

Alternate namesEdit

  • Expanded 5-simplex
  • Stericated hexateron
  • Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)[1]

Cross-sectionsEdit

The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 3-3 duoprisms each.

CoordinatesEdit

The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). This represents the positive orthant facet of the stericated 6-orthoplex.

A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0)

The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:

 
 
 
 
 
 
 
 

Root systemEdit

Its 30 vertices represent the root vectors of the simple Lie group A5. It is also the vertex figure of the 5-simplex honeycomb.

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]
 
orthogonal projection with [6] symmetry

Steritruncated 5-simplexEdit

Steritruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,4{3,3,3,3}
Coxeter-Dynkin diagram          
4-faces 62 6 t{3,3,3}
15 {}×t{3,3}
20 {3}×{6}
15 {}×{3,3}
6 t0,3{3,3,3}
Cells 330
Faces 570
Edges 420
Vertices 120
Vertex figure  
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate namesEdit

  • Steritruncated hexateron
  • Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)[2]

CoordinatesEdit

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

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

This construction exists as one of 64 orthant facets of the steritruncated 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]

Stericantellated 5-simplexEdit

Stericantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,2,4{3,3,3,3}
Coxeter-Dynkin diagram          
or      
4-faces 62 12 rr{3,3,3}
30 rr{3,3}x{}
20 {3}×{3}
Cells 420 60 rr{3,3}
240 {}×{3}
90 {}×{}×{}
30 r{3,3}
Faces 900 360 {3}
540 {4}
Edges 720
Vertices 180
Vertex figure  
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate namesEdit

  • Stericantellated hexateron
  • Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)[3]

CoordinatesEdit

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

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

This construction exists as one of 64 orthant facets of the stericantellated 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]

Stericantitruncated 5-simplexEdit

Stericantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,4{3,3,3,3}
Coxeter-Dynkin diagram          
4-faces 62
Cells 480
Faces 1140
Edges 1080
Vertices 360
Vertex figure  
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate namesEdit

  • Stericantitruncated hexateron
  • Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)[4]

CoordinatesEdit

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

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

This construction exists as one of 64 orthant facets of the stericantitruncated 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]

Steriruncitruncated 5-simplexEdit

Steriruncitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,3,4{3,3,3,3}
2t{32,2}
Coxeter-Dynkin diagram          
or      
4-faces 62 12 t0,1,3{3,3,3}
30 {}×t{3,3}
20 {6}×{6}
Cells 450
Faces 1110
Edges 1080
Vertices 360
Vertex figure  
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate namesEdit

  • Steriruncitruncated hexateron
  • Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)[5]

CoordinatesEdit

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

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

This construction exists as one of 64 orthant facets of the steriruncitruncated 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]

Omnitruncated 5-simplexEdit

Omnitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3}
2tr{32,2}
Coxeter-Dynkin
diagram
         
or      
4-faces 62 12 t0,1,2,3{3,3,3} 
30 {}×tr{3,3} 
20 {6}×{6} 
Cells 540 360 t{3,4} 
90 {4,3} 
90 {}×{6} 
Faces 1560 480 {6}
1080 {4}
Edges 1800
Vertices 720
Vertex figure  
Irregular 5-cell
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal, zonotope

The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedra, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).

Alternate namesEdit

  • Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson)
  • Omnitruncated hexateron
  • Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)[6]

CoordinatesEdit

The vertices of the omnitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive orthant facet of the steriruncicantitruncated 6-orthoplex, t0,1,2,3,4{34,4},            .

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]

PermutohedronEdit

The omnitruncated 5-simplex is the permutohedron of order 6. It is also a zonotope, the Minkowski sum of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.

 
Orthogonal projection, vertices labeled as a permutohedron.

Related honeycombEdit

The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of      .

Coxeter group          
Coxeter-Dynkin                        
Picture      
Name Apeirogon Hextille Omnitruncated
3-simplex
honeycomb
Omnitruncated
4-simplex
honeycomb
Omnitruncated
5-simplex
honeycomb
Facets          

Full snub 5-simplexEdit

The full snub 5-simplex or omnisnub 5-simplex, defined as an alternation of the omnitruncated 5-simplex is not uniform, but it can be given Coxeter diagram           and symmetry [[3,3,3,3]]+, and constructed from 12 snub 5-cells, 30 snub tetrahedral antiprisms, 20 3-3 duoantiprisms, and 360 irregular 5-cells filling the gaps at the deleted vertices.

Related uniform polytopesEdit

These polytopes are a part 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, (x3o3o3o3x - scad)
  2. ^ Klitizing, (x3x3o3o3x - cappix)
  3. ^ Klitizing, (x3o3x3o3x - card)
  4. ^ Klitizing, (x3x3x3o3x - cograx)
  5. ^ Klitizing, (x3x3o3x3x - captid)
  6. ^ Klitizing, (x3x3x3x3x - gocad)

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)". x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

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