Cantellated 5-simplexes

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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-simplex[edit]

Cantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol rr{3,3,3,3} =
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
or CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4-faces 27 6 r{3,3,3}Schlegel half-solid rectified 5-cell.png
6 rr{3,3,3}Schlegel half-solid cantellated 5-cell.png
15 {}x{3,3}Tetrahedral prism.png
Cells 135 30 {3,3}Tetrahedron.png
30 r{3,3}Uniform polyhedron-33-t1.png
15 rr{3,3}Cantellated tetrahedron.png
60 {}x{3}Triangular prism.png
Faces 290 200 {3}
90 {4}
Edges 240
Vertices 60
Vertex figure Cantellated hexateron verf.png
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 names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t02.svg 5-simplex t02 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t02 A3.svg 5-simplex t02 A2.svg
Dihedral symmetry [4] [3]

Bicantellated 5-simplex[edit]

Bicantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2rr{3,3,3,3} =
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
or CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png
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 Bicantellated 5-simplex verf.png
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t13.svg 5-simplex t13 A4.svg
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t13 A3.svg 5-simplex t13 A2.svg
Dihedral symmetry [4] [[3]]=[6]

Cantitruncated 5-simplex[edit]

cantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol tr{3,3,3,3} =
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
or CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
4-faces 27 6 t012{3,3,3}4-simplex t012.svg
6 t{3,3,3}4-simplex t01.svg
15 {}x{3,3}
Cells 135 15 t012{3,3} 3-simplex t012.svg
30 t{3,3}3-simplex t01.svg
60 {}x{3}
30 {3,3}3-simplex t0.svg
Faces 290 120 {3}2-simplex t0.svg
80 {6}2-simplex t01.svg
90 {}x{}2-cube.svg
Edges 300
Vertices 120
Vertex figure Canitruncated 5-simplex verf.png
Irr. 5-cell
Coxeter group A5 [3,3,3,3], order 720
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t012.svg 5-simplex t012 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t012 A3.svg 5-simplex t012 A2.svg
Dihedral symmetry [4] [3]

Bicantitruncated 5-simplex[edit]

Bicantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2tr{3,3,3,3} =
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
or CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png
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 Bicanitruncated 5-simplex verf.png
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t123.svg 5-simplex t123 A4.svg
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t123 A3.svg 5-simplex t123 A2.svg
Dihedral symmetry [4] [[3]]=[6]

Related uniform 5-polytopes[edit]

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)

Notes[edit]

  1. ^ Klitizing, (x3o3x3o3o - sarx)
  2. ^ Klitizing, (o3x3o3x3o - sibrid)
  3. ^ Klitizing, (x3x3x3o3o - garx)
  4. ^ Klitizing, (o3x3x3x3o - gibrid)

References[edit]

  • 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 links[edit]

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