5-simplex
Cantellated 5-simplex
Bicantellated 5-simplex

Birectified 5-simplex
Cantitruncated 5-simplex
Bicantitruncated 5-simplex
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

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 names

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

Coordinates

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

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

Bicantellated 5-simplex

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 names

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

Coordinates

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

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-simplex

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 names

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

Coordinates

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

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

Bicantitruncated 5-simplex

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 names

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

Coordinates

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

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

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)

A5 polytopes

t0
t1
t2
t0,1
t0,2
t1,2
t0,3

t1,3
t0,4
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4

t0,2,4
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,1,2,3,4

Notes

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

References

  • 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
      • (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
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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