5-simplex
Runcinated 5-simplex
Runcitruncated 5-simplex

Birectified 5-simplex
Runcicantellated 5-simplex
Runcicantitruncated 5-simplex
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-simplex

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 names

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

Coordinates

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.

Images

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

Runcitruncated 5-simplex

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 names

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

Coordinates

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.

Images

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

Runcicantellated 5-simplex

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 names

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

Coordinates

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.

Images

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

Runcicantitruncated 5-simplex

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 names

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

Coordinates

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.

Images

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

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)

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

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