Runcinated 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 t03.svg
Runcinated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t013.svg
Runcitruncated 5-simplex
CDel node 1.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 t023.svg
Runcicantellated 5-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-simplex t0123.svg
Runcicantitruncated 5-simplex
CDel node 1.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 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[edit]

Runcinated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,3{3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 47 6 t0,3{3,3,3} 4-simplex t03.svg
20 {3}×{3}
15 { }×r{3,3}
6 r{3,3,3} 4-simplex t1.svg
Cells 255 45 {3,3} 3-simplex t0.svg
180 { }×{3}
30 r{3,3} 3-simplex t1.svg
Faces 420 240 {3} 2-simplex t0.svg
180 {4}
Edges 270
Vertices 60
Vertex figure Runcinated 5-simplex verf.png
Coxeter group A5 [3,3,3,3], order 720
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t03.svg 5-simplex t03 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t03 A3.svg 5-simplex t03 A2.svg
Dihedral symmetry [4] [3]

Runcitruncated 5-simplex[edit]

Runcitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,3{3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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 Runcitruncated 5-simplex verf.png
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t013.svg 5-simplex t013 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t013 A3.svg 5-simplex t013 A2.svg
Dihedral symmetry [4] [3]

Runcicantellated 5-simplex[edit]

Runcicantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3{3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 47
Cells 255
Faces 570
Edges 540
Vertices 180
Vertex figure Runcicantellated 5-simplex verf.png
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t023.svg 5-simplex t023 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t023 A3.svg 5-simplex t023 A2.svg
Dihedral symmetry [4] [3]

Runcicantitruncated 5-simplex[edit]

Runcicantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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 Runcicantitruncated 5-simplex verf.png
Irregular 5-cell
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t0123.svg 5-simplex t0123 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t0123 A3.svg 5-simplex t0123 A2.svg
Dihedral symmetry [4] [3]

Related uniform 5-polytopes[edit]

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)

Notes[edit]

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

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

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