Stericated 5-simplexes

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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-simplex[edit]

Stericated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2r2r{3,3,3,3}
2r{32,2} =
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
or CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png
4-faces 62 6+6 {3,3,3}Schlegel wireframe 5-cell.png
15+15 {}×{3,3}Tetrahedral prism.png
20 {3}×{3}3-3 duoprism.png
Cells 180 60 {3,3}Tetrahedron.png
120 {}×{3}Triangular prism.png
Faces 210 120 {3}
90 {4}
Edges 120
Vertices 30
Vertex figure Stericated hexateron verf.png
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 names[edit]

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

Cross-sections[edit]

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.

Coordinates[edit]

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 system[edit]

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

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t04.svg 5-simplex t04 A4.svg
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t04 A3.svg 5-simplex t04 A2.svg
Dihedral symmetry [4] [[3]]=[6]
Stericated hexateron ortho.svg
orthogonal projection with [6] symmetry

Steritruncated 5-simplex[edit]

Steritruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,4{3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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 Steritruncated 5-simplex verf.png
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t014.svg 5-simplex t014 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t014 A3.svg 5-simplex t014 A2.svg
Dihedral symmetry [4] [3]

Stericantellated 5-simplex[edit]

Stericantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,2,4{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 1.png
or CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png
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 Stericantellated 5-simplex verf.png
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate names[edit]

  • Stericantellated hexateron
  • Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)[3]

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t024.svg 5-simplex t024 A4.svg
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t024 A3.svg 5-simplex t024 A2.svg
Dihedral symmetry [4] [[3]]=[6]

Stericantitruncated 5-simplex[edit]

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

Alternate names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t0124.svg 5-simplex t0124 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t0124 A3.svg 5-simplex t0124 A2.svg
Dihedral symmetry [4] [3]

Steriruncitruncated 5-simplex[edit]

Steriruncitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,3,4{3,3,3,3}
2t{32,2}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
or CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png
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 Steriruncitruncated 5-simplex verf.png
Coxeter group A5×2, [[3,3,3,3]], order 1440
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t0134.svg 5-simplex t0134 A4.svg
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t0134 A3.svg 5-simplex t0134 A2.svg
Dihedral symmetry [4] [[3]]=[6]

Omnitruncated 5-simplex[edit]

Omnitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3}
2tr{32,2}
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 1.png
or CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png
4-faces 62 12 t0,1,2,3{3,3,3}Schlegel half-solid omnitruncated 5-cell.png
30 {}×tr{3,3}Truncated octahedral prism.png
20 {6}×{6}6-6 duoprism.png
Cells 540 360 t{3,4}Truncated octahedron.png
90 {4,3}Tetragonal prism.png
90 {}×{6}Hexagonal prism.png
Faces 1560 480 {6}
1080 {4}
Edges 1800
Vertices 720
Vertex figure Omnitruncated 5-simplex verf.png
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 octahedrons, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).

Alternate names[edit]

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

Coordinates[edit]

The vertices of the truncated 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}, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t01234.svg 5-simplex t01234 A4.svg
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t01234 A3.svg 5-simplex t01234 A2.svg
Dihedral symmetry [4] [[3]]=[6]

Permutohedron[edit]

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.

Omnitruncated Hexateron as Permutohedron.svg
Orthogonal projection, vertices labeled as a permutohedron.

Related honeycomb[edit]

The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png.

Coxeter group
Coxeter-Dynkin CDel node 1.pngCDel infin.pngCDel node 1.png CDel branch 11.pngCDel split2.pngCDel node 1.png CDel branch 11.pngCDel 3ab.pngCDel branch 11.png CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel branch 11.pngCDel 3ab.pngCDel nodes 11.pngCDel 3ab.pngCDel branch 11.png
Picture Uniform apeirogon.png Uniform tiling 333-t012.png Bitruncated cubic honeycomb4.png
Name Apeirogon Hextille Omnitruncated
3-simplex
honeycomb
Omnitruncated
4-simplex
honeycomb
Omnitruncated
5-simplex
honeycomb
Facets Segment definition.svg Omnitruncated 2-simplex graph.png 3-simplex t012.svg 4-simplex t0123.svg 5-simplex t01234.svg

Related uniform polytopes[edit]

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)

Notes[edit]

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

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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / 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