Truncated 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 t01.svg
Truncated 5-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-simplex t12.svg
Bitruncated 5-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in A5 Coxeter plane

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.

Truncated 5-simplex[edit]

Truncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t{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.png
CDel branch 11.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
4-faces 12 6 {3,3,3}Schlegel wireframe 5-cell.png
6 t{3,3,3}Schlegel half-solid rectified 5-cell.png
Cells 45 30 {3,3}Tetrahedron.png
15 t{3,3}Truncated tetrahedron.png
Faces 80 60 {3}
20 {6}
Edges 75
Vertices 30
Vertex figure Truncated 5-simplex verf.png
Tetra.pyr
Coxeter group A5 [3,3,3,3], order 720
Properties convex

The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).

Alternate names[edit]

  • Truncated hexateron (Acronym: tix) (Jonathan Bowers)[1]

Coordinates[edit]

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t01.svg 5-simplex t01 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t01 A3.svg 5-simplex t01 A2.svg
Dihedral symmetry [4] [3]

Bitruncated 5-simplex[edit]

bitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2t{3,3,3,3}
Coxeter-Dynkin diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
4-faces 12 6 2t{3,3,3}4-simplex t12.svg
6 t{3,3,3}4-simplex t01.svg
Cells 60 45 {3,3}3-simplex t0.svg
15 t{3,3}3-simplex t01.svg
Faces 140 80 {3}2-simplex t0.svg
60 {6}2-simplex t01.svg
Edges 150
Vertices 60
Vertex figure Bitruncated 5-simplex verf.png
Triangular-pyramidal pyramid
Coxeter group A5 [3,3,3,3], order 720
Properties convex

Alternate names[edit]

  • Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)[2]

Coordinates[edit]

The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.

Images[edit]

orthographic projections
Ak
Coxeter plane
A5 A4
Graph 5-simplex t12.svg 5-simplex t12 A4.svg
Dihedral symmetry [6] [5]
Ak
Coxeter plane
A3 A2
Graph 5-simplex t12 A3.svg 5-simplex t12 A2.svg
Dihedral symmetry [4] [3]

Related uniform 5-polytopes[edit]

The truncated 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, (x3x3o3o3o - tix)
  2. ^ Klitizing, (o3x3x3o3o - bittix)

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)".  x3x3o3o3o - tix, o3x3x3o3o - bittix

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