Truncated 5-simplexes

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


5-simplex
Truncated 5-simplex
Bitruncated 5-simplex
Orthogonal projections in A5 Coxeter plane

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
Truncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 12 6 {3,3,3}
6 t{3,3,3}
Cells 45 30 {3,3}
15 t{3,3}
Faces 80 60 {3}
20 {6}
Edges 75
Vertices 30
Vertex figure
( )v{3,3}
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
  • Truncated hexateron (Acronym: tix) (Jonathan Bowers)[1]

Coordinates

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

Bitruncated 5-simplex
bitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol 2t{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 12 6 2t{3,3,3}
6 t{3,3,3}
Cells 60 45 {3,3}
15 t{3,3}
Faces 140 80 {3}
60 {6}
Edges 150
Vertices 60
Vertex figure
{ }v{3}
Coxeter group A5 [3,3,3,3], order 720
Properties convex

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

Coordinates

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

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)

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, (x3x3o3o3o - tix)
  2. Klitizing, (o3x3x3o3o - bittix)

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)". x3x3o3o3o - tix, o3x3x3o3o - bittix
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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