Bitruncated_5-simplex

Truncated 5-simplexes

Truncated 5-simplexes

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

More information 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

More information Ak Coxeter plane, A5 ...

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

More information Ak Coxeter plane, A5 ...

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)

More information A5 polytopes ...

Notes

  1. [1]
    Klitizing, (x3x3o3o3o - tix)
  2. [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
More information Family, Regular polygon ...
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