Truncated 5-simplexes

5-simplex	Truncated 5-simplex	Bitruncated 5-simplex
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
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).

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

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.

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
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

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

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.

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

• 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

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
  • Truncated uniform polytera (tix), Jonathan Bowers
• Multi-dimensional Glossary