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

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)

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)

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]

Related uniform 5-polytopes

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)

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 [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)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  x3x3o3o3o - tix, o3x3x3o3o - bittix

External links

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