Truncated 6-simplexes

6-simplex	Truncated 6-simplex
Bitruncated 6-simplex	Tritruncated 6-simplex
Orthogonal projections in A7 Coxeter plane

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.

Truncated 6-simplex

Truncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces	14: 7 {3,3,3,3} 7 t{3,3,3,3}
4-faces	63: 42 {3,3,3} 21 t{3,3,3}
Cells	140: 105 {3,3} 35 t{3,3}
Faces	175: 140 {3} 35 {6}
Edges	126
Vertices	42
Vertex figure	Elongated 5-cell pyramid
Coxeter group	A6, [35], order 5040
Dual	?
Properties	convex

Alternate names

• Truncated heptapeton (Acronym: til) (Jonathan Bowers)

The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Bitruncated 6-simplex

Bitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	2t{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces	14
4-faces	84
Cells	245
Faces	385
Edges	315
Vertices	105
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)

Coordinates

The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Tritruncated 6-simplex

Tritruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	3t{3,3,3,3,3}
Coxeter-Dynkin diagram	or
5-faces	14 2t{3,3,3,3}
4-faces	84
Cells	280
Faces	490
Edges	420
Vertices	140
Vertex figure
Coxeter group	A6, [[35]], order 10080
Properties	convex, isotopic

The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.

The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration: and .

Alternate names

• Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)

Coordinates

The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Symmetry	[4]	[[3]](*)=[6]

Note: (*) Symmetry doubled for A_k graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

Related polytopes

Isotopic uniform truncated simplices

Dim.	2	3	4	5	6	7	8
Name Coxeter	Hexagon = t{3} = {6}	Octahedron = r{3,3} = {31,1} = {3,4} $\left\{\begin{array}{l}3\\3\end{array}\right\}$	Decachoron 2t{33}	Dodecateron 2r{34} = {32,2} $\left\{\begin{array}{l}3, 3\\3 ,3\end{array}\right\}$	Tetradecapeton 3t{35}	Hexadecaexon 3r{36} = {33,3} $\left\{\begin{array}{l}3, 3, 3\\3, 3, 3\end{array}\right\}$	Octadecazetton 4t{37}
Images					60px	60px60px	60px60px
Facets		{3}	t{3,3}	r{3,3,3}	2t{3,3,3,3}	2r{3,3,3,3,3}	3t{3,3,3,3,3,3} 30px
As intersecting dual simplexes	∩	120px ∩	60px ∩	60px60px ∩	∩	∩	∩

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A₆ Coxeter plane orthographic projections.

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.
• Richard Klitzing, 6D, uniform polytopes (polypeta) o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary