Truncated 6-simplexes

File:6-simplex t0.svg 6-simplex Template:CDD	File:6-simplex t01.svg Truncated 6-simplex Template:CDD
File:6-simplex t12.svg Bitruncated 6-simplex Template:CDD	File:6-simplex t23.svg Tritruncated 6-simplex Template:CDD
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
Class	A6 polytope
Schläfli symbol	t{3,3,3,3,3}
Coxeter-Dynkin diagram	Template:CDD Template:CDD
5-faces	14: 7 {3,3,3,3} File:5-simplex t0.svg 7 t{3,3,3,3} File:5-simplex t01.svg
4-faces	63: 42 {3,3,3} File:4-simplex t0.svg 21 t{3,3,3} File:4-simplex t01.svg
Cells	140: 105 {3,3} File:3-simplex t0.svg 35 t{3,3} File:3-simplex t01.svg
Faces	175: 140 {3} 35 {6}
Edges	126
Vertices	42
Vertex figure	File:Truncated 6-simplex verf.png ( )v{3,3,3}
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	File:6-simplex t01.svg	File:6-simplex t01 A5.svg	File:6-simplex t01 A4.svg
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	File:6-simplex t01 A3.svg	File:6-simplex t01 A2.svg
Dihedral symmetry	[4]	[3]

Bitruncated 6-simplex

Bitruncated 6-simplex
Type	uniform 6-polytope
Class	A6 polytope
Schläfli symbol	2t{3,3,3,3,3}
Coxeter-Dynkin diagram	Template:CDD Template:CDD
5-faces	14
4-faces	84
Cells	245
Faces	385
Edges	315
Vertices	105
Vertex figure	File:Bitruncated 6-simplex verf.png { }v{3,3}
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	File:6-simplex t12.svg	File:6-simplex t12 A5.svg	File:6-simplex t12 A4.svg
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	File:6-simplex t12 A3.svg	File:6-simplex t12 A2.svg
Dihedral symmetry	[4]	[3]

Tritruncated 6-simplex

Tritruncated 6-simplex
Type	uniform 6-polytope
Class	A6 polytope
Schläfli symbol	3t{3,3,3,3,3}
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	14 2t{3,3,3,3}
4-faces	84
Cells	280
Faces	490
Edges	420
Vertices	140
Vertex figure	File:Tritruncated 6-simplex verf.png {3}v{3}
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: Template:CDD and Template:CDD.

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	File:6-simplex t23.svg	File:6-simplex t23 A5.svg	File:6-simplex t23 A4.svg
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak Coxeter plane	A3	A2
Graph	File:6-simplex t23 A3.svg	File:6-simplex t23 A2.svg
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 Template:CDD = Template:CDD t{3} = {6}	Octahedron Template:CDD = Template:CDD r{3,3} = {31,1} = {3,4} $\left\{\begin{array}{c}3\\ 3\end{array}\right\}$	Decachoron Template:CDD 2t{33}	Dodecateron Template:CDD 2r{34} = {32,2} $\left\{\begin{array}{c}3,3\\ 3,3\end{array}\right\}$	Tetradecapeton Template:CDD 3t{35}	Hexadecaexon Template:CDD 3r{36} = {33,3} $\left\{\begin{array}{c}3,3,3\\ 3,3,3\end{array}\right\}$	Octadecazetton Template:CDD 4t{37}
Images	File:Truncated triangle.svg	File:3-cube t2.svgFile:Uniform polyhedron-33-t1.svg	File:4-simplex t12.svg File:Schlegel half-solid bitruncated 5-cell.png	File:5-simplex t2.svg File:5-simplex t2 A4.svg	File:6-simplex t23.svg File:6-simplex t23 A5.svg	File:7-simplex t3.svg File:7-simplex t3 A5.svg	File:8-simplex t34.svg File:8-simplex t34 A7.svg
Vertex figure	( )∨( )	File:Octahedron vertfig.svg { }×{ }	File:Bitruncated 5-cell verf.png { }∨{ }	File:Birectified hexateron verf.png {3}×{3}	File:Tritruncated 6-simplex verf.png {3}∨{3}	{3,3}×{3,3}	File:Quadritruncated 8-simplex verf.png {3,3}∨{3,3}
Facets		{3} File:Regular polygon 3 annotated.svg	t{3,3} File:Uniform polyhedron-33-t01.svg	r{3,3,3} File:Schlegel half-solid rectified 5-cell.png	2t{3,3,3,3} File:5-simplex t12.svg	2r{3,3,3,3,3} File:6-simplex t2.svg	3t{3,3,3,3,3,3} File:7-simplex t23.svg
As intersecting dual simplexes	File:Regular hexagon as intersection of two triangles.png Template:CDD ∩ Template:CDD	File:Stellated octahedron A4 A5 skew.png Template:CDD ∩ Template:CDD	File:Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png Template:CDD ∩ Template:CDD	File:Dual 5-simplex intersection graph a5.pngFile:Dual 5-simplex intersection graph a4.png Template:CDD ∩ Template:CDD	Template:CDD ∩ Template:CDD	Template:CDD ∩ Template:CDD	Template:CDD ∩ Template:CDD

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (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. "6D uniform polytopes (polypeta) with acronyms". o3x3o3o3o3o - til, o3x3x3o3o3o - batal, o3o3x3x3o3o - fe Template:Sfn whitelist

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary