Truncated 6-simplexes

Orthogonal projections in A₇ Coxeter plane
File:6-simplex t0.svg 6-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:6-simplex t01.svg Truncated 6-simplex File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
File:6-simplex t12.svg Bitruncated 6-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:6-simplex t23.svg Tritruncated 6-simplex File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png

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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.png
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)^[1]

Coordinates

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
A_k 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]
A_k 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	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3b.pngFile:CDel nodeb.pngFile:CDel 3b.pngFile:CDel nodeb.png
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)^[2]

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
A_k 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]
A_k 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	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png
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: File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png and File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png.

Alternate names

Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)^[3]

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
A_k 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]
A_k 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 File:CDel branch 11.png = File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.png t{3} = {6}	Octahedron File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png r{3,3} = {3^1,1} = {3,4} ${\begin{cases} 3 \\ 3 \end{cases}}$	Decachoron File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.png 2t{3³}	Dodecateron File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 2r{3⁴} = {3^2,2} ${\begin{cases} 3, 3 \\ 3, 3 \end{cases}}$	Tetradecapeton File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 3t{3⁵}	Hexadecaexon File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 3r{3⁶} = {3^3,3} ${\begin{cases} 3, 3, 3 \\ 3, 3, 3 \end{cases}}$	Octadecazetton File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.png 4t{3⁷}
Images	File:Truncated triangle.svg	File:3-cube t2.svgFile:Uniform polyhedron-33-t1.svg	File:4-simplex t12.svgFile:Schlegel half-solid bitruncated 5-cell.png	File:5-simplex t2.svgFile:5-simplex t2 A4.svg	File:6-simplex t23.svgFile:6-simplex t23 A5.svg	File:7-simplex t3.svgFile:7-simplex t3 A5.svg	File:8-simplex t34.svgFile: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.png	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 File:CDel branch 10.png ∩ File:CDel branch 01.png	File:Stellated octahedron A4 A5 skew.png File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png ∩ File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 01ld.png	File:Compound dual 5-cells and bitruncated 5-cell intersection A4 coxeter plane.png File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:Dual 5-simplex intersection graph a5.pngFile:Dual 5-simplex intersection graph a4.png File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png	File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png ∩ File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png

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.

A6 polytopes
File:6-simplex t0.svg t₀	File:6-simplex t1.svg t₁	File:6-simplex t2.svg t₂	File:6-simplex t01.svg t_0,1	File:6-simplex t02.svg t_0,2	File:6-simplex t12.svg t_1,2	File:6-simplex t03.svg t_0,3	File:6-simplex t13.svg t_1,3	File:6-simplex t23.svg t_2,3
File:6-simplex t04.svg t_0,4	File:6-simplex t14.svg t_1,4	File:6-simplex t05.svg t_0,5	File:6-simplex t012.svg t_0,1,2	File:6-simplex t013.svg t_0,1,3	File:6-simplex t023.svg t_0,2,3	File:6-simplex t123.svg t_1,2,3	File:6-simplex t014.svg t_0,1,4	File:6-simplex t024.svg t_0,2,4
File:6-simplex t124.svg t_1,2,4	File:6-simplex t034.svg t_0,3,4	File:6-simplex t015.svg t_0,1,5	File:6-simplex t025.svg t_0,2,5	File:6-simplex t0123.svg t_0,1,2,3	File:6-simplex t0124.svg t_0,1,2,4	File:6-simplex t0134.svg t_0,1,3,4	File:6-simplex t0234.svg t_0,2,3,4	File:6-simplex t1234.svg t_1,2,3,4
File:6-simplex t0125.svg t_0,1,2,5	File:6-simplex t0135.svg t_0,1,3,5	File:6-simplex t0235.svg t_0,2,3,5	File:6-simplex t0145.svg t_0,1,4,5	File:6-simplex t01234.svg t_0,1,2,3,4	File:6-simplex t01235.svg t_0,1,2,3,5	File:6-simplex t01245.svg t_0,1,2,4,5	File:6-simplex t012345.svg t_0,1,2,3,4,5

Notes

↑ Klitzing, (o3x3o3o3o3o - til)
↑ Klitzing, (o3x3x3o3o3o - batal)
↑ Klitzing, (o3o3x3x3o3o - fe)

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

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Klitzing, (o3x3o3o3o3o - til)

[2] Klitzing, (o3x3x3o3o3o - batal)

[3] Klitzing, (o3o3x3x3o3o - fe)

[1]

[2]

[3]

Truncated 6-simplexes

Contents