Truncated 6-simplexes

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

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
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	( )v{3,3,3}
Coxeter group	A₆, [3⁵], order 5040
Dual	?
Properties	convex

Alternate names

Truncated heptapeton (Acronym: til) (Jonathan Bowers)

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
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
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
5-faces	14
4-faces	84
Cells	245
Faces	385
Edges	315
Vertices	105
Vertex figure	{ }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
A_k Coxeter plane	A₆	A₅	A₄
Graph
Dihedral symmetry	[7]	[6]	[5]
A_k Coxeter plane	A₃	A₂
Graph
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	or
5-faces	14 2t{3,3,3,3}
4-faces	84
Cells	280
Faces	490
Edges	420
Vertices	140
Vertex figure	{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: 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
A_k Coxeter plane	A₆	A₅	A₄
Graph
Symmetry	[[7]]^(*)=[14]	[6]	[[5]]^(*)=[10]
A_k Coxeter plane	A₃	A₂
Graph
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} = {3^1,1} = {3,4} $\left\{{\begin{array}{l}3\\3\end{array}}\right\}$	Decachoron 2t{3³}	Dodecateron 2r{3⁴} = {3^2,2} $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$	Tetradecapeton 3t{3⁵}	Hexadecaexon 3r{3⁶} = {3^3,3} $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$	Octadecazetton 4t{3⁷}
Images
Vertex figure	( )∨( )	{ }×{ }	{ }∨{ }	{3}×{3}	{3}∨{3}	{3,3}×{3,3}	{3,3}∨{3,3}
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}
As intersecting dual simplexes	∩	∩	∩	∩	∩	∩	∩

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.

Notes

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, ISBN 978-0-471-01003-6
  - (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

External links

v t e Fundamental convex regular and uniform polytopes
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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 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 • Polytope operations

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatstilhtm_(o3x3o3o3o3o_-_til)]-1] Klitzing, (o3x3o3o3o3o - til).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsbatalhtm_(o3x3x3o3o3o_-_batal)]-2] Klitzing, (o3x3x3o3o3o - batal).

[FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsfehtm_(o3o3x3x3o3o_-_fe)]-3] Klitzing, (o3o3x3x3o3o - fe).

A6 polytopes
t₀	t₁	t₂	t_0,1	t_0,2	t_1,2	t_0,3	t_1,3	t_2,3
t_0,4	t_1,4	t_0,5	t_0,1,2	t_0,1,3	t_0,2,3	t_1,2,3	t_0,1,4	t_0,2,4
t_1,2,4	t_0,3,4	t_0,1,5	t_0,2,5	t_0,1,2,3	t_0,1,2,4	t_0,1,3,4	t_0,2,3,4	t_1,2,3,4
t_0,1,2,5	t_0,1,3,5	t_0,2,3,5	t_0,1,4,5	t_0,1,2,3,4	t_0,1,2,3,5	t_0,1,2,4,5	t_0,1,2,3,4,5