Truncated 6-simplexes

Last updated March 22, 2019

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.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

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.

In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

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)^[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.

In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself.

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)^[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
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)^[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
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	( )v( )	{ }×{ }	{ }v{ }	{3}×{3}	{3}v{3}	{3,3}x{3,3}	{3,3}v{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

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

Related Research Articles

In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

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

In six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.

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

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

In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

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

In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.

In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.

In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.

In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.

In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.

In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.

In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.

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
  - (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 4-polytope	5-cell	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

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[1] Klitzing, (o3x3o3o3o3o - til)

[2] Klitzing, (o3x3x3o3o3o - batal)

[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

Truncated 6-simplexes

Contents

Truncated 6-simplex

Alternate names

Coordinates

Images

Bitruncated 6-simplex

Alternate names

Coordinates

Images

Tritruncated 6-simplex

Alternate names

Coordinates

Images

Related polytopes

Related uniform 6-polytopes

Notes

Related Research Articles

References

External links