Pentellated 6-simplexes

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Orthogonal projections in A₆ Coxeter plane
6-simplex	Pentellated 6-simplex	Pentitruncated 6-simplex	Penticantellated 6-simplex
Penticantitruncated 6-simplex	Pentiruncitruncated 6-simplex	Pentiruncicantellated 6-simplex	Pentiruncicantitruncated 6-simplex
Pentisteritruncated 6-simplex	Pentistericantitruncated 6-simplex	Pentisteriruncicantitruncated 6-simplex (Omnitruncated 6-simplex)

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

There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-simplex is also called an expanded 6-simplex, constructed by an expansion operation applied to the regular 6-simplex. The highest form, the pentisteriruncicantitruncated 6-simplex, is called an omnitruncated 6-simplex with all of the nodes ringed.

Pentellated 6-simplex

Pentellated 6-simplex
Type	Uniform polypeton
Schläfli symbol	t_0,5{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces	126: 7+7 {3⁴} 21+21 {}x{3,3,3} 35+35 {3}x{3,3}
4-faces	434
Cells	630
Faces	490
Edges	210
Vertices	42
Vertex figure	5-cell antiprism
Coxeter group	A₆ [[3,3,3,3,3]], order 10080
Properties	convex

Alternate names

Expanded 6-simplex
Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)^[1]

Coordinates

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

A second construction in 7-space, from the center of a rectified 7-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0,0)

Root vectors

Its 42 vertices represent the root vectors of the simple Lie group A₆. It is the vertex figure of the 6-simplex honeycomb.

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 Coxter-Dynkin diagram.

Pentitruncated 6-simplex

Pentitruncated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,1,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	826
Cells	1785
Faces	1820
Edges	945
Vertices	210
Vertex figure
Coxeter group	A₆, [3,3,3,3,3], order 5040
Properties	convex

Alternate names

Teracellated heptapeton (Acronym: tocal) (Jonathan Bowers)^[2]

Coordinates

The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the runcitruncated 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]

Penticantellated 6-simplex

Penticantellated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1246
Cells	3570
Faces	4340
Edges	2310
Vertices	420
Vertex figure
Coxeter group	A₆, [3,3,3,3,3], order 5040
Properties	convex

Alternate names

Teriprismated heptapeton (Acronym: topal) (Jonathan Bowers)^[3]

Coordinates

The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,1,2,3). This construction is based on facets of the penticantellated 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]

Penticantitruncated 6-simplex

penticantitruncated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,1,2,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1351
Cells	4095
Faces	5390
Edges	3360
Vertices	840
Vertex figure
Coxeter group	A₆, [3,3,3,3,3], order 5040
Properties	convex

Alternate names

Terigreatorhombated heptapeton (Acronym: togral) (Jonathan Bowers)^[4]

Coordinates

The vertices of the penticantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 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]

Pentiruncitruncated 6-simplex

pentiruncitruncated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,1,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1491
Cells	5565
Faces	8610
Edges	5670
Vertices	1260
Vertex figure
Coxeter group	A₆, [3,3,3,3,3], order 5040
Properties	convex

Alternate names

Tericellirhombated heptapeton (Acronym: tocral) (Jonathan Bowers)^[5]

Coordinates

The vertices of the pentiruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,1,2,3,4). This construction is based on facets of the pentiruncitruncated 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]

Pentiruncicantellated 6-simplex

Pentiruncicantellated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1596
Cells	5250
Faces	7560
Edges	5040
Vertices	1260
Vertex figure
Coxeter group	A₆, [[3,3,3,3,3]], order 10080
Properties	convex

Alternate names

Teriprismatorhombated tetradecapeton (Acronym: taporf) (Jonathan Bowers)^[6]

Coordinates

The vertices of the pentiruncicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 7-orthoplex.

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 Coxter-Dynkin diagram.

Pentiruncicantitruncated 6-simplex

Pentiruncicantitruncated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,1,2,3,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1701
Cells	6825
Faces	11550
Edges	8820
Vertices	2520
Vertex figure
Coxeter group	A₆, [3,3,3,3,3], order 5040
Properties	convex

Alternate names

Terigreatoprismated heptapeton (Acronym: tagopal) (Jonathan Bowers)^[7]

Coordinates

The vertices of the pentiruncicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 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]

Pentisteritruncated 6-simplex

Pentisteritruncated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,1,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1176
Cells	3780
Faces	5250
Edges	3360
Vertices	840
Vertex figure
Coxeter group	A₆, [[3,3,3,3,3]], order 10080
Properties	convex

Alternate names

Tericellitruncated tetradecapeton (Acronym: tactaf) (Jonathan Bowers)^[8]

Coordinates

The vertices of the pentisteritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,2,3,4). This construction is based on facets of the pentisteritruncated 7-orthoplex.

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 Coxter-Dynkin diagram.

Pentistericantitruncated 6-simplex

pentistericantitruncated 6-simplex
Type	uniform polypeton
Schläfli symbol	t_0,1,2,4,5{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	126
4-faces	1596
Cells	6510
Faces	11340
Edges	8820
Vertices	2520
Vertex figure
Coxeter group	A₆, [3,3,3,3,3], order 5040
Properties	convex

Alternate names

Great teracellirhombated heptapeton (Acronym: gatocral) (Jonathan Bowers)^[9]

Coordinates

The vertices of the pentistericantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 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]

Omnitruncated 6-simplex

Omnitruncated 6-simplex
Type	Uniform 6-polytope
Schläfli symbol	t_0,1,2,3,4,5{3⁵}
Coxeter-Dynkin diagrams
5-faces	126: 14 t_0,1,2,3,4{3⁴} 42 {}xt_0,1,2,3{3³} x 70 {6}xt_0,1,2,3{3,3} x
4-faces	1806
Cells	8400
Faces	16800: 4200 {6} 1260 {4}
Edges	15120
Vertices	5040
Vertex figure	irregular 5-simplex
Coxeter group	A₆, [[3⁵]], order 10080
Properties	convex, isogonal, zonotope

The omnitruncated 6-simplex has 5040 vertices, 15120 edges,16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-simplex.

Alternate names

Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
Omnitruncated heptapeton
Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)^[10]

Permutohedron and related tessellation

The omnitruncated 6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through the origin and the seven vertices of the 6-simplex.

Like all uniform omnitruncated n-simplices, the omnitruncated 6-simplex can tessellate space by itself, in this case 6-dimensional space with three facets around each hypercell. It has Coxeter-Dynkin diagram of .

Coordinates

The vertices of the omnitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 7-orthoplex, t_0,1,2,3,4,5{3⁵,4}, .

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 Coxter-Dynkin diagram.

Related uniform 6-polytopes

The pentellated 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.

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

Notes

↑ Klitzing, (x3o3o3o3o3x - staf)
↑ Klitzing, (x3x3o3o3o3x - tocal)
↑ Klitzing, (x3o3x3o3o3x - topal)
↑ Klitzing, (x3x3x3o3o3x - togral)
↑ Klitzing, (x3x3o3x3o3x - tocral)
↑ Klitzing, (x3o3x3x3o3x - taporf)
↑ Klitzing, (x3x3x3o3x3x - tagopal)
↑ Klitzing, (x3x3o3o3x3x - tactaf)
↑ Klitzing, (x3x3x3o3x3x - gatocral)
↑ Klitzing, (x3x3x3x3x3x - gotaf)

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, editied 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.
Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o3o3o3x - staf, x3x3o3o3o3x - tocal, x3o3x3o3o3x - topal, x3x3x3o3o3x - togral, x3x3o3x3o3x - tocral, x3x3x3x3o3x - tagopal, x3x3o3o3x3x - tactaf, x3x3x3o3x3x - tacogral, x3x3x3x3x3x - gotaf

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	BC_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	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

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