Pentellated 6-simplexes

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)
Orthogonal projections in A6 Coxeter plane

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 6-polytope
Schläfli symbol	t0,5{3,3,3,3,3}
Coxeter-Dynkin diagram
5-faces	126: 7+7 {34} 21+21 {}×{3,3,3} 35+35 {3}×{3,3}
4-faces	434
Cells	630
Faces	490
Edges	210
Vertices	42
Vertex figure	5-cell antiprism
Coxeter group	A6×2, [[3,3,3,3,3]], order 10080
Properties	convex

Alternate names

• Expanded 6-simplex
• Small terated tetradecapeton (Acronym: staf) (Jonathan Bowers)

Cross-sections

The maximal cross-section of the pentellated 6-simplex with a 5-dimensional hyperplane is a stericated hexateron. This cross-section divides the pentellated 6-simplex into two hexateral hypercupolas consisting of 7 5-simplexes, 21 5-cell prisms and 35 Tetrahedral-Triangular duoprisms each.

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

Ak Coxeter plane	A6	A5	A4
Graph
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak 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.

Configuration

This configuration matrix represents the expanded 6-simplex, with 12 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

Element	fk	f0	f1	f2		f3		f4			f5
	f0	42	10	20	20	20	60	10	40	30	2	10	20
	f1	2	210	4	4	6	18	4	16	12	1	5	10
	f2	3	3	280	*	3	3	3	6	3	1	3	4
		4	4	*	210	0	6	0	6	6	0	2	6
	f3	4	6	4	0	210	*	2	2	0	1	2	1
		6	9	2	3	*	420	0	2	2	0	1	3
	f4	5	10	10	0	5	0	84	*	*	1	1	0
		8	16	8	6	2	4	*	210	*	0	1	1
		9	18	6	9	0	6	*	*	140	0	0	2
	f5	6	15	20	0	15	0	6	0	0	14	*	*
		10	25	20	10	10	10	2	5	0	*	42	*
		12	30	16	18	3	18	0	3	4	*	*	70

Pentitruncated 6-simplex

Pentitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Penticantellated 6-simplex

Penticantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,2,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Penticantitruncated 6-simplex

penticantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Pentiruncitruncated 6-simplex

pentiruncitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,3,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Pentiruncicantellated 6-simplex

Pentiruncicantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,2,3,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak 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.

Pentiruncicantitruncated 6-simplex

Pentiruncicantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,3,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Pentisteritruncated 6-simplex

Pentisteritruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,4,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak 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.

Pentistericantitruncated 6-simplex

pentistericantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,4,5{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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Omnitruncated 6-simplex

Omnitruncated 6-simplex
Type	Uniform 6-polytope
Schläfli symbol	t0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams
5-faces	126: 14 t0,1,2,3,4{34} 42 {}×t0,1,2,3{33} × 70 {6}×t0,1,2{3,3} ×
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)

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

Ak Coxeter plane	A6	A5	A4
Graph
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak 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.

Configuration

This configuration matrix represents the omnitruncated 6-simplex, with 35 permutations of elements. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole polytope. The nondiagonal numbers say how many of the column's element occur in or at the row's element.

Element	fk	f0	f1			f2									f3										f4									f5
	f0	5040	2	2	2	2	2	2	2	1	2	2	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	1	2	1	1	2	2	2
	f1	2	5040	*	*	1	1	1	1	1	0	0	0	0	1	1	1	2	1	1	2	1	0	0	1	1	2	1	2	1	1	1	0	1	2	2
		2	*	5040	*	1	0	0	1	0	1	1	1	0	1	1	2	1	0	1	0	1	1	2	1	2	1	2	1	1	1	0	1	2	1	2
		2	*	*	5040	0	1	1	0	0	1	1	0	1	1	1	0	0	2	1	1	1	2	1	2	1	1	1	1	0	2	1	1	2	2	1
	f2	6	3	3	0	1680	*	*	*	*	*	*	*	*	1	1	1	1	0	0	0	0	0	0	1	1	1	1	1	1	0	0	0	1	1	2
		4	2	0	2	*	2520	*	*	*	*	*	*	*	1	0	0	0	1	1	1	0	0	0	1	1	1	0	1	0	1	1	0	1	2	1
		4	2	0	2	*	*	2520	*	*	*	*	*	*	0	1	0	0	1	0	1	1	0	0	1	0	1	1	1	0	1	1	0	1	2	1
		4	2	2	0	*	*	*	2520	*	*	*	*	*	0	0	1	1	0	1	0	1	0	0	0	1	1	1	1	1	1	0	0	1	1	2
		4	4	0	0	*	*	*	*	1260	*	*	*	*	0	0	0	2	0	0	2	0	0	0	0	0	2	0	2	1	0	1	0	0	2	2
		6	0	3	3	*	*	*	*	*	1680	*	*	*	1	0	0	0	0	0	0	1	1	1	1	1	1	1	0	0	1	0	1	2	1	1
		4	0	2	2	*	*	*	*	*	*	2520	*	*	0	1	0	0	0	1	0	0	1	1	1	1	0	1	1	0	1	0	1	2	1	1
		4	0	4	0	*	*	*	*	*	*	*	1260	*	0	0	2	0	0	0	0	0	0	2	0	2	0	2	0	1	0	0	1	2	0	2
		6	0	0	6	*	*	*	*	*	*	*	*	840	0	0	0	0	2	0	0	0	2	0	2	0	0	0	0	0	2	1	1	2	2	0
	f3	24	12	12	12	4	6	0	0	0	4	0	0	0	420	*	*	*	*	*	*	*	*	*	1	1	1	0	0	0	0	0	0	1	1	1
		12	6	6	6	2	0	3	0	0	0	3	0	0	*	840	*	*	*	*	*	*	*	*	1	0	0	1	1	0	0	0	0	1	1	1
		12	6	12	0	2	0	0	3	0	0	0	3	0	*	*	840	*	*	*	*	*	*	*	0	1	0	1	0	1	0	0	0	1	0	2
		12	12	6	0	2	0	0	3	3	0	0	0	0	*	*	*	840	*	*	*	*	*	*	0	0	1	0	1	1	0	0	0	0	1	2
		12	6	0	12	0	3	3	0	0	0	0	0	2	*	*	*	*	840	*	*	*	*	*	1	0	0	0	0	0	1	1	0	1	2	0
		8	4	4	4	0	2	0	2	0	0	2	0	0	*	*	*	*	*	1260	*	*	*	*	0	1	0	0	1	0	1	0	0	1	1	1
		8	8	0	4	0	2	2	0	2	0	0	0	0	*	*	*	*	*	*	1260	*	*	*	0	0	1	0	1	0	0	1	0	0	2	1
		12	6	6	6	0	0	3	3	0	2	0	0	0	*	*	*	*	*	*	*	840	*	*	0	0	1	1	0	0	1	0	0	1	1	1
		24	0	12	24	0	0	0	0	0	4	6	0	4	*	*	*	*	*	*	*	*	420	*	1	0	0	0	0	0	1	0	1	2	1	0
		12	0	12	6	0	0	0	0	0	2	3	3	0	*	*	*	*	*	*	*	*	*	840	0	1	0	1	0	0	0	0	1	2	0	1
	f4	120	60	60	120	20	30	30	0	0	20	30	0	20	5	10	0	0	10	0	0	0	5	0	84	*	*	*	*	*	*	*	*	1	1	0
		48	24	48	24	8	12	0	12	0	8	12	12	0	2	0	4	0	0	6	0	0	0	4	*	210	*	*	*	*	*	*	*	1	0	1
		48	48	24	24	8	12	12	12	12	8	0	0	0	2	0	0	4	0	0	6	4	0	0	*	*	210	*	*	*	*	*	*	0	1	1
		36	18	36	18	6	0	9	9	0	6	9	9	0	0	3	3	0	0	0	0	3	0	3	*	*	*	280	*	*	*	*	*	1	0	1
		24	24	12	12	4	6	6	6	6	0	6	0	0	0	2	0	2	0	3	3	0	0	0	*	*	*	*	420	*	*	*	*	0	1	1
		36	36	36	0	12	0	0	18	9	0	0	9	0	0	0	6	6	0	0	0	0	0	0	*	*	*	*	*	140	*	*	*	0	0	2
		48	24	24	48	0	12	12	12	0	8	12	0	8	0	0	0	0	4	6	0	4	2	0	*	*	*	*	*	*	210	*	*	1	1	0
		24	24	0	24	0	12	12	0	6	0	0	0	4	0	0	0	0	4	0	6	0	0	0	*	*	*	*	*	*	*	210	*	0	2	0
		120	0	120	120	0	0	0	0	0	40	60	30	20	0	0	0	0	0	0	0	0	10	20	*	*	*	*	*	*	*	*	42	2	0	0
	f5	720	360	720	720	120	180	180	180	0	240	360	180	120	30	60	60	0	60	90	0	60	60	120	6	15	0	20	0	0	15	0	6	14	*	*
		240	240	120	240	40	120	120	60	60	40	60	0	40	10	20	0	20	40	30	60	20	10	0	2	0	5	0	10	0	5	10	0	*	42	*
		144	144	144	72	48	36	36	72	36	24	36	36	0	6	12	24	24	0	18	18	12	0	12	0	3	3	4	6	4	0	0	0	*	*	70

Full snub 6-simplex

The full snub 6-simplex or omnisnub 6-simplex, defined as an alternation of the omnitruncated 6-simplex is not uniform, but it can be given Coxeter diagram and symmetry [[3,3,3,3,3]]⁺, and constructed from 14 snub 5-simplexes, 42 snub 5-cell antiprisms, 70 3-s{3,4} duoantiprisms, and 2520 irregular 5-simplexes filling the gaps at the deleted vertices.

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.

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)". x3o3o3o3o3x - staf, x3x3o3o3o3x - tocal, x3o3x3o3o3x - topal, x3x3x3o3o3x - togral, x3x3o3x3o3x - tocral, x3x3x3x3o3x - tagopal, x3x3o3o3x3x - tactaf, x3x3x3o3x3x - tacogral, x3x3x3x3x3x - gotaf

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary