Pentellated 6-simplexes

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 6-polytope
Schläfli symbol	t_0,5{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)^[1]

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.

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 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.^[2]

f_k	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
f₂	4	4	*	210	0	6	0	6	6	0	2	6
f₃	4	6	4	0	210	*	2	2	0	1	2	1
f₃	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