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.
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:
Note: (*) Symmetry doubled for Ak 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]
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.
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.
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.
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.
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.
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.
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.
Great teracellirhombated heptapeton (Acronym: gatocral) (Jonathan Bowers)[10]
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.
Pentisteriruncicantitruncated 6-simplex (Johnson's omnitruncation for 6-polytopes)
Omnitruncated heptapeton
Great terated tetradecapeton (Acronym: gotaf) (Jonathan Bowers)[11]
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, t0,1,2,3,4,5{35,4}, .
Note: (*) Symmetry doubled for Ak 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.[12]
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.
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, ISBN978-0-471-01003-6[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]
