In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.
There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.
| bgcolor=#e7dcc3 colspan=2 | Runcinated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t0,3 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 70 | |
| 4-faces | 455 | |
| Cells | 1330 | |
| Faces | 1610 | |
| Edges | 840 | |
| Vertices | 140 | |
| Vertex figure | ||
| Coxeter group | A6, [3<sup>5</sup>], order 5040 | |
| Properties | convex |
The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | biruncinated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t1,4 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 84 | |
| 4-faces | 714 | |
| Cells | 2100 | |
| Faces | 2520 | |
| Edges | 1260 | |
| Vertices | 210 | |
| Vertex figure | ||
| Coxeter group | A6, [[35]], order 10080 | |
| Properties | convex |
The vertices of the biruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | Runcitruncated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t0,1,3 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 70 | |
| 4-faces | 560 | |
| Cells | 1820 | |
| Faces | 2800 | |
| Edges | 1890 | |
| Vertices | 420 | |
| Vertex figure | ||
| Coxeter group | A6, [3<sup>5</sup>], order 5040 | |
| Properties | convex |
The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | biruncitruncated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t1,2,4 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 84 | |
| 4-faces | 714 | |
| Cells | 2310 | |
| Faces | 3570 | |
| Edges | 2520 | |
| Vertices | 630 | |
| Vertex figure | ||
| Coxeter group | A6, [3<sup>5</sup>], order 5040 | |
| Properties | convex |
The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | Runcicantellated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t0,2,3 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 70 | |
| 4-faces | 455 | |
| Cells | 1295 | |
| Faces | 1960 | |
| Edges | 1470 | |
| Vertices | 420 | |
| Vertex figure | ||
| Coxeter group | A6, [3<sup>5</sup>], order 5040 | |
| Properties | convex |
The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | Runcicantitruncated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t0,1,2,3 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 70 | |
| 4-faces | 560 | |
| Cells | 1820 | |
| Faces | 3010 | |
| Edges | 2520 | |
| Vertices | 840 | |
| Vertex figure | ||
| Coxeter group | A6, [3<sup>5</sup>], order 5040 | |
| Properties | convex |
The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | biruncicantitruncated 6-simplex | |
|---|---|---|
| Type | uniform 6-polytope | |
| Schläfli symbol | t1,2,3,4 | |
| Coxeter-Dynkin diagrams | ||
| 5-faces | 84 | |
| 4-faces | 714 | |
| Cells | 2520 | |
| Faces | 4410 | |
| Edges | 3780 | |
| Vertices | 1260 | |
| Vertex figure | ||
| Coxeter group | A6, [[35]], order 10080 | |
| Properties | convex |
The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.
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.