These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry.
There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.
| Rectified 10-simplex | ||
|---|---|---|
| Type | uniform polyxennon | |
| Schläfli symbol | t1 | |
| Coxeter-Dynkin diagrams | ||
| 9-faces | 22 | |
| 8-faces | 165 | |
| 7-faces | 660 | |
| 6-faces | 1650 | |
| 5-faces | 2772 | |
| 4-faces | 3234 | |
| Cells | 2640 | |
| Faces | 1485 | |
| Edges | 495 | |
| Vertices | 55 | |
| Vertex figure | 9-simplex prism | |
| Petrie polygon | decagon | |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] | |
| Properties | convex | |
The rectified 10-simplex is the vertex figure of the 11-demicube.
The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.
| Birectified 10-simplex | ||
|---|---|---|
| Type | uniform 9-polytope | |
| Schläfli symbol | t2 | |
| Coxeter-Dynkin diagrams | ||
| 8-faces | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 1980 | |
| Vertices | 165 | |
| Vertex figure | x | |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] | |
| Properties | convex | |
The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.
| Trirectified 10-simplex | ||
|---|---|---|
| Type | uniform polyxennon | |
| Schläfli symbol | t3 | |
| Coxeter-Dynkin diagrams | ||
| 8-faces | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 4620 | |
| Vertices | 330 | |
| Vertex figure | x | |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] | |
| Properties | convex | |
The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.
| Quadrirectified 10-simplex | ||
|---|---|---|
| Type | uniform polyxennon | |
| Schläfli symbol | t4 | |
| Coxeter-Dynkin diagrams | ||
| 8-faces | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 6930 | |
| Vertices | 462 | |
| Vertex figure | x | |
| Coxeter groups | A10, [3,3,3,3,3,3,3,3,3] | |
| Properties | convex | |
The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.