321 Polytope
This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and Coxeter-Dynkin diagram: .
| 321 polytope | |
|---|---|
| Type | Uniform 7-polytope |
| Family | k21 polytope |
| Schläfli symbol | {3,3,3,32,1} |
| Coxeter symbol | 321 |
| Coxeter-Dynkin diagram | |
| 6-faces | 702 total: 126 311 576 {35} |
| 5-faces | 6048: 4032 {34} 2016 {34} |
| 4-faces | 12096 {33} |
| Cells | 10080 {3,3} |
| Faces | 4032 {3} |
| Edges | 756 |
| Vertices | 56 |
| Vertex figure | 221 polytope |
| Petrie polygon | octadecagon |
| Coxeter group | E7, order 2903040 |
| Properties | convex |
In 7-dimensional geometry, the 321 is a uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplex.
For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within a 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The 1-skeleton of the 321 polytope is called a Gosset graph.
Read more about this topic: 3 21 Polytope