Icosahedron (11-dimensional)
| Icosahedron (11-dimensional) | |
|---|---|
| Rank | 3 |
| Dimension | 11 |
| Elements | |
| Faces | 20 triangles |
| Edges | 30 |
| Vertices | 12 |
| Vertex figure | Pentagonal pentagrammic coil |
| Petrie polygons | 6 9-dimensional decagons |
| Holes | 12 pentagonal-pentagrammic coils |
| Related polytopes | |
| Dual | Exists |
| Petrie dual | Petrial icosahedron (11-dimensional) |
| Conjugate | Icosahedron (11-dimensional) |
| Abstract & topological properties | |
| Flag count | 120 |
| Euler characteristic | 2 |
| Schläfli type | {3,5} |
| Orientable | Yes |
| Genus | 0 |
| Properties | |
| Convex | No |
The 11-dimensional realization of the icosahedron is a regular skew polyhedron in 11-dimensional Euclidean space. It is the blend of the convex icosahedron, the great icosahedron and the hemiicosahedron.
It is the simplex realization of the icosahedron and thus has the highest dimension of any realization of the icosahedron.
Vertex coordinates[edit | edit source]
Since the 11-dimensional icosahedron is a simplex realization, its vertex coordinates can be given as those of the 11-simplex. However since it is the blend of 3 pure polytopes, it has two degrees of freedom corresponding to the relative scaling of its components, and as a result has more general coordinates.
Related polytopes[edit | edit source]
: Faithful realization
: Non-faithful realization
: Blend with the icosahedron.
: Blend with the great icosahedron.
: Blend with the hemiicosahedron.
The 11-dimensional icosahedron is a realization of the abstract regular polytope {3,5}. In total there are 6 faithful symmetric realizations of this polytope, of which 2 are pure. There are an additional 2 degenerate symmetric realizations, one of which is pure.