There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.
| bgcolor=#e7dcc3 colspan=2 | Cantellated 120-cell | |
|---|---|---|
| Type | Uniform 4-polytope | |
| Uniform index | 37 | |
| Coxeter diagram | ||
| Cells | ||
| Faces | 4800+3600+720 | |
| Edges | 10800 | |
| Vertices | 3600 | |
| Vertex figure | wedge | |
| Schläfli symbol | t0,2 | |
| Symmetry group | H4, [3,3,5], order 14400 | |
| Properties | convex |
The cantellated 120-cell is a uniform 4-polytope. It is named by its construction as a Cantellation operation applied to the regular 120-cell. It contains 1920 cells, including 120 rhombicosidodecahedra, 1200 triangular prisms, 600 octahedra. Its vertex figure is a wedge, with two rhombicosidodecahedra, two triangular prisms, and one octahedron meeting at each vertex.
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated 120-cell | |
|---|---|---|
| Type | Uniform 4-polytope | |
| Uniform index | 42 | |
| Schläfli symbol | t0,1,2 | |
| Coxeter diagram | ||
| Cells | ||
| Faces | 9120: 2400+3600+ 2400+720 | |
| Edges | 14400 | |
| Vertices | 7200 | |
| Vertex figure | sphenoid | |
| Symmetry group | H4, [3,3,5], order 14400 | |
| Properties | convex |
This 4-polytope is related to the regular 120-cell. The cantitruncation operation create new truncated tetrahedral cells at the vertices, and triangular prisms at the edges. The original dodecahedron cells are cantitruncated into great rhombicosidodecahedron cells.
The image shows the 4-polytope drawn as a Schlegel diagram which projects the 4-dimensional figure into 3-space, distorting the sizes of the cells. In addition, the decagonal faces are hidden, allowing us to see the elemented projected inside.
| bgcolor=#e7dcc3 colspan=2 | Cantellated 600-cell | |
|---|---|---|
| Type | Uniform 4-polytope | |
| Uniform index | 40 | |
| Schläfli symbol | t0,2 | |
| Coxeter diagram | ||
| Cells | 1440 total: 120 3.5.3.5 600 3.4.3.4 720 4.4.5 | |
| Faces | 8640 total: (1200+2400) +3600+1440 | |
| Edges | 10800 | |
| Vertices | 3600 | |
| Vertex figure | isosceles triangular prism | |
| Symmetry group | H4, [3,3,5], order 14400 | |
| Properties | convex |
This 4-polytope has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter diagram by removing one node at a time:
| Node | Order | Coxeter diagram | Cell | Picture | |
|---|---|---|---|---|---|
| 0 | 600 | Cantellated tetrahedron (Cuboctahedron) | |||
| 1 | 1200 | None (Degenerate triangular prism) | |||
| 2 | 720 | Pentagonal prism | |||
| 3 | 120 | Rectified dodecahedron (Icosidodecahedron) |
There are 1440 pentagonal faces between the icosidodecahedra and pentagonal prisms. There are 3600 squares between the cuboctahedra and pentagonal prisms. There are 2400 triangular faces between the icosidodecahedra and cuboctahedra, and 1200 triangular faces between pairs of cuboctahedra.
There are two classes of edges: 3-4-4, 3-4-5: 3600 have two squares and a triangle around it, and 7200 have one triangle, one square, and one pentagon.
| bgcolor=#e7dcc3 colspan=2 | Cantitruncated 600-cell | |
|---|---|---|
| Type | Uniform 4-polytope | |
| Uniform index | 45 | |
| Coxeter diagram | ||
| Cells | ||
| Faces | 8640: 3600+1440+ 3600 | |
| Edges | 14400 | |
| Vertices | 7200 | |
| Vertex figure | sphenoid | |
| Schläfli symbol | t0,1,2 | |
| Symmetry group | H4, [3,3,5], order 14400 | |
| Properties | convex |