120-cell |
Cantellated 120-cell |
Cantellated 600-cell |
600-cell |
Cantitruncated 120-cell |
Cantitruncated 600-cell |
| Orthogonal projections in H3 Coxeter plane | ||
|---|---|---|
In four-dimensional geometry, a cantellated 120-cell is a convex uniform polychoron, being a cantellation (a 2nd order truncation) of the regular 120-cell.
There are four degrees of cantellations of the 120-cell including with permutations truncations. Two are expressed relative to the dual 600-cell.
|
| Cantellated 120-cell | |
|---|---|
| Type | Uniform polychoron |
| Uniform index | 37 |
| Coxeter-Dynkin diagram | |
| Cells | 1920 total: 120 (3.4.5.4) 1200 (3.4.4) 600 (3.3.3.3) |
| Faces | 4800{3}+3600{4}+720{5} |
| Edges | 10800 |
| Vertices | 3600 |
| Vertex figure | wedge |
| Schläfli symbol | t0,2{5,3,3} |
| Symmetry group | H4, [3,3,5] |
| Properties | convex |
The cantellated 120-cell is a uniform polychoron. 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.
| H3 | A2 / B3 / D4 |
|---|---|
[10] |
[6] |
Schlegel diagram. Pentagonal face are removed. |
| Cantitruncated 120-cell | |
|---|---|
| Type | Uniform polychoron |
| Uniform index | 42 |
| Schläfli symbol | t0,1,2{5,3,3} |
| Coxeter-Dynkin diagram | |
| Cells | 1920 total: 120 (4.6.10) 1200 (3.4.4) 600 (3.6.6) |
| Faces | 9120: 2400{3}+3600{4}+ 2400{6}+720{10} |
| Edges | 14400 |
| Vertices | 7200 |
| Vertex figure | sphenoid |
| Symmetry group | H4, [3,3,5] |
| Properties | convex |
The cantitruncated 120-cell is a uniform polychoron.
This polychoron 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 polychoron 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.
| H3 | A2 / B3 / D4 |
|---|---|
[10] |
[6] |
Centered on truncated icosidodecahedron cell with decagonal faces hidden. |
| Cantellated 600-cell | |
|---|---|
| Type | Uniform polychoron |
| Uniform index | 40 |
| Schläfli symbol | t0,2{3,3,5} |
| Coxeter-Dynkin diagram | |
| Cells | 1440 total: 120 3.5.3.5 600 3.4.3.4 720 4.4.5 |
| Faces | 8640 total: (1200+2400){3} +3600{4}+1440{5} |
| Edges | 10800 |
| Vertices | 3600 |
| Vertex figure | isosceles triangular prism |
| Symmetry group | H4, [3,3,5] |
| Properties | convex |
The cantellated 600-cell is a uniform polychoron. It has 1440 cells: 120 icosidodecahedra, 600 cuboctahedra, and 720 pentagonal prisms. Its vertex figure is an isosceles triangular prism, defined by one icosidodecahedron, two cuboctahedra, and two pentagonal prisms.
This polychoron has cells at 3 of 4 positions in the fundamental domain, extracted from the Coxeter-Dynkin diagram by removing one node at a time:
| Node | Order | Coxeter-Dynkin |
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.
| H4 | - |
|---|---|
[30] |
[20] |
| F4 | H3 |
[12] |
[10] |
| A2 / B3 / D4 | A3 / B2 |
[6] |
[4] |
Stereographic projection with its 3600 green triangular faces and its 3600 blue square faces. |
| Cantitruncated 600-cell | |
|---|---|
| Type | Uniform polychoron |
| Uniform index | 45 |
| Coxeter-Dynkin diagram | |
| Cells | 1440 total: 120 5.6.6 720 4.4.5 600 4.6.6 |
| Faces | 8640: 3600{4}+1440{5}+ 3600{6} |
| Edges | 14400 |
| Vertices | 7200 |
| Vertex figure | sphenoid |
| Schläfli symbol | t0,1,2{3,3,5} |
| Symmetry group | H4, [3,3,5] |
| Properties | convex |
The cantitruncated 600-cell is a uniform polychoron. It is composed of 1440 cells: 120 rhombicosidodecahedron, 600 truncated tetrahedra, 720 pentagonal prisms, and 1200 hexagonal prisms. It has 7200 vertices, 14400 edges, and 8640 faces (3600 squares, 1440 pentagons, and 3600 hexagons). It has an irregular tetrahedral vertex figure, filled by two truncated tetrahedra, one rhombicosidodecahedron, and one hexagonal prism.
Schlegel diagram |
| H3 | A2 / B3 / D4 |
|---|---|
[10] |
[6] |
| 120-cell | rectified 120-cell |
truncated 120-cell |
cantellated 120-cell |
runcinated 120-cell |
bitruncated 120-cell |
cantitruncated 120-cell |
runcitruncated 120-cell |
omnitruncated 120-cell |
|---|---|---|---|---|---|---|---|---|
| {5,3,3} | t1{5,3,3} | t0,1{5,3,3} | t0,2{5,3,3} | t0,3{5,3,3} | t1,2{5,3,3} | t0,1,2{5,3,3} | t0,1,3{5,3,3} | t0,1,2,3{5,3,3} |
| 600-cell | rectified 600-cell |
truncated 600-cell |
cantellated 600-cell |
runcinated 600-cell |
bitruncated 600-cell |
cantitruncated 600-cell |
runcitruncated 600-cell |
omnitruncated 600-cell |
| {3,3,5} | t1{3,3,5} | t0,1{3,3,5} | t0,2{3,3,5} | t0,3{3,3,5} | t1,2{3,3,5} | t0,1,2{3,3,5} | t0,1,3{3,3,5} | t0,1,2,3{3,3,5} |