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Regular polytopes
Regular Polytope in 4-space
A stereographic projection of 120-cell in 4-space
This gives a decomposition of 3-sphere into 120 regular icosahedra. Each icosahedron is a fundamental domain of the action of the action of the binary icosahedral group on the 3-sphere and the orbit space is the Poincaré homology 3-sphere.
As was shown by Schläfli in the middle of the 19th century, there are 6 kinds of regular polytopes in 4-space.
| number of 3-cells | number of vertices | shape of 3-cell |
|---|---|---|
| 5 | 5 | tetrahedron |
| 8 | 16 | cube |
| 16 | 8 | tetrahedron |
| 24 | 24 | octahedron |
| 120 | 600 | icosahedron |
| 600 | 120 | tetrahedron |
Models of regular polytopes made by Yuro Otobe
