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 |