Regular polytopes

Regular Polytope in 4-space

polytope.jpg
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-cellsnumber of verticesshape 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

120cells_otobe.jpg

high resolution picture for 120-cell

high resolution picture for 600-cell