| The Swiss mathematician Ludwig Schläfli was the first in 1852 to classify all regular polytopes in higher dimensions. In 4 dimensions, there are six polytopes: the 5-cell, the 8-cell, the 16-cell, the 24-cell, the 120-cell and the 600-cell. These polytopes are also called polychora. In 5 or more dimensions, there are only 3 polytopes, the simplices, the hypercubes and cross polytopes. In this movie, we show animations of these polyopes in 4 and 5 dimensions. In 5 dimensions, the polytopes are called polytera. The pictures were generated in Mathematica 7. The 4-polytopes were photographed in 3 space with the stereographic projection P(x,y,z,w)=(x,y,z)/(1-w). The pictures of the 5-polytopes were obtained by taking two successive stereographic projections. Schläfli also was the first to find the general Euler formula for general n-polytopes. It is v-e+fc=0 for 4-polytopes and v-e+f-c+h=2 for 5 polytopes. |
