On Regular Polytopes

Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension n ≥ −1; now in dim. 2, 3 and 4 there are extra polytopes, while in general dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron exist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the SO(2) = U(1) group being (abelian and) divisible, is related to the existence of arbitrarily-sided plane regular polygons, and the splitting of the Lie algebra of the O(4) group will be seen responsible for the Schläfli special polytopes in 4-dim., two of which percolate down to three. In spite of dim. 8 being also special (Cartanʹs triality), we argue why there are no extra polytopes, while it has other consequences: in particular the existence of the three division algebras over the reals ℝ, complex ℂ, quaternions ℍ and octonions O is seen also as another feature of the special properties of corresponding orthogonal groups, and of the spheres of dimension 0, 1, 3 and 7.