On regular polytopes

Abstract

Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension $n\ge -1$; now in {\rm dim}. 2, 3 and 4 there are \emph{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 $\mathrm{SO}(2)=\mathrm{U}(1)$ group being (abelian and) \emph{divisible}, is related to the existence of arbitrarily-sided plane regular polygons, and the \emph{splitting} of the Lie algebra of the $\mathrm{O}(4)$ group will be seen responsible for the Schläfli special polytopes in 4-dim., two of which percolate down to three. In spite of {\rm dim}. 8 being also special (Cartan's \emph{triality}), we argue why there are no \emph{extra} polytopes, while it has other consequences: in particular the existence of the three \emph{division algebras} over the reals $\mathbb{R}$: complex $\mathbb{C}$, quaternions $\mathbb{H}$ and octonions $\mathbb{O}$ 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.

To appear in the journal "Reports on Mathematical Physics (Poland)"