On regular polytopes

TL;DR

This paper explores the existence and uniqueness of regular polytopes across different dimensions, linking their properties to special features of orthogonal groups and division algebras.

Contribution

It provides a group-theoretic explanation for the existence of regular polytopes in various dimensions and clarifies why certain dimensions have extra polytopes.

Findings

In 2, 3, and 4 dimensions, extra regular polytopes exist due to special group properties.

In higher dimensions, only the hyper-tetrahedron, hyper-cube, and hyper-octahedron exist.

The existence of division algebras correlates with special properties of orthogonal groups and spheres.

Abstract

Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension $n\geq-1$; 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 Schl\"{a}fli 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.