Plato’s Error and a Mean Field Formula for Convex Mosaics

Abstract

Plato claimed that the regular solids are the building blocks of all matter. His views, commonly referred to as the geometric atomistic model, had enormous impact on human thought despite the fact that four of the five Platonic solids can not fill space without gaps. In this paper we quantify these gaps, showing that the errors in Plato’s estimates were quite small. We also develop a mean field approximation to convex honeycombs using a generalized version of Plato’s idea. This approximation not only admits to view convex mosaics in d=3 dimensions as a continuum but we also find that it is quite accurate, showing that Plato’s geometric intuition may have been remarkable.