Abstract

Plato associated regular polyhedra with the classical Elements, in particular, he associated Earth with the cube. His views have been broadly regarded, at best, as metaphors.

However, if we approximate natural fragments by convex polyhedra and count the respective numbers for faces, vertices and edges then in most cases we find averages close to 6,8,12, the values corresponding to the cube.

I will explain this phenomenon by using the theory of convex mosaics, geometric computer simulations of such mosaics, discrete element simulations of fragmentation and, last but not least, field data from over 4000 fragments.

We found [1] that stress fields determine fragmentation patterns (albeit in an averaged sense) and a vast majority of naturally occurring stress fields produce patterns which yield fragments with cuboid averages.

Whether and to what extent these findings vindicate Plato’s views and how they are related to the Gömböc are interesting subjects which I am happy to discuss if time admits.

[1] https://www.pnas.org/content/117/31/18178.short