We explain the general significance of integer-based descriptors for natural shapes and show that the evolution of two such descriptors, called mechanical descriptors (the number _N_(_t_) of static balance points and the Morse–Smale graph associated with the scalar distance function measured from the center of mass) appear to capture (unlike classical geophysical shape descriptors) one of our most fundamental intuitions about natural abrasion: shapes get monotonically _simplified_ in this process. Thus mechanical descriptors help to establish a correlation between subjective and (...) objective descriptors of perceived objects. (shrink)

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$$d=3$$d=3dimensions as a continuum but we also find that it is quite accurate, showing that Plato’s geometric intuition may have been remarkable. (shrink)