A genealogy of convex solids via local and global bifurcations of gradient vector fields

TL;DR

This paper explores how three-dimensional convex bodies can change their critical point configurations through bifurcations, demonstrating that all theoretically possible shape transitions can occur physically, such as via sediment abrasion.

Contribution

It constructs families of convex bodies connecting different classification classes and shows these transitions can be realized through specific bifurcations in gradient vector fields.

Findings

Transitions between classification classes are realizable by saddle-node and saddle-saddle bifurcations.

All combinatorially possible shape transitions can occur in physical processes.

The work provides a complete understanding of shape evolution via bifurcations in convex bodies.

Abstract

Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies these are nondegenerate maxima, minima, and saddle-points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work it was shown that these classifications are complete in that no class is empty. Here we construct 1- and 2-parameter families of convex bodies connecting members of adjacent primary and secondary classes and show that transitions between them can be realized by codimension 1 saddle-node and saddle-saddle (heteroclinic) bifurcations in the gradient vector fields. Our results indicate that all combinatorially possible transitions can be realized in physical shape evolution processes, e.g. by abrasion of sedimentary particles.