A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

Abstract

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 saddlenode 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.

Morse-Smale complexes on convex polyhedra

“…Definition 6. The topological graph G on S 2 , whose vertices are the critical points of ρ K , and whose edges are the edges of the Morse-Smale complex, is called the Morse-Smale graph generated by ρ K [5]. This graph is usually regarded as a 3colored quadrangulation of S 2 , where the 'colors' of the vertices are the three types of a critical point.…”

“…The number of stable, unstable and saddle-type critical points are called first order mechanical descriptors [9,20] whereas Morse-Smale complexes belong to second order mechanical descriptors [4,8]. While the mathematical properties of Morse-Smale complexes associated with a smooth mechanical distance function have been investigated in detail [5,8], so far there has been little experimental data to test this theory. The main obstacle in obtaining field data is the identification of Morse-Smale complexes on scanned images of particles [14].…”

“…We remark that a Morse-Smale complex on a surface is naturally associated to the topological graph whose vertices and edges are those of the complex. This graph, called Morse-Smale graph, often appears in applications [5], and by the above correspondence, information on one provides equivalent information on the other one.…”

Centering Koebe polyhedra via Möbius transformations

“…We remark that the variant of Theorem 3 with respect to the center of mass of the polyhedron, which we state as Problem 1, proves that every combinatorial class has a representative whose every face, vertex and edge contains a static equilibrium point [8]. An affirmative answer to the problem, with many applications in mechanics [8,10,9], would be a discrete version of Theorem 1 in [10], stating that for every 3-colored quadrangulation Q of S 2 there is a convex body K whose Morse-Smale graph, with respect to its center of mass, is isomorphic to Q. These papers also describe possible applications of our problem in various fields of science, from physics to chemistry to manufacturing.…”

A solution to some problems of Conway and Guy on monostable polyhedra

“…In addition to the existence of Gömböc, in their paper [32] Domokos and Várkonyi proved the existence of a convex body with S stable and U unstable equilibrium points for any S, U ≥ 1. This investigation was extended in [15] to the combinatorial equivalence classes defined by the Morse-Smale complexes of ρ K , and in [12] for transitions between these classes. Based on these results, for any S, U ≥ 1 we define the set (S, U ) c as the family of smooth convex bodies K having S stable and U unstable equilibrium points, where K has no degenerate equilibrium point, and at each such point bd(K) has a positive Gaussian curvature.…”