A topological classification of convex bodies

Abstract

Abstract. The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M C of Morse-Smale functions on S 2 . Here we show that even M C exhibits the complexity known for general Morse-Smale functions on S 2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P 2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P 2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.

On the Cover of the Rolling Stone

“…Várkonyi and Domokos [26], settling a conjecture due to Arnold (1995), constructed a smooth convex body (called gömböc) with a unique stable and a unique unstable equilibrium. Their discovery has lead to a systematic study of the topological properties of stable and unstable points [13]. However, it is not obvious how to approximate a smooth convex body by a polyhedron with the same stability: Domokos et al [14] show that uniformly random discretization may introduce additional stable equilibria.…”

Balancing polyhedra

“…Polyhedra may be regarded as purely geometric objects, however, they are also often intuitively identified with solids. Among the most obvious sources of such intuition are dice which appear in various polyhedral shapes: while classical, cubic dice have 6 faces, a large diversity of other dice exist as well: dice with 2, 3,4,6,8,10,12,16,20,24,30 and 100 faces appear in various games [36]. The key idea behind throwing dice is that each of the aforementioned faces is associated with a stable mechanical equilibrium point where dice may be at rest on a horizontal plane.…”

“…In the literature, the family of convex polyhedra having the same face lattice is called a combinatorial class; here we call it a secondary combinatorial class, and discuss it in Section 5. In an entirely analogous manner, one can define also secondary equilibrium classes of convex bodies, for more details the interested reader is referred to [16]. While it is immediately clear that for any polyhedron P we have f ≥ S, v ≥ U, (1.4) inverse type relationships (e.g.…”

A characterization of the symmetry groups of mono-monostatic convex bodies

“…In the proof of Theorem 2 we have shown that for any discrete 2dimensional point group G and any ε > 0, there is an element K ∈ F whose symmetry group is G and which satisfies d H (K, B 3 ) < ε. Nevertheless, Lemma 4 of [7], and the observation that the smoothing subroutine described in its proof preserves the symmetry group of the body, yield that the statement in (ii) of Theorem 2 for F = (1, 1) c holds under the additional requirement that the body K has a C ∞ -class differentiable boundary.…”