Rigidity for sticky discs

Abstract

We study the combinatorial and rigidity properties of disk packings with generic radii. We show that a packing of n disks in the plane with generic radii cannot have more than 2n − 3 pairs of disks in contact.The allowed motions of a packing preserve the disjointness of the disk interiors and tangency between pairs already in contact (modeling a collection of sticky disks). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly 2n − 3 contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma.Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly, et al.[13] on the number of contacts in a jammed packing of disks with generic radii. Sticky disk and framework rigidity A motion of a packing, called a flex, is one that preserves the radii, the disjointness of the disk interiors, as well as tangency between pairs of disks with a corresponding edge in the contact graph. (The last condition makes the disks "sticky".) A packing is rigid when all its flexes arise from rigid body motions; otherwise, it is flexible.Since any packing has a neighborhood on which the contact graph remains fixed along any flex (one must move at least some distance before a new contact can appear), the constraints on the packing are locally equivalent to preserving the pairwise distances between the circle centers. Forgetting that the disks have radii and must remain disjoint and keeping only the distance contraints between the centers, we get exactly "framework rigidity", where we have a configuration * Figure 1: A rigid sticky disk packing and its underlying bar framework. Disks are the green circles with red center points / joints, the bars are blue segments. p = (p 1 , . . . , p n ) of n points in a d-dimensional Euclidean space and a graph G; the pair (G, p) is called a (bar-and-joint) framework. The allowed flexes of the points are those that preserve the distances between the pairs indexed by the edges of G. As with packings, a framework is rigid when all its flexes arise from rigid body motions and otherwise flexible. Given a packing (p, r) with contact graph G, we call the framework (G, p) its underlying framework.Motivations Rigidity of sticky disk packings, and the relationship to frameworks, has several related, but formally different motivations. The first comes from the study of colloidal matter [31], which is made of micrometer-sized particles that interact with "short range potentials" [2, 33] that, in a limit, behave like sticky disks [22] (spheres in 3d).Secondly, there is a connection to jammed packings. Here one has a "container", which can shrink uniformly and push the disks together. In this setting, we do not require any contact graph to be preserved. A fundamen...

Almost‐Rigidity of Frameworks

“…Frameworks are graphs with fixed edge lengths embedded in a Euclidean space. Studied for their mathematical properties for decades, they also arise as models in a great many applications: from engineering macroscale structures, such as sculptures [12], play structures [32], and the Kurilpa bridge in Brisbane [1]; to designing the microstructure of materials with novel properties [4,5,31,33], to studying arrangements of jammed particles [9,21,28], to understanding allostery in biology [35,43]; to studying the properties of molecules [23,36,42]; to analyzing the structure of proteins [25]; to studying origami folding [14]. A question of interest in all of these areas is whether a framework is locally rigid, meaning the only continuous motions of the vertices that preserve the lengths of the edges are rigidbody motions.…”

Homothetic packings of centrally symmetric convex bodies

“…Theorem 1.2. [5] Let P = (G, p, r) be a disc packing. If the set {r v : v ∈ V } is algebraically independent over the rational numbers, then G is a (2, 3)-sparse planar graph.…”

“…It was conjectured in [5] that any (2, 3)-sparse graph is the contact graph of a disc packing with algebraically independent radii; this is equivalent to the conjecture that any (2, 3)-sparse graph is the contact graph of an independent disc packing. We would also conjecture an equivalent result regarding regular symmetric bodies.…”

“…The hope for Theorem 1.5 would be to use the idea of approximating discs with other types of convex bodies in some way to prove the conjecture of Connelly, Gortler and Thurston [5].…”

“…In Section 2 we shall set out the required background material for convex geometry and rigidity theory for general normed spaces, as well as defining most of the notation we shall use for later sections. In Section 3 we shall prove Theorem 1.3; although many of the ideas implemented were originally set out in [5], there are significant technicalities that first need to be addressed. In Section 4 we shall discuss the space of all centrally symmetric convex bodies, its topology and some important subspaces it has that will be required in later sections.…”