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doi:10.1090/tran/2011-363-09

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“…The isostatic theorem with a flat torus as the container was proven in [9] using very general results of Guo [37] on circle patterns in piecewise-flat surfaces. Such methods could also be applied to the tri-cusp setting.…”

Section: (Iii) Relationship To the Flat Torus Isostatic Theoremmentioning

confidence: 99%

Rigidity for sticky discs

2019

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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...

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Section: (Iii) Relationship To the Flat Torus Isostatic Theoremmentioning

confidence: 99%