Almost all circle polyhedra are rigid

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...

show abstract

“…A similar statement and proof to lemma 2.10, phrased in terms of inversive distances [32], was found independently by Bowers et al [33].…”

Section: Remark 211supporting

confidence: 80%

Rigidity for sticky discs

2019

View full text Add to dashboard Cite

show abstract

“…The last statement was first proved by H. Gluck in [23]. Recently, Gluck's proof has been adapted to prove the rigidity of almost all triangulated circle polyhedra, see [8]. The tetrahedron T j with the vertices x 0 , x j , x j+1 , and x n+1 of the suspension is grayed out; the thin line segment x 0 x n+1 is an edge of T j , but is not an edge of the suspension.…”

Section: Simple Polyhedramentioning

confidence: 99%

Recognition of affine-equivalent polyhedra by their natural developments

Alexandrov¹

2021

Preprint

View full text Add to dashboard Cite

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two polyhedra are isometric or not by using their developments only. In this article, we study a similar problem about whether it is possible, using only the developments of two convex polyhedra of Euclidean 3-space, to understand that these polyhedra are (or are not) affine-equivalent.

show abstract

“…Let J (at (p, r)) be the Jacobian of f . Its row corresponding to an edge ij of G has the following pattern [5]…”

Section: Jacobianmentioning

confidence: 99%

“…Our argument in this section is a variation on ideas explored in [5,10]. The idea is to show that J cannot have any non-zero cokernel vector.…”

Section: Jacobianmentioning

confidence: 99%

See 1 more Smart Citation

Packing Disks by Flipping and Flowing

Connelly¹,

Gortler²

2019

Preprint

View full text Add to dashboard Cite

We provide a new type of proof for the Koebe-Andreev-Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge e − of this graph and then continuously flow the disks in the plane, such that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge e − in G.

show abstract

Almost all circle polyhedra are rigid

Cited by 7 publications

References 8 publications

Rigidity for sticky discs

Rigidity for sticky discs

Recognition of affine-equivalent polyhedra by their natural developments

Packing Disks by Flipping and Flowing

Contact Info

Product

Resources

About