The Isostatic Conjecture
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The Isostatic Conjecture Robert Connelly1 · Steven J. Gortler2 · Evan Solomonides1 · Maria Yampolskaya1 Received: 7 August 2017 / Revised: 25 June 2018 / Accepted: 4 December 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses, which have been observed with numerical Monte Carlo simulations. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. Keywords Packings · Square torus · Density · Granular materials · Stress distribution · Koebe · Andreev · Thurston
1 Introduction Granular material, made of small rocks or grains of sand, is often modeled as a packing of circular disks in the plane or round spheres in space. In order to analyze the internal stresses that resolve external loads, there is a lot of interest in the distribution of the
Dedicated to the memory of Ricky Pollack. Editor in Charge: János Pach Robert Connelly [email protected] Steven J. Gortler [email protected] Evan Solomonides [email protected] Maria Yampolskaya [email protected] 1
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
2
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
123
Discrete & Computational Geometry
stresses in that material. See, for example, [18–20,29]. A self stress is an assignment of scalars to the edges of the graph of contacts such that at each vertex (the disk centers) there is a vector equilibrium maintained. One property that has come up in this context is that, when a packing is jammed in some sort of container, there is necessarily an internal self stress that appears. It seems to be taken as a matter of (empirical) fact that when the radii of the circles (or spheres) are chosen generically, there is only one such self stress, up to scaling. In that case, one says that the structure is isostatic. This statement seems to be borne out in many computer simulations, since it is essentially a geometric property of the jammed configuration of circular disks. See, for example, the work of Roux [32], Atkinson et al. [3,4,9]. When the disks all have the same radius, and thus are non-generic, for example, it quite often turns out that the packing is not isostatic. Here, we refer to the (mathematical) statement that when the radii and lattice are generic, the packing has a single stress up to scaling, as the isostatic conjecture. Note that when the radii of the packing disks are chosen generically, this does not imply that the coordinates of the configuration of the centers of the disks are generic. There is a wide literature on the rigidity of frameworks, when the configuration is generic, for example from the basic results in the plane starting with Laman [26], and more generally as described in Asimow and Roth
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