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The Isostatic Conjecture

  • Ricky Pollack Memorial Issue
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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.

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References

  1. Asimow, L., Roth, B.: The rigidity of graphs. Trans. Am. Math. Soc. 245, 279–289 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asimow, L., Roth, B.: The rigidity of graphs. II. J. Math. Anal. Appl. 68(1), 171–190 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkinson, S., Stillinger, F.H., Torquato, S.: Detailed characterization of rattlers in exactly isostatic, strictly jammed sphere packings. Phys. Rev. E 88(6), Art. No. 062208 (2013)

  4. Atkinson, S., Stillinger, F.H., Torquato, S.: Existence of isostatic, maximally random jammed monodisperse hard-disk packings. Proc. Natl Acad. Sci. USA 111(52), 18436–18441 (2014)

    Article  Google Scholar 

  5. Bezdek, A., Bezdek, K., Connelly, R.: Finite and uniform stability of sphere packings. Discrete Comput. Geom. 20(1), 111–130 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blind, G.: Über Unterdeckungen der Ebene durch Kreise. J. Reine Angew. Math. 236, 145–173 (1969)

    MathSciNet  MATH  Google Scholar 

  7. Colin de Verdière, Y.: Un principe variationnel pour les empilements de cercles. Invent. Math. 104(3), 655–669 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Connelly, R.: Rigidity and energy. Invent. Math. 66(1), 11–33 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Connelly, R.: Rigidity of packings. Eur. J. Comb. 29(8), 1862–1871 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Connelly, R., Dickinson, W.: Periodic planar disc packings. Philos. Trans. R. Soc. Lond. A 372(2008), Art. No. 20120039 (2014)

  11. Connelly, R., Funkhouser, M., Kuperberg, V., Solomonides, E.: Packings of equal disks in a square torus. Discrete Comput. Geom. 58(3), 614–642 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Connelly, R., Shen, J.D., Smith, A.D.: Ball packings with periodic constraints. Discrete Comput. Geom. 52(4), 754–779 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Danzer, L.: Finite point-sets on $S^2$ with minimum distance as large as possible. Discrete Math. 60, 3–66 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. de Laat, D., de Oliveira Filho, F.M., Vallentin, F.: Upper bounds for packings of spheres of several radii. Forum Math. Sigma 2, Art. No. e23 (2014)

  15. Dickinson, W., Guillot, D., Keaton, A., Xhumari, S.: Optimal packings of up to five equal circles on a square flat torus. Beitr. Algebra Geom. 52(2), 315–333 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donev, A., Stillinger, F.H., Torquato, S.: Calculating the free energy of nearly jammed hard-particle packings using molecular dynamics. J. Comput. Phys. 225(1), 509–527 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Donev, A., Torquato, S., Stillinger, F.H., Connelly, R.: A linear programming algorithm to test for jamming in hard-sphere packings. J. Comput. Phys. 197(1), 139–166 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Edwards, S.F., Grinev, D.V.: The missing stress-geometry equation in granular media. Physica A 294(1–2), 57–66 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Edwards, S.F., Mounfield, C.C.: A theoretical model for the stress distribution in granular matter. I. Basic equations. Physica A 226(1–2), 1–11 (1996)

    Article  MathSciNet  Google Scholar 

  20. Edwards, S.F., Mounfield, C.C.: A theoretical model for the stress distribution in granular matter. III. Forces in sandpiles. Physica A 226(1–2), 25–33 (1996)

    Article  MathSciNet  Google Scholar 

  21. Fejes Tóth, L.: Lagerungen in der Ebene auf der Kugel und im Raum. Die Grundlehren der mathematischen Wissenschaften, vol. 65. Springer, Berlin (1972)

  22. Guo, R.: Local rigidity of inversive distance circle packing. Trans. Am. Math. Soc. 363(9), 4757–4776 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Heppes, A.: Some densest two-size disc packings in the plane. Discrete Comput. Geom. 30(2), 241–262 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kennedy, T.: A densest compact planar packing with two sizes of discs. arXiv:math/0412418v1 (2004)

  25. Kennedy, T.: Compact packings of the plane with two sizes of discs. Discrete Comput. Geom. 35(2), 255–267 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  27. Luo, F.: Rigidity of polyhedral surfaces. III. Geom. Topol. 15(4), 2299–2319 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Melissen, J.B.M.: Packing and covering with circles = pakken en overdekken met cirkels: (met een samenvatting in het nederlands). PhD Thesis, Universiteit Utrecht (1997)

  29. Mounfield, C.C., Edwards, S.F.: A theoretical model for the stress distribution in granular matter. II. Forces in pipes. Physica A 226(1–2), 12–24 (1996)

    Article  MathSciNet  Google Scholar 

  30. Musin, O.R., Nikitenko, A.V.: Optimal packings of congruent circles on a square flat torus. Discrete Comput. Geom. 55(1), 1–20 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Roth, B., Whiteley, W.: Tensegrity frameworks. Trans. Am. Math. Soc. 265(2), 419–446 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  32. Roux, J.-N.: Geometric origin of mechanical properties of granular materials. Phys. Rev. E (3) 61(6, part B), 6802–6836 (2000)

    Article  MathSciNet  Google Scholar 

  33. Stephenson, K.: Introduction to Circle Packing: The Theory of Discrete Analytic Functions. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  34. Swinnerton-Dyer, H.P.F.: Inhomogeneous lattices. Proc. Camb. Philos. Soc. 50, 20–25 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: Rivin, I., Rourke, C., Series, C. (eds.) The Epstein Birthday Schrift. Geometry and Topology Monographs, vol. 1, pp. 511–549. Geometry and Topology Publications, Coventry (1998)

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Acknowledgements

This work was partially supported by the National Science Foundation Grant DMS-1564493 for Connelly, Solomonides and Yampolskaya, and National Science Foundation Grant DMS-1564473 for Gortler.

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Robert Connelly, Evan Solomonides & Maria Yampolskaya

  2. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

    Steven J. Gortler

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  1. Robert Connelly
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  2. Steven J. Gortler
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  3. Evan Solomonides
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  4. Maria Yampolskaya
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Correspondence to Robert Connelly.

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Connelly, R., Gortler, S.J., Solomonides, E. et al. The Isostatic Conjecture. Discrete Comput Geom 64, 734–758 (2020). https://doi.org/10.1007/s00454-018-00051-0

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  • DOI: https://doi.org/10.1007/s00454-018-00051-0

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