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Frameworks with Forced Symmetry I: Reflections and Rotations

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Abstract

We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results, we develop a new technique for deriving linear representations of sparsity matroids on colored graphs and extend the direction network method of proving rigidity characterizations to handle reflections.

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Notes

  1. Colored graphs in this sense are also called “gain graphs” in the literature, e.g. [36].

  2. This terminology is from [1]. Elissa Ross introduced this class in [23], and we introduced this terminology in light of her contribution. In [23], they are called “constructive periodic orbit graphs”.

  3. The matroid of Ross graphs has more circuits, but these are the ones we are interested in here. See Sect. 2.5.

  4. Recall that here we are using Ross circuit to refer to only one kind of circuit in the Ross matroid. The other type of circuit cannot appear since reflection-Laman graphs do not have (2, 2) blocks with trivial \(\rho \)-image.

References

  1. Berardi, M., Heeringa, B., Malestein, J., Theran, L.: Rigid components in fixed-lattice and cone frameworks. In: Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG) (2011). arXiv:1105.3234

  2. Bölcskei, A., Szél-Kopolyás, M.: Construction of \(D\)-graphs related to periodic tilings. KoG 6, 21–27 (2002)

  3. Borcea, C., Streinu, I., Tanigawa, S.: Periodic body-and-bar frameworks. SIAM J. Discrete Math. 29(1), 93–112 (2015). doi:10.1137/120900265

  4. Borcea, C.S., Streinu, I.: Periodic frameworks and flexibility. Proc. R. Soc. Lond. Ser. A 466(2121), 2633–2649 (2010). doi:10.1098/rspa.2009.0676

  5. Borcea, C.S., Streinu, I.: Minimally rigid periodic graphs. Bull. Lond. Math. Soc. 43(6), 1093–1103 (2011). doi:10.1112/blms/bdr044

  6. Brylawski, T.: Constructions. Theory of Matroids. Encyclopedia of Mathematics and Its Applications, vol. 26, pp. 127–223. Cambridge University Press, Cambridge (1986). doi:10.1017/CBO9780511629563.010

  7. Develin, M., Martin, J.L., Reiner, V.: Rigidity theory for matroids. Comment. Math. Helv. 82(1), 197–233 (2007). doi:10.4171/CMH/89

  8. Edmonds, J., Rota, G.-C.: Submodular set functions (abstract). In: Waterloo Combinatorics Conference, University of Waterloo, Ontario (1966)

  9. Fowler, P.W., Guest, S.D.: A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids Struct. 37(12), 1793–1804 (1999)

  10. Jordán, T., Kaszanitzky, V., Tanigawa, S.: Gain-sparsity and symmetric rigidity in the plane. Technical Report (2012). http://www.cs.elte.hu/egres/tr/egres-12-17

  11. Király, F., Theran, L., Tomioka, R.: The algebraic and combinatorial approach to matrix completion. To appear in J. Mach. Learn. Res. (2015). arXiv:1211.4116. With an appendix by T. Uno

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

  13. Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Math. 308(8), 1425–1437 (2008). doi:10.1016/j.disc.2007.07.104

  14. Malestein, J., Theran, L.: Generic rigidity of frameworks with orientation-preserving crystallographic symmetry. Preprint, arXiv:1108.2518 (2011)

  15. Malestein, J., Theran, L.: Generic rigidity of reflection frameworks. Preprint, arXiv:1203.2276 (2012)

  16. Malestein, J., Theran, L.: Generic combinatorial rigidity of periodic frameworks. Adv. Math. 233, 291–331 (2013). doi:10.1016/j.aim.2012.10.007

  17. Malestein, J., Theran, L.: Generic rigidity with forced symmetry and sparse colored graphs. In: Connelly, R., Weiss, A.I., Whiteley, W. (eds.) Rigidity and Symmetry. Fields Institute Communications, vol. 70, pp. 227–252. Springer, New York (2014). doi:10.1007/978-1-4939-0781-6_12

  18. Malestein, J., Theran, L.: Frameworks with forced symmetry II: orientation-preserving crystallographic groups. Geom. Dedicata 170, 219–262 (2014). doi:10.1007/s10711-013-9878-6

  19. Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27, 294 (1864)

    Google Scholar 

  20. Milnor, J.: Singular points of complex hypersurfaces. Annals of Mathematics Studies, vol. 61. Princeton University Press, Princeton (1968)

    Google Scholar 

  21. Nixon, A., Ross, E.: Periodic rigidity on a variable torus using inductive constructions (2012). arXiv:1204.1349

  22. Rivin, I.: Geometric simulations: a lesson from virtual zeolites. Nat. Mater. 5, 931–932 (2006). doi:10.1038/nmat1792

    Article  Google Scholar 

  23. Ross, E.: Inductive constructions for frameworks on a two-dimensional fixed torus. Preprint, arXiv:1203.6561 (2012)

  24. Ross, E., Schulze, B., Whiteley, W.: Finite motions from periodic frameworks with added symmetry. Int. J. Solids Struct. 48(11–12), 1711–1729 (2011). doi:10.1016/j.ijsolstr.2011.02.018

  25. Sartbaeva, A., Wells, S., Treacy, M., Thorpe, M.: The flexibility window in zeolites. Nat. Mater. 5(12), 962–965 (2006)

    Article  Google Scholar 

  26. Schulze, B.: Symmetric Laman theorems for the groups \({\cal {C}}_{2}\) and \({\cal {C}}_{s}\). Electron. J. Comb. 17(1): Research Paper 154, 61 (2010). http://www.combinatorics.org/Volume_17/Abstracts/v17i1r154.html

  27. Schulze, B.: Symmetric versions of Laman’s theorem. Discrete Comput. Geom. 44(4), 946–972 (2010). doi:10.1007/s00454-009-9231-x

  28. Schulze, B., Tanigawa, S.I.: Linking rigid bodies symmetrically. Eur. J. Comb. 42(0), 145–166 (2014). doi:10.1016/j.ejc.2014.06.002

    Article  MathSciNet  Google Scholar 

  29. Schuze, B., Tanigawa, S.I.: Infinitesimal rigidity of symmetric frameworks. Preprint, arXiv: 1308.6380 (2013)

  30. Streinu, I., Theran, L.: Sparsity-certifying graph decompositions. Graphs Comb. 25(2), 219–238 (2009). doi:10.1007/s00373-008-0834-4

  31. Streinu, I., Theran, L.: Slider-pinning rigidity: a Maxwell-Laman-type theorem. Discrete Comput. Geom. 44(4), 812–837 (2010). doi:10.1007/s00454-010-9283-y

  32. Tanigawa, S-I.: Matroids of gain graphs in applied discrete geometry. Preprint, arXiv:1207.3601 (2012)

  33. Treacy, M.M.J., Foster, M.D., Rivin, I.: Towards a catalog of designer zeolites. Turning Points in Solid State. Materials and Surface Science, pp. 208–220. RSC Publishing, Cambridge (2008)

    Google Scholar 

  34. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM J. Discrete Math. 1(2), 237–255 (1988). doi:10.1137/0401025

  35. Whiteley, W.: Some matroids from discrete applied geometry. In: Bonin, J., Oxley, J.G., Servatius, B. (eds.) Matroid Theory. Contemporary Mathematics, vol. 197, pp. 171–311. American Mathematical Society, Providence, RI (1996)

    Chapter  Google Scholar 

  36. Zaslavsky, T.: Voltage-graphic matroids. Matroid Theory and Its Applications, pp. 417–424. Liguori, Naples (1982)

    Google Scholar 

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Acknowledgments

LT was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 247029-SDModels. JM was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 226135. LT and JM had been supported by NSF CDI-I Grant DMR 0835586 to Igor Rivin and M. M. J. Treacy.

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Authors and Affiliations

  1. Mathematisches Institut der Universität Bonn, Bonn, Germany

    Justin Malestein

  2. Aalto Science Institute and Department of Computer Science, Aalto University, Espoo, Finland

    Louis Theran

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  1. Justin Malestein
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  2. Louis Theran
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Correspondence to Louis Theran.

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Malestein, J., Theran, L. Frameworks with Forced Symmetry I: Reflections and Rotations. Discrete Comput Geom 54, 339–367 (2015). https://doi.org/10.1007/s00454-015-9692-z

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