Abstract
A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms—giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.
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References
Alexandrov, V.: Flexible polyhedra in the Minkowski 3-space. Manuscr. Math. 11(3), 341–356 (2003)
Asimov, L., Roth, B.: The rigidity of graphs. Trans. Am. Math. Soc. 245, 279–289 (1978)
Baker, E.: An analysis of the bricard linkages. Mech. Mach. Theory 15, 267–286 (1980)
Bishop, D.M.: Group Theory and Chemistry. Clarendon Press, Oxford (1973)
Bolker, E.D., Roth, B.: When is a bipartite graph a rigid framework? Pac. J. Math. 90, 27–44 (1980)
Borcea, C.S., Streinu, I.: Periodic frameworks and flexibility. Proc. R. Soc., Math. Phys. Eng. Sci. 466, 2633–2649 (2010)
Bottema, O.: Die Bahnkurven eines merkwürdigen Zwölfstabgetriebes. Österr. Ing.-Arch. 14, 218–222 (1960)
Bricard, R.: Mémoire sur la théorie de l’octaèdre articulé. J. Math. Pures Appl. 5(3), 113–148 (1897)
Cauchy, A.L.: Sur les polygons et les polyèdres. Oevres Complètes d’Augustin Cauchy 2è Série Tom 1, 26–38 (1905)
Connelly, R.: A counterexample to the rigidity conjecture for polyhedra. Inst. Haut. Etud. Sci. Publ. Math. 47, 333–335 (1978)
Connelly, R.: Highly symmetric tensegrity structures. http://www.math.cornell.edu/~tens/ (2008)
Connelly, R.: The rigidity of suspensions. J. Differ. Geom. 13(3), 399–408 (1978)
Connelly, R.: Rigidity and energy. Invent. Math. 66, 11–33 (1982)
Connelly, R., Fowler, P.W., Guest, S.D., Schulze, B., Whiteley, W.: When is a symmetric pin-jointed framework isostatic? Int. J. Solids Struct. 46, 762–773 (2009)
Connelly, R., Jordán, T., Whiteley, W.: Generic global rigidity of body-bar frameworks. Egerváry Research Group on Combinatorial Optimization. Technical Report TR-2009-13 (2009)
Crapo, H., Whiteley, W.: Statics of frameworks and motions of panel structures, a projective geometric introduction. Topol. Struct. 6, 43–82 (1982)
Crapo, H., Whiteley, W.: Spaces of stresses, projections, and parallel drawings for spherical polyhedra. Contrib. Algebra Geom./Beitrage Algebra Geom. 35, 259–281 (1994)
Fowler, P.W., Guest, S.D.: A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids Struct. 37, 1793–1804 (2000)
Graver, J.E., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Mathematics. AMS, Providence (1993)
Guest, S.D., Fowler, P.W.: Symmetry conditions and finite mechanisms. Mech. Mater. Struct. 2(6), 293–301 (2007)
Guest, S.D., Schulze, B., Whiteley, W.: When is a symmetric body-bar structure isostatic? Int. J. Solids Struct. 47, 2745–2754 (2010)
Hall, L.H.: Group Theory and Symmetry in Chemistry. McGraw–Hill, New York (1969)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)
Kangwai, R.D., Guest, S.D.: Detection of finite mechanisms in symmetric structures. Int. J. Solids Struct. 36, 5507–5527 (1999)
Kangwai, R.D., Guest, S.D.: Symmetry-adapted equilibrium matrices. Int. J. Solids Struct. 37, 1525–1548 (2000)
Owen, J.C., Power, S.C.: Frameworks, symmetry and rigidity. Preprint (2009)
Malestein, J., Theran, L.: Generic combinatorial rigidity of periodic frameworks. Preprint, arXiv:1008.1837 (2010)
Ross, E.: Combinatorial and geometric rigidity of periodic structures. Draft Ph.D. thesis, York University, Toronto, ON, Canada. To appear
Ross, E., Schulze, B., Whiteley, W.: Finite motions from periodic frameworks with added symmetry. Preprint, arXiv:1010.5440 (2010)
Roth, B., Whiteley, W.: Tensegrity frameworks, Amer. Math. Soc. 266(2), 419–446 (1981)
Saliola, F.V., Whiteley, W.: Some notes on the equivalence of first-order rigidity in various geometries. arXiv:0709.3354 (2007)
Schulze, B.: Combinatorial and geometric rigidity with symmetry constraints. Ph.D. thesis, York University, Toronto, ON, Canada (2009)
Schulze, B.: Symmetry as a sufficient condition for a finite flex. SIAM J. Discrete Math. 24(4), 1291–1312 (2010)
Schulze, B.: Block-diagonalized rigidity matrices of symmetric frameworks and applications. Beitrage Algebra Geom. 51(2), 427–466 (2010)
Schulze, B.: Injective and non-injective realizations with symmetry. Contrib. Discret. Math. 5(1), 59–89 (2010)
Schulze, B.: Symmetric versions of Laman’s Theorem. Discrete Comput. Geom. 44(4), 946–972 (2010)
Schulze, B., Whiteley, W.: Coning, symmetry, and spherical frameworks (2010, in preparation)
Servatius, B., Whiteley, W.: Constraining plane configurations in CAD: combinatorics of directions and lengths. SIAM J. Algebr. Discrete Methods 12, 136–153 (1999)
Stachel, H.: Zur Einzigkeit der Bricardschen Oktaeder. J. Geom. 28, 41–56 (1987)
Stachel, H.: Flexible cross-polytopes in the Euclidean 4-space. J. Geom. Graph. 4(2), 159–167 (2000)
Stachel, H.: Flexible octahedra in the hyperbolic space. Math. Appl. (János Bolyai memorial volume) 581, 209–225 (2006)
Tarnai, T.: Finite mechanisms and the timber octagon of Ely cathedral. Topol. Struct. 14, 9–20 (1988)
White, N., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Algebr. Discrete Methods 4, 481–511 (1983)
White, N., Whiteley, W.: The algebraic geometry of bar and body frameworks. SIAM J. Algebr. Discrete Methods 8, 1–32 (1987)
Whiteley, W.: Cones, infinity and one-story buildings. Topol. Struct. 8, 53–70 (1983)
Whiteley, W.: Infinitesimally rigid polyhedra I. Statics of frameworks. Trans. Am. Math. Soc. 285(2), 431–465 (1984)
Whiteley, W.: Infinitesimal motions of a bipartite framework. Pac. J. Math. 110, 233–255 (1984)
Whiteley, W.: A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom. 4, 75–95 (1989)
Whiteley, W.: Some matroids from discrete applied geometry. Contemp. Math. 197, 171–311 (1996)
Whiteley, W.: Counting out to the flexibility of molecules. Phys. Biol. 2, 1–11 (2005)
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B. Schulze was supported by the DFG Research Unit 565 ‘Polyhedral Surfaces’.
W. Whiteley was supported by a grant from NSERC (Canada).
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Schulze, B., Whiteley, W. The Orbit Rigidity Matrix of a Symmetric Framework. Discrete Comput Geom 46, 561–598 (2011). https://doi.org/10.1007/s00454-010-9317-5
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DOI: https://doi.org/10.1007/s00454-010-9317-5