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Symmetric Versions of Laman’s Theorem

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Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are “fixed” by various symmetry operations of the framework.

For the group \(\mathcal{C}_{3}\) which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46:762–773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with \(\mathcal{C}_{3}\) symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints.

In addition, we establish symmetric versions of Henneberg’s theorem and Crapo’s theorem for \(\mathcal{C}_{3}\) which provide alternate characterizations of “generically” isostatic graphs with \(\mathcal{C}_{3}\) symmetry.

As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups \(\mathcal{C}_{2}\) and \(\mathcal{C}_{s}\) , in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups \(\mathcal{C}_{2}\) and \(\mathcal{C}_{s}\) which are generated by a half-turn and a reflection in the plane, respectively.


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Correspondence to Bernd Schulze.

Additional information

Research for this article was supported, in part, under a grant from NSERC (Canada), and final preparation occurred at the TU Berlin with support of the DFG Research Unit 565 “Polyhedral Surfaces.”

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Schulze, B. Symmetric Versions of Laman’s Theorem. Discrete Comput Geom 44, 946–972 (2010). https://doi.org/10.1007/s00454-009-9231-x

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  • Generic rigidity
  • Infinitesimal rigidity
  • Bar-and-joint frameworks
  • Symmetric frameworks
  • Laman graphs
  • Henneberg construction
  • 3Tree2 partition