Abstract
We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first paper in this sequence from groups generated by a single rotation to groups generated by translations and rotations. The proof makes use of new families of matroids and submodular functions defined on crystallographic groups.
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Acknowledgments
We thank Igor Rivin for encouraging us to take on this project and many productive discussions on the topic. Our initial work on this topic was part of a larger effort to understand the rigidity and flexibility of hypothetical zeolites, which is supported by NSF CDI-I grant DMR 0835586 to Rivin and M. M. J. Treacy. LT is funded by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 247029-SDModels. JM is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 226135.
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Malestein, J., Theran, L. Frameworks with forced symmetry II: orientation-preserving crystallographic groups. Geom Dedicata 170, 219–262 (2014). https://doi.org/10.1007/s10711-013-9878-6
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DOI: https://doi.org/10.1007/s10711-013-9878-6