Geometriae Dedicata

, Volume 170, Issue 1, pp 219–262 | Cite as

Frameworks with forced symmetry II: orientation-preserving crystallographic groups

  • Justin Malestein
  • Louis Theran
Original Paper


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.


Combinatorial rigidity Discrete geometry Matroids Submodular functions 

Mathematics Subject Classification

52C25 52B40 



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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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Einstein Institute of Mathematics, Givat Ram, The Hebrew UniversityJerusalemIsrael
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany

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