Generic rigidity of frameworks with orientation-preserving crystallographic symmetry

2011/08/11 by Justin Malestein, Louis Theran, Malestein, Justin +1 · 3 citations
Mathematics · #52B40 #52C25 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #math.CO #math.GT #msc:52B40 #msc:52C25

paper · pdf · doi:10.48550/arxiv.1108.2518

70 pages, 9 figures

arxiv created 2012/02/11 · arxiv updated 2015/03/19

Abstract

We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immediately yield a Maxwell-Laman-type combinatorial characterization for frameworks embedded in 2-dimensional cones that arise as quotients of the plane by a finite order rotation.

Cited by

• A Riemannian Variational and Spectral Framework for High-Dimensional Sphere Packing: Barrier-Dynamics Reconciliation, Periodic Rigidity, and Discrete-Time Guarantees
• Frameworks with forced symmetry II: Orientation-preserving crystallographic groups
• Frameworks with forced symmetry I: Reflections and rotations

Related

• Frameworks with forced symmetry I: Reflections and rotations
• Generic rigidity with forced symmetry and sparse colored graphs
• Frameworks with forced symmetry II: Orientation-preserving crystallographic groups
• Non-Euclidean Crystallographic Rigidity
• Frameworks with crystallographic symmetry