Orientation-Reversing Crystallographic Rigidity

Abstract

Abstract This paper provides a combinatorial characterisation for generic forced symmetric rigidity of bar-joint frameworks in the Euclidean plane that are symmetric with respect to the orientation-reversing wallpaper group $$\mathbb {Z}^2\rtimes \mathcal {C}_s$$ Z 2 ⋊ C s , also known as pm in crystallography, under a fixed lattice representation. Corresponding results for the wallpaper groups cm and pg follow directly from this. The method used also provides an inductive construction for the corresponding gain graphs, in terms of Henneberg-type graph operations.