Orientation-Reversing Crystallographic Rigidity

TL;DR

This paper characterizes the combinatorial conditions for the rigidity of symmetric bar-joint frameworks in the plane under orientation-reversing wallpaper groups, providing new inductive construction methods for these structures.

Contribution

It introduces a combinatorial characterization for generic forced symmetric rigidity under orientation-reversing wallpaper groups, including an inductive construction for the associated gain graphs.

Findings

Provides a combinatorial criterion for rigidity in specific wallpaper groups

Offers an inductive method for constructing gain graphs

Extends results to groups $cm$ and $pg$

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 $\mathbb{Z}^2\rtimes\mathcal{C}_s$, 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.