Non-Euclidean Crystallographic Rigidity

Abstract

This paper establishes combinatorial characterisations of forced-symmetric and forced-periodic rigidity (under a fixed lattice) of bar-joint frameworks in non-Euclidean normed planes. In $\ell_q$-planes for $q\in(1,\infty)\backslash\{2\}$, we prove characterisations for forced-periodic rigidity and forced-reflectionally-symmetric rigidity. We also characterise forced-symmetric rigidity in this space with respect to the orientation-reversing wallpaper group $\mathbb{Z}^2\rtimes\mathcal{C}_s$, otherwise known as $pm$ in crystallography. In the $\ell_1$ and $\ell_\infty$-planes, we provide characterisations for forced-periodic rigidity and forced-$\mathbb{Z}^2\rtimes\mathcal{C}_s$-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations.