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 ${\mathbb{Z}}^2⋊{\mathcal{C}}_s$, otherwise known as $pm$ in crystallography. In the ${\ell }_1$ and ${\ell }_{\infty }$-planes, we provide characterisations for forced-periodic rigidity and forced-${\mathbb{Z}}^2⋊{\mathcal{C}}_s$-symmetric rigidity. All of these characterisations are proved by inductive constructions involving Henneberg-type graph operations.

30 pages, 13 figures