Characterization of crystallographic rigidity in dimensions three and higher

Establish combinatorial characterizations of forced symmetric rigidity for bar-joint frameworks in Euclidean dimensions d≥3 and for arbitrary crystallographic symmetry groups.

Background

The paper notes that even generic rigidity of finite bar-joint frameworks in dimension 3 and higher lacks a combinatorial characterization. Consequently, corresponding characterizations for forced symmetric rigidity of bar-joint frameworks are unavailable in all dimensions d≥3 and for all symmetry groups. The paper contrasts this unresolved problem with Tanigawa’s complete characterization for crystallographic body-bar frameworks in fixed-lattice d-space.

References

A combinatorial characterisation of generic rigid (finite) frameworks is not known for dimension 3 and higher. Thus, there are also no characterisations of forced symmetric rigidity for bar-joint frameworks in dimension $d\geq 3$, for any symmetry group.

— Orientation-Reversing Crystallographic Rigidity  (2502.14392 - Esson et al., 20 Feb 2025) in Section 5.3, “Higher dimensions”