Snapping Deployable Toroids for Modular Gridshells

We introduce a novel class of polyhedral tori (PQ-toroids) that snap between two stable configurations - a flat state and a deployed one separated by an energy barrier. Being able to create PQ-toroids from any set of given planar bottom and side faces opens the possibility to assemble the bistable blocks into a thick freeform curved shell structure to follow a planar quadrilateral (PQ) net with coplanar adjacent offset directions. A design pipeline is developed and presented for inversely computing PQ-toroid modules using conjugate net decompositions of a given surface. We analyze the snapping behavior and energy barriers through simulation and build physical prototypes to validate the feasibility of the proposed system. This work expands the geometric design space of multistable origami for lightweight modular structures and offers practical applications in architectural and deployable systems.