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.

Voss nets are surface parametrizations whose parameter lines follow a conjugate network of geodesics. Their discrete counterparts, so called V-hedra, are flexible quadrilateral meshes with planar faces such that opposite angles at every vertex are equal; by replacing this equality condition with the closely related constraint that opposite angles are supplementary, we get so-called anti-V-hedra. In this paper, we study the problem of constructing and manipulating (anti-)V-hedra. First, we present a V-hedra generator that constructs a modifiable (anti-)V-hedron in a geometrically exact way, from a set of simple conditions already proposed by Sauer in 1970; our generator can compute and visualize the flexion of the (anti-)V-hedron in real time. Second, we present an algorithm for the design and interactive exploration of V-hedra using a handle-based deformation approach; this tool is capable of simulating the one-parametric isometric deformation of an imperfect V-hedron via a quad soup approach. Moreover, we evaluate the performance and accuracy of our tools by applying the V-hedra generator to constraints obtained by numerical optimization. In particular, we use example surfaces that originate from one-dimensional families of smooth Voss surfaces – each spanned by two isothermal conjugate nets – for which an explicit parametrization is given. This allows us to compare the isometric deformation of the smooth target surface with the rigid folding of the optimized (imperfect) V-hedron.