Mapping the discrete folding landscape

Folding is emerging as a promising manufacturing process to transform flat materials into functional structures, offering efficiency by reducing the need for welding, gluing, and molding, while minimizing waste and enabling automation. Designing target shapes requires not only to determine cuts and folds, but also folding pathways. Simple combinatorics is impractical as the possibilities grow factorially with the number of folds. To address this, we present a graph-based algorithm for polyhedral shapes. By representing the target shape as a graph, where nodes correspond to faces and edges represent adjacency, the algorithm identifies all possible fold sequences and maps the configuration space into a discrete set of intermediate configurations. This systematic mapping is critical for the design of optimized processes, the simplifying of folding operations, the reduction of failures, and the improvement of manufacturing reliability. Folding is emerging as a promising manufacturing process to transform flat materials into functional structures, with a reduced need for welding, gluing, and molding, while minimizing waste and enabling automation. Here, the authors introduce a graph-based algorithm for mapping the complete discrete folding landscape of three-dimensional structures, including Platonic solids, from two-dimensional templates, optimizing folding pathways and enhancing manufacturing reliability.