Mapping the discrete folding landscape

TL;DR

The paper addresses how to design target 3D polyhedral shapes from flat templates by accounting for folding pathways, not just cuts. It introduces a graph-based algorithm that represents shapes as face graphs and templates as subgraphs, enabling exhaustive mapping of folding paths and intermediate configurations into a discrete folding landscape. Key contributions include a binary-encoded catalog of configurations, a notion of folding states via foldable edges and red links, and isomorphism-based pruning using VF2++; the approach is demonstrated on cube, icosahedron, and dodecahedron and extended to misfolds and multifarious templates. The results show sublinear growth in discovered configurations with more templates, suggesting scalable exploration, and point to broad applicability for data-driven folding design and programmable metamaterials.

Abstract

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.

Figures (11)

• Figure 1: Examples of foldable structures at different scales. From DNA strands origami structures at the nanometer scale Veneziano2020Kim2023, to cell-activated hinges Cells2012, to self-folding robots Felton2014 and to large-scale foldable telescopes.
• Figure 2: Scheme of the algorithm. The folded polyhedron (cube) and its unfolded template (Latin cross) (A) serve as the starting point of the algorithm. Based on these structures, we construct the shell graph and the face graphs of both the template and the folded polyhedron (B). These three structures act as inputs to the algorithm. The process begins by constructing the face graph of the cut, obtained by subtracting the face graph of the template from the face graph of the folded polyhedron. This step also yields the shell graph of the cut (0). The algorithm selects an initial link (edge) on the shell graph, defining the first configuration (1). Adjacent links to this starting link are identified as potential follow-ups (2). The folded polyhedron is assigned as the parent configuration of the starting-link configuration (3). The algorithm checks whether this configuration has been encountered before. If it has, the algorithm moves on to 6; else, it is saved as a new configuration (4). If the configuration is new, the next configurations to evaluate are generated by adding the identified follow up links to the current configuration. These configurations, along with their respective follow-ups, are added to a queue. Each new configuration is designated as a child of the current one in the relational map (5). A new configuration is retrieved from the queue, and the process returns to step (4). When the queue is exhausted, the algorithm loops back to (1) and selects a new starting link. If no new links remain and the queue is empty, the algorithm terminates.
• Figure 3: Catalog of folding states and configurations for the cube and Latin cross. Two configurations differ in the set of cut edges, while folding states differ in the set of foldable edges. (A) There are 24 configurations of the cube where no face can be unfolded (i.e., zero foldable edges), despite having certain edges cut (marked in red). (B) When one foldable edge is present, there are 14 distinct configurations, corresponding to four folding states. However, these four folding states are isomorphic, so they are grouped into a single folding state. (C)–(E) Scheme of the folding states, their isomorphisms, and the corresponding configurations for cases with two, three, and five foldable edges. Notably, no folding state exists with exactly four foldable edges. In (C) we distinguish between folding states obtained through a connected path of cut edges (in blue) and those arising from a combination of disconnected cuts (in green), these folding states are obtained by recursively applying the algorithm.
• Figure 4: Folding landscape for the cube. The algorithm retrieves a network where each node represents a non-isomorphic folding state, and the links indicate possible transitions between states through a sequence of edge cuts. The cube has eleven distinct templates. The structure in the center (cube) has no red links and the templates have five, structures with the same number of red links are organized in concentric circles with ascending order by their number of red links. The folding pathways for the Latin cross are highlighted in red.
• Figure 5: Folding landscape for the cube and a multifarious template with misfolds. Similarly to Figures \ref{['cube_map']} the network of templates, target structures and intermediate folding states is here shown with additional misfolded structures (in yellow and orange) as well. On A we analyze a multifarious template capable of folding into either an octahedron or a tritetrahedral (boat) structure. A multifarious template is one that can fold into more than one closed structure. To construct the folding landscape, we apply the proposed algorithm, starting from both closed configurations of the same template. Structures are organized by their number of red links starting from the top (the template with seven) to the bottom (the target structures with zero). To the octahedron and the tritetrahedral (boat) structure we add three additional misfolds to which the template can evolve to. These structures in orange with no red links were added as the target structure input to the algorithm together with their respective template. Intermediate folding states which only evolve towards misfolds are shown in yellow. On B several templates and the target structure, the cube, are joined by misfolded folding states added as target structures to the algorithm (orange structures).