{"paper_id":"499e22b1-c9f3-4a41-a382-257ba5cd9928","body_text":"Mapping the discrete folding landscape | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Mapping the discrete folding landscape Nuno Araujo, Joao Neves, Bernardo Marques, Cristóvão Dias This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7771844/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 26 Feb, 2026 Read the published version in Communications Physics → Version 1 posted You are reading this latest preprint version 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. Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Statistical physics Physical sciences/Materials science/Soft materials/Self-assembly Folding graph theory algorithm Full Text Additional Declarations There is NO Competing Interest. Cite Share Download PDF Status: Published Journal Publication published 26 Feb, 2026 Read the published version in Communications Physics → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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