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This paper leverages group theory and symmetry classification of linear modes to analyze periodic origami folding, revealing that spatial symmetries (layer groups) dictate folding behavior. By classifying allowed linear shape-periodic modes via irreducible representations, the authors establish a sufficient condition for nonlinear strain modes, unifying symmetry, topology, and deployability. This framework enables the principled design of novel periodic origami structures with desired folding properties.
Origami's rigid periodic folding is fundamentally governed by its spatial symmetries, unlocking a predictive design principle for novel deployable structures.
Broadly studied origami crease patterns such as the Miura-ori are capable of rigid folding that maintains their spatial periodicity. However, periodic crease patterns generically require additional creases and fold into quasi-cylindrical shapes. Here, using group theory, we show that the folding properties of such sheets are largely determined by their spatial symmetries, known as layer groups. Within each symmetry class, the allowed linear shape-periodic modes of sheets are constrained and classified by the irreducible representation they belong to. Such classification further gives a sufficient condition, containing a symmetry constraint and a topological constraint, for the existence of a nonlinear strain mode. In this way, familiar special cases are placed within a unified framework that connects symmetry, network topology, and deployability. Finally, we turn our classification results into a predictive design principle that enables the rapid design of novel sheets with prescribed properties.