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Periodic origami patterns made with repeating unit cells of creases andpanels bend and twist in complex ways. In principle, such soft modes ofdeformation admit a simplified asymptotic description in the limit of a largenumber of cells. Starting from a bar and hinge model for the elastic energy ofa generic four parallelogram panel origami pattern, we derive a complete set ofgeometric compatibility conditions identifying the pattern's soft modes in thislimit. The compatibility equations form a system of partial differentialequations constraining the actuation of the origami's creases (a scalar anglefield) and the relative rotations of its unit cells (a pair of skew tensorfields). We show that every solution of the compatibility equations is thelimit of a sequence of soft modes -- origami deformations with finite bendingenergy and negligible stretching. Using these sequences, we derive a plate-liketheory for parallelogram origami patterns with an explicit coarse-grainedquadratic energy depending on the gradient of the crease-actuation and therelative rotations of the cells. Finally, we illustrate our theory in thecontext of two well-known origami designs: the Miura and Eggbox patterns.Though these patterns are distinguished in their anticlastic and synclasticbending responses, they show a universal twisting response. General soft modescaptured by our theory involve a rich nonlinear interplay between actuation,bending and twisting, determined by the underlying crease geometry.

Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity