Discrete Symmetries Govern Folding of Quadrilateral-based Origami

The past decade has seen origami’s rise as a candidate for elastic metamaterials granting properties such as negative Poisson ratios. While we have recently shown a pairing between rigid body modes and folding motions in periodic triangulations, many applications utilize quadrilateral faces to control the mechanical response, requiring engineered symmetries such as the parallelogram faces used in crease patterns like the Miura-ori, eggbox, Barreto’s Mars, and block fold. Here we present a novel formalism that reduces the conventional 3-vector constraints at each vertex to scalar constraints on each edge that describe the infinitesimal deformations, including face bending, of generic quadrilateral-based patterns. These modes are governed by a constraint matrix — resembling a quantum Hamiltonian — that can be decomposed into disjoint sectors of solutions in the presence of discrete symmetries. We place special emphasis on a class of parallelogram origami, including all of the aforementioned crease patterns, exhibiting a permutation symmetry for which we obtain analytical expressions for their linear deformations, revealing such crease patterns always exhibit a single auxetic mode.