Isometries of Trapezoid-Based Origami

Rigidly foldable origami tessellations require a careful balance between folding degrees of freedom and geometric compatibility constraints. However, such tessellations generically exhibit nonrigid linear isometries that allow the panels to bend without stretching. While these linear isometries can be modeled via a triangulation of the crease pattern, it is challenging to analytically characterize the deformations in all but the simplest geometries. Here, we discuss a recently developed theoretical framework for characterizing the linear isometries of quadrilateral-based origami tessellations that more clearly distinguishes between folding and bending degrees of freedom. We showcase the utility of this framework using examples of trapezoid-based origami (including the special case of parallelograms), from which we deduce capabilities of such crease patterns as mechanical metamaterials.