  Origami-based design holds promise for developing materials whose mechanical
properties are tuned by crease patterns introduced to thin sheets. Although
there has been heuristic developments in constructing patterns with desirable
qualities, the bridge between origami and physics has yet to be fully
developed. To truly consider origami structures as a class of materials,
methods akin to solid mechanics need to be developed to understand their
long-wavelength behavior. We introduce here a lattice theory for examining the
mechanics of origami tessellations in terms of the topology of their crease
pattern and the relationship between the folds at each vertex. This formulation
provides a general method for associating mechanical properties with periodic
folded structures, and allows for a concrete connection between more
conventional materials and the mechanical metamaterials constructed using
origami-based design.
