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.While for hundreds of years origami has existed as an artistic endeavor, recent decades have seen the application of folding thin materials to the fields of architecture, engineering, and material science [1][2][3][4][5][6][7]. Controlled actuation of thin materials via patterned folds has led to a variety of self-assembly strategies in polymer gels [8] and shape-memory materials [4], as well elastocapillary self-assembly [9], leading to the design of a new category of shape-transformable materials inspired by origami design. The origami repertoire itself, buoyed by advances in the mathematics of folding and the burgeoning field of computational geometry [10], is no longer limited to designs of animals and children's toys that dominate the art in popular consciousness, but now includes tessellations, corrugations, and other non-representational structures whose mechanical properties are of interest from a scientific perspective. These properties originate from the confluence of geometry and mechanical constraints that are an intrinsic part of origami, and ultimately allow for the construction of mechanical meta-materials using origami-based design [1-4, 6, 11-13]. In this paper we formulate a general theory for periodic lattices of folds in thin materials, and combine the language of traditional lattice solid mechanics with the geometric theory underlying origami.A distinct characteristic of all thin materials is that geometric constraints dominate the mechanical response of the structure. Because of this strong coupling between shape and mechanics, it is far more likely for a thin sheet to deform by bending without stretching. Strategically weakening a material with a crease or fold, and thus lowering the energetic cost of stretching, allows complex deformations and re-ordering of the material for negligible elastic energy cost. This vanishing energy cost, especially combined with increased control over micro-and nanoscopic material systems, indicates the great promise for structures whose characteristics depend primarily on geometry, rather than material composition.By patterning creases, hinges, or fold...