  Origami is the archetype of a structural material with unusual mechanical
properties that arise almost exclusively from the geometry of its constituent
folds and forms the basis for mechanical metamaterials with an extreme
deformation response. Here we consider a simple periodically folded structure
Miura-ori, which is composed of identical unit cells of mountain and valley
folds with four-coordinated ridges, defined completely by 2 angles and 2
lengths. We use the geometrical properties of a Miura-ori plate to characterize
its elastic response to planar and non-planar piece- wise isometric
deformations and calculate the two-dimensional stretching and bending response
of a Miura-ori sheet, and show that the in-plane and out-of-plane Poisson's
ratios are equal in magnitude, but opposite in sign. Our geometric approach
also allows us to solve the inverse design problem of determining the geometric
parameters that achieve the optimal geometric and mechanical response of such
structures.
