  Mechanical metamaterials exhibit exotic properties at the system level, that
emerge from the interactions of many nearly rigid building blocks. Determining
these emergent properties theoretically has remained an open challenge outside
of a few select examples. Here, for a large class of periodic and planar
kirigami, we provide a coarse-graining rule linking the design of the panels
and slits to the kirigami's macroscale deformations. The procedure gives a
system of nonlinear partial differential equations (PDE) expressing geometric
compatibility of angle functions related to the motion of individual slits.
Leveraging known solutions of the PDE, we present excellent agreement between
simulations and experiments across kirigami designs. The results reveal a
surprising nonlinear wave-type response persisting even at large boundary
loads, the existence of which is determined completely by the Poisson's ratio
of the unit cell.
