Geometric information and rigidity percolation in floppy origami

Floppy origami structures have many folded configurations and are thus natural candidates for storing information geometrically. To address how we might harness this idea, we study the effect of folds and constraints in a planar tessellation inspired by the simplest globally coordinated origami pattern known as Miura-ori. Introducing folds randomly along the diagonals of the quads in Miura-ori makes the rigid structure floppy. We show how the number of degrees of freedom in the system varies with the increase in the density of constraints, first linearly, and then nonlinearly. In the nonlinear regime, mechanical cooperativity sets in via an inherent redundancy that depends on the assignment of constraints, and the degrees of freedom in the system depends on the density of constraints in a scale-invariant manner. The redundancy in the constraints shows a percolation transition at a critical constraint density $\rho_c$. Our work shows how floppy origami can be used to store information in a scale-invariant way, and how we can control its rigidity exquisitely by taking advantage of a percolation transition.