14 - Rigid Origami Theory

Summary

The final chapter considers more theoretical aspects of rigid origami.The first section outlines a proof that deciding whether or not an origami crease pattern can be rigidly folded from the unfolded state using some subset of the creases is NP-hard.Then configuration spaces of rigid origami crease patterns are discussed in more depth than in the previous chapter, including a proof that the germ of single-vertex rigid origami configuration spaces is isomorphic to the germ of a quadratic form.Examples of disconnected rigid origami configuration spaces are also included.The chapter, and book, ends with an introduction to the theory of self-folding, where we imagine that a crease pattern is rigidly folded using actuators on the creases, and we wish for these actuators to fold the crease pattern to a target state and not to some other rigid origami state.The aim is to characterize when simple actuating forces can do this, and we present the current theory behind this as well as its limitations.