6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Spring 2017)

On the unfolding side, we'll cover “vertex unfolding”, which is a variation on edge unfolding kind of like hinged dissections. We'll prove that this type of unfolding exists, even for nonconvex polyhedra, provided every face is a triangle. Then we'll cover recent breakthroughs in general unfolding, for orthogonal polyhedra.

On the folding side, we'll prove Cauchy's Rigidity Theorem: convex polyhedra have exactly one convex realization (viewing faces as rigid and edges as hinges). Then we'll show how to extend this to Alexandrov's Uniqueness Theorem: if you glue up the boundary of a polygon, there's at most one convex polyhedron you can make. (Next lecture we'll see how to actually get one.)

YQ Lu will give a guest lecture about his work in paper sliceforms, as embodied by his software Sliceform Studio [https://www.sliceformstudio.com/]. See his resulting artwork at https://www.sliceformstudio.com/gallery.html.

Then we'll work on problems related to gluings and sphere wrapping.