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

Prof. Erik Demaine; Martin Demaine; Dr. Jason Ku; TAs Adam Hesterberg & Jayson Lynch

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[+] Polyhedron unfolding: Edge unfoldings, general unfoldings, big questions, curvature, general unfolding of convex polyhedra, source unfolding, star unfolding, edge-unfolding special convex polyhedra, fewest nets, edge-ununfoldable nonconvex polyhedra.
This lecture begins our coverage of polyhedron (un)folding. Specifically, we'll talk about how to cut open a polyhedral surface into one piece without overlap so that that piece can be cut out of sheet material and assembled into the 3D surface. What distinguishes this field from origami is that we're not allowed to cover any part of the surface more than once. Two of the biggest open problems in geometric folding are here:
  1. Does every convex polyhedron have an edge unfolding? This is also the oldest problem, implicit back to 1525.
  2. Does every polyhedron (without boundary) have a general unfolding? This is one of my favorite open problems.
I'll describe the various attacks and partial results toward solving the first problem, as well as the reverse situations where we know much more: general unfolding of convex polyhedra and edge unfolding of general polyhedra.

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Handwritten notes, page 1/9 • [previous page] • [next page] • [PDF]

Handwritten notes, page 1/9 • [previous page] • [next page] • [PDF]

Slides, page 1/36 • [previous page] • [next page] • [PDF]

Edge unfolding vs. general unfolding (figures drawn by Erik Demaine)

Slides, page 1/36 • [previous page] • [next page] • [PDF]

The video above should play if your web browser supports either modern Flash or HTML5 video with H.264 or WebM codec. The handwritten notes and slides should advance automatically. If you have any trouble with playback, email Erik.