# 6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2020)

## Prof. Erik Demaine; Martin Demaine; TAs Yevhenii Diomidov & Klara Mundilova

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 [+] Universal hinge patterns: box pleating, polycubes; orthogonal maze folding. NP-hardness: introduction, reductions; simple foldability; crease pattern flat foldability; disk packing (for tree method). This lecture covers two main topics: First, continuing our theme from Lecture 3 on efficient origami design, we'll see how subsets of a single hinge pattern are enough to fold any orthogonal shape made up of cubes, whereas other approaches use a completely different set of creases for each origami model you want. In general, we can fold n cubes from an O(n) × O(n) square of paper. In the special case of “orthogonal mazes”, we can waste almost no paper, with the folding only a small constant factor smaller than the original piece of paper. You can try out this yourself. Second, we'll see a few ways in which origami is hard. Specifically, I'll give a brief, practical introduction to NP-hardness, and prove three origami problems NP-hard: folding a given crease pattern via a sequence of simple folds; flat folding a given crease pattern (using any folded state); optimal design of a uniaxial base, even when the tree is just a star.
 No support for video detected. Install Flash or use an HTML5 browser. Download Video: 360p, 720p Handwritten notes, page 1/7 • [previous page] • [next page] • [PDF] Handwritten notes, page 1/7 • [previous page] • [next page] • [PDF] Slides, page 1/20 • [previous page] • [next page] • [PDF] http://​www.pnas.org/​content/​107/​28/​12441 / http://​www.pnas.org/​content/​suppl/​2010/​06/​25/​0914069107.DCSupplemental / http://​erikdemaine.org/​papers/​Matter_PNAS/​ (covered under the MIT Faculty Open-Access Policy) Slides, page 1/20 • [previous page] • [next page] • [PDF]

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