6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2010)

Prof. Erik Demaine


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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 4 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.

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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