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

Prof. Erik Demaine

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[+] Maekawa and Kawasaki's Theorems Revisited and Extended: Maekawa's Theorem, Kawasaki's Theorem, history (Justin, Huffman, Husimi), Robertson's 1977 paper, manifolds without boundary, strata, volumes, degree of map, higher dimensions, Justin's Theorem, Gauss-Bonnet Theorem, tessellations on a sphere
(guest lecture by Thomas Hull)

The theorems named after Maekawa and Kawasaki are the fundamental theorems of flat origami. They are bread & butter results for modern origami design. What is seldom seen is a result of Jacques Justin that generalizes both of these theorems and shows that they are fundamentally related to each other. We will prove Justin's Theorem in a way that shows how all these flat origami results are just a consequence of the Gauss-Bonnet Theorem. We will also see how Kawasaki's Theorem was actually (kind of) proved previously by a British mathematician named Robertson, except that Robertson's version holds in arbitrary dimension.

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