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

Prof. Erik Demaine


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[+] Polyhedron unfolding: Vertex unfolding, facet paths, generally unfolding orthogonal polyhedra, grid unfolding, refinement, Manhattan towers, orthostacks, orthotubes, orthotrees.
Polyhedron folding: Cauchy's Rigidity Theorem, Alexandrov's uniqueness of folding.

This lecture continues the theme of unfolding polyhedra, and kicks off our coverage of folding polygons into polyhedra.

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.)

Download Video: 360p, 720p

Handwritten notes, page 1/8[previous page][next page][PDF]

Handwritten notes, page 1/8[previous page][next page][PDF]

Slides, page 1/26[previous page][next page][PDF]

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

Slides, page 1/26[previous page][next page][PDF]

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