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Part III: Polyhedra a: Folding Polygons Joseph ORourke Smith College

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Outline: Folding Polygons zAlexandrovs Theorem zAlgorithms yEdge-to-Edge Foldings zExamples yFoldings of the Latin Cross yFoldings of the Square zOpen Problems yTransforming shapes?

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Aleksandrovs Theorem (1941) zFor every convex polyhedral metric, there exists a unique polyhedron (up to a translation or a translation with a symmetry) realizing this metric."

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Alexandrov Gluing (of polygons) Alexandrov Gluing (of polygons) zUses up the perimeter of all the polygons with boundary matches: xNo gaps. xNo paper overlap. xSeveral points may glue together. zAt most 2 angle at any glued point. zHomeomorphic to a sphere. Aleksandrovs Theorem unique polyhedron

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Folding the Latin Cross

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Folding Polygons to Convex Polyhedra zWhen can a polygon fold to a polyhedron? yFold = close up perimeter, no overlap, no gap : zWhen does a polygon have an Aleksandrov gluing?

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Unfoldable Polygon

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Foldability is rare Lemma: The probability that a random polygon of n vertices can fold to a polytope approaches 0 as n 1.

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Perimeter Halving

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Edge-to-Edge Gluings zRestricts gluing of whole edges to whole edges. [Lubiw & ORourke, 1996]

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New Re-foldings of the Cube

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Video [Demaine, Demaine, Lubiw, JOR, Pashchenko (Symp. Computational Geometry, 1999)]

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Open: Practical Algorithm for Cauchy Rigidty Find either a polynomial-time algorithm, or even a numerical approximation procedure, that takes as input the combinatorial structure and edge lengths of a triangulated convex polyhedron, and outputs coordinates for its vertices.

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Two Case Studies zThe Latin Cross zThe Square

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Folding the Latin Cross z85 distinct gluings zReconstruct shapes by ad hoc techniques z23 incongruent convex polyhedra

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23 Latin Cross Polyhedra Sasha Berkoff, Caitlin Brady, Erik Demaine, Martin Demaine, Koichi Hirata, Anna Lubiw, Sonya Nikolova, Joseph ORourke

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Foldings of a Square zInfinite continuum of polyhedra. zConnected space

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Dynamic Web page

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Open: Fold/Refold Dissections Can a cube be cut open and unfolded to a polygon that may be refolded to a regular tetrahedron (or any other Platonic solid)? [M. Demaine 98]

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Koichi Hirata

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