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

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Presentation on theme: "Part III: Polyhedra a: Folding Polygons Joseph ORourke Smith College."— Presentation transcript:

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

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

3 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."

4 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

5 Folding the Latin Cross

6 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?

7 Unfoldable Polygon

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

9 Perimeter Halving

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

11 New Re-foldings of the Cube

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

13 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.

14 Two Case Studies zThe Latin Cross zThe Square

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

16 23 Latin Cross Polyhedra Sasha Berkoff, Caitlin Brady, Erik Demaine, Martin Demaine, Koichi Hirata, Anna Lubiw, Sonya Nikolova, Joseph ORourke

17 Foldings of a Square zInfinite continuum of polyhedra. zConnected space

18 Dynamic Web page

19 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]

20 Koichi Hirata

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