part iii: polyhedra a: folding polygons joseph orourke smith college
TRANSCRIPT
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Part III: PolyhedraPart III: Polyhedraa: Folding Polygonsa: Folding Polygons
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
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Outline: Folding PolygonsOutline: Folding Polygons
Alexandrov’s TheoremAlgorithms
Edge-to-Edge FoldingsExamples
Foldings of the Latin Cross Foldings of the Square
Open Problems Transforming shapes?
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Aleksandrov’s Theorem Aleksandrov’s Theorem (1941)(1941)
“For 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 Alexandrov Gluing (of polygons)polygons)Uses up the perimeter of all the
polygons with boundary matches:No gaps.No paper overlap.Several points may glue together.
At most 2 angle at any glued point.Homeomorphic to a sphere.
Aleksandrov’s Theorem unique “polyhedron”
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Folding the Latin CrossFolding the Latin Cross
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Folding Polygons to Convex Folding Polygons to Convex PolyhedraPolyhedra
When can a polygon fold to a polyhedron? “Fold” = close up
perimeter, no overlap, no gap :
When does a polygon have an Aleksandrov gluing?
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Unfoldable PolygonUnfoldable Polygon
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Foldability is “rare”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 HalvingPerimeter Halving
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Edge-to-Edge GluingsEdge-to-Edge Gluings
Restricts gluing of whole edges to whole edges.
[Lubiw & O’Rourke, 1996]
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New Re-foldings of the New Re-foldings of the CubeCube
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VideoVideo
[Demaine, Demaine, Lubiw, JOR, [Demaine, Demaine, Lubiw, JOR, Pashchenko (Symp. Computational Pashchenko (Symp. Computational Geometry, 1999)]Geometry, 1999)]
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Open: Practical Algorithm for Open: Practical Algorithm for Cauchy RigidtyCauchy 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 StudiesTwo Case Studies
The Latin CrossThe Square
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Folding the Latin CrossFolding the Latin Cross
85 distinct gluingsReconstruct
shapes by ad hoc techniques
23 incongruent convex polyhedra
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23 Latin Cross Polyhedra23 Latin Cross Polyhedra
Sasha Berkoff, Caitlin Brady, Erik Demaine, Martin Demaine, Koichi Hirata, Anna Lubiw, Sonya Nikolova, Joseph O’Rourke
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Foldings of a SquareFoldings of a Square
Infinite continuum of polyhedra.Connected space
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DynamicWeb page
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Open: Fold/Refold Open: Fold/Refold DissectionsDissections
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 HirataKoichi Hirata