part ii: paper b: one-cut theorem joseph orourke smith college (many slides made by erik demaine)
TRANSCRIPT
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Part II: PaperPart II: Paperb: One-Cut Theoremb: One-Cut Theorem
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
(Many slides made by Erik (Many slides made by Erik Demaine)Demaine)
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OutlineOutline
Problem definitionResultExamplesStraight skeletonFlattening
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Fold-and-Cut ProblemFold-and-Cut Problem
Given any plane graph (the cut graph)
Can you fold the piece of paper flat so thatone complete straight cut makes the graph?
Equivalently, is there is a flat folding that lines up precisely the cut graph?
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History of Fold-and-CutHistory of Fold-and-Cut
Recreationally studiedby Kan Chu Sen (1721) Betsy Ross (1777) Houdini (1922) Gerald Loe (1955) Martin Gardner (1960)
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TheoremTheorem [Demaine, Demaine, Lubiw [Demaine, Demaine, Lubiw 1998]1998]
[Bern, Demaine, Eppstein, Hayes [Bern, Demaine, Eppstein, Hayes 1999]1999]Any plane graph can be lined up
by folding flat
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Straight SkeletonStraight Skeleton
Shrink as in Lang’s universal molecule, but Handle nonconvex polygons
new event when vertex hits opposite edge Handle nonpolygons
“butt” vertices of degree 0 and 1 Don’t worry about active paths
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PerpendicularsPerpendiculars
Behavior is more complicated than tree method
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A Few ExamplesA Few Examples
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Flattening PolyhedraFlattening Polyhedra[Demaine, Demaine, Hayes, Lubiw][Demaine, Demaine, Hayes, Lubiw]
Intuitively, can squash/collapse/flatten a paper model of a polyhedron
Problem: Is it possible without tearing?
Flattening a cereal box
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Connection to Fold-and-CutConnection to Fold-and-Cut
2D fold-and-cut Fold a 2D polygon
through 3Dflat, back into 2D
so that 1D boundarylies in a line
3D fold-and-cut Fold a 3D
polyhedronthrough 4Dflat, back into 3D
so that 2D boundarylies in a plane
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Flattening ResultsFlattening Results
All polyhedra homeomorphic to a sphere can be flattened (have flat folded states)[Demaine, Demaine, Hayes, Lubiw] ~ Disk-packing solution to 2D fold-and-cut
Open: Can polyhedra of higher genus be flattened?
Open: Can polyhedra be flattened using 3D straight skeleton? Best we know: thin slices of convex
polyhedra