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Part II: Paper b: One-Cut Theorem Joseph ORourke Smith College (Many slides made by Erik Demaine)

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Outline zProblem definition zResult zExamples zStraight skeleton zFlattening

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Fold-and-Cut Problem zGiven any plane graph (the cut graph) zCan you fold the piece of paper flat so that one complete straight cut makes the graph? zEquivalently, is there is a flat folding that lines up precisely the cut graph?

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History of Fold-and-Cut zRecreationally studied by yKan Chu Sen (1721) yBetsy Ross (1777) yHoudini (1922) yGerald Loe (1955) yMartin Gardner (1960)

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Theorem [Demaine, Demaine, Lubiw 1998] [Bern, Demaine, Eppstein, Hayes 1999] zAny plane graph can be lined up by folding flat

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Straight Skeleton zShrink as in Langs universal molecule, but yHandle nonconvex polygons new event when vertex hits opposite edge yHandle nonpolygons butt vertices of degree 0 and 1 yDont worry about active paths

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Perpendiculars zBehavior is more complicated than tree method

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A Few Examples

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A Final Example

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Flattening Polyhedra [Demaine, Demaine, Hayes, Lubiw] zIntuitively, can squash/ collapse/flatten a paper model of a polyhedron zProblem: Is it possible without tearing? Flattening a cereal box

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Connection to Fold-and-Cut z2D fold-and-cut yFold a 2D polygon xthrough 3D xflat, back into 2D yso that 1D boundary lies in a line z3D fold-and-cut yFold a 3D polyhedron xthrough 4D xflat, back into 3D yso that 2D boundary lies in a plane

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Flattening Results zAll polyhedra homeomorphic to a sphere can be flattened (have flat folded states) [Demaine, Demaine, Hayes, Lubiw] y~ Disk-packing solution to 2D fold-and-cut zOpen: Can polyhedra of higher genus be flattened? zOpen: Can polyhedra be flattened using 3D straight skeleton? yBest we know: thin slices of convex polyhedra

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