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Part II: Paper b: One-Cut Theorem

Part II: Paper b: One-Cut Theorem. Joseph O’Rourke Smith College (Many slides made by Erik Demaine). Outline. Problem definition Result Examples Straight skeleton Flattening. Fold-and-Cut Problem. Given any plane graph (the cut graph )

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Part II: Paper b: One-Cut Theorem

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  1. Part II: Paperb: One-Cut Theorem Joseph O’Rourke Smith College (Many slides made by Erik Demaine)

  2. Outline • Problem definition • Result • Examples • Straight skeleton • Flattening

  3. Fold-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?

  4. History of Fold-and-Cut • Recreationally studiedby • Kan Chu Sen (1721) • Betsy Ross (1777) • Houdini (1922) • Gerald Loe (1955) • Martin Gardner (1960)

  5. Theorem [Demaine, Demaine, Lubiw 1998] [Bern, Demaine, Eppstein, Hayes 1999] • Any plane graph can be lined upby folding flat

  6. Straight 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

  7. Perpendiculars • Behavior is more complicated than tree method

  8. A Few Examples

  9. Flattening a cereal box Flattening Polyhedra[Demaine, Demaine, Hayes, Lubiw] • Intuitively, can squash/collapse/flatten a paper model of a polyhedron • Problem: Is it possible without tearing?

  10. Connection to Fold-and-Cut • 3D fold-and-cut • Fold a 3D polyhedron • through 4D • flat, back into 3D • so that 2D boundarylies in a plane • 2D fold-and-cut • Fold a 2D polygon • through 3D • flat, back into 2D • so that 1D boundarylies in a line

  11. Flattening 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

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