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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

Part II: Paperb: One-Cut Theorem

Joseph O’Rourke

Smith College

(Many slides made by Erik Demaine)

outline
Outline
  • Problem definition
  • Result
  • Examples
  • Straight skeleton
  • Flattening
fold and cut problem
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?
history of fold and cut
History of Fold-and-Cut
  • Recreationally studiedby
    • Kan Chu Sen (1721)
    • Betsy Ross (1777)
    • Houdini (1922)
    • Gerald Loe (1955)
    • Martin Gardner (1960)
theorem demaine demaine lubiw 1998 bern demaine eppstein hayes 1999
Theorem [Demaine, Demaine, Lubiw 1998] [Bern, Demaine, Eppstein, Hayes 1999]
  • Any plane graph can be lined upby folding flat
straight skeleton
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
perpendiculars
Perpendiculars
  • Behavior is more complicated than tree method
flattening polyhedra demaine demaine hayes lubiw

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?
connection to fold and cut
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
flattening results
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