Part II: Paper b: One-Cut Theorem

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# Part II: Paper b: One-Cut Theorem - PowerPoint PPT Presentation

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: Paperb: 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)
• 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
• 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]
• Any plane graph can be lined upby folding flat
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
• Behavior is more complicated than tree method

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
• 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
• 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