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?
History of Fold-and-Cut zRecreationally studied by yKan Chu Sen (1721) yBetsy Ross (1777) yHoudini (1922) yGerald Loe (1955) yMartin Gardner (1960)
Theorem [Demaine, Demaine, Lubiw 1998] [Bern, Demaine, Eppstein, Hayes 1999] zAny plane graph can be lined up by folding flat
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
Perpendiculars zBehavior is more complicated than tree method
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
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
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