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Part III: Polyhedra b: Unfolding. Joseph O’Rourke Smith College. Outline: Edge-Unfolding Polyhedra. History (Dürer) ; Open Problem; Applications Evidence For Evidence Against. Unfolding Polyhedra. Cut along the surface of a polyhedron. Unfold into a simple planar polygon without overlap.
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Part III: Polyhedrab: Unfolding Joseph O’Rourke Smith College
Outline: Edge-Unfolding Polyhedra • History (Dürer) ; Open Problem; Applications • Evidence For • Evidence Against
Unfolding Polyhedra • Cut along the surface of a polyhedron • Unfold into a simple planar polygon without overlap
Edge Unfoldings • Two types of unfoldings: • Edge unfoldings: Cut only along edges • General unfoldings: Cut through faces too
Commercial Software Lundström Design, http://www.algonet.se/~ludesign/index.html
Albrecht Dürer, 1425 Snub Cube
Open: Edge-Unfolding Convex Polyhedra Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]
Open Problem Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron? How about via perimeter-halving?
Cut Edges form Spanning Tree Lemma: The cut edges of an edge unfolding of a convex polyhedron to a simple polygon form a spanning tree of the 1-skeleton of the polyhedron. • spanning: to flatten every vertex • forest: cycle would isolate a surface piece • tree: connected by boundary of polygon
Unfolding the Platonic Solids Some nets: http://www.cs.washington.edu/homes/dougz/polyhedra/
Unfolding the Globe: Fuller’s maphttp://www.grunch.net/synergetics/map/dymax.html
Successful Software • Nishizeki • Hypergami • Javaview Unfold • ... http://www.cs.colorado.edu/~ctg/projects/hypergami/ http://www.fucg.org/PartIII/JavaView/unfold/Archimedean.html
Prismoids Convex top A and bottom B, equiangular. Edges parallel; lateral faces quadrilaterals.
Percent Random Unfoldings that Overlap [O’Rourke, Schevon 1987]
Sclickenrieder1:steepest-edge-unfold “Nets of Polyhedra” TU Berlin, 1997
Open: Edge-Unfolding Convex Polyhedra (revisited) Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?
Open: Fewest Nets For a convex polyhedron of n vertices and F faces, what is the fewest number of nets (simple, nonoverlapping polygons) into which it may be cut along edges? • ≤ F • ≤ (2/3)F [for large F] • ≤ (1/2)F [for large F] • ≤ F < ½?; o(F)?