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Part III: Polyhedra b: Unfolding Joseph ORourke Smith College

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Outline: Edge-Unfolding Polyhedra zHistory (Dürer) ; Open Problem; Applications zEvidence For zEvidence Against

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Unfolding Polyhedra zCut along the surface of a polyhedron zUnfold into a simple planar polygon without overlap

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Edge Unfoldings zTwo types of unfoldings: yEdge unfoldings: Cut only along edges yGeneral unfoldings: Cut through faces too

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Commercial Software Lundström Design,

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Albrecht Dürer, 1425

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Melancholia I

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Albrecht Dürer, 1425 Snub Cube

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Open: Edge-Unfolding Convex Polyhedra Does every convex polyhedron have an edge- unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]

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Open Problem Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron? How about via perimeter-halving?

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Example

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Symposium on Computational Geometry Cover Image

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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. o spanning: to flatten every vertex o forest: cycle would isolate a surface piece o tree: connected by boundary of polygon

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Unfolding the Platonic Solids Some nets:

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Unfolding the Archimedean Solids

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Archimedian Solids

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Nets for Archimedian Solids

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Unfolding the Globe: Unfolding the Globe: Fullers map

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Successful Software zNishizeki zHypergami zJavaview Unfold z...

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Prismoids Convex top A and bottom B, equiangular. Edges parallel; lateral faces quadrilaterals.

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Overlapping Unfolding

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Volcano Unfolding

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Unfolding Domes

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Cube with one corner truncated

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Sliver Tetrahedron

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Percent Random Unfoldings that Overlap [ORourke, Schevon 1987]

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Sclickenrieder 1 : steepest-edge-unfold Nets of Polyhedra TU Berlin, 1997

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Sclickenrieder 2 : flat-spanning-tree-unfold

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Sclickenrieder 3 : rightmost-ascending-edge-unfold

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Sclickenrieder 4 : normal-order-unfold

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Open: Edge-Unfolding Convex Polyhedra (revisited) Does every convex polyhedron have an edge- unfolding to a net (a simple, nonoverlapping polygon)?

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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 F (2/3)F [for large F] (1/2)F [for large F] < ½?; o(F)?

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