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**Part III: Polyhedra b: Unfolding**

Joseph O’Rourke Smith College

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**Outline: Edge-Unfolding Polyhedra**

History (Dürer) ; Open Problem; Applications Evidence For Evidence Against

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**Unfolding Polyhedra Cut along the surface of a polyhedron**

Unfold into a simple planar polygon without overlap

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**Edge Unfoldings Two types of unfoldings:**

Edge unfoldings: Cut only along edges General 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. spanning: to flatten every vertex forest: cycle would isolate a surface piece 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: Fuller’s map http://www. grunch**

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**Successful Software Nishizeki Hypergami Javaview Unfold ...**

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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**

[O’Rourke, Schevon 1987]

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**Sclickenrieder1: steepest-edge-unfold**

“Nets of Polyhedra” TU Berlin, 1997

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**Sclickenrieder2: flat-spanning-tree-unfold**

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**Sclickenrieder3: rightmost-ascending-edge-unfold**

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**Sclickenrieder4: 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 ≤ (2/3)F [for large F] ≤ (1/2)F [for large F] ≤ F < ½?; o(F)?

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