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Part III: Polyhedra b: Unfolding Joseph ORourke Smith College.

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Presentation on theme: "Part III: Polyhedra b: Unfolding Joseph ORourke Smith College."— Presentation transcript:

1 Part III: Polyhedra b: Unfolding Joseph ORourke Smith College

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

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

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

5 Commercial Software Lundström Design,

6 Albrecht Dürer, 1425

7 Melancholia I

8 Albrecht Dürer, 1425 Snub Cube

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

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

11 Example

12 Symposium on Computational Geometry Cover Image

13 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

14 Unfolding the Platonic Solids Some nets:

15

16 Unfolding the Archimedean Solids

17 Archimedian Solids

18 Nets for Archimedian Solids

19 Unfolding the Globe: Unfolding the Globe: Fullers map

20 Successful Software zNishizeki zHypergami zJavaview Unfold z...

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

22 Overlapping Unfolding

23 Volcano Unfolding

24 Unfolding Domes

25 Cube with one corner truncated

26 Sliver Tetrahedron

27 Percent Random Unfoldings that Overlap [ORourke, Schevon 1987]

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

29 Sclickenrieder 2 : flat-spanning-tree-unfold

30 Sclickenrieder 3 : rightmost-ascending-edge-unfold

31 Sclickenrieder 4 : normal-order-unfold

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

33 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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