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Part III: Polyhedra b: Unfolding

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1 Part III: Polyhedra b: Unfolding
Joseph O’Rourke Smith College

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

3 Unfolding Polyhedra Cut along the surface of a polyhedron
Unfold into a simple planar polygon without overlap

4 Edge Unfoldings Two types of unfoldings:
Edge unfoldings: Cut only along edges General 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. spanning: to flatten every vertex forest: cycle would isolate a surface piece tree: connected by boundary of polygon

14 Unfolding the Platonic Solids
Some nets:


16 Unfolding the Archimedean Solids

17 Archimedian Solids

18 Nets for Archimedian Solids

19 Unfolding the Globe: Fuller’s map http://www. grunch

20 Successful Software Nishizeki Hypergami  Javaview Unfold ...

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
[O’Rourke, Schevon 1987]

28 Sclickenrieder1: steepest-edge-unfold
“Nets of Polyhedra” TU Berlin, 1997

29 Sclickenrieder2: flat-spanning-tree-unfold

30 Sclickenrieder3: rightmost-ascending-edge-unfold

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

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