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)?