1. Unfolding and Reconstructing Polyhedra Brendan Lucier University of Waterloo Master’s Thesis Presentation University of Waterloo, Waterloo, Ontario January 2006

2. Outline Unfolding Polyhedra  Introduction  Ununfoldable Polyhedra  Convex Polyhedra Reconstructing Polyhedra

3. Outline Unfolding Polyhedra  Introduction  Ununfoldable Polyhedra  Convex Polyhedra Reconstructing Polyhedra

4. Unfolding Polyhedra We consider the act of cutting a polyhedron along its edges and unfolding it into the plane. The resulting planar figure is an edge unfolding. Two faces of the unfolding overlap if they have a common interior point. An unfolding with no overlap is a simple unfolding, or net. A polyhedron with no net is referred to as ununfoldable.

5. Unfolding with Overlaps Image due to W. Schlickenrieder, “Nets of Polyhedra,” 1999

6. Unfolding with Overlaps Image due to W. Schlickenrieder, “Nets of Polyhedra,” 1999

7. Simple Unfolding Image due to W. Schlickenrieder, “Nets of Polyhedra,” 1999

8. Definitions A convex set is one where the line between two points in the set remains in the set. A face angle of a polyhedron is an interior angle in a face. The total face angle of a vertex is the sum of its face angles. The curvature of a vertex is 360 minus its total face angle. convexnonconvex positive curvature negative curvature

9. Locality of Overlaps An overlap is k-local if the overlapping faces are connected by a path of at most k vertices in the unfolding. 2-local overlap 1-local overlap 3- and 4-local overlaps

10. Outline Unfolding Polyhedra  Introduction  Ununfoldable Polyhedra  Convex Polyhedra Reconstructing Polyhedra  Introduction  Edge Vectors  Edge Lengths

11. History 1999: Bern, Demaine, Eppstein, and Kuo discover an ununfoldable polyhedron with convex faces. This polyhedron has 24 faces. 2002: Grünbaum constructs an ununfoldable convex- faced polyhedron with 13 faces. 2003: Bern, Demaine, Eppstein, Kuo, Mantler, and Snoeyink construct an ununfoldable simplicial polyhedron with 36 faces. Images due to E. Demaine, “Folding and Unfolding,” 2001, and B. Grunbaum, “No-net Polyhedra,” 2002.

12. Describing 1-Local Overlaps A 1-local overlap corresponds precisely to a situation in which a vertex in an unfolding has total face angle greater than 360 o. Convex polyhedra clearly avoid 1-local overlaps, since they contain no vertices with negative curvature. Motivating Question: does every polyhedron admit an unfolding that contains no 1-local overlaps? No! v

13. Four-Pointed Star Our polyhedron is a 4-pointed star. The polyhedron is parameterized:  α (large) is the length of each spike.  β (small) is the height of the center points.

14. Argument Claim: every unfolding of the 4- pointed star contains a 1-local overlap. Main idea: To avoid 1-local overlaps, we must cut at least two opposing edges incident to each B vertex.

15. Argument (con’t) Our cuts must form a spanning tree, so they cannot form a cycle. The cuts must therefore be made as shown in the bottom diagram. There is one final cut that must be made.

16. Argument (con’t) The center vertices have must have at least 2 incident cuts (due to negative curvature). The remaining cut must therefore be incident with both C vertices, but they are not adjacent! It is therefore impossible to cut our 4-pointed star to avoid 1-local overlaps.

17. Creating a Smaller Example The 4-Pointed Star is an ununfoldable polyhedron with 16 triangular faces. We shall now make modifications to create a smaller ununfoldable polyhedron. The result will be an ununfoldable convex-faced polyhedron with 9 faces.

18. Modification #1: 3-Pointed Star

19. Modification #2: Flatten One side of the polyhedron is flattened. Two triangular faces are combined into a single quadrilateral face. Think of the operation as “cutting” the polyhedron into two halves, lengthwise.

20. Modification #3: Perturb the symmetry of the star by increasing the angle between two spikes. This adds a new parameter, φ, indicating the angle change. Perturbing by φ = 10 o is sufficient.

21. Unfolding This Polyhedron At the B vertices with face angles greater than 120, two opposing edges must be cut. The vertices with negative curvature must each have two incident cuts. This leaves a few possibilities for unfolding, which are dealt with by a case analysis.

22. Possible Unfoldings

23. Summary There is a polyhedron with 16 triangular faces for which every unfolding contains a 1-local overlap. There exists an ununfoldable, convex-faced polyhedron with 9 faces.

24. Outline Unfolding Polyhedra  Introduction  Ununfoldable Polyhedra  Convex Polyhedra Reconstructing Polyhedra

25. History 1975: Shepphard conjectures that all convex polyhedra can be unfolded without overlap. 1992-1993: Namiki and Fukuda create a Mathematica package for unfolding polyhedra. Fukuda makes a stronger conjecture about unfolding convex polyhedra. 1997: Schlickenrieder performs an empirical study of various unfolding methods and makes further conjectures about unfolding convex polyhedra.

26. 2-local overlaps Recall that an unfolding of a convex polyhedron will never contain a 1-local overlap. For convex polyhedra, we study 2-local overlaps. 2-local overlap

27. Sufficient Conditions We have no clean characterization of conditions for a 2- local overlap, but we do have a set of sufficient conditions. A 2-local overlap occurs from cut tree T if  a vertex v has an unfolding angle >270 incident with edge (v,w), where w has degree 1, and  the curvature at w is sufficiently small.

28. Disproving Conjectures We use the conditions from the previous slide as follows:  We disprove certain conjectures of the form “every convex polyhedron has a non-overlapping unfolding of the form X.”  For each unfolding type X, we construct a polyhedron such that we obtain a 2-local overlap when we use X.

29. Conjecture #1: Shortest Path Tree Given a polyhedron, a shortest path tree is created as follows:  Choose a vertex v to be the root.  For every other vertex w, take the shortest path from v to w.  The union of all these paths is the shortest path tree. Conjecture (Fukuda): If a convex polyhedron is cut along a shortest path tree, the resulting unfolding will not contain an overlap. We shall disprove this conjecture.

30. Idea of Proof Consider the illustrated graph and shortest path tree. Convert into a convex polyhedron by raising the internal vertices slightly; shortest path tree remains the same. Also, curvatures will be as small as we like, so the conditions for a 2-local overlap are satisfied.

31. Conjecture #2: Steepest Edge Take a polyhedron and choose a direction vector. From each vertex, cut the edge that is directed most toward the chosen vector. This forms the steepest edge cut tree. Conjecture (Schlickenrieder): every convex polyhedron has a steepest edge cut tree that creates a non-overlapping unfolding.

32. Disproving Schlickenrieder: Steepest edge algorithm applied to the graph at right with direction “up.” Get a cut tree that generates a 2-local overlap.

33. Disproving Schlickenrieder: Steepest edge algorithm applied to the graph at right with direction “up.” Get a cut tree that generates a 2-local overlap. In fact, if the direction is chosen to be close enough to up, we get the same cut tree. The given graph eliminates a certain open range of choices for directions.

34. Disproving Schlickenrieder (con’t) Embed our graph in a triangle and turn it into a convex terrain by raising interior vertices. Form a polyhedron by covering the sphere with many instances of this terrain. Place instances in such a way to cover all possible choices of direction vectors.

35. Normal Order Unfoldings

36. Disproving Normal Order Unfoldings

37. Summary We specified conditions on a cut tree that generate a 2-local overlap. We created convex polyhedra for which unfoldings of the following types have overlaps:  Shortest path cut tree  Steepest edge cut tree

38. Outline Unfolding Polyhedra  Introduction  Ununfoldable Polyhedra  Convex Polyhedra Reconstructing Polyhedra

39. Introduction In the thesis we analyze the complexity of two related decision problems: Problem 1: Edge Vectors  Given a set of vectors, does there exist a polygon/polyhedron with those vectors as edges (i.e. edge lengths and orientations)? Problem 2: Edge Lengths  Given a set of values, does there exist a polygon/polyhedron with those values as edge lengths? In this talk we will only discuss the Edge Vector problem.

40. Edge Vector Reconstruction We will look at many different types of edge vector reconstruction problems. We attempt to reconstruct different things:  Polygons  Polyhedra  Convex Polygons/Polyhedra  Orthogonal Polygons/Polyhedra  etc.

41. Partition Partition: given a set of positive integer values with a sum of 2S, is there a subset of the values with a sum of S? Partition is known to be an NP-hard problem. Partition remains NP-complete if we require that all values are unique and that our subset uses exactly half of the input values (shown in thesis). 7 5 3 7 6 1 8 4 7 48 8 7 1 3 5 7 7 6 4 24

42. Review: Computational Complexity We will be showing that some edge vector reconstruction problems are NP-hard. We do this by showing that if we can solve the edge vector problem, then we can solve Partition with just a little more work.  Given some input values (w 1, …, w n ) for the Partition problem, we will create input values (i.e. a set of sticks) for our reconstruction problem.  We will choose the input values so that the answer to our reconstruction problem is “Yes” if and only if the answer to the Partition problem is “Yes.”  This shows that reconstruction must be just as hard as Partition.

43. Problem 1: Degenerate Polygons A degenerate polygon is one in which two incident edges are collinear. The edge vector problem is NP-hard for degenerate polygons.  Proof: Given an instance (w 1, …, w n ) of Partition, we create an instance of the edge vector reconstruction problem with input 11 w1w1 w2w2 wnwn...

44. Problem 2: Degenerate Polyhedra Now extend to polyhedra. The edge vector problem is NP-complete for degenerate polyhedra.  Given an instance (w 1, …, w n ) of Partition, create an instance of Edge Vector with 11 w1w1 w2w2 wnwn... 11 SS 1111

45. Problem 3: Convex Polygons The edge vector problem for convex polygons is NP-complete. We reduce from the version of Partition where the two sets must have the same number of values and all values are unique. Suppose (w 1, …, w n ) is an instance of Partition, with a sum of 2S. Then our decision problem will have the following vectors: n/2 S B1B1 S B2B2

46. Problem 3: Convex Polygons The edge vector problem for convex polygons is NP-complete. We reduce from the version of Partition where the two sets must have the same number of values and all values are unique. Suppose (w 1, …, w n ) is an instance of Partition, with a sum of 2S. Then our decision problem will have the following vectors: n/2 S B1B1 S B2B2 (S,n/2)

47. Problem 4: Convex Polyhedra The edge vector problem is also NP-hard for convex polyhedra. We build our solution upon that for convex polygons. Create two copies of the polygon construction. Add some vectors (0,0,1). Then the only way these vectors can form a convex polyhedron is if they create a prism, with our convex polygon as the base.

48. Summary of all results

49. Conclusion Overarching idea: different ways to create a polyhedron.

50. Thank You