1. Unfolding Polyhedral Surfaces
Joseph O’Rourke
Smith College

2. Outline What is an unfolding? Why study? Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
Orthogonal polyhedra

3. ten1.mov

5. What is an unfolding?
Cut surface and unfold to a single nonoverlapping piece in the plane.

6. What is an unfolding?
Cut surface and unfold to a single nonoverlapping piece in the plane.

7. Unfolding Polyhedra Two types of unfoldings:
Edge unfoldings: Cut only along edges
General unfoldings: Cut through faces too

8. What is an unfolding?
Cut surface and unfold to a single nonoverlapping piece in the plane.

9. Cube with one corner truncated

10. “Sliver” Tetrahedron

11. 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.

12. Polygons: Simple vs. Weakly Simple

13. Nonsimple Polygons

14. Andrea Mantler example

15. Cut edges: strengthening
Lemma: The cut edges of an edge unfolding of a convex polyhedron to a single, connected piece form a spanning tree of the 1-skeleton of the polyhedron.
[Bern, Demaine, Eppstein, Kuo, Mantler, O’Rourke, Snoeyink 01]

16. What is an unfolding?
Cut surface and unfold to a single nonoverlapping piece in the plane.

17. Cuboctahedron unfolding
[Matthew Chadwick]

18. [Biedl, Lubiw, Sun, 2005]

19. Outline What is an unfolding? Why study? Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
Orthogonal polyhedra

20. Lundström Design, http://www.algonet.se/~ludesign/index.html

21. Outline What is an unfolding? Why study? Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
Orthogonal polyhedra

22. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible

23. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible

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

25. Albrecht Dürer, 1425
Melancholia I

26. Albrecht Dürer, 1425
Snub Cube

27. Unfolding the Platonic Solids
Some nets:

28. Archimedian Solids
[Eric Weisstein]

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

30. Sclickenrieder2: flat-spanning-tree-unfold

31. Sclickenrieder3: rightmost-ascending-edge-unfold

32. Sclickenrieder4: normal-order-unfold

33. Percent Random Unfoldings that Overlap
[O’Rourke, Schevon 1987]

34. Classes of Convex Polyhedra with Edge-Unfolding Algorithms
Prisms
Prismoids
“Domes”
“Bands”
Radially monotone lattice quadrilaterals
Prismatoids???

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

36. Overlapping Unfolding

37. Volcano Unfolding

38. Unfolding “Domes”

39. Proof via degree-3 leaf truncation
dodec.wmv
[Benton, JOR, 2007]

40. Lattice Quadrilateral Convex Caps

42. Classes of Convex Polyhedra with Edge-Unfolding Algorithms
Prisms
Prismoids
“Domes”
“Bands”
Radially monotone lattice quadrilaterals
Prismatoids???

43. Unfolding Smooth Prismatoids
[Benbernou, Cahn, JOR 2004]

44. 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
[Spriggs]
≤ (1/2)F
[Dujmenovic, Moran, Wood]
≤ (3/8)F
[Pincu, 2007]

45. Outline What is an unfolding? Why study? Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
Orthogonal polyhedra

46. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible

47. General Unfoldings of Convex Polyhedra
Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net).
Source unfolding [Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87]
Star unfolding [Aronov & JOR ’92]
Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007]
[Poincare?]

48. Shortest paths from x to all vertices

50. Source Unfolding

51. Star Unfolding

52. Star-unfolding of 30-vertex convex polyhedron

54. [Alexandrov, 1950]

55. Geodesics & Closed Geodesics
Geodesic: locally shortest path; straightest lines on surface
Simple geodesic: non-self-intersecting
Simple, closed geodesic:
Closed geodesic: returns to start w/o corner
Geodesic loop: returns to start at corner
(closed geodesic = simple, closed geodesic)

56. Lyusternick-Schnirelmann Theorem
Theorem: Every closed surface homeomorphic to a sphere has at least three, distinct closed geodesics.
Birkoff 1927: at least one closed geodesic
LS 1929: at least three
“gaps” filled in 1978 [BTZ83]
Pogorelov 1949: extended to polyhedral surfaces

57. Quasigeodesic Aleksandrov 1948
left(p) = total incident face angle from left
quasigeodesic: curve s.t.
left(p) ≤ 
right(p) ≤ 
at each point p of curve.

58. Closed Quasigeodesic
[Lysyanskaya, O’Rourke 1996]

59. Grayson Curve Shortening
[Lysyanskaya, O’Rourke 1996]

64. Open: Find a Closed Quasigeodesic
Is there an algorithm
polynomial time
or efficient numerical algorithm
for finding a closed quasigeodesic on a (convex) polyhedron?

65. Exponential Number of Closed Geodesics
Theorem: 2(n) distinct closed quasigeodesics.
[Aronov & JOR 2002]

66. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible

67. Edge-Ununfoldable Orthogonal Polyhedra
[Biedl, Demaine, Demaine, Lubiw, JOR, Overmars, Robbins, Whitesides, ‘98]

68. Spiked Tetrahedron
[Tarasov ‘99] [Grünbaum ‘01] [Bern, Demaine, Eppstein, Kuo ’99]

69. Unfoldability of Spiked Tetrahedron
(BDEKMS ’99)
Theorem: Spiked tetrahedron is edge-ununfoldable

70. Overlapping Star-Unfolding

71. Outline What is an unfolding? Why study? Main questions: status
Convex: edge unfolding
Convex: general unfolding
Nonconvex: edge unfolding
Nonconvex: general unfolding
Orthogonal polyhedra

72. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible

73. Overlapping Source Unfolding
[Kineva, JOR 2000]

74. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible
Orthogonal
polyhedra

75. Orthogonal Polygon / Polyhedron

76. Grid refinement: Orthogonal Polyhedra
[DIL04] [DM04]

77. Types of Unfoldings

78. Gridding Hierarchy

79. Edge-Unf
Vertex-Unf
Not always possible

80. Orthogonal Terrain

81. Four algorithmic techniques
Strip/Staircase unfoldings
Recursive unfoldings
Spiraling unfoldings
Nested spirals

82. Four algorithmic techniques
Strip/Staircase unfoldings
Recursive unfoldings
Spiraling unfoldings
Nested spirals
4x5 grid unfolding Manhattan towers
[Damian, Flatland, JOR ’05]

83. Edge-Unf
Vertex-Unf
Not always possible
All genus-0 o-polyhedra

84. Manhattan Tower

85. Base & Recursion Tree

86. Single Box Unfolding

87. Suturing two spirals

88. Suture Animation rdsuture_4x.wmv
Animation by Robin Flatland & Ray Navarette, Siena College

90. Four algorithmic techniques
Strip/Staircase unfoldings
Recursive unfoldings
Spiraling unfoldings
Nested spirals
-unfolding orthogonal polyhedra
[Damian, Flatland, JOR ’06b]

91. Edge-Unf
Vertex-Unf
Not always possible
All genus-0 o-polyhedra

92. Dents vs. Protrusions

93. Band unfolding tree (for extrusion)

94. Visiting front children

95. Visiting back children

96. Retrace entire path back to entrance point

98. Deeper recursion

99. All strips thickened to fill surface

100. 4-block example
b0b1b2b3b4

101. Result
Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece,
which may be viewed as a 2n x 2n grid unfolding
(and so is, in places, 1/2n thin).
Arbitrary (non-orthogonal) polyhedra: still open.

102. Status of main questions
Shapes
Edge Unfolding?
General Unfolding?
Convex polyhedra
???
Yes: always possible
Nonconvex polyhedra
No: not always possible