2. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Unsolved Problems in Visibility Joseph O’Rourke Smith College  Art Gallery Theorems  Illuminating Disjoint Triangles  Illuminating Convex Bodies  Mirror Polygons  Trapping Rays with Mirrors

3. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Art Gallery Theorems  360º-Guards:  Klee’s Question  Chvátal’s Theorem  Fisk’s Proof  180º-Guards:  Tóth’s Theorem  180º-Vertex Guards:  Urrutia’s Example

4. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Klee’s Question  How many guards, In fixed positions, In fixed positions, each with 360º visibility each with 360º visibility are necessary are necessary and sometimes sufficient and sometimes sufficient to visually cover to visually cover a polygon of n vertices a polygon of n vertices

5. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.

6. Quad’s, Pentagons, Hexagons

7. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Chvátal’s Theorem [n/3] guards suffice (and are sometimes necessary) to visually cover a polygon of n vertices [n/3] guards suffice (and are sometimes necessary) to visually cover a polygon of n vertices

8. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Chvátal’s Comb Polygon

9. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Fisk’s Proof 1. Triangulate polygon with diagonals 2. 3-color graph 3. Monochromatic guards cover polygon 4. Some color is used no more than [n/3] times

10. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Polygon Triangulation

11. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. 3-coloring

12. 180º-Guards Csaba Tóth proved that [n/3] 180º-guards suffice.

13. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. π-floodlights

14. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. 180º-Vertex Guards

15. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Urrutia’s 5/8’s Example

16. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Outline  Art Gallery Theorems  Illuminating Disjoint Triangles  Illuminating Convex Bodies  Mirror Polygons  Trapping Rays with Mirrors

17. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Illuminating Disjoint Triangles How might lights suffice to illuminate the boundary of n disjoint triangles? Boundary point is illuminated if there is a clear line of sight to a light source.

18. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. n=3

24. Current Status  n lights are sometimes necessary  [(5/4)n] lights suffice.  Conjecture (Urrutia): n+c lights suffice (for some constant c).

25. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Outline  Art Gallery Theorems  Illuminating Disjoint Triangles  Illuminating Convex Bodies  Mirror Polygons  Trapping Rays with Mirrors

26. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Illuminating Convex Bodies Boundary point illuminated* if light ray penetrates to interior of object. Status:  2D: Settled  3D: Open

27. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Parallelogram: 2 2 = 4 lights

28. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.

31. Parallelopiped: 2 3 = 8 lights

32. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Open Problem Do 7 lights suffice to illuminate* the entire boundary for all other convex bodies (e.g., polyhedra) in 3D? (Hadwiger [1960])

33. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Outline  Art Gallery Theorems  Illuminating Disjoint Triangles  Illuminating Convex Bodies  Mirror Polygons  Trapping Rays with Mirrors

34. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Mirror Polygon: Illuminable?

35. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Mirror Polygons Victor Klee (1973): Is every mirror polygon illuminable from each of its points? G. Tokarsky (1995): No: For some polygons, a light at a certain point will leave another point dark.

36. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Room not illuminable from x

37. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Tokarsky Polygon

38. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Vertex Model?

39. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Round Vertex Model

40. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Conjectures Under round-vertex model, all mirror polygons are illuminable from each point. Under the vertex-kill model, the set of dark points has measure zero.

41. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Open Question Are all mirror polygons illuminable from some point?

42. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Outline  Art Gallery Theorems  Illuminating Disjoint Triangles  Illuminating Convex Bodies  Mirror Polygons  Trapping Rays with Mirrors

43. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Trapping Light Rays with Mirrors  Arbitrary Mirrors  Circular Mirrors  Segment Mirrors -------------------------  Narrowing Light Rays

44. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Light from x is trapped!

45. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Enchanted Forest of Mirror Trees

46. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Angular Spreading

47. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Ray approaching limit

48. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. 10 Rays; 3 Segments

49. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. 1000 mirrors vs. 1 ray

50. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Conjectures No collection of disjoint segment mirrors can trap all the light from one source. No collection of disjoint circle mirrors can trap all the light from one source

51. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Conjectures (continued) A collection of disjoint segment mirrors may trap only X nonperiodic rays from one source. X =  countable number of  finite number of  zero?

52. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Narrowing Light Rays Rays are narrowed to ε if the angle between any pair or rays that escape to infinity is less than ε > 0.

53. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. 20º → 10 º

54. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. 10º → 5 º

55. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Necklace of Mirrors: 7 Disks

56. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Necklace of Mirrors: 13 Disks

57. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002. Narrowing Theorems Given any ε > 0, the light emitted by a point source can be narrowed by a finite number of disjoint segment mirrors, or circle mirrors.

58. Copyright 2002 Joseph O’Rourke. DIMACS Connect Institute Keynote Address, May 4, 2002.  Art Gallery Theorems  Do [(5/8)n] 180º vertex guards suffice?  Illuminating Disjoint Triangles  Do n+c lights suffice?  Illuminating Convex Bodies  Do 8 lights suffice in 3D?  Mirror Polygons  Is every polygon illuminable from some point?  Trapping Rays with Mirrors  Can segment mirrors trap all rays from one light source? Favorite Open Problems