Unsolved Problems in Visibility Joseph O’Rourke Smith College

1. Unsolved Problems in VisibilityJoseph O’RourkeSmith College • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

2. 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

3. Klee’s Question • How many guards, • In fixed positions, • each with 360º visibility • are necessary • and sometimes sufficient • to visually cover • a polygon of n vertices

5. Quad’s, Pentagons, Hexagons

6. Chvátal’s Theorem [n/3] guards suffice (and are sometimes necessary) to visually cover a polygon of n vertices

7. Chvátal’s Comb Polygon

8. Fisk’s Proof • Triangulate polygon with diagonals • 3-color graph • Monochromatic guards cover polygon • Some color is used no more than [n/3] times

9. Polygon Triangulation

10. 3-coloring

11. 180º-Guards Csaba Tóth proved that [n/3] 180º-guards suffice.

12. π-floodlights

13. 180º-Vertex Guards

14. Urrutia’s 5/8’s Example

15. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

16. 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.

17. n=3

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

24. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

25. Illuminating Convex Bodies Boundary point illuminated* if light ray penetrates to interior of object. Status: • 2D: Settled • 3D: Open

26. Parallelogram: 22 = 4 lights

30. Parallelopiped: 23 = 8 lights

31. Open Problem Do 7 lights suffice to illuminate* the entire boundary for all other convex bodies (e.g., polyhedra) in 3D? (Hadwiger [1960])

32. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

33. Mirror Polygon: Illuminable?

34. 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.

35. Room not illuminable from x

36. Tokarsky Polygon

37. Vertex Model?

38. Round Vertex Model

39. 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.

40. Open Question Are all mirror polygons illuminable from some point?

41. Outline • Art Gallery Theorems • Illuminating Disjoint Triangles • Illuminating Convex Bodies • Mirror Polygons • Trapping Rays with Mirrors

42. Trapping Light Rays with Mirrors • Arbitrary Mirrors • Circular Mirrors • Segment Mirrors ------------------------- • Narrowing Light Rays

43. Light from x is trapped!

44. Enchanted Forest of Mirror Trees

45. Angular Spreading

46. Ray approaching limit

47. 10 Rays; 3 Segments

48. 1000 mirrors vs. 1 ray

49. 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

50. 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?