The Maths of Pylons , Art Galleries and Prisons Under the Spotlight

1. The Maths of Pylons, Art Galleries and Prisons Under the Spotlight John D. Barrow

2. Some Fascinating Properties of Straight Lines O O O O O O O O O

3. O O O O O O O O O O O O O O O O O O Draw 4 lines through all 9 points The pencil must not leave the paper. No reversing O O O O O O O O O O O O O O O O O O

4. Think Outside The Box O O O O O O O O O 1

5. Think Outside The Box 2 O O O O O O O O O 1

6. Think Outside The Box 2 O O O O O O O O O 1 3

7. Think Outside The Box 2 O O O O O O O O O 1 3 4 !

8. Pylon of the Month

9. “PGG 3 and friends gather around National Grid Company Plc's Norwich Main Substation”

10. Rigidity 

11. The triangle is the ONLY rigid polygon

12. ALLconvex polyhedra are rigid

13. What about non-convex polyhedra? Robert Connelly (1978) finds an 18 triangular-sided example That keeps the volume the same BUT “Almost every” non-convex polyhedron is rigid

14. Klaus Steffen’s 14-sided rigid non-convex polyhedron with 9 vertices and 21 edges

15. Guarding Art Galleries

16. The Art Gallery Problem camera How many cameras are needed to guard a gallery and where should they be placed?

17. Simple Polygonal Galleries Regions with holes are not allowed and no self intersections convex polygon one camera is enough an arbitrary n-gon (n corners) ? cameras might be needed

18. How Many Cameras ? n – 2 cameras can guard the simple n-sided polygon. A camera on a diagonal guards two triangles.  no. cameras can be reduced to roughly n/2. A corner is adjacent to many triangles. So placing cameras at vertices can do even better …

19. Triangulate! Triangulate!

20. Triangulation To make things easier, we divide a polygon into pieces that each need one guard Guard the gallery by placing a camera in every triangle Join pairs of corners by non-intersecting lines that lie inside the polygon

21. 3-Colouring the Gallery Assign each corner a colour: pink,green,oryellow. Any two corners connected by an edge or a diagonal must have different colours. n = 19 Thus the vertices of every triangle will be in three different colors. If a 3-colouring is possible, put guards at corners of same colour Pick the smallest of the coloured corner groupings to locate the cameras. You will need at most [n/3] = 6 cameras where [x] is the integer part of x.

22. The Chvátal Art Gallery Theorem For a simple polygon with n corners, [n/3] cameras are sufficient and sometimes necessary to have every interior point visible from at least one of the cameras. For n = 100, n/3 = 33.33 and [n/3] = 33 [x] is the integer part of x Note that [n/3] cameras may not always be necessary Finding the minimum number is computationally ‘hard’.

23. The Worst Case Scenario [n/3] V-shaped rooms A camera can never be positioned so as to watch over two Vs Here, the maximum of [n/3] cameras is required

24. Orthogonal galleries All corners are right angles Only [n/4] guards are needed, and are always sufficient n = 100 needs only 25 guards now

25. Rectangular galleries All adjacent rooms have connecting archways 8 rooms and 4 guards in the arches In a rectangular gallery with r rooms, [r/2] guards are needed to guard the gallery

26. The Double Cover Problem How many guards must be placed in the gallery so that at least m guards are visible from every point in the gallery? m = 2: A polygon which requires 4 guards to provide double coverage. The entire polygon is only visible from the vertex We can find a gallery which can be covered by one guard located at a particular point, but if the guard is placed elsewhere, even arbitrarily close to the first guard, some of the gallery will be hidden when the guard is at the new position.

27. Mobile Guards Counter eg Counter eg Edge guards patrol along the polygon walls Diagonal guards patrol inside the gallery along straight lines between corners In 1981, Toussaint conjectured that except for a small number of polygons, [n/4] edge guards are sufficient to guard a polygon. Still unproven. O’Rourke proved that the minimum number of mobile guards necessary and sufficient to guard a polygon is [n/4]. He also showed that [(3n+4+4h)/16] mobile guards are necessary and sufficient to guard orthogonal polygons with h holes. n = 100 and h = 0 needs [304/16] = 9, whereas with immobile guards it is [n/4] =25

28. A Worst Case [(3n+4+4h)/16] mobile guards required n = 20 h = 4 needs 80/16 = 5 guards

29. An orthogonal gallery divided into rectangular rooms More than [n/2] guards may be needed. Take a central rectangular room with a similar room on each side. One guard can watch the central room and one other. But no two side rooms share a common wall so each need an extra guard. So, five rooms require four guards. For a gallery with c corners and h holes that is divided into r rectangular rooms, we may need [(2r +c - 2h - 4) / 4] guards Here: c = 20, h = 4, r = 5 so [18/4] = [9/2] = 4

30. The Night Watchman’s Problem Find the shortest closed route around the gallery such that every point can be seen at least once

31. The Art Thief’s Problem Find the shortest path around the gallery that is not visible from particular security points

32. The Fortress Problem n/2 corner guards are always necessary and sufficient to guard the exterior of a polygonal fortress with n walls n = 4 example 4/2 = 2 x is smallest integer  x So  = 4 and 2 = 2

33. n/2  corner guards orn/3 point guards (ie located anywhere) are always sufficient and sometimes necessary to guard the polygonal exterior of a fortress with n corners n = 7 needs  7/3  = 3 point guards and 4 corner guards

34. Prisoner Cell H Block Problem 4 1 For orthogonal fortresses with n Corners: 1 + n/4 corner guards are necessary and sufficient 2 3 12-sided H block will need 1 + 3 = 4

35. The Prison Yard Problem

36. The Prison Yard Problem • Suppose you want to guard the interior and the exterior • n/2 corner guards are always sufficient and may be necessary for a convex polygon with n corners. It is [n/2] if non convex. • Eg n = 101: 51 for convex and 50 for non convex • [5n/12] + 2 corner guards or [(n+4)/3] point guards are always sufficient for an orthogonal prison with holes • Eg n = 100: 43 corner guards or 34 point guards suffices

37. It is now time for you to return to your cells