2. Pylon of the Month

3. Rigidity 

4. Division leads to more boundary  cut Division Can be Dangerous

5. Fire Storms Ignition of dust produces explosive spread of fire

6. Why use the Fosbury Flop?

7. High Jump Launch Kinetic Energy = M  g  H Centre Of Gravity Centre of gravity H = height of centre of gravity is raised

8. Why Do Tightrope Walkers Carry Long Poles ?

9. High inertia Slow to move Low inertia Quick to move Mass M Average radius R mass spread C Inertia = C  Mass  (Radius) 2 Inertia high low

10. Large inertia means you wobble more slowly

13. StableUnstable

14. MIR Space Station Disaster 1977

15. Damaged Hit by the cargo Spaceship! What are MIR’s Principal Moments of Inertia ??

16. Rigging Rowing Eights

17. Rowing Has Its Moments +N -N F F then Forces on boat

18. Moment on boat  = -Ns + N(s+r) - N(s+2r) + N(s+3r) = +2Nr Then, half a stroke later…N reverses to –N and.. Moment on boat = -2Nr The Wiggling Boat s

19. Moment = +Ns - N(s+r) - N(s+2r) + N(s+3r) = 0 The Italian Rig No wiggle!

20. Three no-wiggle rigs for eights ‘Italian tandem Rig ’ ‘German Rig’

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

22. 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

23. Triangulate!

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

25. 3-Colouring the Gallery Assign each corner a colour: pink, green, or yellow. Any two corners connected by an edge or a diagonal must have different colours. Thus the vertices of every triangle will be in three different colors. A 3-colouring is always 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] cameras. Here it is [19/3] = 6.

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

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

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

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

31. The Global Village 57 Asians 21 Europeans 14 Western hemisphere 8 Africans 70 are non-white 30 are white 6 possess 60% of all wealth – all from the USA 30 live in substandard housing 70 are illiterate 50 are malnourished 1 owns a computer 1 has a college education