1. Mike Paterson Uri Zwick Overhang

2. Mike Paterson Uri Zwick Overhang

3. The overhang problem How far off the edge of the table can we reach by stacking n identical blocks of length 1? J.G. Coffin – Problem 3009, American Mathematical Monthly (1923). Real-life 3D versionIdealized 2D version

4. The classical solution Harmonic Piles Using n blocks we can get an overhang of

5. Is the classical solution optimal? Obviously not!

6. Inverted pyramids?

7. Unstable!

8. Inverted pyramids?

9. Diamonds? The 4-diamond is stable

10. Diamonds? The 5-diamond is …

11. Diamonds? The 5-diamond is Unstable!

12. Is that so?

13. yes!

14. Why is this unstable?

15. … and this stable?

16. Equilibrium F 1 + F 2 + F 3 = F 4 + F 5 x 1 F 1 + x 2 F 2 + x 3 F 3 = x 4 F 4 + x 5 F 5 Force equation Moment equation F1F1 F5F5 F4F4 F3F3 F2F2

17. Forces between blocks Assumption: No friction. All forces are vertical. Equivalent sets of forces

18. Stability Definition: A stack of blocks is stable iff there is an admissible set of forces under which each block is in equilibrium.

19. Stability Definition: A stack of blocks is stable iff there is an admissible set of forces under which each block is in equilibrium. 11 3

20. Checking stability

21. F1F1 F2F2 F3F3 F4F4 F5F5 F6F6 F7F7 F8F8 F9F9 F 10 F 11 F 12 F 13 F 14 F 15 F 16 F 17 F 18 Equivalent to the feasibility of a set of linear inequalities:

22. Stability and Collapse A feasible solution of the primal system gives a set of stabilizing forces. A feasible solution of the dual system with a negative objective function describes an infinitesimal motion that decreases the potential energy.

23. Small optimal stacks Overhang = 1.16789 Blocks = 4 Overhang = 1.30455 Blocks = 5 Overhang = 1.4367 Blocks = 6 Overhang = 1.53005 Blocks = 7

24. Small optimal stacks

26. Overhang = 2.14384 Blocks = 16 Overhang = 2.1909 Blocks = 17 Overhang = 2.23457 Blocks = 18 Overhang = 2.27713 Blocks = 19

27. Support and balancing blocks Principal block Support set Balancing set

28. Support and balancing blocks Principal block Support set Balancing set

29. Principal block Support set Stacks with downward external forces acting on them Loaded stacks Size = number of blocks + sum of external forces.

30. Principal block Support set Stacks in which the support set contains only one block at each level Spinal stacks

31. Loaded vs. standard stacks Loaded stacks are slightly more powerful. Conjecture: The difference is bounded by a constant.

32. Cheating…

33. Optimal spinal stacks

34. … Optimality condition:

35. Spinal overhang Let S (n) be the maximal overhang achievable using a spinal stack with n blocks. Let S * (n) be the maximal overhang achievable using a loaded spinal stack on total weight n. Theorem: A factor of 2 improvement over harmonic stacks! Conjecture:

36. 100 blocks example Spine Shadow Towers

37. Are spinal stacks optimal? No! Support set is not spinal! Overhang = 2.32014 Blocks = 20

38. Optimal weight 100 construction Overhang = 4.20801 Blocks = 47 Weight = 100

39. Brick-wall constructions

41. Parabolic constructions 5-stack Number of blocks:Overhang: Stable!

42. Using n blocks we can get an overhang of (n 1/3 ) !!! An exponential improvement over the O(log n) overhang of spinal stacks !!!

43. Parabolic constructions 5-slab 4-slab 3-slab

44. r-slab

45. 5-slab

46. r-slab 5-slab

47. r-slab 5-slab

49. Vases Weight = 1151.76 Blocks = 1043 Overhang = 10

50. Vases Weight = 115467. Blocks = 112421 Overhang = 50

51. Oil lamps Weight = 1112.84 Blocks = 921 Overhang = 10

52. Open problems Is the (n 1/3 ) construction tight? Yes! Shown recently by Paterson-Peres-Thorup-Winkler-Zwick What is the asymptotic shape of vases? What is the asymptotic shape of oil lamps? What is the gap between brick-wall constructions and general constructions? What is the gap between loaded stacks and standard stacks?