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Mike Paterson

Mike Paterson. Overhang bounds. Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler. The classical solution. Using n blocks we can get an overhang of. Harmonic Stacks. Is the classical solution optimal?. Obviously not!. Inverted triangles?. Balanced?. ???.

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Mike Paterson

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  1. Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler

  2. The classical solution Using n blocks we can get an overhang of Harmonic Stacks

  3. Is the classical solution optimal? Obviously not!

  4. Inverted triangles? Balanced?

  5. ???

  6. Inverted triangles? Balanced?

  7. Inverted triangles? Unbalanced!

  8. Inverted triangles? Unbalanced!

  9. Diamonds? Balanced?

  10. Diamonds? The 4-diamond is balanced

  11. Diamonds? The 5-diamond is …

  12. Diamonds? … unbalanced!

  13. What really happens?

  14. What really happens!

  15. How do we know this is unbalanced?

  16. … and this balanced?

  17. Equilibrium F1 F2 F3 F4 F5 Force equation F1 + F2 + F3 = F4 + F5 Moment equation x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5

  18. Checking balance

  19. Checking balance F5 F6 F2 F4 F3 F1 F8 F11 F12 F7 F10 F9 F14 F13 F15 F16 Equivalent to the feasibilityof a set of linear inequalities: F17 F18

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

  21. Blocks = 17 Overhang = 2.1909 Blocks = 16 Overhang = 2.14384 Blocks = 19 Blocks = 18 Overhang = 2.27713 Overhang = 2.23457 Small optimal stacks

  22. Support and balancing blocks Principalblock Balancing set Support set

  23. Support and balancing blocks Balancing set Principalblock Support set

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

  25. Spinal stacks Stacks in which the support set contains only one blockat each level Principalblock Support set

  26. Optimal spinal stacks … Optimality condition:

  27. 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: Conjecture: A factor of 2 improvement over harmonic stacks!

  28. Optimal weight 100 loaded spinal stack

  29. Optimal 100-block spinal stack

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

  31. Optimal 30-block stack Blocks = 30 Overhang = 2.70909

  32. Optimal (?) weight 100 construction Weight = 100 Blocks = 49 Overhang = 4.2390

  33. “Parabolic” constructions 6-stack Number of blocks: Overhang: Balanced!

  34. “Parabolic” constructions 6-slab 5-slab 4-slab

  35. r-slab r-slab

  36. r-slab within an (r+1)-slab

  37. So with n blocks we can get an overhang of cn1/3 for some constant c!!! An exponential improvementover theln noverhang of spinal stacks !!! Note: cn1/3 ~ e1/3 ln n Overhang, Paterson & Zwick, American Math. Monthly Jan 2009

  38. What is really the best design? Some experimental results with optimised “brick-wall” constructions Firstly, symmetric designs

  39. “Vases” Weight = 1151.76 Blocks = 1043 Overhang = 10

  40. “Vases” Weight = 115467. Blocks = 112421 Overhang = 50

  41. then, asymmetric designs

  42. “Oil lamps” Weight = 1112.84 Blocks = 921 Overhang = 10

  43. Ωn is a lower bound for overhang with n blocks? Can we do better? Not much! Theorem: Maximum overhang is less than Cn1/3 for some constant C Maximum overhang, Paterson, Perez, Thorup, Winkler, Zwick, American Math. Monthly, Nov 2009

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

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