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Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick

Maximum Overhang. Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick. The overhang problem. How far off the edge of the table can we reach by stacking n identical blocks of length 1 ?

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Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick

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  1. MaximumOverhang Mike PatersonYuval PeresMikkel ThorupPeter WinklerUri Zwick

  2. The overhang problem How far off the edge of the table can we reach by stacking n identical blocks of length 1? J.B. Phear – Elementary Mechanics (1850)J.G. Coffin – Problem 3009, American Mathematical Monthly (1923). No friction Length parallel to table “Real-life” 3D version Idealized 2D version

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

  4. Is the classical solution optimal? Obviously not!

  5. Inverted triangles? Balanced?

  6. ???

  7. Inverted triangles? Balanced?

  8. Inverted triangles? Unbalanced!

  9. Diamonds? The 4-diamond is balanced

  10. Diamonds? The 5-diamond is …

  11. Diamonds? The 5-diamond isUnbalanced!

  12. What really happens?

  13. What really happens!

  14. Why is this unbalanced?

  15. … and this balanced?

  16. 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

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

  18. 1 1 3 Balance Definition: A stack of blocks is balanced iff there is an admissible set of forces under which each block is in equilibrium.

  19. Checking balance

  20. 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 Static indeterminacy:balancing forces, if they exist, are usually not unique!

  21. Balance, Stability and Collapse Most of the stacks considered are precariously balanced, i.e., they are in an unstable equilibrium. In most cases the stacks can be made stableby small modifications. The way unbalanced stacks collapse can be determined in polynomial time

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

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

  24. Support and balancing blocks Principalblock Balancing set Support set

  25. Support and balancing blocks Balancing set Principalblock Support set

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

  27. Spinal stacks Stacks in which the support set contains only one blockat each level Principalblock Support set Assumed to be optimal in: J.F. Hall, Fun with stacking Blocks, American Journal of Physics 73(12), 1107-1116, 2005.

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

  29. Optimal spinal stacks … Optimality condition:

  30. 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!

  31. Optimal 100-block spinal stack Towers Shield Spine

  32. Optimal weight 100 loaded spinal stack

  33. Loaded spinal stack + shield

  34. spinal stack + shield + towers

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

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

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

  38. Brick-wall constructions

  39. Brick-wall constructions

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

  41. Using n blocks we can get an overhang of (n1/3) !!! An exponential improvementover theO(log n)overhang of spinal stacks !!!

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

  43. r-slab r-slab

  44. r-slab within a (r+1)-slab

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

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

  47. Forces within “vases”

  48. Unloaded “vases”

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

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