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Mike Paterson Uri Zwick

Overhang. Mike Paterson Uri Zwick. 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 version. Idealized 2D version. The classical solution.

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Mike Paterson Uri Zwick

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  1. Overhang Mike PatersonUri 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.G. Coffin – Problem 3009, American Mathematical Monthly (1923). “Real-life” 3D version Idealized 2D version

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

  4. Is the classical solution optimal? Obviously not!

  5. Inverted pyramids?

  6. Inverted pyramids? Unstable!

  7. Diamonds? The 4-diamond is stable

  8. Diamonds? The 5-diamond is …

  9. Diamonds? The 5-diamond isUnstable!

  10. What really happens?

  11. What really happens!

  12. Why is this unstable?

  13. … and this stable?

  14. 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

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

  16. 1 1 3 Stability Definition: A stack of blocks is stable iff there is an admissible set of forces under which each block is in equilibrium.

  17. Checking stability

  18. Checking stability 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

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

  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 block at each level Principalblock Support set

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

  27. Optimal spinal stacks … Optimality condition:

  28. 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!

  29. 100 blocks example Towers Shadow Spine

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

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

  32. Brick-wall constructions

  33. Brick-wall constructions

  34. “Parabolic” constructions 5-stack Number of blocks: Overhang: Stable!

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

  36. “Parabolic” constructions 5-slab 4-slab 3-slab

  37. r-slab 5-slab

  38. r-slab 5-slab

  39. r-slab 5-slab

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

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

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

  43. Open problems • Is the (n1/3) 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 constructionsand general constructions? • What is the gap between loaded stacks and standard stacks?

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