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Mike Paterson

Academic Sponsors’ Day, MSRI, March 2, 2012. Overhang. Uri Zwick. Peter Winkler. Mikkel Thorup. Yuval Peres. Mike Paterson. A Crow Problem:. How long does it take to drive off the crow?. n. -n. -2. -1. 0. 1. 2. Simple random walk: about n 2 throws.

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Mike Paterson

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  1. Academic Sponsors’ Day, MSRI, March 2, 2012 Overhang Uri Zwick Peter Winkler Mikkel Thorup Yuval Peres Mike Paterson

  2. A Crow Problem:

  3. How long does it take to drive off the crow? n -n -2 -1 0 1 2 Simple random walk: about n2throws. One way to see that: consider the probability distribution of crow’s location; its variance goes up by 1after each throw.

  4. A new problem, brought to MSRI in spring ’05 by Zwick: the crow comes back…

  5. …at night! Now what---your first stone will hit the crow and dislodge him, but after that you’re increasingly unsure where he is. You can certainly get him off the wall in order n3 throws, and you certainly still need at least n2. Which is the truth?

  6. Theorem: Order n3throws are necessary. Proof: Uses two different potential functions, each for the wrong problem. -n 0 n An unusual case of two wrongs making a right.

  7. The overhang problem How far off the edge of the table can we reach by stacking n identical blocks of length 1? “Real-life” 3D version Idealized 2D version

  8. Back in time with the overhang problem… John F. Hall, Fun with Stacking Blocks, Am. J. Physics December 2005. Martin Gardner - Scientific American’s “Mathematical Games” column, 1969. J.G. Coffin – Problem 3009, American Mathematical Monthly, 1923. George M. Minchin, A Treatise on Statics with Applications to Physics, 6th ed. (Clarendon, Oxford, 1907), Vol. 1, p. 341.William Walton, A Collection of Problems in Illustration of the Principles of Theoretical Mechanics 2nd ed. (Deighton, Bell, Cambridge, 1855), p. 183.J.B. Phear, Elementary Mechanics (MacMillan, Cambridge, 1850), pp. 140–141.

  9. The classical solution Using n bricks we can get an overhang of “Harmonic Stack”

  10. Is the classical solution optimal? Apparently not. How can we improve the construction?

  11. Inverted pyramids? Claimed to be stable in Mad About Physics, by Chris Jargodzki and Franklin Potter, but…

  12. They are unbalanced, when the number of layers exceeds 2.

  13. Diamonds? The 4-diamond is balanced…

  14. But the 5-diamond is …

  15. not.

  16. What really happens?

  17. What really happens!

  18. Why is this unbalanced?

  19. … and this balanced?

  20. 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

  21. Forces between bricks Assumption: No friction.All forces are vertical. Equivalent sets of forces

  22. 1 1 3 Balanced Stacks Definition: A stack of bricks is balanced iff there is an admissible set of forces under which each brick is in equilibrium.

  23. How can we tell if a stack is balanced?

  24. Checking for 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

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

  26. Bricks = 4 Overhang = 1.16789 Bricks = 7 Overhang = 1.53005 Small optimal stacks Bricks = 5 Overhang = 1.30455 Bricks = 6 Overhang = 1.4367

  27. Small optimal stacks

  28. Small optimal stacks

  29. Bricks = 19 Overhang = 2.27713 Small optimal stacks Bricks = 17 Bricks = 16 Overhang = 2.1909 Overhang = 2.14384 Bricks = 18 Overhang = 2.23457

  30. Support and counterweight bricks Counter-weights Support set These examples are “spinal”: support stack has only one brick per level, so overhang increases with height. Spinal stacks can achieve overhang S(n) ~ log n.

  31. 100 bricks example

  32. But are spinal stacks optimal? No! When # bricks reaches 20 . . . Support set is not spinal. Overhang = 2.32014, slightly exceeding S(20).

  33. Optimal weight 100 construction Weight = 100 Bricks = 47 Overhang = 4.20801

  34. Brick-wall constructions

  35. Brick-wall constructions

  36. “Parabolic” construction 5-stack Number of bricks: Overhang: Stable!

  37. Thus: n bricks can achieve an overhang of order n1/3 ... an exponential improvementover theorderlog noverhang of spinal stacks.

  38. Mayan from 900 BC---no keystone

  39. The Upper Bound Is order n1/3 best possible?? Yes! Argument is based on the idea that laying bricks is like stoning crows. Each additional brick… spreads forces the same way that throwing a stone (at night) spreads the crow’s probability bar.

  40. A generalized version of the “stoning crows” analysis shows that it takes order n3 bricks to get the stack to lean out by n . In particular, a stack of only nbricks cannot overhangby more than 6n1/3brick lengths. The parabolic construction gives overhang (3/16)1/3 n1/3 ~ .572357121 n1/3, so we have the order right but the constant is off by an order of magnitude. Simulations suggest that the constant can be improved by adjusting the shape of the brick wall construction…

  41. “Vases” Weight = 1151.76 Bricks = 1043 Overhang = 10

  42. “Vases” Weight = 115467. Bricks = 112421 Overhang = 50

  43. “Oil lamps” Weight = 1112.84 Bricks = 921 Overhang = 10 giving overhang of about 1.02 n1/3.

  44. How about using the third dimension? Our upper bound proof makes no use of the fact that bricks cannot overlap in space! Hence, the 6n1/3 bound applies even in 3D, as long as there are no non-vertical forces. However, the constant can be improved in space by skintling, Effectively increasing the brick length to (1+w)1/2.

  45. Open problems • What is the correct constant in the maximum overhang, in the rectilinear case? In the general 3-dimensional case? • 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? • Can the proof be extended to cover non-vertical forces (if, indeed, they are possible for 3D bricks)? • How much friction is needed to change the 1/3 exponent for overhang?

  46. Thank you for your attention. Happy stacking…

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