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Mike Paterson Uri Zwick Overhang

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Mike Paterson Uri Zwick Overhang

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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). Real-life 3D versionIdealized 2D version No friction Length parallel to table

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

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Is the classical solution optimal? Obviously not!

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Inverted triangles? Balanced?

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???

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Inverted triangles? Balanced?

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Inverted triangles? Unbalanced!

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Inverted triangles?

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Diamonds? The 4-diamond is balanced

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Diamonds? The 5-diamond is …

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Diamonds? The 5-diamond is Unbalanced!

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What really happens?

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What really happens!

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Why is this unbalanced?

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… and this balanced?

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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

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

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

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Checking balance

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

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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.

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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 stable by small modifications. The way unbalanced stacks collapse can be determined in polynomial time

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

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Small optimal stacks

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

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Support and balancing blocks Principal block Support set Balancing set

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Support and balancing blocks Principal block Support set Balancing set

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

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

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

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Cheating…

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Optimal spinal stacks

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… Optimality condition:

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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:

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Optimal 100-block spinal stack Spine Shield Towers

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Optimal weight 100 loaded spinal stack

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Loaded spinal stack + shield

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spinal stack + shield + towers

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

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Optimal 30-block stack Overhang = 2.70909 Blocks = 30

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

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Brick-wall constructions

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Parabolic constructions 6-stack Number of blocks:Overhang: Balanced!

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

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Parabolic constructions 6-slab 5-slab 4-slab

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r-slab

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

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Vases Weight = 1151.76 Blocks = 1043 Overhang = 10

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Vases Weight = 115467. Blocks = 112421 Overhang = 50

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Forces within vases

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Unloaded vases

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Oil lamps Weight = 1112.84 Blocks = 921 Overhang = 10

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Forces within oil lamps

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Brick-by-brick constructions

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Is the (n 1/3 ) the final answer? Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick Maximum Overhang Yes!

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1 0123-3-2 Splitting game Start with 1 at the origin How many splits are needed to get, say, a quarter of the mass to distance n? At each step, split the mass in a given position between the two adjacent positions

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Open problems What is the asymptotic shape of vases? What is the asymptotic shape of oil lamps? What is the gap between brick-wall stacks and general stacks? What is the gap between loaded stacks and standard stacks?

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