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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.G. Coffin – Problem 3009, American Mathematical Monthly (1923). Real-life 3D versionIdealized 2D version

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The classical solution Harmonic Piles 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 pyramids?

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Unstable!

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Inverted pyramids?

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Diamonds? The 4-diamond is stable

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Diamonds? The 5-diamond is …

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Diamonds? The 5-diamond is Unstable!

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Is that so?

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yes!

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Why is this unstable?

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… and this stable?

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

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

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Checking stability

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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:

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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 with a negative objective function describes an infinitesimal motion that decreases the potential energy.

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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

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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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100 blocks example Spine Shadow Towers

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

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

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Brick-wall constructions

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Parabolic constructions 5-stack Number of blocks:Overhang: Stable!

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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 5-slab 4-slab 3-slab

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r-slab

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5-slab

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r-slab 5-slab

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r-slab 5-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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Oil lamps Weight = 1112.84 Blocks = 921 Overhang = 10

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Open problems Is the (n 1/3 ) 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 constructions and general constructions? What is the gap between loaded stacks and standard stacks?

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