mike paterson uri zwick
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Mike PatersonUri 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 version Idealized 2D version
![Page 3: Mike Paterson Uri Zwick](https://reader036.vdocuments.us/reader036/viewer/2022062309/56813ec7550346895da92f12/html5/thumbnails/3.jpg)
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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Inverted pyramids?
Unstable!
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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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What really happens?
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What really happens!
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Why is this unstable?
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… and this stable?
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Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
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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.
1 1
3
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Checking stability
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Checking stability
F1F2 F3 F4 F5 F6
F7F8 F9 F10
F11 F12
F13F14 F15 F16
F17 F18
Equivalent to the feasibilityof 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 describes an infinitesimal motion that
decreases the potential energy.
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Small optimal stacks
Overhang = 1.16789Blocks = 4
Overhang = 1.30455Blocks = 5
Overhang = 1.4367Blocks = 6
Overhang = 1.53005Blocks = 7
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Small optimal stacks
Overhang = 2.14384Blocks = 16
Overhang = 2.1909Blocks = 17
Overhang = 2.23457Blocks = 18
Overhang = 2.27713Blocks = 19
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Support and balancing blocks
Principalblock
Support set
Balancing
set
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Support and balancing blocks
Principalblock
Support set
Balancing
set
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Principalblock
Support set
Stacks with downward external
forces acting on them
Loaded stacks
Size =
number of blocks
+ sum of external
forces.
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Principalblock
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
1
1
Loaded stacks are slightly more powerful.
Conjecture: The difference is bounded by a constant.
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Optimal spinal stacks
…
Optimality condition:
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Spinal overhangLet 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 = 2.32014Blocks = 20
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Optimal weight 100 construction
Overhang = 4.20801Blocks = 47
Weight = 100
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Brick-wall constructions
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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 (n1/3) !!!
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
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 (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 constructions
and general constructions?● What is the gap between loaded stacks
and standard stacks?