Mike Paterson Uri Zwick
description
Transcript of Mike Paterson Uri Zwick
![Page 1: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/1.jpg)
Mike PatersonUri Zwick
Overhang
![Page 2: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/2.jpg)
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 version Idealized 2D version
No frictionLength parallel to table
![Page 3: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/3.jpg)
The classical solution
Harmonic Stacks
Using n blocks we can get an overhang of
![Page 4: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/4.jpg)
Is the classical solution optimal?
Obviously not!
![Page 5: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/5.jpg)
Inverted triangles?
Balanced?
![Page 6: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/6.jpg)
???
![Page 7: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/7.jpg)
Inverted triangles?
Balanced?
![Page 8: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/8.jpg)
Inverted triangles?
Unbalanced!
![Page 9: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/9.jpg)
Diamonds?
The 4-diamond is balanced
![Page 10: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/10.jpg)
Diamonds?
The 5-diamond is …
![Page 11: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/11.jpg)
Diamonds?
The 5-diamond is Unbalanced!
![Page 12: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/12.jpg)
What really happens?
![Page 13: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/13.jpg)
What really happens!
![Page 14: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/14.jpg)
Why is this unbalanced?
![Page 15: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/15.jpg)
… and this balanced?
![Page 16: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/16.jpg)
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
![Page 17: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/17.jpg)
Forces between blocks
Assumption: No friction.All forces are vertical.
Equivalent sets of forces
![Page 18: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/18.jpg)
Balance
Definition: A stack of blocks is balanced iff there is an admissible set of forces under which each block is in equilibrium.
1 1
3
![Page 19: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/19.jpg)
Checking balance
![Page 20: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/20.jpg)
Checking balance
F1F2 F3 F4 F5 F6
F7F8 F9 F10
F11 F12
F13F14 F15 F16
F17 F18
Equivalent to the feasibilityof a set of linear inequalities:
Static indeterminacy:balancing forces, if they exist, are usually not unique!
![Page 21: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/21.jpg)
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
![Page 22: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/22.jpg)
Small optimal stacks
Overhang = 1.16789Blocks = 4
Overhang = 1.30455Blocks = 5
Overhang = 1.4367Blocks = 6
Overhang = 1.53005Blocks = 7
![Page 23: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/23.jpg)
Small optimal stacks
Overhang = 2.14384Blocks = 16
Overhang = 2.1909Blocks = 17
Overhang = 2.23457Blocks = 18
Overhang = 2.27713Blocks = 19
![Page 24: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/24.jpg)
Support and balancing blocks
Principalblock
Support set
Balancing
set
![Page 25: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/25.jpg)
Support and balancing blocks
Principalblock
Support set
Balancing
set
![Page 26: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/26.jpg)
Principalblock
Support set
Stacks with downward external
forces acting on them
Loaded stacks
Size =
number of blocks
+ sum of external
forces.
![Page 27: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/27.jpg)
Principalblock
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.
![Page 28: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/28.jpg)
Loaded vs. standard stacks
1
1
Loaded stacks are slightly more powerful.
Conjecture: The difference is bounded by a constant.
![Page 29: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/29.jpg)
Optimal spinal stacks
…
Optimality condition:
![Page 30: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/30.jpg)
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:
![Page 31: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/31.jpg)
Optimal 100-block spinal stack
Spine
Shield
Towers
![Page 32: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/32.jpg)
Optimal weight 100 loaded spinal stack
![Page 33: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/33.jpg)
Loaded spinal stack + shield
![Page 34: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/34.jpg)
spinal stack + shield + towers
![Page 35: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/35.jpg)
Are spinal stacks optimal?
No!
Support set is not spinal!
Overhang = 2.32014Blocks = 20
Tiny gap
![Page 36: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/36.jpg)
Optimal 30-block stack
Overhang = 2.70909Blocks = 30
![Page 37: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/37.jpg)
Optimal (?) weight 100 construction
Overhang = 4.2390Blocks = 49
Weight = 100
![Page 38: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/38.jpg)
Brick-wall constructions
![Page 39: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/39.jpg)
Brick-wall constructions
![Page 40: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/40.jpg)
“Parabolic” constructions
6-stack
Number of blocks: Overhang:
Balanced!
![Page 41: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/41.jpg)
Using n blocks we can get an overhang of (n1/3) !!!
An exponential improvement over the O(log n) overhang of
spinal stacks !!!
![Page 42: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/42.jpg)
“Parabolic” constructions
6-slab
5-slab
4-slab
![Page 43: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/43.jpg)
r-slab
r-slab
![Page 44: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/44.jpg)
r-slab within a (r+1)-slab
![Page 45: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/45.jpg)
![Page 46: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/46.jpg)
“Vases”
Weight = 1151.76
Blocks = 1043
Overhang = 10
![Page 47: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/47.jpg)
“Vases”
Weight = 115467.
Blocks = 112421
Overhang = 50
![Page 48: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/48.jpg)
Forces within “vases”
![Page 49: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/49.jpg)
Unloaded “vases”
![Page 50: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/50.jpg)
“Oil lamps”
Weight = 1112.84
Blocks = 921
Overhang = 10
![Page 51: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/51.jpg)
Forces within “oil lamps”
![Page 52: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/52.jpg)
Brick-by-brick constructions
![Page 53: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/53.jpg)
Is the (n1/3) the final answer?
Mike PatersonYuval Peres
Mikkel ThorupPeter Winkler
Uri Zwick
MaximumOverhangYes!
![Page 54: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/54.jpg)
1
0 1 2 3-3 -2 -1
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
![Page 55: Mike Paterson Uri Zwick](https://reader036.fdocuments.us/reader036/viewer/2022062314/5681411b550346895daceae9/html5/thumbnails/55.jpg)
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?