Mike PatersonUri Zwick
Overhang
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
Harmonic Piles
Using n blocks we can get an overhang of
Is the classical solution optimal?
Obviously not!
Inverted pyramids?
Inverted pyramids?
Unstable!
Diamonds?
The 4-diamond is stable
Diamonds?
The 5-diamond is …
Diamonds?
The 5-diamond is Unstable!
What really happens?
What really happens!
Why is this unstable?
… and this stable?
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
Forces between blocks
Assumption: No friction.All forces are vertical.
Equivalent sets of forces
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
Checking stability
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:
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.
Small optimal stacks
Overhang = 1.16789Blocks = 4
Overhang = 1.30455Blocks = 5
Overhang = 1.4367Blocks = 6
Overhang = 1.53005Blocks = 7
Small optimal stacks
Overhang = 2.14384Blocks = 16
Overhang = 2.1909Blocks = 17
Overhang = 2.23457Blocks = 18
Overhang = 2.27713Blocks = 19
Support and balancing blocks
Principalblock
Support set
Balancing
set
Support and balancing blocks
Principalblock
Support set
Balancing
set
Principalblock
Support set
Stacks with downward external
forces acting on them
Loaded stacks
Size =
number of blocks
+ sum of external
forces.
Principalblock
Support set
Stacks in which the support set contains
only one block at each level
Spinal stacks
Loaded vs. standard stacks
1
1
Loaded stacks are slightly more powerful.
Conjecture: The difference is bounded by a constant.
Optimal spinal stacks
…
Optimality condition:
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:
100 blocks example
Spine
Shadow
Towers
Are spinal stacks optimal?
No!
Support set is not spinal!
Overhang = 2.32014Blocks = 20
Optimal weight 100 construction
Overhang = 4.20801Blocks = 47
Weight = 100
Brick-wall constructions
Brick-wall constructions
“Parabolic” constructions
5-stack
Number of blocks: Overhang:
Stable!
Using n blocks we can get an overhang of (n1/3) !!!
An exponential improvement over the O(log n) overhang of
spinal stacks !!!
“Parabolic” constructions
5-slab
4-slab
3-slab
r-slab
5-slab
r-slab
5-slab
r-slab
5-slab
“Vases”
Weight = 1151.76
Blocks = 1043
Overhang = 10
“Vases”
Weight = 115467.
Blocks = 112421
Overhang = 50
“Oil lamps”
Weight = 1112.84
Blocks = 921
Overhang = 10
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?