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?