Mike Paterson Uri Zwick

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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.

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Overhang

Mike PatersonUri 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

Using n blocks we can get an overhang of

Harmonic Piles

Diamonds?

The 4-diamond is stable

Diamonds?

The 5-diamond is …

Diamonds?

The 5-diamond isUnstable!

Equilibrium

F1

F2

F3

F4

F5

Force equation

F1 + F2 + F3 = F4 + F5

Moment equation

x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5

Forces between blocks

Assumption: No friction.All forces are vertical.

Equivalent sets of forces

1

1

3

Stability

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

Checking stability

F5

F6

F2

F4

F3

F1

F8

F11

F12

F7

F10

F9

F14

F13

F15

F16

Equivalent to the feasibilityof a set of linear inequalities:

F17

F18

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.

Blocks = 4

Overhang = 1.16789

Blocks = 7

Overhang = 1.53005

Blocks = 6

Overhang = 1.4367

Blocks = 5

Overhang = 1.30455

Small optimal stacks

Blocks = 17

Overhang = 2.1909

Blocks = 16

Overhang = 2.14384

Blocks = 19

Blocks = 18

Overhang = 2.27713

Overhang = 2.23457

Small optimal stacks

Support and balancing blocks

Principalblock

Balancing set

Support set

Support and balancing blocks

Balancing set

Principalblock

Support set

Stacks with downward external forces acting on them

Principalblock

Size= number of blocks + sum of external forces.

Support set

Spinal stacks

Stacks in which the support set contains only one block at each level

Principalblock

Support set

Loaded stacks are slightly more powerful.

Conjecture: The difference is bounded by a constant.

Optimal spinal stacks

Optimality condition:

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:

Conjecture:

A factor of 2 improvement over harmonic stacks!

100 blocks example

Towers

Spine

Are spinal stacks optimal?

No!

Support set is not spinal!

Blocks = 20

Overhang = 2.32014

Optimal weight 100 construction

Weight = 100

Blocks = 47

Overhang = 4.20801

“Parabolic” constructions

5-stack

Number of blocks:

Overhang:

Stable!

Using n blocks we can get an overhang of (n1/3) !!!

An exponential improvementover theO(log n)overhang of spinal stacks !!!

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 constructionsand general constructions?
• What is the gap between loaded stacks and standard stacks?