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
Using n blocks we can
get an overhang of
Harmonic Stacks
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
F4
F3
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
F5
Force equation
Moment equation
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
F2
F1
F3
F4
F5
F11
F7
F8
F9
F10
F13
F14
F15
F16
F6
F12
Equivalent to the feasibility
of 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.
Small optimal stacks
Blocks = 4
Overhang = 1.16789
Blocks = 5
Overhang = 1.30455
Blocks = 6
Overhang = 1.4367
Blocks = 7
Overhang = 1.53005
Small optimal stacks
Blocks = 16
Overhang = 2.14384
Blocks = 17
Overhang = 2.1909
Blocks = 18
Overhang = 2.23457
Blocks = 19
Overhang = 2.27713
Support and balancing blocks
Balancing
set
Principal
block
Support
set
Support and balancing blocks
Balancing
set
Principal
block
Support
set
Loaded stacks
Stacks with
downward external
forces acting on them
Size
=
Principal
block
number of
blocks
+
sum of external
forces.
Support
set
Spinal stacks
Stacks in which the
support set contains
only one block at each
level
Principal
block
Support
set
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 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 of total weight n.
Theorem:
Conjecture:
A factor of 2 improvement
over harmonic stacks!
100 blocks example
Towers
Shadow
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
Brick-wall constructions
Brick-wall constructions
Brick-wall constructions
“Parabolic” constructions
5-stack
Number of blocks:
Stable!
Overhang:
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?