Mike Paterson Uri Zwick Overhang

Mike Paterson Uri 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.B. Phear – Elementary Mechanics (1850) J.G. Coffin – Problem 3009, American Mathematical Monthly (1923). Real-life 3D versionIdealized 2D version No friction Length parallel to table

The classical solution Harmonic Stacks Using n blocks we can get an overhang of

Is the classical solution optimal? Obviously not!

Inverted triangles? Balanced?

???

Inverted triangles? Balanced?

Inverted triangles? Unbalanced!

Inverted triangles?

Diamonds? The 4-diamond is balanced

Diamonds? The 5-diamond is …

Diamonds? The 5-diamond is Unbalanced!

What really happens?

What really happens!

Why is this unbalanced?

… and this balanced?

Equilibrium F 1 + F 2 + F 3 = F 4 + F 5 x 1 F 1 + x 2 F 2 + x 3 F 3 = x 4 F 4 + x 5 F 5 Force equation Moment equation F1F1 F5F5 F4F4 F3F3 F2F2

Forces between blocks Assumption: No friction. All forces are vertical. Equivalent sets of forces

Balance Definition: A stack of blocks is balanced iff there is an admissible set of forces under which each block is in equilibrium. 11 3

Checking balance

F1F1 F2F2 F3F3 F4F4 F5F5 F6F6 F7F7 F8F8 F9F9 F 10 F 11 F 12 F 13 F 14 F 15 F 16 F 17 F 18 Equivalent to the feasibility of a set of linear inequalities: Static indeterminacy: balancing forces, if they exist, are usually not unique!

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.

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

Small optimal stacks Overhang = Blocks = 4 Overhang = Blocks = 5 Overhang = Blocks = 6 Overhang = Blocks = 7

Small optimal stacks

Overhang = Blocks = 16 Overhang = Blocks = 17 Overhang = Blocks = 18 Overhang = Blocks = 19

Support and balancing blocks Principal block Support set Balancing set

Support and balancing blocks Principal block Support set Balancing set

Principal block Support set Stacks with downward external forces acting on them Loaded stacks Size = number of blocks + sum of external forces.

Principal block 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), , 2005.

Loaded vs. standard stacks Loaded stacks are slightly more powerful. Conjecture: The difference is bounded by a constant.

Cheating…

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: A factor of 2 improvement over harmonic stacks! Conjecture:

Optimal 100-block spinal stack Spine Shield Towers

Optimal weight 100 loaded spinal stack

Loaded spinal stack + shield

spinal stack + shield + towers

Are spinal stacks optimal? No! Support set is not spinal! Overhang = Blocks = 20 Tiny gap

Optimal 30-block stack Overhang = Blocks = 30

Optimal (?) weight 100 construction Overhang = Blocks = 49 Weight = 100

Brick-wall constructions

Parabolic constructions 6-stack Number of blocks:Overhang: Balanced!

Using n blocks we can get an overhang of (n 1/3 ) !!! An exponential improvement over the O(log n) overhang of spinal stacks !!!

Parabolic constructions 6-slab 5-slab 4-slab

r-slab

r-slab within a (r+1)-slab

Vases Weight = Blocks = 1043 Overhang = 10

Vases Weight = Blocks = Overhang = 50

Forces within vases

Unloaded vases

Oil lamps Weight = Blocks = 921 Overhang = 10

Forces within oil lamps

Brick-by-brick constructions

Is the (n 1/3 ) the final answer? Mike Paterson Yuval Peres Mikkel Thorup Peter Winkler Uri Zwick Maximum Overhang Yes!

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

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?