Mike Paterson Uri Zwick Overhang. Mike Paterson Uri Zwick Overhang.
Post on 31-Mar-2015
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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 version Idealized 2D version
No frictionLength parallel to table
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
Balance
Definition: A stack of blocks is balanced iff there is an admissible set of forces under which each block is in equilibrium.
1 1
3
Checking balance
F1F2 F3 F4 F5 F6
F7F8 F9 F10
F11 F12
F13F14 F15 F16
F17 F18
Equivalent to the feasibilityof a set of linear inequalities:
Static indeterminacy:balancing forces, if they exist, are usually not unique!
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 = 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
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
Assumed to be optimal in:
J.F. Hall, Fun with stacking Blocks, American Journal of Physics 73(12), 1107-1116, 2005.
Loaded vs. standard stacks
1
1
Loaded stacks are slightly more powerful.
Conjecture: The difference is bounded by a constant.
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:
Using n blocks we can get an overhang of (n1/3) !!!
An exponential improvement over the O(log n) overhang of
spinal stacks !!!
Is the (n1/3) the final answer?
Mike PatersonYuval Peres
Mikkel ThorupPeter Winkler
Uri Zwick
MaximumOverhangYes!
1
0 1 2 3-3 -2 -1
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
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