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 versionIdealized 2D version
The classical solution Harmonic Piles Using n blocks we can get an overhang of
Is the classical solution optimal? Obviously not!
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:
Stability and Collapse A feasible solution of the primal system gives a set of stabilizing forces. A feasible solution of the dual system with a negative objective function describes an infinitesimal motion that decreases the potential energy.
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:
Open problems Is the (n 1/3 ) 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?