The Optimal Can: An Uncanny Approach

Abstract
We examine a novel variation of the classic cylindrical can optimization problem encountered in almost every first-semester calculus course: to minimize the cost of the material used to make the can, which amounts to minimizing the surface area given a fixed volume. The result is a height equal to the diameter of the circular top (square front profile).
We add a distribution/storage ("ship+store") cost (based on real data) to the material cost to get a total cost to minimize. We show two things numerically:

• For low ship+store costs, the material cost dominates and the can has the classic square profile.

• For ship+store costs above a critical value, the symmetry of the square is broken and the can takes on a rectangular profile resulting from the rectangular geometry of the distribution truck or shelf/cabinet area for storage.

Interestingly, the can dimensions remain constant (at different values) above and below the critical ship+store cost, exactly as with a Landau "meanfield" phase transition in statistical mechanics. The parameter for the cost of ship+store appears exactly as an inverse-temperature in magnetic systems, and this Landau-type transition has the same basic phenomenology as a Landau approximation to a magnetic system that is non-magnetic above a critical temperature (paramagnetic) but magnetizes below a critical temperature (ferromagnetic), much like a Curie temperature for a discontinuous phase transition.