d by dy

Why are coke cans cylindrical?  Why do milk cartons have square cross sections?

Coke cans are cylindrical because the cylinder is the second-most optimal shape for minimizing surface area under a volume constraint.  The most optimal shape is the sphere, but you cannot stack spheres.  The minimization problem ensures that the weight of the containers, and hence the cost of their manufacturing, are minimized.

That is simple enough.  However, coke cans are actually not perfectly cylindrical.  It is thinner at the top than at the side.  Furthermore, its bottom surface is curved, shaped like an three-dimensional arch bridge.  The reason again has to do with weight minimization.  But in the real world there is the added constraint that the container must be strong enough to withstand pressurized interiors.  Does this also explain why the height of coke cans exceed the diameter of its cross section?  According to basic calculus, the surface area of a cylinder is minimized when the height equals the diameter.

So why are milk cartons not cylindrical?  I think the answer lies in the fact that milk cartons are made from paper, rather than alloys used in coke cans.  Paper is strong enough to hold non-pressurized liquid, and vastly cheaper.  Thus, for milk cartons, whatever small cost savings in manufacturing are offset by increased costs of shelf space.  To see the point of shelf space, consider the fact that you cannot fill up an entire sheet of paper with circles;  there would be gaps and wasted space.

Alloy is used in coke cans because of its strength.  Alloy and paper are not the only common materials, however.  One question remains:  why are bigger soft drink products contained in plastic bottles rather than cans?  Does plastic actually become more cost effective as the size of the container increases?

* * *

Coincidentally, after I read and thought about coke cans last Sunday, I put a can of diet Pepsi in the freezer to cool it down.  When I forgot to take it out before I went to bed, the can’s side surface broke.  That is empirical evidence that the reinforced top and the concave bottom are stronger.

Feed for this entry. Leave a response, or trackback from your own site.