I no longer remember how I came to be aware of this paper. No matter. Here is Paul Rojas’s The straight line, the catenary, the brachistochrone, the circle, and Fermat. It is about a set of optimization problems, in this case, attempts to find the shortest path something can follow.
The talk of the catenary and the brachistochrone give away that this is a calculus paper. The catenary and the brachistochrone are some of the oldest problems in calculus as we know it. The catenary is the problem of what shape a weighted chain takes under gravity. The brachistochrone is the problem of what path a beam of light traces out moving through regions with different indexes of refraction. (As in, through films of glass or water or such.) Straight lines and circles we’ve heard of from other places.
The paper relies on calculus so if you’re not comfortable with that, well, skim over the lines with symbols. Rojas discusses the ways that we can treat all these different shapes as solutions of related, very similar problems. And there’s some talk about calculating approximate solutions. There is special delight in this as these are problems that can be done by an analog computer. You can build a tool to do some of these calculations. And I do mean “you”; the approach is to build a box, like, the sort of thing you can do by cutting up plastic sheets and gluing them together and setting toothpicks or wires on them. Then dip the model into a soap solution. Lift it out slowly and take a good picture of the soapy surface.
This is not as quick, or as precise, as fiddling with a Matlab or Octave or Mathematica simulation. But it can be much more fun.