A catenary is the shape assumed by a hanging chain. More precisely, it is the solution of the following mathematical problem: of all plane curves of a fixed length joining two fixed points, which has the least potential energy in a uniform gravitational field? (Or, alternatively, which curve has the lowest center of mass?) If the two fixed endpoints are at the same height, the catenary joining them looks roughly like a U or a parabola; however, it is not a parabola. While parabolas have the shape of the curve y = x², catenaries have the shape of the graph of the hyperbolic cosine function, y = cosh(x).

Curiously, catenaries also arise in an unrelated problem: if two circles lie in parallel planes, both centered on an axis perpendicular to the planes, what is the surface of least area joining them? The surface can be found experimentally by making...