Constant Mean Curvature Surfaces

The "mean curvature" at a point on a surface is defined as the average $(\kappa_{1} + \kappa_{2})/2$ of its two principal curvatures. We can construct surfaces of revolution of constant mean curvature by rotating a special type of curve around the $z$ axis. Such curves are either "catenary curves","elliptic catenary curves", or "hyperbolic catenary curves". We obtain such a curve by rolling a conic section along a fixed line, while tracing the path of its focal point. The curve through which the focal point passes is our desired curve. The three cases correspond to rotating ellipses, parabolas, and hyperbolas respectively. These curves belong to the broader family of curves obtained by following a point on the boundary of a shape as it is rolled along, called "roulettes". See here for an interesting account of catenary curves.

A minimal surface can be thought of as a surface which locally minimises its area. More concretely, a minimal surface has constant zero mean curvature. The study of minimal surfaces began with Lagrange's work in 1762, where he investigated the problem of finding a surface of minimal area which stretches across a given boundary curve [1]. This is known as Plateau's problem, named after Belgian physicist Joseph Plateau. Plateau's problem is explored in detail in Oskar Singer's book [2].

Everyday examples of minimal surfaces can be made using soap films. Alongside their aesthetic appeal, soap films offer a neat solution to the "motorway problem" (the problem of trying to join several points using the shortest possible total length of roads). For more on this surprising application, check out this excellent video. Using a mathematical "knot" (a closed loop of string, which may intertwine) as a frame for a soap film, it is possible to create minimal surfaces which can give insight into knot theory. Use the links to the left to explore models of surfaces with constand mean curvature.

Ideas pertaining to minimal surfaces can be used in architecture, to great effect. For example, Costa's minimal surface is a particularly intriguing minimal surface, inspiring snow and stone sculptures [3]. It is a relatively modern surface, discovered by Brazilian mathematician Celso Costa in 1982. Another fun minimal surface is the gyroid, a periodic minimal surface discovered by Alan Schoen in 1970. Alan Schoen's YouTube channel has some fun videos on minimal surfaces and soap films (here, periodic means that is repeats itself in the same way as the familiar graphs of $sin$ and $cos$). The gyroid has inspired a large sculpture, through which visitors to its exhibition can climb. See here for many artworks based on minimal surfaces, or see here for a concrete furniture exhibition, all of  which are minimal surfaces. A particularly exciting minimal surface installation is Chris Bosse's "green void".

References

[1] J. Lagrange, Essai d'une nouvelle méthode pour détérminer les maxima et les minima des formules intégrales indéfinies (1760-1761), http://gallica.bnf.fr/ark:/12148/bpt6k2155691/f385

[2] Oskar Singer, SINGER'S LOCK: The Revolution in the Understanding of Weather, http://weather.org/singer/chapt08.htm

[3] http://stanwagon.com/public/InvisibleHandshakeHyperseeingSubmit.pdf