Minimal Surfaces

Did you know that bubbles are actually really good at math? Everytime an object is dipped in bubble solution, the bubble has to solve a differential equation to figure out the shape it will take while lying on that object. That differential equation is: \[ (1 + u_x^2)u_{yy} -2u_xu_yu_{xy} + (1 + u_y^2)u_{xx} = 0 \] This differential equation is incredibly difficult for any human to solve, and as a result there are are not that many well defined examples of minimal surfaces.

A More Digestible Defintion

The differential equation shown above is great as it defines a minimal surface as any solution to that equation. The problem is, the equation is incredibly hard to solve, and it provides no real insight towards what properties these types of surfaces actually have. Another definition of a minimal surface is defined by the curvature at all points. If the mean curvature is 0 at all points, then surface is a minimal surface. To better explain this take a look at this surface:

This is one concrete example known as the catenoid. It is the solid of revolution created from the catenary curve. The catenary curve is an interesting phenomenon but will not be explained in this blog. For the mean curvature of all points to be 0, the surface can be viewed from some symmetric point. In this case, let's place the symmetric point in the very miiddle of the catenoid; this would be the point in the middle of the smallest circle of the catenoid. At this point, if one curve were taken from the surface, there exists another curve on the opposite side of the symmetry point that has equal but opposite curvature. This observation gives a much better idea as to what a minimal surface can look when compared to the differential equation. From this observation, there is one surface that can immediately be added to the list of minimal surfaces: the plane. To calculate the mean curvature for a surface, the curvature at all points must be found using calculus in the form of the equation: \[ \frac{1}{2\pi} \int_{0}^{2\pi} \kappa (\theta) \,d\theta \] This function integrates over the circumference of the circle and takes the curvature for every angle theta. Since curvature can be negative, the integral can sum to 0, making the mean curvature 0.

Creating Minimal Surfaces

Bubble soap, being the mathemetician it is, will create minimal surfaces when covering a structure of some kind. In a typical bubble blower, this is most often a circular section of the plane. If the frame that was dipped in bubble soap were more complex, the surface created by the bubble film will actually be a minimal surface. In this manner, there are an infinite amount of minimal surfaces that can be created in practice, but are difficult to define using mathematics. What we do know is that the surface created is a solution to the differential equation, and the surface has a mean curvature of 0. The frame I have created looks like:

This frame will be dipped in bubble solution to see what minimal surface is created as a result. My guess is that the surface will be totally symmteric around the center point. The surface will also consist of 4 planes that bow in towards the origin point. The reason I chose this frame was because it seems intuitive that a bubble would want to make a sphere shape. This frame consists of two perpindicular sections of that sphere, but despite this I know that this frame will not create a sphere. I am curious to see the minimal surface in practice and that is why I chose this frame.