Minimal Surfaces

Introduction

Imagine you work for a company that manufactures cake stands like the one pictured below.

You are in charge of packaging and decide to wrap the stand with shrink wrap. However, to save money you want the amount of shrink wrap between the top plate and bottom stand to be as small as possible.

This is an example of a minimal surface: the surface between two frames with the minimum possible area. This post will explore a mathematical definition of these surfaces and examine some more interesting examples.

A Mathematical Definition of Minimal Surfaces

Even though we know minimal surfaces are the surface of minimal area between a given frame, what does this mean mathematically? Although there are several ways to define a minimal surface, we will use a definition involving curvature.

Curvature is a concept you may have learned in a calculus course. In two dimensions, it is a value that measures how curvy a line is at a given point. The calculation is: \[ K = \frac{\| \vec{T}(t) \|}{\| \vec{r}(t) \|}\ \] in which \( \vec{r}(t) \) is the parametrization of the curve and \( \vec{T}(t) \) is the unit tangent to \( \vec{r}(t) \).

Notably, this equation works for a two-dimensional line- how would we apply it to 3 dimensions? For our 3-dimensional surface, what we aim to do is find the average curvature of a point on our surface. To do this, we imagine that we slice our surface through that point. The cross section will be a line for which we can calculate the curvature at our point. If we do this infinite times and take the average, we will get the average curvature of that point. As we do this, we want to make sure we only make slices that result in non-trivial cuts. For example, if we were to take the slice through the point that is perpendicular to the surface, our cross-section would only be a point, for which we cannot take the curvature. Thus, to avoid any of these weird cases, we start our calculations by finding any plane that is perpendicular to our surface at the point and rotating this to find all the feasible cuts through that point. Thus, if we define \( \theta \) as the angle our starting normal plane is rotated, the mean curvature can be calculated as follows: \[ \frac{1}{2 \pi}\ \int_{0}^{2 \pi} K( \theta ) \, d\theta\ \]

Now, how does this relate to minimal surfaces? A surface is minimal when the mean curvature we calculate above is 0 for every point on the surface.

Minimal Surfaces as Bubbles

One easy way to visualize minimal surfaces is to use bubble solution. When a frame is dipped in the solution, it automatically forms the smallest surface between the frames. The bubble frame I decided to use were two stars above each other, but with the top one rotated so that the points were offset. The frame looks as depicted below:

I predict that the minimal surface will be stretched inward between the stars, and pulled outward to reach the points of the stars. In other words, I expect it to look similiar to the image below:

Why this example?

I chose this frame because between the interesting shapes of the top and bottom frames and the rotation of the top frame, the model had plenty of interesting characteristics that seemed like they would lend themselves well to forming an interesting minimal surface. I particularly liked that the frames jutted out in different directions, which I imagine would pull the minimal surface in interesting ways.

Author: Sarah Bombrys