Minimal Surfaces

Minimal surfaces are almost exactly as they sound, it is the surface that minimizes the area around it. Think back to when you were younger, and you learned about area and surface area in math class. The methods you learned then were set equations for area, but the goal then was not to minimize the surface. Now, as we get more complex, and we want to cover as much surface as we can with as little material as possible and the way we accomplish this is with minimal surfaces.

Many people have attempted to explain minimal surfaces and how they work over the past several hundred years, but personally, one idea makes the most sense to me and is the easiest to understand. It is the idea that, in order to be considered a minimal surface, the mean curvature at every point on the surface must equal 0. And if you think it through, it makes perfect sense. If the mean curvature is zero, that means all of the curves balance each other out and when the area is taken, there won't be any extra space for the area to take up, hence a minimal surface.

Now you may need a refresher on how to calculate curvature. The equation for curvature is as follows:

$k=\frac{\parallel \stackrel{\to \; <!--\; rightwards\; arrow\; -->}{T}\text{'}\left(t\right)\parallel }{\parallel \stackrel{\to \; <!--\; rightwards\; arrow\; -->}{r}\text{'}\left(t\right)}$

In this equation, k is curvature, r is the parameterization of the curve, and T is the unit tangent vector to r. The tangent plane can be found by finding the normal vector at the point in question and then finding a plane that the normal vector goes through. Once you have curvature, you can find the mean curvature of the entire surface, by calculating the integral of all curvatures and dividing by 2pi. If the mean curvature is zero, we have a minimal surface.

A more exciting way to figure out if a surface is a minimal surface is with bubbles. They are a simple, real-world way to check for minimal surfaces, as they will fill the area with as little bubble solution as possible.

For example, I created a shape similar to that of a basic bubble wand (circle), but I distorted it in the z direction by cos(5x) to give it a zigzag look as seen below:

I believe that even though the edge of the circle goes up and down, the bubble will form and will pull towards the center in order to minimize the amount of bubble juice used. I chose this shape because even though the edges are unique, the minimal surface should be fairly basic, or minimal.

I also wanted to experiment with a helicoid because I think the minimal surface on it will form a sort of slide-like bubble that wraps around the edges when dipped in the bubble juice. The picture below shows my second example of a helicoid:

I made and tested versions of both of these shapes with pipe cleaners and I am interested to see if the harder surface forms the same minimal surfaces as the pipe cleaners. You'll notice both of my shapes also have an extend line attached to them and I have added this to make the bubble process slightly less messy, but it should not affect the minimal surface effect.

I believe these objects should both print fairly easy with minimal supports and complications, but as always, we will see.