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Minimal Surfaces

Introduction

Minimal surface is the smallest area that gets formed for a given perimeter. Let's look at a quick simple example, the catenoid. This example contains two circles on top of each other. In the picture below you can see the minimal surface of this example. As you can see it is not a cylinder. In the actual minimal surface, you can see the surface gets pulled in towards the middle. The easiest way to find the minimal surface is by bubbles. When you dip a frame into bubble solution a surface is formed between the frames. This surface is the minimal surface for the frame. Using bubble solution, we can easily see the minimal surface for surfaces that aren't even mathematically found yet.


Minimal Surface Definition

There are a couple of ways to define a minimal surface. We will be looking at the definition using mean curvature. Curvature is defined by \[K=\frac{ \Vert \vec{T} \ '(t)\Vert}{\Vert \vec{r}\ '(t) \Vert}\] In the equation K stands for curvature. Where \(\vec{r}(t)\) stands for the parametrization of the curve and \(\vec{T}(t)\) stands for the unit tangent to the parametrization of the curve. Using this formula the curvature would have to equal 0 at every point then only the surface created would be a minimal surface. To compute for curvature, we would intersect the plane containing the normal vector perpendicular with the surface. We would then rotate the plane by an angle \(\theta\). The angle \(\theta\) is in the standard plane that contains the normal vector. This would then give us our entire curve. We know we can find the average of a function f(x) in [a, b] by \(\frac{1}{b-a} \int_{a}^{b} f(x)dx\). We can now use this to find the mean curvature of a surface \[\frac{1}{2\pi}\int_{0}^{2\pi}K(\theta)d\theta\] After looking at this long definition we find out a surface can only be the minimal surface if and only if the mean curvature is equal to 0.

My Example

I have chosen an example to show minimal surfaces. I have created a frame that can be dipped into bubble solution to form the minimal surface for the frame. From class I created a simple frame and I recreated it with openscad. The frame consisted of a square and a circle. Trying to find different shapes I realized that this would have been a boring example and would not form an interesting minimal surface. So, I changed both of the shapes. The shapes that I ended up choosing for my frame can be seen in the image below.



We can see the chosen shapes are a hexagon and a 6-sided star. I wanted the two of my shapes to be different. We can see the star has much more sides than the hexagon. I wanted to try and see if a minimal surface bubble can still be formed between the two shapes even though they seem very different. The minimal surface I think that will be formed for my frame will look a bit like the catenoid that I first talked about. The middle of the bubble film will be pulled in but towards the edges we will be able to see both shaped distinctively. When I print this model out it will be 2 X 2 X 2 inches. I will be changing the dimensions of my model until I can find something that will form the minimal surface for my frame.

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