Introduction
The next topic of discussion is a rather interesting one that has been periodically studied throughout history. In short, a minimal surface is just a surface that locally minimizes its area. Now, this may sound rather confusing, but there have been quite a few real-world examples that mathematicians have discovered. The helicoid, the catenoids, and the saddle tower are some of the more easily understood examples. Minimal surfaces are super interesting to visualize, and they tie into a lot of other concepts of mathematics too. Today, we will be further understanding minimal surfaces by creating an object and dipping the object into a soap solution to show the minimal surface.
Minimal Surfaces?
As we discussed before, the object that is created will be dipped into a soap solution to show the minimal surface. This is actually really fascinating to think about. Soap actually has a unique property that it creates a soap film when dealing with minimal surfaces. Soap film is the idea of having thing layers of liquid (soap) that is surrounded by air. This will create the minimal surface. This is exactly what we will be performing with the object that we will be designing.
Now that we further understand the process of what we will be doing with our object to find the minimal surface, lets further get into exactly what a minimal surface is and how it is defined. So, we will define the surface as \(S\). A surface \(S \subset R^3\) is minimal only if its curvature is zero at every point. Curvature? Mean curvature of our surface is defined as the curvature of a surface in some ambient space. Now, the mean curvature can actually be found. The mean curvature is the average of the signed curvature over all angles \[C=\frac{1}{2\pi}\int_0^{2\pi} K(\theta)d(\theta)\] This further proves the statement that we made about that a surface is minimal only if its curvature is zero at every point. The image below is a gyroid that was discovered in 1970 by Alan Schoen. This is an infinitely connected triply periodic minimal surface.
My example will not be as sophisticated as this one, but I included this image to show just how unique and fascinating minimal surfaces are. Now that we have a further understanding of minimal surfaces we can get into our example.
Minimal Surface Example
For my minimal surface example, I wanted to continue from my example that I created in class with the pipe cleaners. My example consisted of a cube that was missing one of its sides. The problem that I had in class was that when my object was dipped in the soap solution it was sideways which caused the structural integrity to be a little off. Being able to 3D print my object will allow the structural integrity to be better and easier to see the minimal surface.
The above is my object! It is rather simple and looks pretty lame; however, when it is dipped into the soap solution it creates a rather interesting minimal surface. The minimal surface that it should create will look like a parabola from a 2-dimensional view, or a saddle from a 3-dimensional view. The only problem that I see when going to print this is that the object might need to be a little bit more thickness to it. However, I believe that this object is super cool and that it will create a nice minimal surface to look at!
Why This Example?
I chose this example for a few reasons. First, I believe that the frame/shape I selected was a rather unique example. This example was also chosen, for someone who is just learning the topic will easily be able to understand the concept better. Lastly, this was the example that I chose in class, and I wanted to create a more accurate representation of what I had in mind of what a good minimal surface would be.
The next topic of discussion is a rather interesting one that has been periodically studied throughout history. In short, a minimal surface is just a surface that locally minimizes its area. Now, this may sound rather confusing, but there have been quite a few real-world examples that mathematicians have discovered. The helicoid, the catenoids, and the saddle tower are some of the more easily understood examples. Minimal surfaces are super interesting to visualize, and they tie into a lot of other concepts of mathematics too. Today, we will be further understanding minimal surfaces by creating an object and dipping the object into a soap solution to show the minimal surface.
Minimal Surfaces?
As we discussed before, the object that is created will be dipped into a soap solution to show the minimal surface. This is actually really fascinating to think about. Soap actually has a unique property that it creates a soap film when dealing with minimal surfaces. Soap film is the idea of having thing layers of liquid (soap) that is surrounded by air. This will create the minimal surface. This is exactly what we will be performing with the object that we will be designing.
Now that we further understand the process of what we will be doing with our object to find the minimal surface, lets further get into exactly what a minimal surface is and how it is defined. So, we will define the surface as \(S\). A surface \(S \subset R^3\) is minimal only if its curvature is zero at every point. Curvature? Mean curvature of our surface is defined as the curvature of a surface in some ambient space. Now, the mean curvature can actually be found. The mean curvature is the average of the signed curvature over all angles \[C=\frac{1}{2\pi}\int_0^{2\pi} K(\theta)d(\theta)\] This further proves the statement that we made about that a surface is minimal only if its curvature is zero at every point. The image below is a gyroid that was discovered in 1970 by Alan Schoen. This is an infinitely connected triply periodic minimal surface.
Minimal Surface Example
For my minimal surface example, I wanted to continue from my example that I created in class with the pipe cleaners. My example consisted of a cube that was missing one of its sides. The problem that I had in class was that when my object was dipped in the soap solution it was sideways which caused the structural integrity to be a little off. Being able to 3D print my object will allow the structural integrity to be better and easier to see the minimal surface.
Why This Example?
I chose this example for a few reasons. First, I believe that the frame/shape I selected was a rather unique example. This example was also chosen, for someone who is just learning the topic will easily be able to understand the concept better. Lastly, this was the example that I chose in class, and I wanted to create a more accurate representation of what I had in mind of what a good minimal surface would be.
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