Introduction: what is a minimal surface?
A minimal surface is a surface of smallest area for a given perimeter. To better understand this definition, let's consider how to think of a surface. A surface is a homeomorphism, meaning that if you are an ant standing on the surface, you'll think you're on a plane. That's the same reason that when we stand on Earth's surface, we see it as flat despite the fact that it curves.
We can describe minimal surfaces using some pretty nasty differential equations, or we can describe them a way I think is easier. Take \(\vec{r}(t)\) as the parameterization of a curve and \(\vec{T}(t)\) as the unit tangent to \(\vec{r}(t)\). Then, the curvature is \(K=\frac{||\vec{T'}(t)||}{||\vec{r'}(t)||}\). Curvature is simply how quickly the direction changes at various points along the curve. To see a demonstration of the concept of curvature, refer to the image below!
(Image from math.stackexchange.com)
For a surface "M" in \(\mathbb{R^3}\), there is a tangent plane at each point. At every point on M, there is a normal vector perpendicular to the tangent plane. This is like our ant standing at this point on the surface and pointing straight up and down. Now, we want to intersect any plane containing the normal vector with the surface. Doing this produces a curve! We can use the equation given earlier to compute its curvature.
Note that the average of a function \(f(x)\) on \([a,b]\) is \[\frac{1}{b-a} \int_{a}^{b} f(x) dx. \] This \(f(x)\) is our curvature function, K. The mean curvature of the surface M at a point "p" is then \[\frac{1}{2\pi} \int_{1}^{2\pi} K(\theta) d\theta, \] where \(\theta\) is an angle from a starting plane that contains the normal vector.
Given all of this information, we finally arrive at a concise definition! A surface is a minimal surface if and only if the mean curvature at every point is zero!
The shape of my frame and best guess of what the bubble will look like
In my ruled surface blog post, I chose a helicoid because of my interests in biochemistry and DNA-like structures. I wanted to make my minimal surface to be a helicoid as well to continue exploring this cool shape. I was able to modify my ruled surface code to remove the ruled lines and only have the perimeter to use as my frame. The bubble will form the smallest possible area given this perimeter, as stated in the introduction. The frame and bubble will look something like this:
(Image from https://en.wikipedia.org/wiki/Minimal_surface)
Why I chose the design I did
Again, I chose this design to continue exploring the helicoid I used in a previous project. Additionally, I anticipate a helicoid will fit inside the wide-mouth mason jar our bubble solution will be in. My print will be about 1 x 2 inches, and I'm really hoping the bubble forms! If not, I'll work on changing the dimensions until my design works. I thought this design was unique as well because it differed from examples given in class, such as saddles and catinoids. A helicoid is actually the only ruled minimal surface besides the plane, which makes it especially interesting! (Source: shortinformer.com)
Word count: 509
A minimal surface is a surface of smallest area for a given perimeter. To better understand this definition, let's consider how to think of a surface. A surface is a homeomorphism, meaning that if you are an ant standing on the surface, you'll think you're on a plane. That's the same reason that when we stand on Earth's surface, we see it as flat despite the fact that it curves.
We can describe minimal surfaces using some pretty nasty differential equations, or we can describe them a way I think is easier. Take \(\vec{r}(t)\) as the parameterization of a curve and \(\vec{T}(t)\) as the unit tangent to \(\vec{r}(t)\). Then, the curvature is \(K=\frac{||\vec{T'}(t)||}{||\vec{r'}(t)||}\). Curvature is simply how quickly the direction changes at various points along the curve. To see a demonstration of the concept of curvature, refer to the image below!
For a surface "M" in \(\mathbb{R^3}\), there is a tangent plane at each point. At every point on M, there is a normal vector perpendicular to the tangent plane. This is like our ant standing at this point on the surface and pointing straight up and down. Now, we want to intersect any plane containing the normal vector with the surface. Doing this produces a curve! We can use the equation given earlier to compute its curvature.
Note that the average of a function \(f(x)\) on \([a,b]\) is \[\frac{1}{b-a} \int_{a}^{b} f(x) dx. \] This \(f(x)\) is our curvature function, K. The mean curvature of the surface M at a point "p" is then \[\frac{1}{2\pi} \int_{1}^{2\pi} K(\theta) d\theta, \] where \(\theta\) is an angle from a starting plane that contains the normal vector.
Given all of this information, we finally arrive at a concise definition! A surface is a minimal surface if and only if the mean curvature at every point is zero!
The shape of my frame and best guess of what the bubble will look like
In my ruled surface blog post, I chose a helicoid because of my interests in biochemistry and DNA-like structures. I wanted to make my minimal surface to be a helicoid as well to continue exploring this cool shape. I was able to modify my ruled surface code to remove the ruled lines and only have the perimeter to use as my frame. The bubble will form the smallest possible area given this perimeter, as stated in the introduction. The frame and bubble will look something like this:
Why I chose the design I did
Again, I chose this design to continue exploring the helicoid I used in a previous project. Additionally, I anticipate a helicoid will fit inside the wide-mouth mason jar our bubble solution will be in. My print will be about 1 x 2 inches, and I'm really hoping the bubble forms! If not, I'll work on changing the dimensions until my design works. I thought this design was unique as well because it differed from examples given in class, such as saddles and catinoids. A helicoid is actually the only ruled minimal surface besides the plane, which makes it especially interesting! (Source: shortinformer.com)
Word count: 509
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