A minimal surface is the surface with minimal area for a given perimeter. As such, what is the minimal surface of a perimeter is am easy question with a hard answer. More formally, a surface, \(S\), is minimal if and only if for every point of \(S\) has a neighborhood bounded by a simple closed curve which has the least area among all surfaces with the same boundary. Soap bubbles are often used when discussing minimal surfaces because the soap naturally creates a minimal surface given a perimeter—note, an actual soap bubble that is a sphere is not a minimal surface as it does not have a boundary perimeter. As simple minimal surface to begin thinking about is the catenoid, shown below.
The curved surface of the catenoid represents the minimal surface for our given perimeter—i.e. the two circles at the top and bottom. If you think back to the last time you played with bubbles, you might remember seeing either this shape or other similar ones. See the image below to see that soap bubbles will in fact form the catenoid as the minimal surface between two perimeter circles.
There are a number of ways to formally define what a minimal surface is, one of which we touched upon earlier. Now, however, I am going to give what I feel is a more useful, but perhaps less intuitive, definition for a minimal surface:
We will define curvature by \[ \kappa = \frac{ \| \vec{T} '(t) \| }{ \| \vec{r} '(t) \|} \] where \( \kappa\) is curvature, \(\vec{r} (t)\) is the parameterization of a curve, and \(\vec{T} (t)\) is the unit tangent vector to \(\vec{r} (t)\). For a surface \(M\) in \( ℝ^3 \), there is a tangent plane at each point. Every point on \(M\) has a normal vector perpendicular to the tangent plane. Intersect any plane containing the normal vector with the surface in order to get a curve, defined by \(f(x)\). Note that the average of the curve defined by \(f(x)\) over \([a,b]\) is \[ \frac{1}{b-a} \int^{b}_{a} f(x) dx. \] As we rotate the plane always containing the normal vector by the angle \(\theta\) the curvature varies so we must compute the mean curvature of the surface \(M\) at point \(p\) as follows \[ \frac{1}{2\pi} \int^{2\pi}_{0} \kappa(\theta) d\theta. \] The surface is a minimal surface if and only if the mean curvature at of every point \(p\) on the surface \(M\) is zero (0).
Following the general theme of this course we will be creating a 3D example; but how does one model a minimal surface in three dimensions? As previously stated, minimal surfaces are easy questions with hard answers, and also previously stated, soap bubbles will naturally create a minimal surface given a perimeter. So, we will be creating a perimeter, which will be called a frame when referring to the actual 3D model, that we can dip into soap bubble solution and then watch the soap bubbles form the minimal surfaces. This allows us to ask the easy question of what will our minimal surface be and get the tool of the bubble solution to do the hard work of answering the question for us.
As you will be able to tell from the title of this post, I will be attempting to determine what the minimal surface of a trefoil knot is via our frame and bubbles. An image of the frame I will be 3D printing can be found below, note this frame will be approximately 2 inches square when printed.
I was originally concerned that using a trefoil knot would be seen as repetitive, especially considering our next topic is also knots, but I decided that the surface the trefoil knot and its frame would produce were too interesting to pass up. The minimal surface formed will not be too revolutionary in all but one aspect. The simple aspect of the surface is that it will be a fairly average smooth surface with the exception being that it will intersect itself. Since a knot "passes through" itself so will the minimal surface. I find the terribly interesting as some of the points \(p\) on our surface \(M\) will have two normal vectors at the points of intersection, and those normal vectors will have two tangents vectors that are equal, and are the same vector in actuality. All while the mean curvature at these points \(p\) is zero.
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