Skip to main content

Minimal Surfaces

Did you know that bubbles are actually really good at math? Everytime an object is dipped in bubble solution, the bubble has to solve a differential equation to figure out the shape it will take while lying on that object. That differential equation is: \[ (1 + u_x^2)u_{yy} -2u_xu_yu_{xy} + (1 + u_y^2)u_{xx} = 0 \] This differential equation is incredibly difficult for any human to solve, and as a result there are are not that many well defined examples of minimal surfaces.

A More Digestible Defintion

The differential equation shown above is great as it defines a minimal surface as any solution to that equation. The problem is, the equation is incredibly hard to solve, and it provides no real insight towards what properties these types of surfaces actually have. Another definition of a minimal surface is defined by the curvature at all points. If the mean curvature is 0 at all points, then surface is a minimal surface. To better explain this take a look at this surface:
This is one concrete example known as the catenoid. It is the solid of revolution created from the catenary curve. The catenary curve is an interesting phenomenon but will not be explained in this blog. For the mean curvature of all points to be 0, the surface can be viewed from some symmetric point. In this case, let's place the symmetric point in the very miiddle of the catenoid; this would be the point in the middle of the smallest circle of the catenoid. At this point, if one curve were taken from the surface, there exists another curve on the opposite side of the symmetry point that has equal but opposite curvature. This observation gives a much better idea as to what a minimal surface can look when compared to the differential equation. From this observation, there is one surface that can immediately be added to the list of minimal surfaces: the plane. To calculate the mean curvature for a surface, the curvature at all points must be found using calculus in the form of the equation: \[ \frac{1}{2\pi} \int_{0}^{2\pi} \kappa (\theta) \,d\theta \] This function integrates over the circumference of the circle and takes the curvature for every angle theta. Since curvature can be negative, the integral can sum to 0, making the mean curvature 0.

Creating Minimal Surfaces

Bubble soap, being the mathemetician it is, will create minimal surfaces when covering a structure of some kind. In a typical bubble blower, this is most often a circular section of the plane. If the frame that was dipped in bubble soap were more complex, the surface created by the bubble film will actually be a minimal surface. In this manner, there are an infinite amount of minimal surfaces that can be created in practice, but are difficult to define using mathematics. What we do know is that the surface created is a solution to the differential equation, and the surface has a mean curvature of 0. The frame I have created looks like:
This frame will be dipped in bubble solution to see what minimal surface is created as a result. My guess is that the surface will be totally symmteric around the center point. The surface will also consist of 4 planes that bow in towards the origin point. The reason I chose this frame was because it seems intuitive that a bubble would want to make a sphere shape. This frame consists of two perpindicular sections of that sphere, but despite this I know that this frame will not create a sphere. I am curious to see the minimal surface in practice and that is why I chose this frame.

Comments

Post a Comment

Popular posts from this blog

Knot 9-31

Knot 9-31 A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into. The Crossing Number For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings! The Unknotting Number The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still...
Post a Comment
Read more

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...
Post a Comment
Read more

Middle of Mass Measured from Moment

The center of mass of an object is the point that lies in the average position of mass in that object. For a normal polygon, that point is in the middle of the shape. However, if the shape is irregular, has concave angles, or has holes, the center of mass can be far from the middle of the object. Two things come into factor when determining the center of mass of an object. Weight and distance. Obviously, if one side of an object has more mass than the other, the center of mass will be on the heavier side. But, distance plays a role, too. If two sides of an object have the same mass, but one is farther from the middle, the center of mass will be on the side that has farther mass. Multiplying both of these factors together gives the moment of an object. Dividing the moment by the overall mass gives the location of the center of mass.   The reason I chose my shape is to mimic how those toy balancing birds work. The toy is a model bird with its wings outstretched. The ...
Post a Comment
Read more