Skip to main content

Soapy Surfaces

Mathematics and physics are strongly related. Certain events in physics can be easily explained away with an equation. The shape soap film makes between two curves, however, isn't so easy to explain. Soap film always forms minimal surfaces due to the physical properites of soapy water. A minimal surface is a surface where every point on that surface has a mean curvature of zero. What does that even mean? Well, curvature is a measurement of a one dimensional curve. It tells how fast the curve is changing direction at any point on the curve. The formula is \(\left|\left|\frac{d\vec{T}}{ds}\right|\right|\). But surfaces are two dimensional. So, to take the measure of curvature at a point on a surface, you find the mean curvature of every curve inside the planes containing the normal line to that point on the surface. Because that is an infinite number of measurements, we use an integral to find the mean of the curvatures. If the mean curvature of every point on the surface is zero, then that is a minimal surface. Now, only the mean of every point has to be zero. If every curvature of every point was zero, that would be a plane. But what does a mean curvature of zero actually mean for a point on a surface? It essentially means that the area immediately around the point is going both up and down. The measurement of the degree to which the area is changing averages out to zero. An example of a minimal surface is a hyperbolic paraboloid. Every point has the area around it going up and down, averaging out to zero. To make a hyperbolic paraboloid using soap film, you would need a ring whose height is a sine function that has two crests and two troughs.
The frame I chose to make is made up of three semicircles. The middle semicircle is 90 degrees and 45 degrees away from the other two semicircles. The frame was made by using the drawRay function to create short, straight lines that make up a semicircle. I then created three of the semicircles and rotated one of them -90 degrees and the other one 45 degrees. Then, I made a ray on the edge of the semicircles as the handle to the frame. I expect there to be two films between the three semicircles. The 45 degree one will have film that bends inwards between the two semicircles instead of being composed of straight lines. The 90 degree one will be the same except the film will bend more inwards. The minimal surface made between two rings is composed of the shape a rope makes when held at two points. I think it is possible that the inward bend between the semicircles are also composed of that same shape. This would explain why the semicircles further apart from one another have a more inward bend. I chose this frame to demonstrate how different minimal surfaces can be based on the distance the two curves are from one another.

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more