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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.

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