MTH401
Riemann Surfaces
For the print this week, we made Reimann surface by running different equations of z=r*e^(i*theta) through Mathematica and generating a 3d object. Mine was the result of taking (z^3 - 1)^(1/3), which resulted in an object that looks like a lillypad in the shape of a bowl.
First, broke down the problem into five pieces. The x-direction and y-direction converted to polar coordinates, the equation for z in polar, the equation for my object in terms of z, the real part of that equation, and the imaginary part, as broken down by Stephen Fischer [1, p.1-3]. This made it easy to manipulate the objects and see what would look nice, using the Extrude argument to add thickness.
Next, I had to select what parts of the output of the z equation I would use. I could use the real part, the imaginary part, or both with one superimposed on another. The imaginary part looked appealing, however seemed much too difficult to print, as the origin has a sharp edges that meet. Next, I looked at the real part, which looked much easier to print. I tried seeing how it would look with both on one image, however it did not look appealing, so I just exported the real part.
Finally was the process of printing the actual object. My object was printed using the makerbot and had no issues. This is primarily due to my selection of the real part over the imaginary part, as the object had no sharp edges. No glue or additional supports were needed to print, besides the default markerbot raft. Overall, it was a very smooth experience.
This assignment was a nice demonstration of complex variables and their mappings. While I haven't had much experience in it, this did help me visualize the two outputs of a function of complex numbers. It was also a good exercise in separating out variables in a Mathematica notedbook, which is something that will be valuable in future assignments, such as the data modeling print.
1) Fisher, S. D. (1999). Complex variables. New York: Dover.
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