Z^3 – Z^2 Real and Complex Parts

thingiverse

Z^3 – Z^2 Real and Complex Parts Marlanna Bozicevich 11/29/2021 George Mason University MATH401: Mathematics Through 3D Printing Riemann Surface Background In complex analysis, a Riemann surface is a configuration in the complex plane with several sheets. Each sheet covers one repetition of the complex function. We use many sheets to avoid the repetition of the functions specific path on one specific plane. The use of multiple sheets in different planes on the z axis is permitted because the values of the corresponding regular branches coincide with one another from one iteration to the next. Riemann surfaces can be thought of as deformed renditions of the complex plane. If you look at a singular point, the “surface” looks like patches of the complex plane, even though from a global scale, the overall shape is much different. My Code For my function, I chose f(z) = z^3 – z^2. For my 3D print I decided to create two objects, one for the real valued function and the other for the imaginary. If we expand this function out in terms of it’s real and complex parts, where z= x+iy, we have the following: (x+iy)^3 – (x+iy)^2 (x^2 + 2ixy + i^2y^2)*(x+iy) – (x+iy)^2 (x^3 + 2ix^2y + i^2xy^2 + ix^2y + 2i^2xy^2 + i^3y^3) – (x^2 + 2ixy + i^2y^2) (x^3 + 3ix^2y + 3i^2xy^2 + i^3y^3) – (x^2 + 2ixy + i^2y^2) x^3 – x^2 – 3xy^2 + y^2 + i(3x^2y – 2xy – y^3) This leaves the real function as f(x,y) = x^3 – x^2 – 3xy^2 + y^2 and the imaginary part as f(x,y) = 3x^2y – 2xy – y^3. I graphed these in Mathematica using ParametricPlot3D. Additionally, due to the uneven nature of the functions, I added stands to both real and imaginary parts so that the figures could stand alone. This was done using the Graphics3D Cylinder command and Plot3D square root functions. Screenshots of my code are shown in the images. Another way that I could have expresses these functions was in terms of its modulus and argument, where f(re^(iθ)) = (re^(iθ))^3– (re^(iθ))^2 = r^3e^(i3θ) – r^2e^(i2θ) = r^2e^(i2θ) (re^(iθ) – 1). Print During my print slot, I had a few issues with my Mathematica code involving PlotRange, as the generated STL files were ignoring this command in my code. After fixing this issue, I also came across a few printing issues involving supports and therefore I have not seen my final object yet and unfortunately do not have any pictures of it to add here on Thingiverse. However, the STL files for both the real and imaginary functions are attached to this post to get an idea of what the 3D prints actually look like. Citations https://www.youtube.com/watch?v=MegHwU8ywCA https://math.berkeley.edu/~teleman/math/Riemann.pdf https://en.wikipedia.org/wiki/Riemann_surface http://xahlee.info/math/what_is_riemann_surface.html