Riemann Surface of Arctan (z)

Riemann Surface of Arctan (z) – by Katharine Downey

The following post is a part of George Mason University Math 401, Mathematics Through 3D Printing.

3D representation of the imaginary values of Arctan(x+iy), where z = x+iy, x = r*cos(θ), and y = r*sin(θ) for theta out to 12π. 

Figure 1: These images are of the printed Reimann surface of arctan(x+iy) for two different orientations

Brief History and Background:

            In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.  These surfaces were first studied by and are named after Bernhard Riemann.  Riemann surfaces can be thought of as deformed versions of the complex plane.  Locally near every point they look like patches of the complex plane, but the global topology can be quite different.  For example, they can look like a sphere or a torus or several sheets glued together. 

            The main interest in Riemann surfaces is that holomorphic functions may be defined between them.  A holomorphic function is a complex-valued function of one or more complex variables that is, at every point of its domain, complex differentiable in a neighborhood of the point (Ahlfors and Sario 1960).  The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal, locally, to its own Taylor series.  Holomorphic functions are the central objects of study in complex analysis for this reason.  Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm function (Jost 2006).

            Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure, specifically a complex structure, which is needed for the unambiguous definition of holomorphic functions.  A two-dimensional real manifold can be turned into a Riemann surface usually in several inequivalent ways if and only if it is orientable and metrizable (Weyl 2009).  The sphere and torus admit complex structures, for example, but the Möbius strip, Klein bottle and real projective plane do not.

Coding Process

            The process was conducted in a program called Mathematica.  The first thing that needed to be done was to define the variables x, y, z, and w, where x and y are the cartesian conversions from the polar coordinate system.  The variable z is the real values of the arctan(x+iy) Reimann surface, this variable was renamed from earlier where z = x+iy.  Similarly, the variable w is the imaginary values of the arctan(x+iy) Reimann surface.  This is illustrated below in Figure 2.

Figure 2: These are the variables that were defined before the surface could be plotted in Mathematica.

            In order to produce a 3D plot of the arctan(x+iy) Reimann surface, a parametric plot must be made with thick enough walls in order for the object to print correctly and withstand wear-and-tear from surroundings after it’s printed.  The ParametricPlot3D command was used to plot the surface and a thickness of 0.1 was included to ensure an adequate thickness of the print.  The radius was out to r = 2 and theta was out to θ = 12π.  The function in its completeness and the initial plot is shown in Figure 3.  The last step is to export the 3D plot to a .stl file in for the 3D printer to read in and generate the 3D version to print.  Different angles of the surface are shown in Figure 4 or illustrative purposes.

Figure 3: This is the plot of the arctan(x+iy) Reimann surface in Mathematica before it was 3D printed

Figure 4: Different orientations of the arctan(x+iy) imaginary surface for illustrative purposes of the shape of the surface.

3D Printing

            I printed the model of the imaginary values of the arctan(x+iy) Reimann surface on an Ultimaker Mini.  Regular PLA filament was used and a specific kind of support was used.  The support type is special to the Ultimaker 3D printer family called tree supports.  These supports are automatically generated for any place on the object that has an angle of 60 degrees or more to the build plate, this includes overhang.

Sources:

Ahlfors, Lars and Sario, Leo (1960), Riemann Surfaces (1st ed.), Princeton, New Jersey:     Princeton University Press

Jost, Jürgen (2006), Compact Riemann Surfaces, Berlin, New York: Springer-Verlag, pp. 208–219,            ISBN 978-3-540-33065-3

Weyl, Hermann (2009) [1913], The concept of a Riemann surface (3rd ed.), New York: Dover       Publications, ISBN 978-0-486-47004-7, MR 0069903