MATH 401: Math Through 3D Printing

Taylor Fountain

Fall 2019, Section 001

1 Period of w = arcsin(z)

Functions, surfaces, and transformations are 3 of several mathematical ideas that are introduced strictly on the field of real numbers: a real number undergoes an operation, outputting some amount of other real numbers, that are often represented on a graph with 2 or 3 real axes. But, when generalizing these concepts to the set of complex numbers, graphing becomes significantly harder: to map these functions from the complex plane onto the complex plane, 4 dimensions are required, while our feeble human minds can only visualize in 3. In the 19^th century, Bernhard Riemann introduced a workaround for this: complex valued functions can be shown with two 3-dimensional plots: 1 from the complex plane to the real axis that represents the real component of the function, and 1 from the complex plane to the imaginary axis that represents it’s imaginary component, producing two Riemann surfaces.

The main difficulty in graphing these surfaces is taking the complex valued function and separating it into the real and complex components. For the inverse sine function ( w=sin^-1(z) ), the separation process starts with inverting the function ( sin(w) = z ) and rewriting the sine function as in it’s exponential form, resulting in

by rearranging the equation into the form of a polynomial where u = e^iw and using the quadratic equation, we can rewrite this equation as

Note that the +/- is kept in only when using the multi-valued inverse sine function. When using the single-valued inverse sine function, only the positive branch is shown. Then, using the complex logarithm identity,

Solving for w and simplifying, we get

To find the real component of this, we again use the complex logarithm identity to rewrite the function as

And, because, when z is strictly real, Arg[z] = 0 and the magnitude is equal to the function itself, we can reduce the equation to

Next came the process of plotting the function, where I used the ParametricPlot3D function in Mathematica, using the polar definitions of x and y with Euler’s formula to get it in a form that Mathematica can render. The resulting graph will only show one period of oscillation; however, due to the multivalued nature of theta in the complex plane, the functions can be stacked to form a complete (and consequently infinite) representation.

 

To simplify printing, the plot was broken into 2 components, which were identical due to the functions symmetry. Two copies of the piece were rotated onto their side to reduce print time, and were printed with rafts and supports in 1hr 30min on the Makerbot; no problems arose. The pieces were then glued to form the real Riemann surface of w = sin^-1(z).

Object - 2 were printed and glued to form Riemann surface