The Magic of Riemann Surfaces by Connie Quezada

The purpose of this project was to obtain a deeper understanding of the complex numbers and the complex plane. For centuries equations like x^2 +1 had no solution since it does not cross the x-axis, however the problem at the time was that mathematicians did not have all the tools they needed to solve this problem. Now, with current knowledge we know imaginary number exist, where the imaginary number i = sqrt(-1). Complex numbers, usually denoted as ‘z’, are numbers that are of the form z = a+ib,where a and b are real numbers and ‘a’ is known as the real part of the complex number and ‘bi’ is imaginary part of the complex number. 

So how do real numbers and imaginary numbers fit together? It turns out that the real number plane is perpendicular to the complex number plane, so this explains why we don’t see the solutions for equations like x^2+1on the real number plane, the solutions are complex numbers! So in order to visualize the solutions we look at the Riemann surface of the complex function. 

A Riemann surface is a surface-like configuration that covers the complex plane with several “surfaces”. The surfaces could either be simple or complicated structures and interconnections. The surface is actually a 3D Projection of the 4D figure. The Riemann surface is constructed from two copies of the complex plane, and the idea is that each input value of z lies directly below its corresponding points on each layer of our Riemann Surface.

Extending x^2+1 into the complex plane to find the solutions of the equation.

For the Surface z^2 +r, where r is a real number, appears to be a ‘pringle shape. At the center of the shape is our real parabola x^2+1where the imaginary part of our complex function is zero and then above and below the center or the part of the parabola that look like “wings” would be where the imaginary part of our complex function is nonzero. 

z^2+r

Finding the solutions the parabola x^2+1
 which can seen as the blue line,
 and the purple points are the solutions.

For the surface z^2 + i,this equation has no values where the equation would be purely real. With this being said the 3D projection looks like it is intersecting itself, however in 4 Dimensions it is technically not intersecting itself but more like ‘passing’ over itself. 

Both of the print for the Riemann surfaces were designed in Mathematica and took about 2 hours to print each on a Makerbot Printer. Under each surface I placed a simple ‘Disk-like’ base I designed in OpenScad, and I simply sliced the two STL files together to get the final object that I would print. 

Mathematica code for z^2+r

z^2+r print

Mathematica code for z^2+i

Print of z^2+i