Riemann Surface of R^1/3 by Milton100

Last crawled date: 3 years ago

Object:
Riemann Surface of function f(z)= z^(1/3)
Created By:
Hope Roberts
Date, School & Place:
3/04/2016
George Mason University located in Fairfax, Va.
The creation took place in Mathematica for a class project for Math 493, Math Through 3D Printing.
Background of Riemann Surfaces
Riemann surfaces are named after Bernhard Riemann who was the first to study them.
Riemann surfaces are one way to show multi-valued functions. Showing that an element from the domain can map to multiple places in the codomain. They resemble surface like structures consisting of infinite sheets that are separated by vertical distance. They are configured in the complex plane, which has vectors 1, and i where i is imaginary and these vectors span the complex numbers. All complex numbers match to unique points in the complex plane. (mathworld). Another way to show the multi-valued functions is branch cuts, which is to take lines or line-segments of the multi-valued functions.
Representing a multivalued function is to ascribe not one point but instead infinite points to the domain. The sheets represented by the mappings from the origin and are all interconnected because it is undefined at the origin, which makes the origin, point not a part of the domain. The functions can be oriented in different ways by multiplying by the complex number.
My function used for the Riemann surface was z^(1/3) and the surface was mapped using polar coordinates with the branches of the surface the tubes on top.
The branches also called branch cuts that are on my surface representing where the function is not continuous and is also non differentiable. The branches show when a sheet of the surface connects another sheet of the surface, which can be seen by the changing colors in surface in the mathematica code. As you increased distance in the polar coordinates you continue crossing branch cuts where the sheets are meeting. This is seen in my code and in my print represented by the tube overlay.
The second picture shows the surface without the tube overlay.
The third picture shows the tubes separate.
The fourth shows the combination.
And the final picture shows the stl file loaded onto MakerBot.