Riemann Surface

grabcad

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. 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.In this example the Riemann surface is f(z)= z^(1/3) and the surface is mapped using polar coordinates with the branches of the surface.The branches also called "branch cuts" represent where the function is not continuous and is also non differentiable. By increasing distance in the polar coordinates you continue crossing "branch cuts" where the sheets are meeting. This can be visible in my 3D rendered images.