Using Mathematica to Graph the Knotted Torus and Möbius Strip

Leonhard Euler and Karl Gauss were the pioneers in leading the way to the understanding of parametric surfaces. These men and many others paved the way for Mathematica, the "world's most powerful general computation system"[1]. Mathematica was first invented in 1988 by a group of mathematicians at Wolfram Research led by Stephen Wolfram. Even though I learned that Mathematica's syntax can be very tricky, once it is learned properly Mathematica's capabilities are very, very powerful.

To begin, I will give a formal definition on exactly what it means to parameterize a surface. A parametric surface is simply a mapping from the 2D plane to the 3D space. For curves in a two-dimension plane only one parameter is needed. Since surfaces are in three-dimension, two parameters, u and v, will be used instead of one. So the points (x, y, z) in the  plane will be parameterized by each point (u, v) in the  plane. So each point (u, v) will be sent to  and will be defined by: r(u,v) = x(u, v)i + y(u, v)j + z(u, v)k, which is called a parametric surface. The parametric equations of the surface are as follows, x = x(u, v), y = y(u, v), z = z(u, v).