So it's been a while since I last posted anything to this blog, and I really want to change that. I recently created a series of mathematical animations, all based on the same function. But I wanted to do more than just share these animations, I wanted to explain how I made them and the interesting features I discovered in the process. I thought about making a video for this purpose, as there are plenty of educational channels on YouTube nowadays. But I’m not that great at public speaking, and I did not want my presentation to detract from the animations themselves. Thus I decided to write this blog post.
In this blog entry I attempt to explain how you can build a polynomial up from its roots. That is the points where the function equals zero. I then explain how this same technique can be used to build rational functions, and what this means in the complex plane. This is something I use to generate a lot of the abstract mathematical art that I share on Tumblr, Flickr, and Deviant Art. So if you are interested in duplicating my art style, or just reading the rest of my blog, this entry is essential.
Root finding methods are an important tool to have in any mathematics library. For those who don't know, root finding methods are, by definition, methods that find the zeros (or roots) of some arbitrary function. That is, given a function and some initial parameters, theses methods will find a solution that when plugged back into the function yields zero.
This might sound like a rather specific case, but this problem of finding zeros can be generalized to more exotic problems. For example, computing the inverse of a given function, or finding the intersection point between two curves, can both be obtained using root finding. Furthermore, despite their practical applications, root finding methods can generate beautiful fractals when applied to the complex plane.
One thing that my math teachers always said: "You can't take the logarithm of a negative number". Well it turns out that you can, but the answer is complex, in every sense of the word. I’m going to assume you already know a bit about complex numbers and logarithms.
In order to take the log of a complex number, we must re-write the number in it's polar form. However this form is not unique. In particular, the parameter 'n' can be any integer value, and we still get a valid result. Furthermore, the result of the log function is dependent on which value of 'n' we choose. This means that the log function is multivalued. In fact an infinite number of complex values could satisfy the log function.