# Möbius Dynamics

For my final project in pattern formation, I investigated a new form of Newton iteration by combining it with Möbius transformations (NB: there’s a very nice video by Doug Arnold and Jonathan Rogness that explains these intuitively). These transformations are functions $$f \colon \mathbb{C} \rightarrow \mathbb{C}$$ given by $$f(z) = \frac{az + b}{cz + d}$$, where $$a, b, c, d \in \mathbb{C}$$ satisfy the condition $$ad – bc \neq 0$$. In Newton iteration, we approximate the roots of a polynomial $$p(z)$$ by constructing the sequence $$(z_n)$$ given by

$$\displaystyle z_{n+1} = p(z_n) \ – \frac{p(z_n)}{p'(z_n)}$$

However, I considered the modified sequence

$$\displaystyle z_{n+1} = p(z_n) \ – \frac{a\frac{p(z_n)}{p'(z_n)} + b}{c\frac{p(z_n)}{p'(z_n)} + d}$$

In other words, I took a Möbius transformation of the term $$\frac{p(z_n)}{p'(z_n)}$$. By modulating the parameters $$a$$, $$b$$, $$c$$, and $$d$$, many different fractal patterns emerge. Here is a gallery of my results:

I was especially interested in how to move continuously from fractal to fractal. Hence, I generated a video that demonstrates how several of these images relate to one another:

As another avenue of pattern generation, I also used the Mandelbrot set as input: Here it is in video form, inverted:

Please also check out my fellow students’ final projects at our course website!