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

However, I considered the modified sequence

\displaystyle z_{n+1} = p(z_n) - \frac{a\frac{p(z_n)}{p

In other words, I took a Möbius transformation of the term \frac{p(z_n)}{p. By modulating the parameters abc, 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:

mobius_mandelbrot

Here it is in video form, inverted:

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