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!