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 given by , where satisfy the condition . In Newton iteration, we approximate the roots of a polynomial by constructing the sequence given by

However, I considered the modified sequence

In other words, I took a Möbius transformation of the term . 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!