I have always been interested in the visualization of mathematics. In high school I spent too much time solving interesting problems with my TI-86 graphing calculator and not enough time with the actual cirriculum. Those experiences however have made me a bit of a tinkerer when it comes to interesting visualizations. Below are some results of a recent project I completed to learn more about Julia Sets.

In these images below the colors indicate the velocity at which an imaginary number when provided to an equation escapes towards velocity. Each pixel in these images can be considered to represent an imaginary number z, where the x-axis denotes its real number component, while the y-axis denotes its imaginary component. When this number z, is continually fed into an equation like z⁴ + 0.434 + 0.75i, we can observe how quickly that resulting number speeds off towards infinity. In this image below, we see red areas that indicate imaginary numbers who when fed into this equation quickly reach infinity. We also observe other points, denoted with orange, yellow, etc, which do not escape to infinity quickly.

The most challenging part of my experiment was not drawing the sets, but coloring them in such a way that was aesthetically pleasing. This meant selection of interesting color palettes (http://kuler.com/), finding a way to map them to products of my equations using bilinear interpolation, and the elimination of banding of colors through continuous color smoothing. This is a topic for another blog post and once I clean up the source code for the project I created, I’ll post it to Github. I came out of this little adventure with some interesting images and a lot more knowledge about Julia sets than I’d previously had.