Fractal Viewer

Here, I give a brief overview of my fractal viewer application. So, fractals can be characterized as infinitely complex shapes, some of which have finite area but infinite roughness and perimeter! Many commonly seen fractals are often derived from very simple equations, like \(z_{new}=z^2+c\), which describe which points are in or out the fractal. We start with \(z=0\) and \(c=\) "an imaginary number" and check whether repeated iterations using the \(z_{new}\) values lead to unbounded behavior (if so, the starting point is not in the fractal, otherwise, it is). For example, for the \(z_{new}=z^2+c\) rule, \(1+0i\) is NOT in the fractal because:

1. \(z_{new}=z^2+c\rightarrow z_{new}=0+1=1\)
2. \(z_{new}=z^2+c\rightarrow z_{new}=1+1=2\)
3. \(z_{new}=z^2+c\rightarrow z_{new}=2+1=3\)

As you keep re-plugging the output back into the formula, the numbers continue to grow larger and larger, but for \(-1+0i\) things are a little different:

1. \(z_{new}=z^2+c\rightarrow z_{new}=0-1=-1\)
2. \(z_{new}=z^2+c\rightarrow z_{new}=1-1=0\)
3. \(z_{new}=z^2+c\rightarrow z_{new}=0-1=-1\)

The numbers are always bounded, never escaping off. From this very simple rule, we can create images like:

Figure 1: The Mandelbrot Set

It's crazy what incredible shapes we can get from such a simple rule! So, my goal was to create a fractal viewer program in order to create a bunch of awesome fractal shapes!

Figure 2: The Burning Ship Fractal.

Figure 2 shows the burning ship fractal, which fittingly looks like a ship engulfed in plooms of pillaring flames! Crazy, especially since it is generated with this simple equation: \(z_{new} = (|Real(z)|+i|Imaginary(z)|)^2+c\)! Check out some of my clips zooming out of fractals on my YouTube channel!