Buddhabrot fractal

Let's imagine there is a number i for which i^2 = -1. Even staring long and hard at this, we find that we can't "solve" this exquation for i. But that doesn't mean that i is entirely useless, even though René Descartes (who coined the term imaginary number to make fun of it) back in the 17th century thought so.

When we play around with i and use it as a multiplication factor, we soon notice some potentially useful or interesting patterns.

For instance, we know that (3 * i)^2 = -9. In the past, when we had an equation like (x - 1)^2 = -9 and tried to solve for x, we would have been stumped. But adding i into the mix, we quickly realize that x can be expressed as (3 * i) + 1 (as well as (-3 * i) + 1).

Playing around with this concept a bit more, inventing more advanced "impossible" equations and solving them with the help of i, we find that all our solutions tend to follow the same pattern. They tend to look like (a * i) + b. The square root of any negative number gains an i, while i becomes -1 when squared, and in the end we almost always get some imaginary part plus (or minus) a real part. In a way, they form a unit, which mathematicians have been calling a complex number.

By convention complex numbers are expressed as a+bi but that's just reordering such that the imaginary part to come after the real.

The Mandelbrot fractal is made up of the set of complex numbers c for which the formula f_c(z) = z^2 + c does not diverge to infinity when iterated from z = 0.

The outer shape of the well-known fractal already appears after the first iteration.

Plotting all the points for which the absolute of f_c(f_c(f_c(0))) remains below a certain threshold, typically 2.

To be continued…