The Buddhabrot fractal is a twist to the Mandelbrot 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…*

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