How this fractal was created
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Imagine a canvas, 4 x 4. Every point has its left-right and up-down position (x and y from the center).
Put an 'arrow' from the center to the point.
In math, arrows (vectors) can be multiplied, even by themselves (squared), giving a new longer arrow. (What were you doing in 10th grade math class?) Another number (small arrow actually) is added to the multiplication result. [arrow = arrow^2 + c]
The new arrow can be processed again to give yet another longer arrow.
Repeat this squaring over and over until:
(a) arrow length reaches a specified value (here I used 4), or
(b) a maximum # of repeats is reached (used 80)
The final arrow length is used to pick the paint color for that point. For all points, giving the left image:
The right image: Before picking the paint color, additional adjustments to final length were made. Example:
L = L + Cosine(1.0 + 10 * (currentY - previousY)) + Sine(previousY^2)
(gave the 'swirling' effect in certain areas)
I used five such adjustments -- this is where the creativity, exploration, and perseverance come in!
A surprising feature is that zooming down into a fractal (e.g., a larger canvas) keeps revealing new features, never ending. Like the explanation of what's holding up a Flat Earth - 'It's turtles all the way down'.
It is amazing that a few fairly simple equations can give such greatly complex and beautiful images, some of which mimic actual nature.
Benoit Mandelbrot speculated that there was a reason for that, essentially the 'fractal roughness' of nature.
He won several prizes for developing fractal geometry and passed away just a few years ago (2010).