Fractals: Gods Artwork, Part II

A fractal set; a set of numbers, or points, that when iterated through even the simplest of  non-linear equations do not diverge off to infinity. The shape of the set itself has the remarkable property that no matter how close you inspect its boundary, you will continue to see a beautifully complex outline. The real fun starts when you start to consider how quickly those points just outside the set diverge.

When a point (pixel) is put through the z_n+1 = z_n² + c equation again and again it moves around and around. If the original point is inside the set it will eventually ‘converge’ to a single location (or oscillate periodically between 2 or more points, more of that later…). If it’s outside the set then it will eventually move further and further away from the set until it gets an infinite distance away. The closer a point is to the edge of the set the more iterations/steps it takes to diverge. Points in this space just outside the set, the fractal boundary, are wrenched between attraction to the set and ultimate divergence away from it.  Hey, let’s color the points outside the set by the number of iterations they take to diverge. Let’s start simple and color those points that take an even number of iterations to diverge white, and those that take an odd number of iterations to diverge blue:

Now zoomed into the top most part:

The complexity seen the closer you get to the set basically means that it’s impossible to say exactly how long a point will take to diverge based on it’s distance away from the set. There is a great sensitivty, one pixel might take 253 iterations to diverge, one just next to it might take 25,589. This sensitive dependence pops up all over fractals, non-linear dynamics and chaos theory generally. It’s most commonly known as the butterfuly effect. (Now that really is a fluid dynamics issue and deserves a blog thread all of it’s own, more on that some other time… (this blogger is in it for the long haul 🙂 ))

OK, back to color piccies (please note that I am English but have elected to adopt the US spelling of color out of a deep sense of respect and friendship…). Instead of an odd/even coloring, let’s chose a color ‘map’ that is smoother. Let’s also do some zooming into the boundary. Each color still refers to how long it takes for a pixel to diverge when iterated, there’s no ‘legend’ relating color to number of iterations to diverge, the change in color just reflects the complexity of the boundary:

All this complexity from such a simple equation. This geometry is encoded into the fabric of our reality, beautiful rendering indeed.

9th July, Hampton Court