Newton proposed a method of approximating the solution to a function that is
difficult to solve explicitly. You start with a guess, then use the first
derivative of the function to interpolate towards the solution. In
nongibberish, that means you figure out what value your formula has at your
"guess" value, draw a line that touches the function (mathematically derived),
and move along the straight line towards the axis. (This is all easier to
explain with diagrams - I'll add some soon.)
All right, Newton's method is explained, but how does that apply to fractals?
Well, let's take the equation z*z*z = 1. Anybody can see that the
answer to this is 1, because 1 times itself any number of times is still one.
The catch is, that this equation also has two other solutions that are
complex numbers (-0.5 +/- 0.5*sqrt(3)i) - these are spaced 120 and 240 degrees
around a circle centered at the origin (1 is also on this circle, at zero
degrees). And if you use Newton's method to "guess" at a solution, you don't
always end up at the same one. You might think that you'd end up with
whichever answer was closest to your initial guess, and this is usually true...
but when you get close to the areas where two answers are almost equally
distant, you get some really really weird stuff - not a simple pattern, but a
fractal one. Like ordinary Julia fractals, though, you can zoom
forever, but the detail you see is just like the detail at the full-size
image; it doesn't vary. In this Newton fractal picture, color indicates
which solution each point homes in on; brighter spots take longer to get there
than darker points:
Enter Paul Derbyshire and his puttering around. He decided to alter the equation a bit, just for the fun of it. Each time you evaluate your z, to try to get closer to a solution, he adds in a c value - constant for a Julia type, varying for a Mandelbrot type. And this makes for some really wild looking things. Some of the shapes are familiar, some are not. He actually has several types of fractals in his "Nova" family, but so far I've only been exploring his first - the original "Nova" type. It looks like this:
