Animated Newton

Newton's method is a technique for solving equations that have several "roots"--the solutions to the equation. Typically, the roots are fixed points, like +/-1 and +/-i, the four roots of z⁴ - 1 = 0. To animate a Newton fractal, I decided to use four roots and to have them move around during the animation. In this case, the four roots move around on four different rose curves. The coloring is the typical iteration band coloring, just using black and white. And, to keep the visual noise down, I used a small number of iterations.

(See a higher-quality version on YouTube. While you're there, check out a variation based on circles.)

The technique has promise and can be expanded to different root paths and different functional forms. Here, I used a quartic polynomial because it's quick and easy, but there's no reason why rational or transcendental functions couldn't be used as well.

Phyllotaxis Square Midget

"Square" midgets are found in embedded Julia sets in the Mandelbrot set. They are so called because they generally exhibit outer structure with four-degree rotational symmetry. That structure is sometimes warped into a squarish shape, too. However, being from embedded Julia sets, they have similar structure as the Julias, which have structure similar to the part of the Mandelbrot set from which they're taken. Or, in other words, more of the same spirals.


In that respect, this one is no different. :-) I like it, though, because it shows the phyllotaxis patterns that aren't often seen with Mandelbrot and Julia sets. They're there, you just have to know where to look. This one was pulled from an embedded Julia off of the 8/21 disk of a midget off the 8/21 disk of the main cardioid. Also, I was able to clean up the "background" a bit so that the phyllotaxis shows up most prominently. If you want to go exploring around here, this image is centered at -0.388301367376806743/0.596673282113460201 with a magnification of 2E14.

What to do with 1 million iterations?

I often like to return to the Mandelbrot set for inspiration and the new year seemed like a good time. Lately, I've been playing with high-iteration, but not-deep-zoom fractals--those which require a lot of iterations but at a reasonable magnification (deep zoom on Ultra Fractal kicks in around a magnification of 1E16). This gives me the chance to look at/for new structures and to do so without waiting days for the image to appear.


This is one of my explorations. It's based on the fraction 2/9, chosen to celebrate the year 2009. I like that there's very little "background" (black) in the image; it's almost completely filled in with structure. This can give you a sense of how the boundary of the Mandelbrot set has a fractal dimension of 2: as you zoom in, the structure of the boundary becomes so dense that it fills in an area. The zoom here is a paltry 24 million, but you can already sense the space filling in.

One other feature: the inside color is Mimosa, Pantone's Color of the Year for 2009 (242/181/11 in RGB space, as near as I can tell).