MANDELBROT SETS

This group of patterns represent graphs of the iterated function z = z^2 + c in the complex plain.

When such a function is itererated starting from z=0, depending on the initial values of C, the value of Z may increase without bound towards infinity, or it may stay bounded. Generally, if the value of |Z| increases beyond 2, it is unbounded and will increase towards infinity.

The set of values for which |Z| remains bounded are known as the Mandelbrot set, named for the "father of fractals", Benoit B. Mandelbrot.

The MAND() function in Pixel Magic uses the "Level set Method" which outputs 0 if |Z| remains bounded after a fixed number of iterations, and otherwise, returns the number of iterations it took for |Z| to escape.

The various patterns in this collection use the output of MAND() in different ways to provide different color schemes. These colorization schemes are described for each one.

Most of them are set up so that they will either zoom in, or zoom out with successive renderings, allowing the creation of zoom movies. For movies that Zoom in (that is Zoom is < 1) you can generally determine the "ending" width of the movie by looking at the precision of the cx & cy coordinates. For example, if cx is .12425 you can zoom in about 5-6 orders of magnitude, so you would set your starting width to 4, and your ending width to .00004. Many of the presets are set at a narrower starting width, which starts where the movie starts to get "interesting".

You can get a preview of the movie in Pix Magic if you have "quad render" turned on. Just keep hitting the "Cmd-R" key.

I've done most of my explorations using the accompanying program "MZoom", and then used Pixel Magic to generate zoom movies into interesting coordinates discovered with MZoom.

Amoeba I call these types of pictures "Amoebas" or "Grains". This preset has a zoom value of 1.4 so you can generate a movie showing the location of this particular grain within the Mandelbrot set.

The easiest way to find grains is to look for smaller copies of the mandelbrot set. These smaller copies are connected to larger copies by filaments, which in some cases appear to be made up of smaller copies of the M-set. Emanating from these smaller copies are often smaller filaments or "hairs", and these smaller filaments on closer inspection appear to be a connected series of "grains".

Amoeba 2 These grains generally closely resemble the corresponding Julia Set which can be constructed using the same seed coordinates. Thus far, I've always found a small copy of the Mandelbrot set at the symetrical center of these grains.

So in general, it seems that most filaments appear to be consructed of small copies of the mandelbrot set, some surrounded by an elaborate "grain" which resembles a julia set.

Coral Reefs These coordinates provide a nice fractal zoom movie, which can zoom to the limits of the computer's precision (about 10^-15).

This particular set of coordinates highlights the complexity of the recursive spiral structures that can be found everywhere in the Mandelbrot set.

Jason's path

Mandel Color In this image, the red, green and blue components are controlled by out of phase sine waves. The sine waves are approximately 120 degrees out of phase, creating an even rainbow spectrum.

These coordinates provide enough precision to make a zoom movie to the limits of the computer's resolution (about 10^-15). This particular movie zooms into the tail of a "seahorse", revealing one of the miniature mandelbrot sets that can be found where two tails meet.

Mandel Color Stripes Sine waves of differing, and relatively high frequencies provide a "striping" effect.

Mandel Color 2

Mandel Cycle 2 Another set of coordinates which go to the limits of the program's precision, providing a long zoom movie. This one zoomes into the "elephant walk" area on the "butt" of the main cardiod.

Mandel Zebra The ZEBRA() function generates color ramps which have alternating stripes. These stripes help highlight the contours of the level sets which surround the Mandelbrot set, and also can make for some pretty pictures.

In this particular case, the ramp has 32 steps, and alternates with colors which are 4 steps away. The red, green and blue components are 45 degrees out of phase with each other, creating alternating tan and blue bands, and resulting in an overall "chrome" effect.

Sauron's Eye This pattern creates a continuous color ramp by cycling the different hue, saturation and value components of each pixel at different speeds, using sine waves of differing frequencies.

Seahorse This is a grayscale colorization, which provides a continuous color ramp from black to white.

Sparse Zebra A gray scale colorization which provides a stripe for every seventh level set, the intent being to isloate the path taken by individual contours.

Web Wars The intent here was to create a striped mapping that could be represented in a very small number of colors. There are only 16 colors used, including black.