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.

Jason's path |

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

Mandel Color 2 |

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. |