## Binary Decomposition

These images were made by the Fracture screensaver.

These images are all based upon the various methods described in previous categories (Mandelbrot, Julia, Dragon and Basin sets). One additional coloring trick is used in their generation, however, called "binary decomposition". This results in regions that would normally be given a single solid color being divided into smaller "tiles" of different colors. The resulting images can be reminiscent of Piet Mondrian's rectilinear subdivisions, Frank Stella's more whimsical abstract color tilings, or of stained glass or crystalline structures.

Mathematically, what is going on here is that the test criteria used in image generation are modified slightly. Normally, the test used would be "is z outside the radius of escape, such that it will deterministically converge to infinity?" (for a Mandelbrot, Julia or Dragon image), or "is z inside the radius of attraction, such that it will deterministically converge on a given root?" (for an Attraction Basin image). In binary decomposition, points that pass that given test then have one additional test applied to them: is their y-value (their imaginary component) above or below the center of the disc used in the test? Those points that are above the disc center are given one color, those below are given a different color. This divides the region that is adjacent to the test disc into two semicircular regions of different colors. Because of the way that the mathematics of fractals "stretches and folds" the complex plane, the further you get from the test disc, the more subdivided these regions become, approaching an infinite subdivision as you approach those points that never converge on the test disc.

**All of these images are copyright © 2001 Ben Haller**. Personal use of these images is allowed; all other use, including any kind of redistribution or reproduction of these images, is forbidden without the express, written consent of Ben Haller.

Inferno

Ishmael

Nautilus

Stained Glass