## Periodicity Analysis

These images were made by the Fracture screensaver.

These images are generated by the same formula as that used to generate Julia and Dragon sets, but with a different sort of analysis of the results. For the standard images of those sets, remember, a point z was iterated, and if it escaped to infinity, it was colored according to the speed of its escape, while if it remained bounded, it was considered to be "in the set" and was colored some solid color. In these images, instead, all of the points that escape to infinity are colored black, while the points that do not escape are analyzed further and colored according to that analysis.

That further analysis is essentially identical to that done for the Attraction Basin images. Each Julia or Dragon Set can be considered to have "roots" -- points that are attractors for other points when the generating formula is iterated. In periodicity analysis, an attempt is made to determine which attractive point a given point z is converging upon, in the same way that with Attraction Basins an attempt is made to determine which root a given point z converges upon after iteration of the root-finding method being used.

The results of that periodicity analysis may be divided into "genera" and "species", in a way inspired by Plate C5 of Benoit Mandelbrot's classic book __The Fractal Geometry of Nature__. The colors chosen for particular attractive points (and the points attracted to them) depend upon the genus and species that that attractive point belongs to. This corresponds visually to a sort of grouping of related sections of the set in a very pleasing manner. The exact method by which this division is performed was my (re-)invention; although Mandelbrot presumably invented a similar method to generate that plate, I have not seen his or any other method published anywhere.

**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.

Poetry

Rainbow

Crystal

Flame