Mira Fractals
Stumbling across the Gumowski-Mira equation in 1992, I quickly created a simple program to explore the equation. Since then, I've found few examples of Gumowski-Mira fractals / attractors ('Mira Fractals') either on the Internet or within fractal applications. Despite Mira Fractals not possessing the complexity or visual impact of some other types of fractals, they can still be beautiful in their own way - sometimes creating patterns with striking similarities to forms present in the natural world (such as single-celled organisms, feathers, and citrus fruit halves). The neglect of this intriguing fractal variant seems a pity.
The Gumowski-Mira equation is: 
The equation was developed at the CERN research centre in 1980 by I. Gumowski and C. Mira - not to create fractals, but to calculate the trajectories of sub-atomic particles.
Screenshots from the Mira Fractal generator:
» download 
Mira 1.2 (Win 2k/XP/Vista; 64kB ZIP) - non-GUI version, adjustable parameters (coded for fast rendering - not usability)
» download Mira 1.2 (Linux; 55kB ZIP)
» download 
Mira 2.0 (Win 98/2k/XP; 11kB ZIP) - GUI version (resizable, small, easy-to-use, slow)
The freeware applications contained in the ZIP files on this page are standalone executables (McAfee report). Please note the OS version compatibility list beside each Windows application.
Some screensavers fail to initiate on my Windows PCs while the Mira 1.2 (and Hopalong Blue, below) executable is running / minimised. As yet, no issues have been reported to me. Nevertheless, I have to state a disclaimer: this software is supplied as is, and the author accepts no liability for damages, direct or consequential, which may result from the use of this software.
One of the main advantages of compiled executables is fast screen rendering. However, if you wish to avoid executables, there are alternatives in a SmallBASIC Mira (very simplistic; runs within the SmallBASIC freeware application), or in the other generators below.
Other Mira Fractal Generators
Flash Mira by James Hayes
'Expanding' Mira Applet by Stephen C. Ferguson - impressive
Mira Applet by Ulli - configurable
Further Information
Gumowski-Mira Formula and Simulation
Mira 'fruit slice' fractal growth (165kB GIF)
Further Fractals
'Hopalong' Barry Martin Fractals
» download 
Hopalong Blue (Win 2k/XP/Vista; 60kB ZIP)
» download 
Tiny Hopalong (Win 98/2k/XP; 5kB ZIP)
» download 
Screensaver (Win 2k/XP/Vista; 57kB ZIP)
Maurer Rose 'Spirograph'
» download 
Maurer Rose (Win 98/2k/XP; 7kB ZIP)