Apophysis Fractal Flame (click to enlarge)I've seen the flames - fractal flames - and they are amazing. Originally an outgrowth of the ideas developed in Symmetry in Chaos by Field and Golubitsky, fractal flames are related to the Chaos Game because they are created by tracking the iterates of starting points as they are mapped into other points. As in the Chaos Game, the iterates reveal, over time, the structure of the attractor associated with the map. Flame fractals use a combination of non-linear maps and very inventive approaches to coloring/visualization.
Flame mathematics is interesting enough on its own, but the intrigue of flames is the ability to generate images of surreal organic beauty. In the following excerpt from The Fractal Flame Algorithm by Scott Draves for the Cosmic Recursive Fractal Flames site, aesthetics are essential:
The Fractal Flame algorithm is a member of the Iterated Function System (IFS) class of fractal algorithms. A two-dimensional IFS creates images by plotting the output of a chaotic attractor directly on the image plane. The fractal flame algorithm is distinguished by three innovations over text-book IFS: non-linear functions, log-density display, and structural coloring. In combination with standard techniques of anti-aliasing and motion blur the result is striking image variety and quality.
The guiding principle of the design of the algorithm is to expose and preserve as much of the information content of the attractor as possible. We found that preserving information maximizes aesthetics.
There's plenty of software available to download for creating flame fractals, including UltraFractal as well as a lot of freeware. Some interesting UltraFractal images are located at the blog site Where Art and Math Collide, which is apparently the name of flame fractal created by the blog's author. The most interesting images I have seen are at the Fractal World Gallery of Cory Ench and the webshot site of Roger Johnston. (click on the image at left and view Johnston's work). It seems that the software Apophysis , a freeware fractal flame editor, is the main rendering tool of choice for these beautiful images. (The image at the top of this post was created using Apophysis).
To go back in time a bit (circa 1995), read the Field and Golubitsky article Symmetric Chaos How and Why from the notices of the AMS.
I want to credit Ian Stewart for turning me on to symmetric chaos. I originally read his Christmas in the House of Chaos piece in Scientific American back in 1992, which is a clever piece featuring the Tractor family (John Tractor, Pointer Tractor, and Lorenza Attractor) making Christmas ornaments iteratively. (Unfortunately I have not been able to find this article online).