Carl Weiman is a retired NASA scientist whose insights and suggestions have been invaluable in helping me create many interesting fractal animations. He writes:
In touring the San Diego Art Institute Century exhibition three years ago I was drawn to a stunning print of a Julia set by Ivan Freyman. Although this fractal has been rendered hundreds of times, the composition and tone of this one set it apart from others I had seen. It perfectly embodied a quote from Bertrand Russell's Mysticism and logic "Mathematics, rightly viewed, possesses not only truth, but supreme beauty ...". I sought out the artist and had the pleasure of meeting him and discussing his techniques and his rich interests in music and art.
New territory in science and art is often staked out with simple definitive examples. If the idea catches on, the territory is rapidly filled in by pioneers who extend, combine, enrich, and develop from these outposts. Mandelbrot's set is the outpost which launched fractals. Ivan's recent movies in this section are an example of enrichment of the field. Mandelbrot sets for z-> z^2 + c, and z->z^3 +c are well known. Though both are enormously jagged curves, there are some uncanny similarities in their bays and frills. I wondered if there was some underlying continuity between the curves with exponents 2 and 3. So, I suggested to Ivan that he program z->z^t+c and let t vary smoothly in small increments rather than just jump from 2 to 3. His program ran for days and saved thousands of large images. When played in sequence as a movie, the results are dazzling. Bays warp, expand and are swallowed up like the evolution of galaxies. At certain critical values of t, a whole new tree of shapes explodes into view. Yet throughout, the fractal character of the whole is preserved. Artists can revel in the dynamic flow and mathematicians can explore and interpret critical events by manipulating the "play" cursor of the Quicktime control slider and freezing frames of interest.
Stay tuned to this website, I am sure there is much more to come! Carl Weiman, March 2008; http://carl_weiman.tripod.com
Thanks for your help, Carl!