Tony Hanmer wrote:
Wow, such a quick response to my request for Kleinian groups in Fractint!
The quick response came from the fact the formula was already there in slightly different form to use Fractint's orbit window (accessed by <ctrl>+<o>). I only had to rip out some (now unnecessary) loop logic - and voila!
Something up your sleeve, hmmm, Gerald? Much appreciated, and fascinating to play with. I see what you mean about its limitations, but the variety of images it makes available (again, due to passes=o) is astounding. (Are they all K-groups?) [...]
I just implemented what's described here (PDF file!)... http://klein.math.okstate.edu/IndrasPearls/tools/twogen.pdf ...simplyfying it to the parabolic case (that's what the "Parab" means in the formula name). As I understand it, these could be described as (generalized) circle inversions, basically taking (four) circles touching one another and looking for the set of points remaining invariant under all possible inversions. In the "KleinGroupTest" par changing p2 from (1.95, 0.04) to (2.0, 0.0) produces such an Apollonian Gasket. But since circles (and lines) can be represented by operations with Hermitean matrices - and circle inversions by Moebius transformations... http://klein.math.okstate.edu/~wrightd/INDRA/Hcircles/ http://klein.math.okstate.edu/~wrightd/INDRA/MobiusonCircles/ ...one can abandon the notion of tangential circles and start to tweak the Moebius transformations to deform the resulting invariant point set. I have other formulas for this (PDF file!): http://klein.math.okstate.edu/IndrasPearls/tools/jorgensen.pdf and some of those: http://klein.math.okstate.edu/IndrasPearls/limitsets/puncture.html but the more complicated the process, the more defective the pictures produced by the naive random method I am using. About two years ago Morgan L. Owens came up with an idea to implement Kleinian Groups as escape time fractals, but this applies to circle inversions, not the more general transformations. ...and as Ken Childress has pointed out, Jos Leys gallery of Kleinian Group images is a "must-see". Regards, Gerald