I do indeed greatly appreciate the speedup. Thanks a million.

always happy to make a contribution! those opportunities are few and far between for me :/

about fractals and maths: i recently had a chat with a good friend of mine (who is one hell of a fractalist) about this. he could also would benefit from a stronger mathematical grounding, however that's hardly hindered him! it turns out that when working with fractals, quite often you can pretty much do what you like and still get awesome looking results

(an obvious corollary is that researchers tend to have boring images

***dodges a thousand hurled chalkboard dusters***.) such a truth is completely unavoidable, so the only defense i could make for poor old mathematics at that stage is "knowing which rules you're breaking can give you more perspective and therefore power". however, it occurs to me now (returning to the matter at hand) that it has a further use, namely simplifying expressions*.

i referred him to this great website i found some years ago, S.O.S math (lol):

http://www.sosmath.com/algebra/algebra.htmlfeel like you dozed too much during high school maths classes? get in line!

the above site is really nice and concise, highly recommended. it goes deeper than just algebra and i still find some use for it now and then (it's great to finally master things i floundered with earlier).

* things like sqrt(x) < const, and the simplification given earlier are two such examples.

Actually math and science were my favorate, and best, subjects in high school, so they were the subjects where I was most awake. Still, I don't have much call for dealing with complex numbers or solving for derivatives in day-to-day life, so those skills have gotten a little rusty with the passing years. Your insight is appreciated.

That site is great. It goes from elementary to intermediate level in nice easy steps. I found the discusssion on polar coordinates for complex numbers very helpful, for example. That's a concept I haven't been clear on until now.

This thread is actually related to another thread I started in the Mandelbrot/Julia area. I wonder if you folks could chime in there. Specifically, I posted a question about the log-log approach to distance estimation as compared to the distance estimator algorithm documented in the book "The Science of Fracal Images." Here's the thread:

How are B&W images from "Beauty of Fractals" created?specifically my post "reply #8", dated 5/1/07.

Duncan C