AdamG>How was the fractal generated? The primordia of a Helianthus sunflower are positioned at sqrt(n)*exp(2*pi*i*n/phi) or sqrt(n)*exp(2*pi*i*n/phi²), depending on whether you want a clockwise or anticlockwise sunflower. Sincerely, Adam P. Goucher This is part of Julian's first email to Edmark (which I hope is OK to post) subsequent to an impromptu hack session following the meeting: I tried to find a way to vary theta, but it seems to not work very well. The problem is that when we made the Fibonacci spirals, we specified that the largest circle was tangent to the (8+1)th and (13+1)th largest. Why 8 and 13? Clearly because they're consecutive Fibonacci numbers, but why not 13 and 21? Or 377 and 610? We use 8 and 13 because it makes the picture look particularly nice–with 233 and 377, the largest bunch of circles would be practically indistinguishable, and the entire picture would look like concentric rings of circles, with the actual structure appearing on closer inspection (also, it would be hard to make because we would need thousands of circles to fill up the area). We don't use 2 and 3 because small numbers make the circles get small too quickly, so that only a few are visible. But this is just for the particular angle of 137.5º. How do we generalize this to other angles? The way to do it would be to pick numbers such that we end up close after that many rotations (numerators of convergents of 2π/theta). There are two problems with this. One of them is that some numbers don't give approximations with numerators in the right range to produce nice pictures, but these numbers are somewhat rare and can be avoided. The other, more serious problem is that if we try to vary theta too much (which really isn't very much at all, about half a degree for the spirals we were doing), any particular pair of numbers stops working. This means that we would either need to have a jump and switch to another pair of numbers, or not keep the circles tangent all the time, or use small numbers which make the pictures less pretty. I also thought some about different shapes to use. The only other "logical" shape of piece I could come up with was a fractal one, being similar to the structure made by putting together all of the pieces. The whole thing would then be self-similar, as either two copies (one being the new piece, one being the rest of the pieces put together) or infinitely many (all of the pieces) of itself scaled and rotated. This produces about what one would expect, although I have not yet figured out exactly how to make it (the pieces being added must be rotated by some unknown amount, but I can get an idea of the shape without knowing). [...] --Julian --rwg PS, apologies to users of Chrome and Squirrel Mail, at least, for GMail screwing up URLs containing spaces, after showing me they tested successfully. (Just copy and paste the whole lines into the URL field.) Such a gross bug can't be a simple oversight--there must be some problematic conflict between design constraints.