On 15/05/02, Morgan L. Owens wrote... over 9x2^10 octets of stuff! 8-0 Thanks for that mate - my head hurts so much! :S As regards the 4D generalisations of M and J sets... aren't they just simple surfaces of rotation of the ordinary complex versions? (Hmmm... some day I've GOT to download the POVRay source code and find out how the hell it renders 3D slices of the 4D Julias...) While I'm here, I thought I'd just report some of my recent fractal antics. I've been busily revising for my final year exams, but in between I've done some fractal work... Not content with Julia sets, after much experimentation I've persuaded WinAmp's AVS to draw (linear) IFS! (Figuring out the correct blit settings was highly non-trivial I can tell you!) As with the Julia sets, the trick is finding good patterns to apply the mapping to... I found that if I draw white graphics, and apply different colour filters to each transformation of the system, the results are really quite pretty! I also did some work with POVRay, drawing graphs for forwards and backwards iterations of the quadratic mapping (with Z-depth indicating iteration count), trying to work out where those complex patters derrive from. I mean, z => z^2 is a very simple function; it's just a big troff! (Oh dear... too much UNIX :S) The Julia set reflects this - it's a circle. Adding the c term only moves the center of the thing... and yet this creates amazingly complicated patterns. Why? Through my WinAmp experiments, it has become clear that ANY patters that has the quadratic mapping applied to it turns up these wonderful patterns. (Unsuprising really...) Also, my POVRay work has shown that the adjasent points on the graph which look so like they are images of each other, are in fact derrived from distant points. (Internally, the Julias seem to be a lot less symmetric than you might think!) I've come up with the "hall of mirrors" analogy. Have you ever been to a department store where they have mirrors on the columns supporting the roof? If you stand between two columns (assuming both have mirrors), you can see an infinite number of reflections of yourself! But, of course, the mirrors are always slightly greenish (never any other colou... mmm....) and never completely flat. The Julia sets are the same. It seems that the infinite progression of spirals/forks/dust are mirror images of one single such feature. (To look at it the other way - the orbits of distant points are ending up in the same vicinity. That's the fun part - the forward transformation of the individual orbits can be seen as the reverse transformation of the bailout boundery...) Erm... am I making sense here? 8-| OK, well anyway, I've also been investigating the Julia sets embedded in M. (Apparently these are quite a well known feature - as I discovered when I first mentioned them on this list!) It seems many an explorer knows that if you tunnel into the center of just a mini Julia, you find 2, 4, 8, 16... groups of the pattern outlining the equipotential curves of a minibrot (how deep depending on how close to the parent minibrow you found this minijulia in the first place). Well... I've been exploring the minijulias of the largest minibrot (the lowest-period one on the negative stem on the M set). Here every minijulia is scuered with a beautiful curving line. And at every major junction of this line, a minijulia sits. The one in the center leads (eventually) to a minibrot, as described above. However If you zoom into other parts of the minijulia, the minijulias you find dotted around the various junctions (and it's a nightmare to try to index them!) don't follow the equipotential curves of a minibrot. The outer most ring isn't a ring. Typically that are S-shaped groups, or just chaotic (but beautiful) "clusters", who's underlying pattern is difficult to understand. I'm still a long way off finding the pattern in all this... To anyone who thinks that M has been completely explored and holds no mysteries any more, I say KNICKERS! B^P OK, now I *am* jibbering... I'm gonna stop typing now... might send a PAR of the mini Julia patterns in a while tho... Thanks. Andrew. _________________________________________________________________ Join the worlds largest e-mail service with MSN Hotmail. http://www.hotmail.com