Im almost used to the routine by now, which is that every time I believe I have the Tset formula written down properly, I see a way to refine it. With this latest and last little lick though, I think it's done My first efforts to resolve the set in 2D, were mostly to demonstrate its existence, and hopefully inspire someone to render it in 3D. I used an HP SX48 calculator, and it was a simple matter to construct a program with the 3D vectors that are built-in. The skewed, truncated Msets that this method produced seemed a reasonable result, although not quite what Id hoped for. Then, a few weeks ago, Tim Wegner directed my attention to the Fractint list and the Morgan Owens formula posted on 12/11/02. At the time, I was still focused on a full 3D rendering, so I wasnt that excited about seeing a 2D set in Fractint. Besides, the Owens formula looked about right, but it produces images quite distinct from those on the HP, and I thus suspected that the Fractint parser wasnt up to the job. But just for drill I guess, I tweaked the formula to bring it a little closer to the HP version and voila the images Id found earlier were there. Well that must be right, I thought, reckoning that if one machine confirmed the other, how could it be otherwise but in a short while I saw that the imaginary value for c2 (which the HP's vectors had snuck in on me) was not appropriate in this case. The simple reason is that triternions are based on the order six cyclic group, and there are no 4th roots of unity in C6 (or for that matter, in the other order six group, D3). I saw that simply squaring c2 would rectify the imaginary term, and the images that this change produced were far more intricate and fun to work with. Moreover, Jim Muths flip of c1 produced a nice two-fold symmetry (and incidentally z1=t1-c1*c1 works well with that too), so it looked like wed arrived. Upon further reflection, however, it became clear that in c2*c2, the term also acts upon itself as a scalar, and that this disrupts its proportionality with c1. The proper solution then is c2*(0,1) or (better) c2=imag(pixel) *(0,1) in the definition line, to minimize calculations. There it is then: Weve watched the Tset go from embryo to fetus to now a newborn, and guess what Cuzzins, Its A Girl! You can go to p.12 of my paper, where shes posing for a picture, dressed all pretty in pink: http://fibonacci-arrays.com/Triternions.pdf Or you can render her yourself with TGirl { c1=real(pixel),c2=imag(pixel)*(0,-1),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3, t2=z3*z3+2*z1*z2, t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2+c2,z3=t3+c3 z=z1+z2+z3 z < 64 } TGirl is easily persuaded to produce Julia sets: TJul { z1=real(pixel),z2=imag(pixel)*(0,1),z3=p3: t1=z1^p1+2*z2*z3, t2=z3^p1+2*z1*z2, t3=z2^p1+2*z3*z1 z1=t1+p1,z2=t2+p2,z3=t3+p3 z=z1*z2+z2*z3+z3*z1 z < 64 } Try, say, z1=-0.75 with z2 and/or z3 at or near 0.1. Looking forward to seeing where this goes Ciao, Russell _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus
Dear reader I have read most of the things about triternions. And rendered some of the frms/pars. A lot of things are coming into my head. Some of them I try to write down here. I hope I will get some comment First of all why is a formula like Tgirl important/interesting? Surely it's a nice girl but that's not enough, is it? Is it because there are second order girls and maybe there are girls everywhere in the image (I didn't check that) or are there other reasons? As a reference for the rest I include two frms TGirl and TMan TGirl { c1=real(pixel),c2=imag(pixel)*(0,-1),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3, t2=z3*z3+2*z1*z2, t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2+c2,z3=t3+c3 z=z1+z2+z3 z < 64 } TMan (XAXIS) { c1=real(pixel),c2=imag(pixel)*(0,1),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2+c2,z3=t3+c3 z=(z1^2+z2^2+z3^2)^.5 z < 16 } Then I don't understand the use of z=z1+z2+z3<64. If I interpret the frm-interpreter right it means the real part of z (as a complex number, so in 2 dimensions), smaller then 64. If you want see things as orbits in CxCxC, so in 6 dimensions I should expect sqr(|z1|+|z2|+|z3|) <64. And certainly not sqr (z1^2+z2^2+z3^2), as in TMan. Strange enough this doesn't work properly. But |z|=|z1+z2+z3| works nice. Then the bailout value. Why 64? Why 16? (In TGirl and TMan) Then the confusion about dimensions. Well, you can see the orbit of (z1,z2,z3) as an orbit in CxCxC, so in a 6 dimensional space, but a simpler, and I believe a much more fruitful idea which is also consistent with a bail out rule like |z1+z2+z3|<number is this: The iteration of z1 and z2 and z3 gives 3 orbits in the complex plane. These orbits influence each other. After every iteration of all three you look if the distance of there sum is greater then the bail out value. I can say that slightly different: You start with a configuration of three points (in TGirl and Man that's three times the same point (0,0)) Then there is a rule (function) witch calculate three new points. The place of the new points is influenced by the places of all 3 old points. Then there is a bailout rule on with you decides if the configuration has escaped or not. With this way of looking, of witch I don't know it's new, I feel two things very strongly 1. Tgirl is not the end but a start of an important development 2. Defining fractals with the aid of orbits of configurations will lead to a whole class of objects with much the same properties the Mandelbrot has! For a start I gave two frm's The first one A_General_T gives all kinds of Mandellike sets, half Mandelsets and also lots of real Mandelbrot sets (at least at first sight). A lot of combinations of p1, p2 and p3 give results The second T_Sisters gives for real(p1) and imaginary(p1) between 0 and 1 different TGirls, also less pretty I think. The surroundings of some of the basins of attraction give a wealth of beautiful fractals. Next times I can give some pars For real(p1)=.7 you find secondary basins of attraction and maybe more? T_General (XAXIS) { ;Jos Hendiks,2003 c1=Pixel,c2=Pixel,c3=Pixel z1=z2=z3=0: t1=z1*z1+2*z2*z3, t2=z3*z3+2*z1*z2, t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2+p2*c2,z3=t3+p3*c3 z=z1+z2+z3 |z|<100 } T_Sisters {;Jos Hendriks,2003 c1=real(pixel),c2=imag(pixel)*(0,1),z1=z2=0: t1=z1*z1+real(p1)*z2 t2=z2*z2+imag(p1)*z1 z1=t1+c1,z2=t2+c2 z=z1+z2 |z|<8 } With these two formulas (as a start) I have really the idea I found a whole new world. I think it's possible to generalise A_General_T much more, so it includes TGirl as well. But then there is the problem of not enough parameters. I shall try that later. At last: Dimensions and the use of c1, c2 etc seems to me much more complicated. For instance the role of c1 and c2 in TGirl is different from that of c3 With pleasure, Jos Hendriks jos@hexaedre-fr.com
participants (2)
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jos hendriks -
Russell Walsmith