Russell Walsmith wrote:

On Tue, 21 Jan 2003 16:36:20 Morgan L. Owens wrote:

...

...though I still say TMan and TJul need a little prophylactic code to stop them from spilling out into six dimensions - unless that's desirable; mainly from the line z=z1+z2+z3.

Right on, Morgan, you da Man! That statement and your Hilbert space reference of a while back helped me recall that a metric space of n dimensions is defined when (a^2+b^2+...n^2)^.5 measures the distance from a point in that space to the orgin

...well, a Euclidean space...

Clearly, the appropriate statement is z=(z1^2+z2^2+z3^2)^.5

Mm. What I was thinking was a situation like this: With the initial conditions: pixel = (.1, .2) p1 = .3 p2 = .4 p3 = .5 the initialiser sets z1..3 to: z1 = .1 z2 = (0, .2) z3 = .5 which, after one iteration, have become: z1 = (.31, .2) z2 = (.65, .04) z3 = (.56, 0) I think my punctuation went a bit astray in my sentence: if the - and ; were swapped, the intention ought to be a bit clearer.

z=(z1^2+z2^2+z3^2)^.5 z < 16 }

As an optimisation, consider instead z = |z1+z2+z3| z < 256 }

TJul (XAXIS) {;Try z1=-0.75 z1=real(pixel),z2=imag(pixel)*(0,1),z3=p3: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+p1,z2=t2+p2,z3=t3+p3 z=(z1^2+z2^2+z3^2)^.5 z < 64 }

Since you've already summoned the Dobiasovsky genie, Morgan, maybe you can wish him on to these formulas too...

BTW, I don't mean to diss your D3Man when I tell you this (he was, after all, a sort of midwife/godfather to TMan), but he's not really based on D3. The reason is that D3, and in fact a dihedral group of any order, is non-Abelian: Therefore, for some term(s) in the number triplet, say j, the outcome of j^2 will not be equal to -j^2. A fractal formula based on D3 will thus require conditional statements in the initialization line...

The origins of D3Man largely came from comparing the group structure underlying complex numbers (Z2+Z2) and hypercomplex numbers (Z4+Z2), I actually intended to look at Z3+Z2. But as it happens, that's Z6 - which of course is Abelian. Nevermind, I ended up back there anyway. What I've got is not D3, it's true. What I wanted some genuine 3-dimensional numbers with some group structure, and Z3 is the only order-3 group. By tacking on a Z2 as a "sign flag", I got something which I could have got just as easily from Z6 ... the collapse from non-Abelian to Abelian comes when the group operation becomes mapped to complex multiplication; it's why in D3Man the main iteration goes z1*z1+z2*z3+z3*z2, for example, instead of z1*z1+2*z2*z3 - because (in D3) z2*z3!=z3*z2. I didn't get around to changing the name, though (I should've been calling it something along the lines of Z6Man or even Z3Man), and I guess, since your initial images matched what I had, I didn't really check. I suspended the whole "Groups for Fractint" thing until I had some unifying approach (and some software to do the tedious (and error-prone!) legwork). It's why I didn't even bother with rotations of the cutting plane (colour-cycling Dobiasovsky's pars - highlighting each cross-section in turn - gives an idea of what I had been seeing). But, needless to say, the important thing here are the visual results, and the TMan/TJul stable is certainly proving fruitful. Morgan L. Owens "Does that make me a bridesmaid?"