THE SPANKY FRACTAL DATABASE

Info on the Quartet Fractal

The Quartet formula is one from a series of interesting new fractals that I have been exploring. It is related to the mandelbrot only in the fact that they are both complex deterministic formula using escape time criteria for bailout but that is where the similarity stops. On closer inspection the images reveal a much different geometry than the mandelbrot. The classic mandelbrot shape is not to be found and the fractal structure tends to iterate to infinite points. The Quartet formula and the related series of fractal formulae that I have discovered are called Inductive Fractals. This name was chosen as all the formulae use an inductive self-reference in the iterative loop. By this I mean storing the results from previous iterations and re-introducing them into equations on the next or subsequent iterative cycles. In a sense, the Mandelbrot and most fractals do this already as in the formula :

        Z = Z**2 + C

Where the Z term is resubmitted on the next iteration to influence the new value. I carry this principle one step further by temporarily holding the value of Z for at least another iteration before reintroducing it back into the equation. Here is a simple example:

        TEMP = Z
	Z = Z**2 + ZP
	ZP = TEMP

You can view a variety of images created with this formula that I have stored in the [pub.fractals.images.quartet] directory.

In appearance the quartet fractal is actually closer to the lambda-sin fractal shown in "The Science of Fractal Images", around page 160-162. There is a definite relationship between them although I haven't proved it mathematically. The lambda-sin formula gives a repeating fractal outline while my quartet formula gives a similar outline but fills the inside with very interesting fractal detail. This added detail is due to an inversion introduced in the iterative loop which constantly turns the dynamics upside-down like turning an hourglass. The effect on the fractal image is dramatic. It no longer has an inside or an outside but shows a fractal that approaches a limit-set which I believe is the border of the lamda sin function. Varying the initial conditions of the lamda-sin causes the fractal outline to skew and this happens in mine as well. What is fascinating is watching the internal structure change with different initial parameters. The set seems to be an infinite extension of a repeating fractal geometry along the real axis. I haven't checked this very far as Fractint only has a max real ordinate of +-32.00 or so, but it seems logical that this would be true from looking within those limits. +- pi is a full period.

It isn't that rewarding looking beyond a single period as the set is repeating. If there are any variations they are very minor. It is more interesting to change the initial conditions and observe a new geometry. It is essentially choosing a new julia set and the variations can be great. The trick is finding starting parameters that give good fractal detail. The p1 parameter is the starting point and can take on any complex value but the most interesting ones tend to be less than 1.0 (both real and imaginary) Actually, some of the most interesting are as these values approach zero.

There seems to be a critical set of numbers where values less than these will produce image sets with large areas hitting the max iteration count. Numbers slightly larger produce very intricate fractals. Any ideas on the shape of the critical set? I think it may either be points from the border of the lamda-sin function or maybe some other function although the lambda-sin seems most likely. I haven't got enough points to really be sure of anything here. This is all very speculative.

Here is the formula preceded by a couple of related ones.

The quartet formula is a variation on an earlier fractal equation of mine called the quatro fractal. (named from it's four point geometry)

 quatro(XYAXIS)  {
; Forget the squared term and iterate a trig function
        z=pixel, zp = (0,0):
          temp = z
          z = sin(z) - zp
          zp = temp,|zp| <= 4 }

It combines that formula with a variant which introduces the scaled inversion. The first formula of this type was the HT formula.

    ht(XAXIS) = {
; Ah! You can use a variable in the inversion but now check for overflow.
; Good results are found when the real part of p1 is in the range 0.1->1.0
; With some sort of special value aprox. 0.148148...
; Setting the imaginary part as well causes very strange fractals

      z = pixel,zp=(0,0),huge=1.0e32:
      temp = z
      z = z*z + zp
      zp = p1/temp, (|zp| <= 64) && (|z| <= huge) }

I was so happy with this fractal that I tried the technique on many of my earlier formula and turned up the quartet formula which is very fascinating indeed. The quartet formula is:

    quartet1 {
; The quatro-HT variation.    From Noel Giffin
; Adding the inversion causes what looks like an infinite extension along
; the real axis. P1 should be non-zero. Use the Imaginary component of P1 as 
; well to create some interesting fractals. The real part of p2 is used 
; as one of the bailout criteria and it must not be zero. Make it larger to 
; diminish the disks that appear on the periphery.
       z=pixel, zp = (0,0), bail=real(p2), huge=1.0e32:
       temp = z
       z = sin(z) - zp
       zp = p1/temp,(|zp| <= bail &&|z| <=huge) }

As the previous term zp is constantly being inverted and fed back into the equation an interesting dynamic develops. A very small value of zp tends to a very large value on the next iteration. Both the current term z and the previous term zp will oscillate in this manner with a period depending on the scaling parameter p1 and the dynamics of the point being evaluated. With certain initial values, the iterative process damps itself and thus will iterate for many cycles while other points swing wildly and escape after a few iterations. It would be simpler and probably mathematically more correct to make the bounds check for the previous and current term to the same value to reflect this symmetry. That is test for:

       (|zp| <=huge && |z| <=huge) }

These equations are taken from my complete formula file which explore these inductive fractals in some detail and shows the logical development of this formula type from its inception. It can be copied from spanky.triumf.ca in the [pub.fractals.formulas]noel.frm file.

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