🚀 go-pugleaf

	I've had a few inquiries about some of the images I have posted to
the alt.binaries.pictures.fractals newsgroup lately and as I was answering
them I thought I would post it here as well. Some of this has been posted
here before, but I have made improvements to the formula making it more
general and also improving the bailout conditions. I also think my
understanding is a little clearer now so pardon me if I reiterate. That is
what we do best in this group anyway. My focus lately has been on
the quartet fractal and it is responsible for most of my recent images.
Someone asked if if was related to the mandelbrot.

	The quartet fractal isn't all that close to the mandelbrot
although, I admit it looks that way from some of the images. It is similar
in some ways. It is a complex deterministic formula using escape time
criteria for bailout but that is where the similarity stops. It's another
in a series of fractal formulae that I call inductive fractals due to
an inductive self-reference in the iterative loop. 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 is due to the inversion 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 formula 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 the mandelbrot although the lambda-sin
seems more likely. I haven't got enough points to really be sure of anything
here. This is all very speculative.

	The P2 parameter is used to control one of the bailout conditions.
Because I use a scaling inversion factor (P1), in the equation, and I have
a pair of complex variables to control, I have to keep tabs on both of them
or one could go to infinity while the other gets really small. They tend to
see-saw back and forth due to the P1 inverted scaling, but if this number is 
less than 1.0 then the period can be many iterations and this gives the other
variable a good chance to overflow. When the equation gets chaotic these
values change drastically. Thats why one of the bailouts is a large
number aprox 1/2 my floating max value. The other bailout controls the fractal
detail. Normally I use 64.0 but any value greater than zero can be used. 
Note that fractint defaults this value to zero and this is exactly wrong
for this formula. You can see that this parameter controls the size of the
disks that appear in the fractal periphery. A higher value has better detail
(smaller disks) but takes longer to compute. The max iteration bailout can
very anywhere from 64 to 32768 or whatever your max integer can be and as you
can imagine it has the greatest impact on compute time, but some initial
conditions, (those close to the critical values) require very high iteration
limits to resolve the fractal.

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.
; Adding the inversion causes what looks like an infinite extension along
; the real axis. Using the Imag component will add some nice asymmetry.
       z=pixel, zp = (0,0), bail = real(p2), huge = 1.0e32:
       temp = z
       z = sin(z) - zp
       zp = p1/temp, (|zp| <= bail && |z| <= huge) }


	These equations are taken from my formula file which can be copied
from spanky.triumf.ca in the [pub.fractals.formulas]noel.frm file via
anonymous ftp or the www URL: http://spanky.triumf.ca/

	Images produced with this formula can be seen on that internet node as
well. Look in [pub.fractals.images.noel] directory. The following images
are relevant:

	rosette1.gif, scarab2.gif, capilary.gif, redfluff.gif, stargate.gif,
bigpink.gif, quartet1.gif, quartet2.gif, quartet3.gif, quartet4.gif

	These should give a reasonable indication of the variations that this
formula is capable of.

						Cheers,
						Noel Giffin

						Noel@triumf.ca