Fractal visionaries:

A few days ago Ray Girvan posted a formula that drew distorted Newton's images, only to be told that this approach had already been investigated.  After trying trying the formula, I realized that the images were strangely familiar.  I had seen similar images somewhere before.  Examining my Newton's formula file, I discovered a formula that contains within itself the ability to duplicate Ray's formula.  The formula is attached to the end of this letter.  (Set real p1 to 1, imag p1 to 3, real p2 to 3, imag p2 to 1, real p3 to 1, and imag p3 to 0.0001 and the formula draws the power 3 Newton's figure.)

The formula is still in the testing stage.  When I get it finalized, I'll eliminate some of the variables to make it easier to control.  When I work with this formula, I start with the above parameters, which draw the power 3 figure, and change them one at a time.  Many settings give a blank screen.  When this happens, I search out the exact setting where the screen goes blank.  Something unusual is almost always found there.  When I get a blank screen that just won't do anything, I always try the bof60 inside fill before giving up.

Today's fractal is all inside fill.  At first glance, it looks nothing at all like a Newton's figure.  But it is.  Tomorrow, I'll post another fractal made with this formula, which doesn't even look like a fractal.

Before I go, I'd like to pose a challenge.  A while back, I was discussing fractals with a non-mathematical co-worker.  The conversation drifted to four-dimensional fractals, and this person asked me to explain in simple language the difference between quaternions and hypercomplex numbers.  I couldn't.  Can anyone out there answer this person's question in easy-to-understand layman's terms?

Jim Muth
jamth@mindspring.com


START PAR FILE FOR 19.6=========================================

Eternal_Triangle   { ; time=0:00:36.30-SF5 on P4-2000
  reset=1960 type=formula formulafile=basicer.frm
  formulaname=NewtonTest26
  center-mag=-0.118626/0.258174/0.5706667/1/-30/0
  params=1/6/3/1/1/2.27e-032 float=y maxiter=90
  bailout=100 inside=bof60 logmap=yes periodicity=0
  colors=000IzAIv9It9Iq8Io8Il7Ik7KgALdDMbFN`IPZKQWNR\
  UPSSSTQVUOXVM_WKaYIdZFf_Di`Bka9nc9oeApgAqiBskBtmBu\
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  zRCzUIzWOyZTw`Zucdteip`blXXiTQeOKaKDaNFaRHaUJaXL`_\
  N`cP`fR`iTInCInCImBImBIlA }

frm:NewtonTest26 {; Jim Muth
a=real(p1), b=imag(p1), c=real(p2),
d=imag(p2), k=real(p3), f=imag(p3),
z=(pixel):
zx=z^b-a
zy=c*z*z
z=z-(d*zx/zy)^k,
|zx| >= f }

END PAR FILE FOR 19.6===========================================