May 20, 1997: Eternal Triangle | May 19 | May 21 | 1997 | FOTD Home |

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\

oCvqCwsCxuDzwDzyEzzEzQNsLQwHSzzh9zfAydBybCx`ExZFxX\

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JveKviLulMupNwgQx_TzRWzIZnbibvseuqhsolrmoqkroiungu\

eltYqtPvqQumRtjSsfUrcVq_WpXXoTYnRVkPRiMOfKKdIHaSap\

`uzJozOlzTixYgwbdugaslZqqXptWivWbxVWzVPzUIzUBByybq\

qzijygevd`rbWn`QkYLgWGkVOnTVrSbuRjyPqzOyczqzVzkjZS\

y7ezrhylkvgnsaqoWtlQwiLzfFyiLykRxnXxqcwtiwvovyuvzz\

sRKwRSzSZzSfzYbzcZyiVwnQutMszIqzErzIrxLstPsoStkWtg\

Zodbk`gfYkaUoYRtTNxRSnQYeObWRcYTd`WebYge`hgbijejlW\

GQYOT_WWbc_dkbfsehzhmq`rgUwZNzPGzF9zH8zJ8zL7zN6zO6\

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===========================================