April 24, 1997: Solar Symbol | April 23 | April 25 | 1997 | FOTD Home |
Fractal
visionaries:
Fractals are things of many mysteries. They raise one
question after another. During the ten years I've been
working with them, these questions have almost constantly been in my
mind.
Perhaps the most immediate question is: how can an infinity of detail
be encoded in such a simple formula?
This question is relatively easy to answer. The answer lies
in the recursive nature of the fractal formula. The fractal
formula as written does not contain infinite detail, but the formula
tells which way to go to see the infinite detail that is waiting there.
Reiterating the Mandelbrot formula is like motoring along a
highway. The instruction to reiterate the formula Z^2+C for
every point of the complex plane is the same as the instruction to
start at Atlantic City, keep traveling west on Route 40, and observe
the sights as you go. Everyone who follows the same simple
fractal instructions will see the same glorious fractal scenery for the
same reason that everyone who follows the same simple highway
directions will see the same glorious real world scenery.
A more difficult question is: what are fractals -- did they exist
before computers were invented?
The simplest answer to this question is that fractals are infinitely
complex images which appear on computer screens when complex formulas
are reiterated and mapped to the complex plane. Since such images
obviously did not exist before computers, it could be said that
fractals did not exist before computers.
But a fractal is not a mere physical image on a computer screen, it is
a mathematical abstraction, an archetype with no more physical
existence than the naked number seven without seven objects of some
kind to give it reality. Since fractal arche
types have no real physical existence, it is meaningless to speak of a
time before computers, when fractal archetypes had even less
existence. The fractal archetypes always existed in
a kind of virtual state, even as they do now. We simply were
unaware of them before computers, because we lacked the means to
manifest them.
There are even more curious aspects to fractals. Behind the
world of fractal images lies the ultimate reality of mathematical
formulas. When no one is looking, nothing is
there but the fractal formulas. Behind the world of everyday
objects lies the ultimate reality of the quantum world -- a world of
incredibly complex mathematical formulas. Quantum theorists
tell us that when no one is looking, nothing is there but the quantum
formulas. What does this curious similarity
imply? Could the world of everyday objects also be a mere
image?
Unfortunately, for now this mystical question must remain unanswered,
for I'm still pondering it. Meanwhile, it's time
to get to today's fractal.
I named the formula and picture Gamut because they covers such a wide
range of effects. With six variable parameters to tweak,
one could spend years before they discovered all this formula can
do. The formula is a bit more complex than I prefer, but
it creates such unusual pictures that it's worth it. As
always, the finished image is posted to ABPF for those who prefer not
to tie up their machines. For those who wish to watch the
picture come to life on their screens, it takes XXXX to build on a
486-100mhz.
Jim Muth
jamth@mindspring.com
START FORMULA===================================================
Solar_Symbol
{ ; time=0:00:03.52-SF5 on P4-2000
reset=2004 type=formula formulafile=basicer.frm
formulaname=gamut center-mag=-4.114731896495253/\
0/131.3539/1/-90/0 params=3/1/0.5/1/-1/5e-031
float=y maxiter=430 inside=0 periodicity=0
colors=00013i42U61FXBNR8HK5BE25xxYfcMPKBcgNETFIY77\
iNKoTVqcesnkotyxzVPOK96MJ6OT7Pa7Rk8Tu8zz7yq8xh8v`9\
uS9tJAsABs1Bj19b18U26L24D23421CA4LH7TPA`WChcFqjIyr\
LkjLXcMJWM4OM5MK5JJ6HH7FG7CE8AC87B959ylQk`JZODLC6O\
OWIGLD8A_16M03xhHjXCYM8LB46b169r9pF8Q71sJLOfIIWECL\
B6AvnQiaJXPDKC6apoUcbNQQFDDS7RN5KI3DD16sBug8fW5TK2\
Elrwd`wUOcJCKUojv95jF0Qy`CyiIjOOW3A0a8Fc6Ud3gf1vgD\
ofOhd_accUYfMSjEMm7GswZceaOOdFTQ7Lq9c2wRwjKhYDUL6F\
hiq_YcQNRHBDhi0_Y0QN0HB04xT6UEsyUcdKOKARYR5t76a47J\
2fPTVGJJ890sh4SMOwrbrUNRFX1jK0NITuEJaB9Jmzmbj`TVPI\
FCIwYTybOiTIVJDF9B`w55t72SX3kR2_K1OE0CWxtKUSXu5pIo\
eDbU9QJ4D3DY66HiYi_PYRHNH8BSL1NG1IA0D50Zj0QV0HF0on\
0da0UP0JC03VZ6FHAMa9EP87Cc0ZO0Hq1yV0VIIsD9SwSleIWP\
9GBV95Cl68W74GoMHdGDUB8J54vOTtkscW`OGIKD6dU7XM5OF3\
G710jQ4NDzxBgc7QK31py4Zd6HK9HY8BM85Bh760BcG8YC4HzP\
LlJGZCAL654b`6QO7DCgotXpy }
frm:Gamut {; Jim Muth
z=g=pixel, a=real(p1), b=imag(p1), c=real(p2),
d=imag(p2), e=real(p3), f=imag(p3):
h=z^a+(g-b)*z-g
j=c*z^d+g
z=z-e*h/j,
f <= |h| }
END PARAMETER FILE==============================================