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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\
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  Elrwd`wUOcJCKUojv95jF0Qy`CyiIjOOW3A0a8Fc6Ud3gf1vgD\
  ofOhd_accUYfMSjEMm7GswZceaOOdFTQ7Lq9c2wRwjKhYDUL6F\
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  eDbU9QJ4D3DY66HiYi_PYRHNH8BSL1NG1IA0D50Zj0QV0HF0on\
  0da0UP0JC03VZ6FHAMa9EP87Cc0ZO0Hq1yV0VIIsD9SwSleIWP\
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  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==============================================