Fractal visionaries:

It was a cold, damp, dreary, foggy day at the fractal place, with a temperature of 45F 7C and a visibility of 1/8km -- a perfect day for hunting fractals.

The image I bagged is a scene in the fractal that results when portions of Z^3 and Z^(-3) are combined and added to C.   I named it "Beyond the Spike" because it is typical of Mandelbrot midgets located just beyond the tip of the negative tails of larger midgets.   I briefly considered naming it "Beyond the Tail", but thought that this name would give the impression that I was commenting on the current political situation in the USA.

Sub-midgets in the area just beyond the tail-tips of larger midgets have straight spikes that are still prominent, but have become floating, cut-off fragments.   This is not at all surprising, since the Julia set with a C-value of -2.1 is exactly such a series of cut-off line segments.   In fact, by examining the characteristics of the Julia sets centered near to but outside the Mandelbrot set, one can get a rough idea of the appearance of midgets in the corresponding vicinity of a larger midget.   The features of the overall fractal may vary widely according to the individual formula, but those features will be twisted into the familiar local features of the larger midget that they are a part of.

The image, which I did soup up a bit in Photoshop, has been posted in GIF format to:and to the FOTD Website at:

For a week or so, I've been building up for another outburst of philosophy.   I've listed below some of the things I've been pondering while watching the fractals appear on my screens.

Fractals are pleasing to look at, interesting to listen to, and always fun to explore, but what do they mean?   Even more importantly, what are they?   Do they really exist, or were they invented by Mr. Julia and Mr. Mandelbrot.   Are they a matter of faith or a new kind of reality?   Is the concept of reality an invention of man?   How are fractals connected with reality?

In my opinion, the most disturbing situation in our modern western society is the disagreement between science and religion.   This split developed because science, with the most accurate means of determining truth yet devised, has consistently failed to find the slightest evidence of the transcendence that is necessasy for religion to have a basis in fact.   Without its life beyond death and its all good and powerful god, religion is reduced to a system of arbitrary moral standards with outmoded superstition attached.

Must scientific progress inevitably lead to Atheism?   Great thinkers such as Carl Sagan and Isaac Asimov said yes.   These writers sometimes mentioned God in their writings, but privately admitted that they considered the ideas of religion to be wishful thinking.

What would an advanced race think of us if they visited earth?   Would they see the crowds attending churches and pity the way so many of us still need to console ourselves with childish fairy tales?   Or would they see the scientists in their laboratories and pity the way so many of us are apparently oblivious of the unseen worlds around us?   Or, as I suspect, would they see the two groups and pity the way they are studying the same thing from different directions and failing to recognize it?

Ever since I began working with fractals, I have had a gut feeling that the truth lies somewhere between science and religion, and that when we finally recognize it, we will find fractals somehow involved.   I'll have much more to say on this important topic in the near future, but for now it's time to shut down the fractal shoppe and feed the cats.

Until next time, take care, and keep the fractal faith.


Jim Muth
jamth@mindspring.com

START FORMULA================================================

MandelbrotMix5     {; Jim Muth
  a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f,
  h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1,
  l=imag(p3)+1, c=pixel:
  z=k*((a*(z^b))+(d*(z^f)))+c^l,
  |z| < 100
}

END FORMULA==================================================

START PARAMETER FILE=========================================

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  formulaname=MandelbrotMix5 float=y maxiter=12000
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END PARAMETER FILE===========================================

START 19.6 PARAMETER-FORMULA FILE============================

Beyond_the_Spike      { ; 25min on a 486-100, 640x480
  reset=1960 type=formula formulafile=critical.frm
  formulaname=MandelbrotMix5 float=y maxiter=12000
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frm:MandelbrotMix5    {; Jim Muth
  a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f,
  h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1,
  l=imag(p3)+1, c=pixel:
  z=k*((a*(z^b))+(d*(z^f)))+c^l,
  |z| < 100 }

END 19.6 PARAMETER-FORMULA FILE==============================