Fractal visionaries and enthusiasts:

Sebastien recently sent me an e-mail in which he mentions the little white cloud on the northeast side of the large minibrot in the 'No Discontinuities' FOTD image of March 11, the 'pimple' that held the 'Outrageous Colors' image of March 12.  This white cloud is what I call an area of anti-bifurcation.  The fractals drawn by the "FinDivBrot-2"-type formulas are filled with such areas, though I have never seen them drawn by any other type of formula.

I sometimes mention anti-bifurcation in my FOTD discussions.  There are two kinds of this oddity that I currently am aware of.  The more common variety occurs in the fractal patterns around minibrots that lie in the basins of attraction of larger minibrots, when the constantly doubling number of elements
approaches infinity some distance from the minibrot, then resets to a much smaller power of two and resumes doubling as we continue diving inward toward the minibrot.  This resetting can occur several times on the way to the minibrot.

The other, much rarer, variety of anti-bifurcation is illustrated in today's image, where everything is backwards and the number of elements is cut in half as the center of repulsion is approached.  No minibrot lies at the center.  Instead, the center is the lowest iteration peak of a fractal mountain.  The difference is made immediately obvious by the colors, which flow in the wrong direction when cycled.

In today's image the outermost blue ring consists of 64 blue elements; the red ring holds 32 elements; the green holds 16; the purple holds 8; and the tiny turquoise area at the center holds 4.  Beyond this, and beyond resolution, lies an area of 2 elements, and then . . . who knows.  Does the series continue 1, 1/2, 1/4...?  The answer lies well beyond the range of resolution, but I assume the series comes to an end at 1, with a minimum of around 40 iterations.

The image rates an 8, about 1-1/2 points of which is due to the mathematical interest.

The name "Anti-Bifurcation" simply describes the image.

The calculation time of 45 seconds is blazingly fast, mostly due to the unusually small iteration count of most of the elements in the scene.

Another gorgeous day graced Fractal Central today, with hazy sunshine and a temperature of 75F 23C.  The fractal cats were warm enough without the full sun, and found comfort in the watered-down sun that filtered through the clouds.  The fractal humans found satisfaction in the typical routine day, in which nothing went obviously wrong.  This is the best typical earth-bound humans can ask for.  The next FOTD will be posted in 24 hours.  Until then, take care, and an activist is one who goes out and protests the un-affordable price the greedy big-oil conglomerates are charging for their products.

Jim Muth
jimmuth@earthlink.net


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

Anti-Bifurcation   { ; time=0:00:45.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=FinDivBrot-2 function=recip
  center-mag=-1.296298258719479/+0.4418325064639432/\
  1.548763e+012/1/-77.5/0 params=5/2.5e+017/0/0
  float=y maxiter=128 inside=0 outside=tdis logmap=46
  periodicity=6 mathtolerance=0.05/1
  colors=000mzzmzzmzzmzzUzzKzzKczKczKmzzzzzzzzzzzzzz\
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  0070`50d20p000000000000000000000000000000000000000\
  000000000000000000000000000000000000000000000Yz_Wz\
  _Wz_Wz_Wz_VzYVzYVzYVzYTzYTzYTzYTzYzyzzzdzyczvazq_z\
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  20qQ2ugJvxYxzjyzdzzazzYzzVzyQzxMzzVzz_zzgzzlzzszzx\
  zzzzzzzzzzzzmzzT7Oz5Wz2az }

frm:FinDivBrot-2   { ; Jim Muth
z=(0,0), c=pixel, a=-(real(p1)-2),
esc=(real(p2)+16), b=imag(p1):
z=(b)*(z*z*fn1(z^(a)+b))+c
|z| < esc }

END PARAMETER FILE=========================================