Fractal visionaries and enthusiasts:

Start with Z^2, subtract Z^3, add Z^4, subtract Z^5 then add C on each iteration.  The result is a fractal with a mis-shapen main bay and several large sub-bays.  Large period-2 buds lie on both the east and west extremities of the parent, with spikes extending outward along the X-axis from both buds.  Today's image is located on the spike extending west from the western large bud.

Luckily, one of the critical points of today's iterated expression lies at zero, which would have saved some tedious calculation if I had not been curious about other critical points.  (A second critical point lies at +0.729323..., which draws a very similar parent fractal.  A point of inflection lies at +0.436117..., but this also draws a very similar image.)

The name "Diminishing Returns" refers to the nature of fractals.  The most interesting Mandeloid is created by raising Z to the power of 2.  As the exponent of Z is increased, the Mandeloids start to look ever more similar, and by the time we reach Z^100, there is almost no difference in the overall appearance or in the smaller details.  In short, the law of diminishing returns kicks in, and the fractals start to all look the same.

The same law of diminishing returns applies to the magnitude.  As we go deeper into a fractal such as the Mandelbrot set, the minibrots we find grow ever more interesting at first, but at a magnitude around 10^25 they begin to all look the same, and by the time we reach 10^100, with some few exceptions, little but nearly identical concentric circles surround the minibrots.

The same diminishing returns appear as we pile on various powers of Z.  Today's parent fractal combines four different powers of Z in each iteration, yet the image falls far short of some of the images created by the MandAutoCritInZ formula, which combines only 2 different powers.  Out of curiosity, I have written formulas that pile on up to 15 different powers, only to discover that, once again, the fractals all start to look the same.

The rating of a 7 is near FOTD average, but the calculation time of 12 minutes is too high a price to pay for a merely average image.  Luckily, rescue from boredom may be found at the FOTD web sites.

Another day of uneventful weather prevailed here at Fractal Central today, with mostly cloudy but dry skies, a little sun and an unremarkable temperature of 43F +6C.  The fractal cats were satisfied that no draft fell on their shelf by the window.

The humans, FL and I, had another in a long string of similar routine days, which are quiet enough, but leave little to report.  The next FOTD, the last of the year, will be posted in 24 hours.  I'm trying to think of a fractal theme for the month of January, so far without much luck.  Until next FOTD, take care, and only 357 days remain before the world ends.

Jim Muth
jimmuth@earthlink.net


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

DiminishingReturns { ; time=0:12:00.00 SF5 at 2000MHZ
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  formulaname=MandelbrotMix5way center-mag=-0.764203\
  8506481917/+0.00004122174900695/1.586226e+012/1/\
  -83.25/0 params=1/2/-1/3/1/4/-1/5/0/0 float=y
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frm:MandelbrotMix5way {; Jim Muth
z=p5, c=pixel,
a=real(p1), b=imag(p1), d=real(p2), f=imag(p2),
g=real(p3), h=imag(p3), j=real(p4), k=imag(p4):
z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c,
|z| <= 100 }

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