Fractal visionaries and enthusiasts:

The complex logarithm is multi-valued, so how then do we know which value is the correct one?   The answer is that they are all 'correct'.   The square roots of 4 are plus-2 and minus-2, and they are both 'correct'.   We use the root that works best in the situation at hand, which is almost always the plus-2 root.   The same is true with the complex logarithms, but the other values are always there, and they also make fractals.

The MandelbrotBC3 formula makes use of these other values of the complex logarithm.   Each value draws a different fractal, even when the exponent of Z remains unchanged.   In today's image we have gone 30 levels up the logarithmic ladder to a level where the Z^(1.5)+C parent Mandeloid is totally changed.   Today's image is located in the 'Seahorse Valley' of its parent.

Since most of the effect is a result of the coloring, which I put at least 10 minutes work into, I have added the usual 1/2 point bonus for the coloring, bringing the overall rating to a less-than-inspiring 6-1/2.   The name "Unity and a Half" refers to the exponent of Z.

The calculation time of a shockingly fast 44 seconds will bring no discomfort to even the most fastidious.   The trip out to the FOTD web site at:to see the finished image there will be equally enjoyable.

The showers moved out of the Fractal Central area early Thursday night, leading to a foggy start for Friday.   By 10am the fog cleared out however, and the rest of the day was near perfect, with blue skies, puffy clouds, and a temperature of 77F 25C.   The fractal cats enjoyed the weather conditions, but they were not happy when a lovesick tomcat wandered by and sprayed the wall right beneath their favorite window.   A quick squirt from the hose removed the offensive odor and restored the cats' good moods.   My day was busier than expected, in fact it was barely under control.

Unless FL gets an antiquing urge, the next FOTD will be posted in 24 hours.   Until then, take care, and be conservatively liberal.


Jim Muth
jamth@mindspring.com
jimmuth@aol.com

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

Unity_and_a_Half-2 { ; time=0:00:44.80-SF5 on P4-2000
  reset=2004 type=formula formulafile=basic.frm
  formulaname=MandelbrotBC3 function=ident logmap=105
  center-mag=+0.1641879378341295/+0.775442975243285/\
  200823.9/1/-5/0 float=y maxiter=1800 periodicity=10
  params=1.5/0/30/0 inside=0
  colors=0000HU000K0U0000KU000N0U0000HU000K0Ul0fi1cf\
  2`c3Y`4VY5SV6QS7NP8KM9HJAEWU2UR4SP5QN6OL7NJ8LH9JFA\
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  OQCMMCKICIECz5Az6Bz6Bz6Bv6Br7Bm7Bh7Bc7BZ8BU8BP8BK8\
  BJ9CI9CI9CI9CIACHACHACHACGBCGBCGBCf5Ga6FY7ET8EP9DK\
  ACH09G2AG3AG5BG6BG7BG9CGACPu`OrZOpYNnXNlWMiVMgTMeS\
  LcRL`QKZPKXNJVMJSLJQKIOJIMHHJGHHFGFEGDDF5mG6lH7lJ9\
  lKAkMBkNDkOEkQFjRHjTIjUJiVLiXMiYNh_Ph`QhaRhm`kxjnv\
  hotgosfoqeopdoncombokaoj`oh_ogZoeYocXpbWp`Vp_UpYTp\
  XSpVRpUQpSPpROpPNpOMpQOnRQlSSjTUhUWfVYdW_bXa`Zc__e\
  Y`gWaiUbkScmQdoOeqMfrLeqKeqKepJdpJdpJdoIdoIcoIcnHc\
  nHcmGbmGbmGblFblFalFakEakEakEdzMezOezPezQfzRfzSfzT\
  gzUgzVgzXhzYhzZhz_iz`izaizbizc`zYSzSJzM0z7LzuJzeHz\
  RZzEXzDWzDVzDTzDSzDRzDQzD }

frm:MandelbrotBC3   { ; by several Fractint users
  e=p1, a=imag(p2)+100
  p=real(p2)+PI
  q=2*PI*fn1(p/(2*PI))
  r=real(p2)+PI-q
  Z=C=Pixel:
    Z=log(Z)
    IF(imag(Z) > r)
      Z=Z+flip(2*PI)
    ENDIF
    Z=exp(e*(Z+flip(q)))+C
  |Z| < a }

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