Fractal of the Day
by Jim Muth

They All Look the Same © (#2)
Jim Muth's fractal image in GIF format (640x480).

FOTD -- August 29, 2006   (Not Rated)

Fractal visionaries and enthusiasts:

Today's harmless little scene appears in the Z^sqrt(sqrt(2))+C Mandeloid as it appears 59 levels up the complex logarithmic ladder.   (The logarithmic ladder is explained below.)   The scene is so harmless in fact that I did not bother giving it a rating.   I did give it a name however.   After not too much thought, I named it "They All Look the Same", which sounds a bit biased, but really is not.

A quick glance at the image will reveal why I gave it that name.   Indeed, all midgets in the Mandeloids with an exponent of Z between 1 and 2 do seem to look pretty much the same.   They all resemble splashes, or starbursts as they are sometimes called.   I spend countless hours searching, hoping that I will find a midget in this range that is truly different.   So far I have had minimal success.

The calculation time of today's parameter file is under 19 minutes on my machine.   This is a bit slow for an image of such questionable worth.   For relief, I recommend downloading the completed image from the FOTD web site at:
The 'complex logarithmic ladder' is a phrase I invented.   I sometimes refer to the same thing by the name 'logarithmic hyperspiral'.   Neither phrase will be found on the internet.   I invented the phrases because the complex logarithmic function is multi-valued, and can have an infinity of solutions.   Each solution gives a unique four-dimensional Julibrot fractal, complete with both Mandelbrot and Julia slices.   In my mind, I simply stacked these four-dimensional fractals onto one another until they formed a single many-layered hyper-object of 5 or so dimensions.   It is the stack of all the Julibrot fractals possible to create with a single exponent of Z that I call the logarithmic ladder.   Today's parent fractal for example exists on the 59th level of this ladder.   The ladder could perhaps be more conveniently thought of as the 59th floor of an infinitely tall skyscraper.

In the MandelbrotBC2 formula, the p1 parameter determines the exponent of Z, while the p2 parameter determines the level of the ladder we wish to explore.

Cloudy and muggy but dry weather prevailed here at New Fractal Central on Monday.   The fractal cats seemed not to notice.   They were too busy chasing each other up and down the hallway.   The big event of their day came when I opened a bottle of soda.   When Nico heard the hiss of the escaping gas, he hissed back at it.   Cassie watched Nico make a fool of himself.

The next FOTD is due in 24 hours.   If all goes well, it will appear on schedule.   Until then, take care, and mysteries just happen.

Jim Muth

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

TheyAllLookTheSame-2 { ; time=0:18:51.96--SF5 on a P200
  reset=2004 type=formula formulafile=allinone.frm inside=0
  formulaname=MandelbrotBC2 float=y passes=1 periodicity=10
  9203e+007 params=1.1892/0/59/0 outside=tdis maxiter=25000
  odumdziexgexe0Ez0Ez0Ez0Ez }

frm:MandelbrotBC2     { ; by several Fractint users
  e=p1, a=imag(p2)+100
    IF(imag(Z) > r)
  |Z| < a }

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

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