FOTD -- December 07, 2005 (Rating 4) Fractal visionaries and enthusiasts: Today's fractal is the central part of the much larger fractal that results when 0.4 parts of Z^(-0.8) are subtracted from Z^2, and (1/C) is added. The entire bloated fractal extends far to the northeast of the small part that appears on the screen, though most of this extended part holds little of interest. The image consists of a Mandelbrot set, this much is obvious. But the set is distorted to such a degree that its various parts are hard to identify. East Valley, which lies at the bottom of the main bay, is easy to find, but there are two buds of nearly the same size, making it difficult to tell which is the large period-2 bud. A check of the filaments reveals that the large bud on the left is the actual period-2 bud, and the filament curving outward and downward from it is the main stem. We know this is the true main stem because it does not split. The broken filament extending to the northwest of the bud is a secondary one, which has become blown up beyond its normal importance. The large bud on the right of the main bay is actually the north period-3 bud. Its filament splits into two main branches, and the point of the split has 3 arms radiating from it, indicating that the bud has a periodicity of 3. This point is the lowest order true 'star'. (I do not count the 'straight' 2-armed star of the main filament.) Today's distorted M-set may be explored just like the true M-set. All the familiar features and sub-midgets are there in their proper places, with their familiar patterns mostly intact, but with the normally flat iteration bands enhanced by the pattern present at that point of the parent fractal. The true Mandelbrot set is 'connected'. All its midgets are connected to the main bay by infinitely thin filaments. But today's distorted set is obviously not connected. Parts of it are clearly separate from the main bay. I am not sure if these disconnected parts actually are connected to the main bay in the underlying four-dimensional Julibrot. My instinctive guess would be that they are connected, though I am unaware if this has ever been determined. These disconnected parts are filled with small midgets, just as the connected parts are. And as would be expected, the patterns around the disconnected midgets are also disconnected. Instead of being surrounded by the expected connected and splitting features, the midgets are surrounded by scattered bits and pieces of debris arranged in groups of 2,4,8.... These 'discon- nected' midgets are usually rather bland. I rarely find much of interest in them. The fractal is 'critical, meaning Z was automatically initial- ized by the M-Mix4 formula to a critical point of the calculated expression. Its midgets are therefore intact. But the generat- ing expression, (Z^2)-0.4(Z^-0.8), has more than one critical point. There is a second Mandeloid connected with another criti- cal point hidden almost invisibly in the image. Notice that the lower right part of the image is filled with holes, and the largest hole, located at the end of a prominent filament, has a purplish valley showing through it. This valley is the only part of the almost totally obscured second Mandeloid that is visible. A quick check will reveal that it has not been calculated at its critical point. But this 'ghost' Mandeloid is important. The midgets of the top Mandeloid will always cut through the ghost beneath, but the top midgets will be surrounded by features that mirror the Julia sets of the part of the 'ghost' Mandeloid over which they lie. It is in areas where the interesting parts of both Mandeloids overlap and mix together that the best scenes exist. In the next few FOTD's I will show several of these scenes found in the lower right part of today's image. I named today's image "The Fractal", which is stating the obvious. Unfortunately, after careful consideration, I could rate the image no higher than a sub-standard 4. The above- average images will come in the next few days. The render time of 1-1/3 minutes is fast even on slow machines. And for the convenience of those with handicapped computers, the completed GIF image has been posted to the FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> Heavy clouds all morning and light snow in the afternoon kept the fractal cats confined to quarters all day Monday here at Fractal Central. Needless to say, they were a bit spoiled over the weekend, and sulked when they did not get all the tuna they could eat. The snow ended on Tuesday and the sky partly cleared, but with a temperature barely above freezing, and two inches (5cm) of snow on the ground, the duo never left the porch. My day was at the usual degree of activity. The next FOTD will appear in 24 hours. Until then, take care, and do your fractal shopping early. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER FILE======================================= The_Fractal { ; time=0:01:19.58--SF5 on a P200 reset=2004 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=1.5457/0.342387/1.078531/1/87.5/-2.5056\ 3458870090017e-014 params=1/2/-0.4/-0.8/0/0 float=y maxiter=512 inside=0 logmap=yes periodicity=10 colors=000G00K00O00S00W02_04c06i28mEApPCs`EvkGocIh\ WKaOMVGPO9QNARNASNAUNAXNA_NAbNAeP8hR6kT4mU2or9qkGr\ dNsYUtR`uWYv_WwdTxhRylPzqMzuKzyIziMzVPzWQzXRzYSzZT\ zZUzSTzMTzFQz9MzhKziKzjKzkKzlKzlCz`CzQCzECz3Cz6Ez8\ GzAIzCJzFLzHNzJPzLQzdIzxBziKzVTzHazNXzTSzYNzcIziLz\ nOztRzyUzqXzi_zaayVcxUewTgvSjuRmnQptPqrPoqPmpQkoRj\ nSimTilUkkVmjWoiXqhYsgPufHwe9ycFzaLz_QzXMzUIzRFzPB\ zQ8zRHzPQzOYzNHzdIzYJzRQzWXz`czejzj`zeRz`HzXczVgzR\ kzNozJpzHqzFrzDszCvzBxzBzzBhzHRzMAzRIzSQzSTzxQziOz\ V5z4EzA7zsMzItzgkz`bzUUzN8zmdzw_zlVzaQzR`zATzDHzQK\ zLQzHPzGOzGNzGmzXfzS_zOTzKazrYzhUzZQzPez6`z9WzBRzE\ JzIKzHLzGHz1Jz5Kz9LzDlzHZzGPzAXztrzPizMazKUzIXzfRz\ jPz_NzQZzlSzWazaUzRgzEkzAbzCUzExz_dzQGzvJz`yzEkzIb\ zHUzGpzMhzK`zJTzHdzP_zMVzKQzI9zZEzSIzM`ztXziTz_PzQ\ 6zuCzgHzU6z6CzAHzDcz1cz9cz1Uz5Rz9OzDtzBbzEjzdazWUz\ OgzkazcXzWRzONzNrzSszLtzF } frm:MandelbrotMix4 {; 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)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END PARAMETER FILE=========================================