; Date: Tue, 06 Dec 2005 22:09:25 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 07-12-05 (The Fractal [4]) ; Id: <1.5.4.16.20051206221121.2d776032@pop.mindspring.com> ; --------- ; ; 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: ; ; ; ; 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========================================= ; ;