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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: For today's flashback I chose the 'FrankJim04' formula, a formula I wrote quite a while ago, played with a few days, then promptly filed away and forgot about. Unlike nearly all of my formulas, it has no changeable parameters. When a real number is raised to a power and the same root is extracted, common sense tells that the result is a return to the original number. But complex numbers are strange. Some complex functions, such as the logarithm function, are multi-valued. This means that the calculation might take a wrong turn, and we might not always get back to the exact place where we started. The iterated expression in today's formula is ((z^z)^(1/z))^2, which takes Z, raises it to the power of Z, then to the power of 1/Z, and finally squares the result before adding C. One would expect that this would be just a more complicated way of calculating the Z^2+C Mandelbrot set, but those multi-values do strange things. The parent fractal of today's image bears some resemblance to the Mandelbrot set, but it most certainly is not the M-set we know so well. Some parts of the original set are missing, and a good bit of extra stuff has been added, while the iteration bands have broken apart into total chaos. Today's scene is located in the extra stuff found in the East Valley of the parent almost-M-set. I named the picture "The Same Root" and rated it at a 5. If I had put more effort into the coloring, I might have produced a 6 or even 7-rated image. But the work has been a bit on the heavy side lately, and I did not want to fall behind, which would mean another of those much-hated FOTD hiatuses. The render time of 9 minutes is a bit slow, but the finished image is available for perusal and downloading on the FOTD web site at: The clouds hung tough all day Tuesday here at Fractal Central, even dropping a few small drops of rain from time to time. But the temperature remained around 75F 24C most of the day, which is right in the comfort zone of the cats. Despite the dampness they spent several hours in the yard, and needed little tuna to end the day happily. Today is starting about the same. The next FOTD will appear in 24 hours. Until then, take care, and open your eyes to what lies before them. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START PARAMETER FILE=======================================
The_Same_Root { ; time=0:09:04.26--SF5 on a P200
reset=2004 type=formula formulafile=jim.frm
formulaname=FrankJim04 passes=1 float=y maxiter=950
center-mag=+0.25597317684299750/-0.000200259583298\
25/5.400547e+007/1/-162.5/-1.00481606299385184e-008
inside=0 outside=tdis periodicity=10
colors=000CpkArl8tl6ulsNirLfqKdpJapI_oHXnGVmFSmEQl\
DNkCLjBIh9EjAGkAIlAJmBLnBNoBOpCQrCSsCTtDVuDXvDYwE_\
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lTRoUOrVLuWJxXGzYEwZFuZGrZHpZImZJk_Kh_Lf_Mc_Na_OZ`\
PX`QU`RS`SP`TN`UIkjGmkEok }
frm:FrankJim04 {; by Frank and Jim
z=c=pixel:
z=((z^z)^(1/z))^2+c,
|z| <= 100 }
END PARAMETER FILE=========================================
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times.
