; Date: Wed, 05 Oct 2005 12:24:19 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 05-10-05 (The Same Root [5]) ; Id: <1.5.4.16.20051005122534.2a87c708@pop.mindspring.com> ; --------- ; ; FOTD -- October 05, 2005 (Rating 5) ; ; 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 calcula- ; ting 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_\ xEayEbvGasIapKamMajNagPadRaaT`ZV`WW`TY`Q_`Na`Kb`J`\ YI_WHYUGXRFWPEUNETKDRICQGBPDANB9M95N49L7hKAcJDZHGW\ GJSFMWEP_CScBVfAYj8`n7cr6fy2lu5iq7fnAdjCagE_hHaiKc\ jNdjQfkThlWimZkmalndnogppjqpmsqpursvsvxsxymtthppcl\ lYhhTddO``I_WJXXKVXLTYMRYNPZOMZPK_QI_RG`X1eRE`LQXG\ bSAnO5zK7xN8wQ9vSAuVBtYCs_DrbEqeFpgGoj9umHnlOhkWaj\ bWijPhqJgxDftHcpKalNZhQXdUV`XSY_QUbOQeLMiJIlHEoEAr\ C7uACvIGvQLvYPweUwmYwtYfqYRnb7pYBkUEgPHbLKZHNVLJ`O\ FeJ9jRCjZFjfIjpJlmLjkNhhPffRdcTbaV`ZW_XYYU_WSaUPcS\ NeQKgOGiLIhNKhPMgROgTQgVSfXUfZUtcWf_XTWYFSpaSZ1Pa3\ Nc5Me7Lh9KjBJlDIoFHqHGsJFuLErKHoKKlKNiJQfJTcJW`JZY\ IaVIdSIgPHjMHmJHpGHsJJpMKmPLkSMhVNeYOc`P`cQZfRWiST\ 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========================================= ; ;