; Date: Fri, 15 Aug 2003 10:19:49 EDT ; From: JimMuth@aol.com ; Subject: [Fractint] FOTD 15-08-03 (Dual Pentagons [5]) ; Id: <5a.2090da93.2c6e4605@aol.com> ; --------- ; ; FOTD -- August 15, 2003 (Rating 5) ; ; Fractal visionaries and enthusiasts: ; ; The world of fractals holds many mysteries that will never be ; discovered, much less solved. But the mystery of today's image ; is one that has been discovered and, if I decide it is worth ; the effort, will be solved. ; ; At risk of being considered repetitive, today's image shows what ; happens to yesterday's image when the angle of the slice is ; double-rotated merely 0.0000685 of one degree from the Julia ; orientation. (Double rotation is a motion that can happen only ; in spaces of four or more dimensions.) ; ; When I began the rotation, I expected the octagon that appeared ; at the center of yesterday's image to become distorted. I did ; not expect it to break apart into two separate symmetrical holes ; shaped like near-perfect pentagons. But this is what happened. ; ; Now the question arises, which of the two pentagonal holes is a ; slice of the original rectangle that appears in the Mandelbrot ; aspect of the scene? And also, what happens to the other hole ; as we turn toward the Mandel direction? ; ; My intuition is that these questions are irrelevant, for there ; is only a single hole in the four-dimensional Julibrot figure, ; but this hole is shaped so irregularly that it can be intersec- ; ted in more than one place by certain slices, such as today's ; slice. Switching the functions in today's formula shows a very ; distorted aspect of the hole that would seem to prove this. ; ; I named the image "Dual Pentagons". I was ready to rate it at ; another 8, but then I realized that it is basically the third ; identical image in a row, and decided that a rating of 5 is more ; appropriate. Those who enjoy the art side of fractals rather ; than the math side might be growing a bit bored. ; ; Those who are feeling the beginnings of boredom, as well as ; those who lack the time, may escape the 15-minute wait of ; rendering the image from the parameter file by downloading the ; completed GIF image from one of the FOTD web sites at: ; ; ; ; and at: ; ; ; ; A really unusual thing happened Thursday here at Fractal Central ; -- for the first time in almost 4 weeks, it did not rain. True, ; a few showers were in the vicinity, but none reached F.C. The ; fractal cats, who have made up after yesterday's quarrel, were ; quite happy, and despite an oppressive temperature of 91F 33C, ; enjoyed several hours in the yard. ; ; Today is due to be a repeat, so I expect a repeat of the cats' ; good moods. Until next FOTD, take care, and it is sometimes ; easier to see the light when one is in the dark. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ Dual_Pentagons { ; time=0:15:53.78--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=flip/ident passes=1 center-mag=-0.00006842222800720/-0.000036\ 15877643450/1535.55/1/-45/3.88578058618804789e-016 params=90.0000685/89.9999315/2.003/0/0/0/-1.743504\ 25848689/6.977862219e-006 float=y maxiter=7200 inside=0 logmap=641 periodicity=10 colors=000voOtmMrkKpiKngJleJicJfaJd_JaYJZWJWTITRIR\ PIOMILKIIIIEFFGGIHHKJIMKIOMJQNKTOKVQLXRMZTN`UNcWOe\ XPgYPi_Qk`RnbSpcSreTtfUvgUxfXwfZve`vebueetdgtdisdk\ sblqalo`lm_llZljYlhWlfVleUlcTlaSm_RmZQmXOmVNmTMmSL\ mQKmOJmMImLKkOMiQOhSQfUSdWUcYWa_Y_a_ZdaXfcVheUjgSl\ iQnkPpmNrnLwoMtlMqiNneNk_NhVNeRMbNL_JLXFKUBJSCLRDO\ RERQFUQGXPH_PIdOJhOKnNLrNJwOLqNMlMOfMPaLQXLSTKTQKU\ NJWJJXGIYDIXGMWJQVMUUPYUSaTVeSYiR`mO`tRbpUdmXej_gg\ bhdejahkYkmVnnSqpPtqMwsJytGulNrdToXZlPdiHjfApdIocP\ nbWnacm`jm_qlZxl`wjawhbwfcvedvcevafv_guZhuXiuVjuTk\ tSltQmtOntNnkcnbslcrkcqjdpidohengemfelefkdfjcgibgh\ aggeaahXWkRQnMLrGFvB9z64s88iAC`CGSEKKFOIOKGWHEcEDk\ BClEClHBlKBlNAlQAlT9lW9lZ8la8ld7lg7lj6lm6lp6ls9kmC\ jhEibHhYJgTKeOLdJMbENa9O`4Rb7UdAWfDZhG`iJckMemPhoS\ kqVmrYpt`rvcuxfwyhuuetqbsn_rjYqgVpcSo_PmXNlTKkQHjM\ EiIChF9gB6f84eC8eFBmjSnjQ } frm:multirot-XY-ZW-new {; draws 6 planes and rotations ;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J ;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3, c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4: z=z^(p2)+c, |z| <= 36 } ; END PARAMETER FILE==================================