; Date: Thu, 14 Aug 2003 11:51:44 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 14-08-03 (Mandelbrot Octagon [8]) ; Id: <1.5.4.16.20030814115202.2997a118@pop.mindspring.com> ; --------- ; ; FOTD -- August 14, 2003 (Rating 8) ; ; Fractal visionaries and enthusiasts: ; ; The joke of the day is an unusual e-mail from Taiwan that I ; found when I checked my e-mail box this morning. The sender is ; having a sale on guano and wanted to know how much I need. The ; letter closed by telling me that if I did not need guano, to ; delete the message -- which I did without raising a stink. ; ; Today's fractal is also quite unusual, and that's no . . . umm ; . . . stuff. It is the Julia set of one of those rectangles ; that appear where the arms branch out in yesterday's image. ; ; It is obvious that the Julia set of a Mandelbrot rectangle will ; be another rectangle . . . at least I thought it was obvious ; until I actually calculated a Julia set. The Julia set of the ; Mandelbrot rectangle I chose is not another rectangle; it is an ; octagon. ; ; The figure in which yesterday's rectangles are embedded has also ; changed. Well, it hasn't actually changed, but rather it looks ; different when sliced in the Julia direction. It has many more ; arms, in addition to smaller octagons where the arms branch out. ; No! Wait a minute! Those are not octagons where the arms ; branch out; they are rectangles. We have a Julia figure with a ; large octagon at the center, surrounded by smaller rectangles. ; ; Actually, I have seen this doubling before. I once checked the ; Julia set of the center of a 5-pointed star of the Mandelbrot ; set and found a 10-pointed star. And as I recall, there were ; also 20-pointed stars in the vicinity. I don't know if any ; 16-sided holes lie near today's octagon, but I am certainly ; going to check. And I wonder what will happen to today's image ; when I rotate the plane of the slice ever so slightly . . . ; ; Some of these questions may be answered by the time the next ; FOTD appears 24 hours or so from now. ; ; I named today's image "Mandelbrot Octagon" for obvious reasons. ; I rated it at an exceptional 8 mostly because of its mathemati- ; cal interest. The render time of 17 minutes can be avoided by ; downloading the completed GIF image from: ; ; ; ; or from: ; ; ; ; Wednesday started pleasant enough here at Fractal Central, with ; sun and a temperature of 86F 30C, but at 2pm, a heavy thunder- ; shower moved in, cutting short the dynamic duo's time in the ; yard and putting them in a bad mood for the rest of the day. ; They must have been quite cranky while I was not around, because ; when I gave them their evening treat of tuna, I noticed a ; new scratch on Tippy's ear, which means they had had a fight. ; Thomas looked totally innocent of course. ; ; Today is starting in a similar manner. Hopefully, the midday ; shower will not be repeated, and the cats will have a happy day. ; For me it will be just another in an unending stream of similar ; days. Until next time, take care, and be alert for polygons ; with multiplying sides. ; ; ; Jim Muth ; jamth@mindspring.com ; jimmuth@aol.com ; ; ; START PARAMETER FILE================================ Mandelbrot_Octagon { ; time=0:17:31.33--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=ident/flip passes=1 center-mag=+0/+0/1878.375/1/-45/3.8857805\ 8618804789e-016 params=90/90/2.003/0/0/0/-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================================== ;