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: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> 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 20.0 PAR-FORMULA 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 20.0 PAR-FORMULA FILE==================================