From: JimMuth@aol.com Subject: (fractint) C-FOTD 01-06-01 (A Fractal Jewel [8]) Date: 01 Jun 2001 09:30:46 EDT Classic FOTD -- June 01, 2001 (Rating 8) Fractal visionaries and enthusiasts: Today's image, the first of the month of June, is a real jewel. So I named it "A Fractal Jewel". A jewel of such magnitude certainly deserves some recognition, which I feel that the rating of a superior 8 supplies. To achieve the effect, I uncorked the outside=fmod option, which, though it is somewhat difficult to use, can produce some stunning effects with certain fractals. The good-old MandelbrotMix4 formula did the footwork as it iterated the expression -3.5Z^(0.7)+3.5Z^(-0.7)+(1/C). This expression draws a parent fractal that at first appears to be a total failure, since it consists of no more than a few tiny patches of color on a black background. Today's scene lies along the shoreline of a mis-shapen bud at the edge of one of these tiny patches. All good things take time, and today's image is no exception. The parameter file takes 22-1/2 minutes to render. But unlike so many natural processes, the rendering of today's image can be bypassed by visiting Paul's web site at: or Scott's site at: and downloading the already-rendered image from there. The fractal weather today was near perfect but bland, with hazy sun and a temperature of 72F (22C). The fractal cats, who are habitually bland, approved blandly. As for me, I've got to get going on more profitable things. Until next time, take care, and row your boat down the stream. Life is but a dream. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ A_Fractal_Jewel { ; time=0:22:37.10--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+0.01550644258372/-0.0139521159263/4.35\ 8122e+009/1/-160 params=-3.5/0.7/3.5/-0.7/0/5000 float=y maxiter=3600 inside=0 proximity=0.5 outside=fmod periodicity=10 colors=000000000QMUPOWPQZPS`PUaTVdWXf_YicZkf_niapl\ cqpetsfvvhyzjzzkzzlzznzzozzoxzpuzqtzqqzsoztlztkzui\ zvfzvdzxazy`zyZzzWzzUzzTzyQzvPytOvqMtpLqnKnkJkiIih\ GfeFdcEa`D__BccIfeMjhTljYnjanjfnjjnjonjtnjxnjznjzj\ lzfnzdoy`pvYqtVssRtpOunLvkIxiEyfBzeFxdIudKtdMqdPoc\ RncUkcWicZhc`eacdaeaah_ajZalW`oU`qT`tQ`vP`uQ_tR_tT\ ZsUZsVYqWYqYWpZWo_Wo`VnaVncUldUleTkfTjhTjiRijRikQh\ lQhnPfoPfpPdnOakO`iMZfMWeMVcLT`LRZLPWKMVKLTJJQJGOJ\ FLIDKIBII9FG6DG5AF39F06F04E02E00E02I02L32O53R83V93\ YB4`E4dG4fJ5jL5nM5pL8oLAoLDoKEnKGnKJnKKnJMlJPlJQlJ\ TlIVkIWkIZkI`kGajGdjGfjGhjGhlehldhkUcKL_JK_KKZKKYK\ JYKJWLJVLIULIUMITMGRMGRMGQOFPOFOOFOPEMPELPELPGPKIR\ GJVBKY8L`3Mc0Q_2UY4YV6`T9dPBhMEkKGoIJsELvBOz9Qz6Tz\ 4UzKczZlzluzjpzhlzehzcdy`_xZWvWRuUOtRJsPFqOBlLGhJL\ cIPZFUUDYPBaK9fF6jA5o53s00x00z00y00x00v00u00t00s22\ q22p32o32n32l42k42j53i53h } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 02-06-01 (Fractal Feathers [8]) Date: 02 Jun 2001 11:08:04 -0400 (EDT) Classic FOTD -- June 02, 2001 (Rating 8) Fractal visionaries and enthusiasts: Though it's not a work day, I seem to have a hundred tasks that need to be done. So this FOTD will have to be hasty. Today's fractal is the second consecutive one to rate an 8. Either I'm becoming a better fractalist, or more likely, I'm having a string of good luck. Actually, most of the worth of today's image lies in the color palette. A minute or two trying to find better colors will show this. To produce an image (not this particular image, which I was not yet aware of), I took Z^1.333 and subtracted Z^1.618 from it before adding C. I named the picture "Fractal Feathers" because the pattern reminds me of feathers. I rated it an 8 because I like it. But pictures speak louder than words, and the way to see the picture is to run the parameter file and wait 25 minutes, or to give Paul and Scott a chance to render and post the image, and then download it in one minute. The image file will be available for downloading at the web sites: and: The fractal weather was cloudy with heavy rain and all the unpleasant things that accompany heavy rain. The cats complained and the basement took on water, etc. The temperature of 64F (18C) was irrelevant. I'll return with more fractal stuff in about 12 hours. Until then, take care, and stay sharp. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Fractal_Feathers { ; time=0:24:50.13--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=ident passes=1 center-mag=+5.32964478731053700/+0.155415801184001\ 90/2.561639e+007/1/135 params=1/1.333/-1/1.618/0/0 float=y maxiter=3200 inside=0 logmap=460 periodicity=10 colors=0000D60D60E60E60F60F60G60G41H41H41I41I41J41\ J41K41L31M31N31O31P31Q31R30S01T11U31V61W73XA3YC3ZF\ 3_G4`J4`L4aO4bP6cS6dT6eX6fY7g_7ha7ic7jf9kg9lj9ml9n\ oAopApsAquArxCsyCuzCvzCvzAxzAxzAxzAxzAxzAxzAxz9xz9\ xz9xz9xz9xz9xz9xz7xz7xz7xz7xz7xz7xz7xz6xz6xz6xz6xz\ 6xz6xz6xz4xz4xz4xz4xz4xz4xz4xz4xz7vzAuzCuzFszIsxJr\ uMrrPpoQpmTojVogYmd`maal_dlXgjTijQliPmiMpgJsgGufDx\ fAzd7zd4zc1zc0za0za0za0z`0z`0z`0z`0y_0x_0x_0v_0vY0\ uY0uY0sX0rX0rX0pX0pV0oU0oT0mS0lR0lQ0jP0jO0iN0iM0gL\ 0fK0fJ0dI0dH0cG0cF0aE0`E0`E0_E0_E0YE0YF0XG0YI0XK0V\ K0TK1SJ1QJ3PJ4OI4QI6VI7aG9jG9sFAzFCzDCzDDvDFoCFgCG\ TAIM9JF6JD3L90M30M10O00P00P00Q00S00T00T00V00X00Y00\ a00f00j00o00s40xA0zG0zM0zQ0zS0zS0zS0zS0zT0zT0zT0zT\ 0zT0zV0zV0zV0zV0z_0zd0zj0zp0zv0zv0zv0zv0zv0zv0zv0z\ v0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0\ zv0zv0zv0zv0zv0zv0zv0zv0z } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Thaddaeus Parker Subject: RE: (fractint) C-FOTD 02-06-01 (Fractal Feathers [8]) Date: 02 Jun 2001 09:54:03 -0700 Jim: I would name it Peacock, but still the same it is a fantastic rendering. Keep it up. Love the antics of the Fractal Cats, got one myself and know exactly how they react to the weather and such. Keep up the good work. Thaddaeus Parker San Diego CA ICQ# 3304633 -----Original Message----- [mailto:owner-fractint@lists.xmission.com]On Behalf Of Jim Muth Sent: Saturday, June 02, 2001 8:08 AM Cc: philofractal@lists.fractalus.com Classic FOTD -- June 02, 2001 (Rating 8) Fractal visionaries and enthusiasts: Though it's not a work day, I seem to have a hundred tasks that need to be done. So this FOTD will have to be hasty. Today's fractal is the second consecutive one to rate an 8. Either I'm becoming a better fractalist, or more likely, I'm having a string of good luck. Actually, most of the worth of today's image lies in the color palette. A minute or two trying to find better colors will show this. To produce an image (not this particular image, which I was not yet aware of), I took Z^1.333 and subtracted Z^1.618 from it before adding C. I named the picture "Fractal Feathers" because the pattern reminds me of feathers. I rated it an 8 because I like it. But pictures speak louder than words, and the way to see the picture is to run the parameter file and wait 25 minutes, or to give Paul and Scott a chance to render and post the image, and then download it in one minute. The image file will be available for downloading at the web sites: and: The fractal weather was cloudy with heavy rain and all the unpleasant things that accompany heavy rain. The cats complained and the basement took on water, etc. The temperature of 64F (18C) was irrelevant. I'll return with more fractal stuff in about 12 hours. Until then, take care, and stay sharp. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Fractal_Feathers { ; time=0:24:50.13--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=ident passes=1 center-mag=+5.32964478731053700/+0.155415801184001\ 90/2.561639e+007/1/135 params=1/1.333/-1/1.618/0/0 float=y maxiter=3200 inside=0 logmap=460 periodicity=10 colors=0000D60D60E60E60F60F60G60G41H41H41I41I41J41\ J41K41L31M31N31O31P31Q31R30S01T11U31V61W73XA3YC3ZF\ 3_G4`J4`L4aO4bP6cS6dT6eX6fY7g_7ha7ic7jf9kg9lj9ml9n\ oAopApsAquArxCsyCuzCvzCvzAxzAxzAxzAxzAxzAxzAxz9xz9\ xz9xz9xz9xz9xz9xz7xz7xz7xz7xz7xz7xz7xz6xz6xz6xz6xz\ 6xz6xz6xz4xz4xz4xz4xz4xz4xz4xz4xz7vzAuzCuzFszIsxJr\ uMrrPpoQpmTojVogYmd`maal_dlXgjTijQliPmiMpgJsgGufDx\ fAzd7zd4zc1zc0za0za0za0z`0z`0z`0z`0y_0x_0x_0v_0vY0\ uY0uY0sX0rX0rX0pX0pV0oU0oT0mS0lR0lQ0jP0jO0iN0iM0gL\ 0fK0fJ0dI0dH0cG0cF0aE0`E0`E0_E0_E0YE0YF0XG0YI0XK0V\ K0TK1SJ1QJ3PJ4OI4QI6VI7aG9jG9sFAzFCzDCzDDvDFoCFgCG\ TAIM9JF6JD3L90M30M10O00P00P00Q00S00T00T00V00X00Y00\ a00f00j00o00s40xA0zG0zM0zQ0zS0zS0zS0zS0zT0zT0zT0zT\ 0zT0zV0zV0zV0zV0z_0zd0zj0zp0zv0zv0zv0zv0zv0zv0zv0z\ v0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0zv0\ zv0zv0zv0zv0zv0zv0zv0zv0z } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Morgan L. Owens" Subject: (fractint) Re: [philofractal] C-FOTD 02-06-01 (Fractal Feathers [8]) Date: 03 Jun 2001 12:00:58 +1200 At 03:08 03/06/2001, Jim Muth wrote: >The fractal weather was cloudy with heavy rain and all the >unpleasant things that accompany heavy rain. The cats >complained and the basement took on water, etc. The temperature >of 64F (18C) was irrelevant. Eeh, you're really going to have to do something about that basement. Next thing, you'll have your house rotting out from under you. Morgan L. Owens "Just be glad your house isn't made of polystyrene." Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 03-06-01 (Julia mandelbrot [6]) Date: 02 Jun 2001 23:03:29 -0400 (EDT) Classic FOTD -- June 03, 2001 (Rating 6) Fractal visionaries and enthusiasts: Three FOTD images in a row, all rating an 8, is too much to hope for. So I had no hopes of producing another 8-rated image when I sat down for my daily search for fractals this evening. Then, when the day's effort yielded only a 6-rated image, I was not disappointed. But a rating of 6 is still above average, and well worth the 7 minutes required to render the scene from the parameter file. The formula that created the image is totally whimsical -- 1.3Z^(1.3)+0.13Z^(-13)+(1/C). I found the parameters by turning off my thinking (quite an easy task) and letting my fingers roam where they wished. The name "Julia Mandelbrot" came to me as I studied the picture. The pattern around the central midget is that of julia sets located in the East Valley area of Mandelbrot midgets, yet the midget itself is pure Mandel stuff. The combination inspired the name. The rating of 6 is honest, since the image has too many imperfections for a higher rating. Other than running the parameter file, an alternate way of viewing the image is to download the GIF file from the Web at: or at: The fractal weather today was variably cloudy but so far dry, with a temperature of 77F (25C). The fractal cats must have approved, for they spent much time in the yard, trying to stay out of trouble. Now it's my turn to try to stay out of trouble. It's hard to imagine how I could get into trouble watching a junky old sci-fi movie, so that's what I'll do. Until next time, take care, and take two fractals at bedtime for a good night's sleep. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Julia_Mandelbrot { ; time=0:06:47.82--SF5 on a p200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-2.485951411296012/+0.00097323098757938\ /3.036155e+007/1/102.5 params=1.3/1.3/0.13/-13/0/0 float=y maxiter=850 inside=0 logmap=-137 periodicity=10 sound=off colors=000F00J00L00S80WB0_F0cJ0gM1kQ5oU8sZCudHwjLy\ pPzvSzzWozVhvV`qUUjUMdUF_S7US0PS0RR0SQ0UP0VO0WM0YL\ 0ZK0_K0`J0bH0cG0dF0eE555555555555555555656757858A5\ AB5BH7APAARCCSFESGLWHKYKUZKUZLR_MR`MS`OFbOEcPCcPC_\ RBWSASV8PW7MZ7J_6Fb5Bc48d4Ac7BbBCbFE`JF`MF_PGZSHZW\ JY_KYcLWgLWiMZcO_ZPbUPcOQeJRgERh8QjAQmAPrAPOAPdAOc\ AOcAObAM`AM`AM_AL_ALZALZALZALYBLWBMWBMVBMUBMUCHeOC\ sZBqY8nW7kU6iS4gR2dQ1bO0_M0YL0VK0UL0UL0UM0UM1UO2UO\ 4UO4UP5UQ6UQ7UR7URAYQC`PFdPHhOKkOMoMOrMkHznKzoMzrO\ zsQzvSzwUzyWzzZzz_zzbzzczzeswgjribnjUjkMYnGLoAY00Z\ 50ZB5_HB_MG_SM`ZR`cYbibbohbtmWveRv_MvUGvMBvG6vA1v4\ 8tHFmLbeOk_QrSSrLVrFYrGVrGUrHSrHRrJQrJPrJOrKMrKLrL\ KrLJrMHkMGeMF_PFRUFRZFRcFRhFRmERrEQrEQrEQrEQrEQrEQ\ rEQrEPrEPrEPrEPrEPrEMrEPrEOrEOrEOrEOrEOrEOrEMrEMrE\ MrEMrEMrEMrEor0nr0nr0nr0mr0mr0mr0mr0kr0kr0kr0kr0jr\ 0jr0jr0ir0ir0ir0ir0hr0hr0 } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Andrew Coppin" Subject: Re: (fractint) C-FOTD 31-05-01 (Fractal Fractles [7]) Date: 03 Jun 2001 09:01:10 -0000 >From: Jim Muth >Reply-To: fractint@lists.xmission.com >To: fractint@lists.xmission.com >CC: philofractal@lists.fractalus.com >Subject: (fractint) C-FOTD 31-05-01 (Fractal Fractles [7]) >Date: Thu, 31 May 2001 10:59:51 -0400 (EDT) > > >Classic FOTD -- May 31, 2001 (Rating 7) Erm... Wow! That is one *trippy* image! I really really dIg those colours! Very nice work, Jim! Thanks. Andrew. PS. One day (when I have web space) I think I may start doing a fractal of the month... _________________________________________________________________________ Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com. Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Andrew Coppin" Subject: Re: (fractint) C-FOTD 03-06-01 (Julia mandelbrot [6]) Date: 03 Jun 2001 09:06:06 -0000 >From: Jim Muth >Reply-To: fractint@lists.xmission.com >To: fractint@lists.xmission.com >CC: philofractal@lists.fractalus.com >Subject: (fractint) C-FOTD 03-06-01 (Julia mandelbrot [6]) >Date: Sat, 2 Jun 2001 23:03:29 -0400 (EDT) > > >Classic FOTD -- June 03, 2001 (Rating 6) I'd rate this one higher than that, perhaps 7, on account of the elegant simplicity of the image and the very nice, harmonious colours. Thanks. Andrew. _________________________________________________________________________ Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com. Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) Re: Wet basement Date: 03 Jun 2001 09:55:03 -0400 (EDT) At 12:00 PM 6/3/01 +1200, Morgan Owens wrote: >Eeh, you're really going to have to do something about that >basement. Next thing, you'll have your house rotting out from >under you. As long as it's not brain rot, I can live with it. The brick house is in pretty good shape -- foundation sitting on solid rock -- and the water comes in only with very heavy rain, like the 10cm in 8 hours downpour we had Friday. To fix the problem, it would be necessary to dig up my yard and the garden of the elderly widow next door, which would upset not only the widow, but the cats as well. I have installed a sump pump, and that keeps the wetness down to a wet floor, so I guess I can live with the dampness a while -- just as long as some alien fractal fungus doesn't develop and start growing down there. Jim M. Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 04-06-01 (Cometary Impression [8]) Date: 04 Jun 2001 01:24:32 -0400 (EDT) Classic FOTD -- June 04, 2001 (Rating 8) Fractal visionaries and enthusiasts: The hot streak continues unabated. Not only is my fractal philosophy once again active, but my fractal images are active also. And all this action piled up on a day with so many mundane chores needing to be done. With today's image we're back up to a superior rating of 8. If the present trend continues, we may reach one of the very rare 9's, or possibly even an unheard-of 10 before long. To create today's fractal I entered the parameters for -2Z^(-1.1)-0.0002Z^(-11)+(1/C) into the MandelbrotMix4 formula, and let her rip. Today's scene lies at the end of a curving filament extending from a bud, very near the point where many filaments converge into a starlike object. I named the image "Cometary Impression" when I saw the eight elements around the midgets and imagined eight circling comets. The parameter file render time of 13 minutes is slow. The download is fast, and will be available shortly on the Web at: and at: The fractal weather today was variably cloudy and breezy. The wind and temperature of 75F (24C) kept the fractal cats indoors most of the day, frequently checking the door to see whether it was still breezy outside. Later in the afternoon the wind died down and the cats went out, scolding me for not turning down the wind earlier. The time is now after 1am -- most certainly time to shut down the fractal shoppe and throw the big switch. Until next time, take care, and have faith in your fractals. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ CometaryImpression { ; time=0:13:25.10--SF5 on a p200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-6.312863840603067/-8.071658728774926/1\ 80586.5/1/147.5 params=-2/-1.1/-0.0002/-11/0/300 float=y maxiter=1500 inside=0 logmap=261 periodicity=10 colors=000002000000000000000000zzz000zzc000zzI20Lm\ cP52UcUW97`99cCCgEEkGGmGIrILtLNyLPzWZrgekrkczrWzm`\ zkczggzejz`mzZozUtzSvzNzzLzzGzzEzz9zz7zz2zz0zz0zz0\ zz0zz0zz0zz0zt0zm0ze5zZ9zPGzINz9Uz2Zz0Pz0Gz07v00m0\ 0g02c05W07S09L0CG2E97G5CL5GP5LU5PZ5Uc5Wg5`k5eo5jt5\ my5zz7oz5`z2Lv25t00o00o00m00m00k00k00k00j00j20j20g\ 20g20e50e50e50c70c70c70`70`90Z90Z90ZC0WC0WC0ZE0WC0\ U90S90P70P52N55L27I09G0EG0GE0IC0L90P90S70U50W20Z00\ c00e00g00j00k00g00e00c00`00Z00W00S00P00N00L00I00G0\ 0000E00U00g27v2Cz5Iz5Nz5SzLLt`E`o7Ez00z00z00z00z00\ z00z00z00z00z00z02z07z09y0Et0Gr0Lo0Nk0Sj0Ue0Zc0`Z0\ eW0cP0gU0kZ0m`0re0vg7ykEzmIzrPztWzyczzjzzozztzzvzz\ vzzyzyyzvzvvzttzorzkozgmzcmzZkzUjzSgzNezIezEcz9`z5\ Zz0Wz0Pz0Wz7czGgzPmzZtzgyzozzvzzyzzzzzzzzzzzzzzzzz\ zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\ zzzzzzzzzzzzzzzzzzzzzzzzz } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 05-06-01 (A Raspy Old Midget [9]) Date: 04 Jun 2001 22:54:11 -0400 (EDT) Classic FOTD -- June 05, 2001 (Rating 9) Fractal visionaries and enthusiasts: Things have been going well here at fractal central lately. I'm in one of those periods where I seem to automatically pick the right places to look for those hard-to-find midgets, and equally important, good color palettes seem to appear almost by magic. I named today's image "A Raspy Old Midget". The eight rasp-like elements surrounding the midget inspired the name. Undecided as to what rating I might bestow upon the picture, I decided on a rather liberal 9. An 8-1/2 might be more accurate, but we need at least a few 9's in the archives. The formula -5(Z^(-1.15))-0.02(Z^(-11.5))+(1/C) drew the parent fractal, which is a rather interesting but oversized figure by itself, with a prominent fan-like element extending eastward from the origin. Today's midget lies on the south border of this fractal fan. The 11-minute parameter file is a bit slow. I advise visiting Paul's web site or Scott's site, and downloading the GIF image file from there. Paul's FOTD site can be found at: Scott's is at: The fractal weather today was virtually perfect, with sunny skies, a temperature of 77F (25C), and happy cats. And having found such a fine fractal, I'm happy too. Until next time, take care, and I wonder whether fractal seeds will grow. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ A_Raspy_Old_Midget { ; time=0:11:41.29--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.77096187026955700/-2.487403802992336\ 00/3.864825e+009/1/92.499/0.003 params=-5/-1.15/-0.02/-11.5/0/0 float=y maxiter=1200 inside=0 logmap=205 periodicity=10 colors=000zSuzTrzUuzVuzWuzXuzYvzZvz_vz`vzayzbyzcyz\ dyzeyzfzzgzzhzzizzjzzkzzlzzmzznzzozzpzzqzzrzzszytz\ yuzyvzvwzvzzuzzuzzuzzrzzrzzpzzpzzuzzpzznzzmzzkzzfz\ zezzcyyavy_rvWpuUmrRkpPfnMemKakI_kGWiEUfAPe8Oc7Ka5\ I_1EY0CY08W07U03R01P00O00M00K00M00K00K00I00I00I00G\ 00G01E13E53E75C87CC8AEAAGAAKC8ME8OG7PG7UI7WK5YM5aO\ 3cO3eP3iR1kU1mW0pW0rY0u_0ya0zc0zc0ze0zf0zi0zf0zi0z\ i0zk0zk0zm1zm5yn7vn8rpAppCnrGmrIiuKfuMeuOcvRavUYyW\ WyYUz_RzcOzeMzfKziIzkEznCzpAzr8zu5zz7zv7zu7zr7zp7y\ m7vk7ui7rf7nc7ma7k_7iY7eU7cR7aP7_O7YM8UI8RG8PE8OC8\ K88I78G58E38A0880870850810800800800500800A00E00G00\ K10M30P30R50U70Y80_A1cC3eC5iE7kG8nIApKCrMEvMGyOIzP\ KzRMzUOzWPzWPzYOzYOz_Oz_MzaMzaMzcKzcKzeKyeIyfIvfIv\ iGuiGrkGrkEpmEpmEnnCmnCmpCkpAkrAirAfu8fu8ev8ev7cy7\ ay7az5_z5cz3_z5Yz5Wz5Uz7Rz7Pz7Oz8Mz8Kz8IzAGzAEzACz\ CAzC8zC7zE5zE3zE1zG0zG0zG } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 06-06-01 (Golden Chariot [8]) Date: 06 Jun 2001 08:38:24 -0400 (EDT) Classic FOTD -- June 06, 2001 (Rating 8) Fractal visionaries and enthusiasts: The formula -0.6(Z^(-1.3))-0.006(Z^(-3))+(1/C), when calculated with an escape radius of 900, draws an hourglass-shaped figure composed of almost total chaos. Two out-zooms are needed to see the entire figure, which is a Mandeloid sitting on its nose, with the tiny hourglass lying in the center of the main bay. The figure is unusual in that the filaments extending from the buds end in rings instead of simply petering out in ever-smaller details. Today's midget lies deep within the end-ring of the main southwest filament of the main bud, which in today's fractal lies on the south shore of the main bay. I named the picture "Golden Chariot" because when I saw it, I had an immediate impression of a chariot wheel. I briefly considered a name such as "Chariot of the Gods", but a similar name has already been used in a well-known book. The image rates an 8. The rating might be a bit liberal, but it's a strange scene in a fractal with even stranger scenes. I'll present some of these other scenes in the next few FOTD's, the first of which will be tomorrow's. A good feature of today's image is that it renders in only 3-1/2 minutes. In an hour or so, the image may also be seen by visiting the web sites of Paul and Scott. The URL's of those sites are: and: The fractal weather, which everyone is waiting for with bated breath, was average today. The partly cloudy skies were partly sunny during the daylight hours and partly starry after dark. The temperature of 80F (26.5C) was average, and the fractal cats had an average day on the porch and in the yard. And it's the start of another average day for me. I've got about the average amount of work to finish before I can relax and turn to tomorrow's FOTD, which is all goes well, will be far above average. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Golden_Chariot { ; time=0:03:30.04--SF5 on a p200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+7.63095735774279/-41.9285774906314/3.1\ 96408e+008/1/162.5 params=-0.6/-1.3/-0.006/-3/0/800 float=y maxiter=380 inside=0 logmap=65 periodicity=10 colors=000045D45I45N45Q49V6GYCLbHSgMYkRdpVks`pxevz\ jzzozztzztzztzzpzwpmpjviepdZiYPb0PYlXTseOwjJzmEzr9\ zw5zw5zw5zw5zw9trEloLbkQffWgcai`gkVllSsnPx_NzLMz6K\ z0Kz2FzEAzQ5za4zT4zL4zD4z45z05z2Az8DzDIzIMzLPzXUng\ XEkZLlZQpZVqZ_tZdwZix`pz`uz`yz`zz`zz`zzezzjzzoxzrq\ zwlzwgzwbxw_fwWOwT6wQ0wO0wQ0rQ0cS0ZS0YT1XV4`W8eWBh\ YEk_Io_LtaOwbRwdVwdYwfawgdwgfwkgwliwniwpkwqlwunwvn\ sxpoyqkzqgzpczp_zpXznUznUynUylUylUylUxkUxkUxkUxiUv\ iZvicvghvgmufqufuufuufwudwsdwsdwsbwsbwqbwqawqawqaw\ p_wp_wp_wpYwnYwnYwnXwnXwlXwlVwlVwlVwkTwkTwkTwkRwiR\ wiRwiQwiQwgQwgOwgOwgOwfNwfNwfNwfLwdLwdLwgJwiJwkJwl\ IwnIwpIwqGwsGwu_wvYwuYwu_wuawucwsewsgusiuskuqmuqow\ qqwqswpuwpwwpwwpwwnwwnwwnywnzwlzwlzwlzwlzwkzwkzwkz\ wkzwkzwizwizwizwizwgzwgzwgzwgzwfzwfzwfzwfzwdzwdzwd\ zwdzwbzwbzwbzwbzwazwazwazwazwkzwizwizwixwixwixwixw\ iwwiwwiwwitwitwitwitwiR45 } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Ricardo M. Forno" Subject: (fractint) Algorithmic Composer Date: 06 Jun 2001 11:49:39 -0300 There are some algorithmic composers over the Web. I think the one I wrote compares favorably to most of them. You can download it from: http://www.geocities.com/rmforno/index.html It does not use a fractal algorithm, but anyway I think this mail is on topic. Hope you like the music. Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Jim Muth Subject: (fractint) C-FOTD 07-06-01 (Charged Midget [8]) Date: 07 Jun 2001 10:23:24 -0400 (EDT) Classic FOTD -- June 07, 2001 (Rating 8) Fractal visionaries and enthusiasts: When a package or roll of photographic film is opened carelessly in a darkroom in dry weather, black tree-like figures sometimes appear on the film when it is developed. The figures are caused by sparks of static electricity that are created as the layers are peeled apart too roughly. These figures have a name, which eludes me at the moment, though I think it begins with the letter 'l'. If I could remember that word, I would have the perfect title for today's image, which reminds me of the sheets of ruined film that I have thrown away over the years. But, being unable to remember, I have named the image "Charged Midget". The scene of today's picture lies in the same fractal as yesterday's, though in a totally different part. Whereas yesterday's scene was located at the end of a filament, today's is located deep in an almost featureless valley on the WNW shore of the main bay. And speaking of fractal bays, this one has that strange hourglass-shaped chunk of total chaos sitting in the middle of it. Usually, such totally chaotic areas hold nothing of interest, but this one is worth at least a cursory investigation. If I find something, it will be tomorrow's FOTD; if not, I'll turn somewhere else. Today's image rates an 8, keeping the run of exceptional images unbroken. The total lack of reds is intentional. Sometimes fewer colors can have greater impact than more. The image was rendered with the 'passes equal b' option, which I rarely use, but which sometimes is the fastest of all. With today's 3-3/4 minute parameter file, the difference of a few seconds is inconsequential however. As always, the GIF image will soon be available on the web for those who would rather not run the parameter file. The URL's are: and: The fractal weather today featured a temperature of 81F (27C), changeable skies, with nearly every cloud type imaginable, and a light sprinkle at noon to wet those going to lunch. It also wet the fractal cats, who scowled as they retreated indoors until the rain stopped. And it's now time for me to retreat to the task of accomplishing the day's work. Until next time, take care, and will we ever find out who is right? Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Charged_Midget { ; time=0:03:42.60--SF5 on a p200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=b center-mag=-14.16793212758799/-11.36345363439138/2\ 6147.93/1/19.999 params=-0.6/-1.3/-0.006/-3/0/800 float=y maxiter=1500 inside=255 logmap=43 periodicity=0 colors=0000Mp0Ho0Co09m09l69jD9iH9gM9jPElSJmXOo`Trc\ Xsgaujgvmkyrqzuvzxyzrxzmsxinudhr`coWZlRVjMRgHNdDKa\ 8GZ3CW09U09S0AR0AP0BO0AM0BL0BJ0AH0CG0CE0FD0HB0IA0L\ 80M60N50P30P10Q00Q00U00V00X00_00`00b00d00e00e00e00\ e00e03e06e0Ae0De0Ge0Je0Me0Pe0Se0We0Ze0ae0de0ge0je0\ le0cj0Wm0Op0Gs08v10y10z30z30z60zA0yB0xE0vG0sJ0rM0p\ O0oR0mS0lW0iZ0g`0ec1dd3bg3`i6_j8_jA_lD_lE_mG_mHYoL\ YoMYpOYpPYrSYrUXsWXsXXu`XuaXvcXvdXvgYuiYsjYslYrmYp\ oYppYorYosYmu_lv_lx_jy_jz_iz_gz_gz_fz_fziaxrZsrWos\ SlsPdizXXzR9zSCzSEzSFzSHzSJzSKzSMzSNzSPzSQzSSzSUzS\ VzUXzUYzUYzU_zW`zW`zWbzWbzXdzXezXezXgzZizZizZjzZjz\ cizgizlizpizuizygzzgzzgzzgzzgzzezzdzzdzzbzzbzz`zz_\ zz_zzYzzYzzXzzVzzVzzUzzUzzSzzSzzQzzPzzNzzMzzKzzJzz\ HzzFzzEzzCzzAzz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9z\ z9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9zz9\ zzAzzAzzAzzKzzSzz_zziz000 } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Multiple Bogeys" Subject: (fractint) Some intriguing stuff. Date: 07 Jun 2001 22:16:27 -0400 ------=_NextPart_001_0000_01C0EF9F.7F321100 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable These formulas make it possible to investigate the escape behavior of the= Henon map: x -> a - by + x^2 y -> x both for the case where x and y are real, and the case where they may be = complex. The formulas whose names end in 'J' are Julia-like. Henon_J1 will vary y = over the screen while fixing x. The parameters fix a, b, and z. Henon-J2 will vary real x and real y over the screen; setting t= heir imaginary parts to zero is accomplished by setting p3 to zero. If p1 and p2 are real (e.g. t= he classic 1.4 and -0.3) the basin of attraction of the familiar real Hen= on map appears. The formulas whose names end in 'M' are Mandelbrot-like. Henon-M3 is most= Mandelbrot-like; b varies over the screen while a and initial x and y (a= ll complex) are set with parameters. No known choice of initial x and y a= cts like the "critical" value 0 for the classic Mandelbrot set, so the result always seems to look pertur= bed. (I looked long and hard with the evolver/explorer to find a good app= roximation to a "critical" value; nothing seems to cut it, perhaps becaus= e there is no "critical" value for such multi-recurrence maps as this. The Jacobian has= nonzero determinant if b is not zero -- if b is zero, both x and y do the usual Mandelbrot set with x =3D 0, y =3D anything "cr= itical". However, the imaginary parts of initial x and y should be zero t= o get "Mandelbrot-like" it seems.) Henon-M2 varies real a and real b over= the screen. The result resembles Lyaounov space! Henon-M4 is like Henon-= M2, but it automatically composites the maps obtained for nine choices of= initial real x and y. These iterate the real Henon map -- imaginary x, y= , a, and b are identically zero. Henon-M5 is complex instead of real, lik= e Henon-M3, but is a composite like Henon-M4 using nine values of initial= real x and y. Henon-M2, M3, and M5 are best viewed with inside =3D per, = outside =3D numb, outside =3D 0, maxiter > 255, and a color map that make= s colors 0 and 255 different and varies a lot in the early colors (try th= e default VGA palette modified to invert the last 32 colors or so). A zoo= m down M5's seahorse valley is interesting. The "halo" of color 255 about= the Set in M5 and in the lower right quadrant of M4 represents the regim= es where strange attractors can occur in the system. The set of points fo= r which actual strange attractors occur is buried among "islands" of orde= r and pockets of escape -- the former look like Lyapunov swallows in M4 a= nd presumably would appear to be mini Mandelbrots in M5 if "critical" val= ues for initial real x and y could be found. WARNING: The below probably has the dreaded "3D" disease, no thanks to MS= N Exploder. If it does, IIRC someone published a utility to strip the enc= rufting^H^H^H^H^H^H^Hoding on here a year or so ago. Henon_M2 { ; p1 is a point in the Henon map. Screen coordinates are param= eters 'a' and 'b'. ; Escape pixels escaped radius 1000. Try non-standard inside o= ptions with ; periodicity checking enabled. a =3D real(pixel), b =3D imag(pixel), z =3D p1: z =3D a - b*imag(z) - sqr(real(z)) + (0,1)*real(z), |z| < 1000000 } Henon_M3 { ; Henon map on complex numbers. ; p1 and p2 are a point in the Henon map. Screen coordinates a= re parameter 'b'. ; Escape pixels escaped radius 1000. Try non-standard inside o= ptions with ; periodicity checking DISabled. a =3D 1.4, b =3D pixel, z =3D p1, w =3D p2: z1 =3D a - b*w - sqr(z) w =3D z z =3D z1, lastsqr+|w| < 1000000 } Henon_M4 { ; Screen coordinates are parameters 'a' and 'b'. ; Escape pixels escaped radius 1000. Try non-standard inside o= ptions with ; periodicity checking enabled. Composite view. a =3D real(pixel), b =3D imag(pixel), z =3D 0, r =3D 0, done =3D 0: z =3D a - b*imag(z) - sqr(real(z)) + (0,1)*real(z), IF (|z| > 1000000) IF (r =3D=3D 0) z =3D 1 r =3D 1 ELSEIF (r =3D=3D 1) z =3D (0,1) r =3D 2 ELSEIF (r =3D=3D 2) z =3D -1 r =3D 3 ELSEIF (r =3D=3D 3) z =3D -(0,1) r =3D 4 ELSEIF (r =3D=3D 4) z =3D 1+(0,1) r =3D 5 ELSEIF (r =3D=3D 5) z =3D 1-(0,1) r =3D 6 ELSEIF (r =3D=3D 6) z =3D -1+(0,1) r =3D 7 ELSEIF (r =3D=3D 7) z =3D -1-(0,1) r =3D 8 ELSE done =3D 1 ENDIF ENDIF done =3D=3D 0 } Henon_M5 { ; Henon map on complex numbers. Screen coordinates are paramet= er 'b'. ; Escape pixels escaped radius 1000. Try non-standard inside o= ptions with ; periodicity checking DISabled. Composite view. a =3D 1.4, b =3D pixel, z =3D 0, w =3D 0, r =3D 0, done =3D 0: z1 =3D a - b*w - sqr(z) w =3D z z =3D z1, IF (lastsqr+|w| > 1000000) IF (r =3D=3D 0) z =3D 1 w =3D 0 r =3D 1 ELSEIF (r =3D=3D 1) z =3D 0 w =3D 1 r =3D 2 ELSEIF (r =3D=3D 2) z =3D -1 w =3D 0 r =3D 3 ELSEIF (r =3D=3D 3) z =3D 0 w =3D -1 r =3D 4 ELSEIF (r =3D=3D 4) z =3D 1 w =3D 1 r =3D 5 ELSEIF (r =3D=3D 5) z =3D 1 w =3D -1 r =3D 6 ELSEIF (r =3D=3D 6) z =3D -1 w =3D 1 r =3D 7 ELSEIF (r =3D=3D 7) z =3D -1 w =3D -1 r =3D 8 ELSE done =3D 1 ENDIF ENDIF done =3D=3D 0 } Henon_J1 { ; Henon map on complex numbers. ; p1 and p2 are a and b. Screen coordinates are y. p3 is x. ; Escape pixels escaped radius 1000. Try non-standard inside o= ptions with ; periodicity checking DISabled. a =3D p1, b =3D p2, z =3D p3, w =3D pixel: z1 =3D a - b*w - sqr(z) w =3D z z =3D z1, lastsqr+|w| < 1000000 } Henon_J2 { ; Henon map on complex numbers. ; p1 and p2 are a and b. Screen coordinates are real(x), real(= y). p3 is imag(x), imag(y). ; Escape pixels escaped radius 1000. Try non-standard inside o= ptions with ; periodicity checking DISabled. a =3D p1, b =3D p2, z =3D real(pixel) + (0,1)*real(p3), w =3D imag(pixe= l) + (0,1)*imag(p3): z1 =3D a - b*w - sqr(z) w =3D z z =3D z1, lastsqr+|w| < 1000000 }
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------=_NextPart_001_0000_01C0EF9F.7F321100 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable


These = formulas make it possible to investigate the escape behavior of the Henon= map:
 
x -> a - by + x^2
y ->= ; x
 
both for the case where x and y are real= , and the case where they may be complex.
 
Th= e formulas whose names end in 'J' are Julia-like. Henon_J1 will vary = ;y over the screen while fixing x. The parameters fix a,
b, an= d z. Henon-J2 will vary real x and real y over the screen; sett= ing their imaginary parts to
zero is accomplished by setting p= 3 to zero. If p1 and p2 are real (e.g. the classic 1.4 and -0.3) the basi= n of attraction of the familiar real Henon map appears.
 =
The formulas whose names end in 'M' are Mandelbrot-like. Heno= n-M3 is most Mandelbrot-like; b varies over the screen while a and initia= l x and y (all complex) are set with parameters. No known choice of initi= al x and y acts like the "critical" value 0
for the classic Ma= ndelbrot set, so the result always seems to look perturbed. (I looked lon= g and hard with the evolver/explorer to find a good approximation to a "c= ritical" value; nothing seems to cut it, perhaps because there is no
"critical" value for such multi-recurrence maps as this. The = Jacobian has nonzero determinant if b is not zero -- if b is zero,
=
both x and y do the usual Mandelbrot set with x =3D 0, y =3D anythin= g "critical". However, the imaginary parts of initial x and y should= be zero to get "Mandelbrot-like" it seems.) Henon-M2 varies real a and r= eal b over the screen. The result resembles Lyaounov space! Henon-M4 is l= ike Henon-M2, but it automatically composites the maps obtained for nine = choices of initial real x and y. These iterate the real Henon map -- imag= inary x, y, a, and b are identically zero. Henon-M5 is complex instead of= real, like Henon-M3, but is a composite like Henon-M4 using nine values = of initial real x and y. Henon-M2, M3, and M5 are best viewed with inside= =3D per, outside =3D numb, outside =3D 0, maxiter > 255, and a color = map that makes colors 0 and 255 different and varies a lot in the ea= rly colors (try the default VGA palette modified to invert the last 32 co= lors or so). A zoom down M5's seahorse valley is interesting. The "halo" = of color 255 about the Set in M5 and in the lower right quadrant of M4 re= presents the regimes where strange attractors can occur in the system. Th= e set of points for which actual strange attractors occur is buried among= "islands" of order and pockets of escape -- the former look like Lyapuno= v swallows in M4 and presumably would appear to be mini Mandelbrots in M5= if "critical" values for initial real x and y could be found.
 
WARNING: The below probably has the dreaded "3D" disea= se, no thanks to MSN Exploder. If it does, IIRC someone published a utili= ty to strip the encrufting^H^H^H^H^H^H^Hoding on here a year or so ago.
 
Henon_M2 { ; p1 is a point in the Henon map. = Screen coordinates are parameters 'a' and 'b'.
    = ;       ; Escape pixels escaped radius 1000= . Try non-standard inside options with
     &= nbsp;     ; periodicity checking enabled.
  a= =3D real(pixel), b =3D imag(pixel), z =3D p1:
  z =3D a - b*imag= (z) - sqr(real(z)) + (0,1)*real(z),
  |z| < 1000000
}
=
 
Henon_M3 { ; Henon map on complex numbers.
&nb= sp;          ; p1 and p2 are= a point in the Henon map. Screen coordinates are parameter 'b'.
 = ;          ; Escape pixels e= scaped radius 1000. Try non-standard inside options with
  &= nbsp;        ; periodicity checking DI= Sabled.
  a =3D 1.4, b =3D pixel, z =3D p1, w =3D p2:
  z= 1 =3D a - b*w - sqr(z)
  w =3D z
  z =3D z1,
  la= stsqr+|w| < 1000000
}
 
Henon_M4 { ; Scr= een coordinates are parameters 'a' and 'b'.
    &n= bsp;      ; Escape pixels escaped radius 1000. T= ry non-standard inside options with
     &nbs= p;     ; periodicity checking enabled. Composite view= .
  a =3D real(pixel), b =3D imag(pixel), z =3D 0, r =3D 0, done = =3D 0:
  z =3D a - b*imag(z) - sqr(real(z)) + (0,1)*real(z),
&= nbsp; IF (|z| > 1000000)
    IF (r =3D=3D 0)
&nbs= p;     z =3D 1
      r =3D= 1
    ELSEIF (r =3D=3D 1)
    &= nbsp; z =3D (0,1)
      r =3D 2
 &nbs= p;  ELSEIF (r =3D=3D 2)
      z =3D -1      r =3D 3
    ELSEIF (r = =3D=3D 3)
      z =3D -(0,1)
  &= nbsp;   r =3D 4
    ELSEIF (r =3D=3D 4)
&n= bsp;     z =3D 1+(0,1)
    &nb= sp; r =3D 5
    ELSEIF (r =3D=3D 5)
  &nbs= p;   z =3D 1-(0,1)
      r =3D 6    ELSEIF (r =3D=3D 6)
     = z =3D -1+(0,1)
      r =3D 7
  =   ELSEIF (r =3D=3D 7)
      z =3D -1-(0,= 1)
      r =3D 8
    ELSE      done =3D 1
    ENDIF  ENDIF
  done =3D=3D 0
}
 
= Henon_M5 { ; Henon map on complex numbers. Screen coordinates are paramet= er 'b'.
           ;= Escape pixels escaped radius 1000. Try non-standard inside options with<= BR>           ; periodi= city checking DISabled. Composite view.
  a =3D 1.4, b =3D pixel,= z =3D 0, w =3D 0, r =3D 0, done =3D 0:
  z1 =3D a - b*w - sqr(z)=
  w =3D z
  z =3D z1,
  IF (lastsqr+|w| > 100= 0000)
    IF (r =3D=3D 0)
    &n= bsp; z =3D 1
      w =3D 0
  &nb= sp;   r =3D 1
    ELSEIF (r =3D=3D 1)
&nbs= p;     z =3D 0
      w =3D= 1
      r =3D 2
    ELSEIF= (r =3D=3D 2)
      z =3D -1
  &= nbsp;   w =3D 0
      r =3D 3
&n= bsp;   ELSEIF (r =3D=3D 3)
      z = =3D 0
      w =3D -1
   &nb= sp;  r =3D 4
    ELSEIF (r =3D=3D 4)
 &nbs= p;    z =3D 1
      w =3D 1      r =3D 5
    ELSEIF (r =3D= =3D 5)
      z =3D 1
   &nb= sp;  w =3D -1
      r =3D 6
 &nb= sp;  ELSEIF (r =3D=3D 6)
      z =3D -1<= BR>      w =3D 1
    &nbs= p; r =3D 7
    ELSEIF (r =3D=3D 7)
   = ;   z =3D -1
      w =3D -1
&nbs= p;     r =3D 8
    ELSE
 &n= bsp;    done =3D 1
    ENDIF
  E= NDIF
  done =3D=3D 0
}
 
Henon_J1 {= ; Henon map on complex numbers.
      &= nbsp;    ; p1 and p2 are a and b. Screen coordinates are y= . p3 is x.
          = ; ; Escape pixels escaped radius 1000. Try non-standard inside options wi= th
           ; peri= odicity checking DISabled.
  a =3D p1, b =3D p2, z =3D p3, w =3D = pixel:
  z1 =3D a - b*w - sqr(z)
  w =3D z
  z =3D= z1,
  lastsqr+|w| < 1000000
}
 
Henon_J2 { ; Henon map on complex numbers.
    &n= bsp;      ; p1 and p2 are a and b. Screen coordi= nates are real(x), real(y). p3 is imag(x), imag(y).
   =         ; Escape pixels escaped radius= 1000. Try non-standard inside options with
    &n= bsp;      ; periodicity checking DISabled.
&n= bsp; a =3D p1, b =3D p2, z =3D real(pixel) + (0,1)*real(p3), w =3D imag(p= ixel) + (0,1)*imag(p3):
  z1 =3D a - b*w - sqr(z)
  w =3D= z
  z =3D z1,
  lastsqr+|w| < 1000000
}
 


Get Your Private,= Free E-mail from MSN Hotmail at http:= //www.hotmail.com.

------=_NextPart_001_0000_01C0EF9F.7F321100-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Thierry B." Subject: Re: (fractint) Some intriguing stuff. Date: 08 Jun 2001 07:25:49 +0000 > These formulas make it possible to investigate the escape behavior of t= he Henon map: =20 > x -> a - by + x^2 > y -> x I've also a few research on the mappin of H=E9non diagram. http://la.buvette.org/fractales/map_henon.euh Sorry, this is only a Fortran source, but I can write some explanation in english this weekend. =20 --=20 Thierry, 42++ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: Guy Marson Subject: Re: (fractint) Some intriguing stuff. Date: 08 Jun 2001 09:57:44 +0200 At 07:25 08/06/01 +0000, you wrote: >> These formulas make it possible to investigate the escape behavior of the Henon map: >=20 >> x -> a - by + x^2 >> y -> x > > I've also a few research on the mappin of H=E9non diagram. > http://la.buvette.org/fractales/map_henon.euh > Sorry, this is only a Fortran source, but I can write some > explanation in english this weekend. mais pas dans la buvette, avec des Jupiler s.v.p. (hickkk..)=20 > >=20 >--=20 >Thierry, 42++ > cheers,=20 guy 47+ Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: JimMuth@aol.com Subject: (fractint) C-FOTD 08-06-01 (Too Much Fractal [7]) Date: 08 Jun 2001 09:15:51 EDT Classic FOTD -- June 08, 2001 (Rating 7) Fractal visionaries and enthusiasts: It seems strange to denigrate a fractal with a rating two points above average, but that's what I feel I must do with today's image. Sometimes, even in the world of fractals, it's possible to have too much of a good thing. Today's image is an example of such fractal excess. The image simply goes too far with too little. The midget at the center is too small to act as a center of attention, leaving the surrounding decorations with nothing to decorate. And the color is excessive. Vibrant color can be spectacular when it is done properly. When it's not done right, the result is boring gaudiness. Today's image just doesn't give me that "this color is right" feeling. Oh, the scene has a surface glitter that brings its rating up to a 7, but it lacks the depth that could bring a rating of 8 or 9. I named the image "Too Much Fractal" in response to my feelings about it. Today's scene lies in another valley, directly across the bay from yesterday's, though it is some distance back from the shoreline, and at a considerable greater depth. It is actually located at the center of a figure-8 ring, which the area is filled with. The 4-1/2 minute render time is marginal, making it the viewers choice whether to go online and download the GIF image from: or from: The fractal weather today started with rain, but the rain ended in mid-morning, and the sun returned in mid-afternoon, sending the temperature up to 75F (24C). The fractal cats celebrated by venturing cautiously into the still-wet grass. It's now time to get busy on other things, so until next time, take care and check the fractal on the cover of the latest issue of "Skeptic" magazine. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Too_Much_Fractal { ; time=0:04:25.66--SF5 on a P200 reset=2001 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=b center-mag=+10.09705566264443000/-15.5438271495980\ 4000/1.007928e+011/1/166.953/-0.502 params=-0.6/-1.3/-0.006/-3/0/1000 float=y maxiter=1500 inside=255 logmap=133 periodicity=10 colors=0800DL0DL0DO0DR0DU0DX0D_0Db0De0Dh0Dk0Dn0Dq6\ FtGGwPDzZIzhQzrXzzdzzlzztzzwzwzwYjlnZrzQyzGzZIf0IN\ 0QW0V`0bf0hl0pt0vz0wz1wz1wz8wzGwzNrzWjzbbzjVzrOzzG\ zzDzzDzzDzzDzzDzzDvzDnzDfyD`tDUpGLlIGhM8dO1`R0YT0W\ V0ZX0bX0dX0hX0jX0nX1rX4tX8yXAzXEzXHzXJzwJzwPywUvwY\ tw`vwYywWywRzwPzwLzwJzwGzwEzwCzwGzwHztJzpNzjPzdRz`\ WzVYzQZzM`zOdzQfzRjzTlzVnzXrzZtz`yzbzzbzzfzzhzzlzz\ nzzrzztzzwzzwzlnzWbnETZ0IL0D80F40I14M0EQ0LT0UX0bb0\ jf0rj0zn0zr0zv0zt0zr0zp0zp0z`0jO0PD06D0ED0JD0RD2YG\ 6dMAjQErVHyZLzbNvfPphPhlRbnRYrRPtUJwUEwU6wW1wW0wW0\ rN0bG0O80T00Z00d00j00n00r00v06w0Cw0Jw0Rw0Yw0dw0lw0\ rw0fw4YwUNwrEwzCwzCwzCvzCtzArzApzAnzAlzAlzHvzNwzWw\ z`wzfwzbwz`wzZwzWnzUdzRVzYRzbOzfKzlGzpDtvDpzDjzDdz\ DZzDUzDNzDJzDLzDNrIPfTRWbULlWAvY0wZ0w`0wb0wb0wW2wN\ JfH`OApD4rD4tD6vD6CwfGwdHwbLwZNwYRwWUwUZwWdwYhwZnw\ `rwbywdzwflwnYwtHwz2wz0D0 } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE================================== Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Multiple Bogeys" Subject: (fractint) Hairy Newton Date: 08 Jun 2001 21:29:56 -0400 ------=_NextPart_001_0000_01C0F062.2AD5F4A0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Yesterday evening I set out to find an interesting family of Newton-based= Mandelbrot mappings. The result was the collection of formulae below. You'll note the lack of 3D disease -- false alarm. All my posts to anothe= r listserv had it, but I now think it must be that listserv rather than M= SN Exploder. Or rather, some interaction between the two (since some post= s from that listserv don't have it), like the weird "email laser" that ha= ppened with this listserv last week (lots of peoples' messages were being= duplicated, but for some reason mine were showing up in sets of five or = six!)...Anyone who's an expert on listservs care to speculate further on = what might be going on? One thing is clear: recent mail software and list= servs have unnecessary complexity, and while we like complexity in our fr= actals, we could do without chaos erupting in the mail system we depend o= n to communicate here... The hrynewt_j and hrynewt_m formulae iterate Newton's method for p(z) =3D= (z^n - 1)(z^3 - az - 1). Both n and a are parameters, as is the toleranc= e, an inverse bailout radius about the roots of p. The Mandelbrot variant= has a vary over the screen while initial z is zero; this is a critical p= oint but not a root of p for n real and greater than 2. You can plug in o= ther values of n -- arbitrary negative or even complex values -- but won'= t generally be able to find minibrots unless n has a positive real part g= reater than two. If n is not an integer, there will be branch cuts in bot= h the Mandelbrot and the Julia variants. The hrynewtnnn_j and _m formulae are optimized versions with specific val= ues for n, mostly small positive integers. They avoid a slow arbitrary ex= ponentiation, and for the smaller values of n re-use powers that are used= on both sides of the polynomial or its derivative. The hrynewt2_m formul= a also has the feature of using a critical point for initial z, instead o= f zero (which is *not* a critical point for n =3D 2). The result is a pro= per Mandelbrot view, but it has a branch cut due to a square root in the = calculation of the critical point, which is a-dependent. The branch cut h= as been intentionally manipulated to put it in a fairly unobtrusive place= , but can't be eliminated; the full Mandelbrot for this one lives on a tw= o-layer Riemann sheet like that of the square root function. The hrynewtnnn_m formulae also use an (XAXIS) symmetry declaration. (The = generic hrynewt_m can't use this without trashing the output for non-real= values of n.) Observations: * Certain choices of n produce three-fold-symmetric Mandelbrot sets. Find= out which! * Mangled and occasionally also intact Mandelbrots can be extracted when = n is "strange" but has a real part greater than 2. * You get radial petals with n real, concentric patterns with n imaginary= , and logarithmic spirals with complex n; the ratio of real to imaginary parts determines whether the spiral is steep (n clos= e to real) or shallow (n close to imaginary). * The Mandelbrots are always quadratic -- for real n > 2, the critical po= int at zero is nondegenerate, and the critical point pair for hrynewt2_m is degenerate only at one specific value of a. The formula file begins with an extensive comment that details the mathem= atical constructions that informed their design. comment { We want a Newton's method with a large number of basins, most of which = are fixed and predictable. This is accomplished by choosing a polynomial function to solve compose= d of two factors, one with many fixed roots, the other with a few mobile ones: p(z) =3D (z^n-1)(z^3-az-1). The Newton iteration is: z -> r(z) where r(z) =3D z - p(z)/p'(z) =3D (zp'(z) - p(z))/p'(z) We easily discover p'(z) to be p'(z) =3D (z^n-1)(3z^2-a) + (nz^(n-1))(z^3-az-1) =3D (3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a so (zp'(z) - p(z)) =3D (z^n-1)(3z^3-az)+(nz^n)(z^3-az-1) - (z^n-1)(z^3-az-= 1) =3D (z^n-1)(2z^3+1)+(nz^n)(z^3-az-1) =3D (2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1 and r(z) =3D ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)/((3+n)z^(n+2)= - a(n+1)z^n - nz^(n-1) - 3z^2 + a) Using the quotient rule the numerator of r'(z) is ((3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a)((n+2)(n+3)z^(n+2) - an= (n+1)z^n - n(n-1)z^(n-1) - 6z^2) - ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)((n+2)(n+3)z^(n+1) - = an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) which factors into ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) and ((3+n)z^(n+3) - a(n+1)z^(n+1) - nz^n - 3z^3 + az) - ((2+n)z^(n+3) - anz= ^(n+1) - (n-1)z^n - 2z^3 - 1) which simplifies to z^(n+3) - az^(n+1) - z^n - z^3 + az + 1 Note that p(z) =3D z^(n+3) - az^(n+1) - z^n - z^3 + az + 1. Thus the critical points of r(z) are the roots of p(z) and the roots of q(z) :=3D ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) These latter are the "interesting" critical points, as the other critic= al points of r(z) are all superattracting. Note that q(z) is divisible by z, so 0 is an "interesting" critical poi= nt of r(z), for n not one of 2, 1, or -1. This is the critical point used in the below hrynewt_m formulas except = for hrynewt2_m. For n =3D 2, q(z)/2 =3D 10z^3 - 3(a+1)z - 1 Put z =3D y + (a+1)/10y to get q(z)/2 =3D y^6 - y^3/10 + (a+1)^3/1000 so 2y^3 =3D 1/10 +/-sqrt(1/100 - 4(a+1)^3/1000), y =3D ((1/10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) and z =3D ((1/10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) + (a+1)/((1/10 +/= -sqrt(1/100 - 4(a+1)^3)/1000)/2)^(1/3) } hrynewt_j { ; p1 is Julia parameter, p2 is exponent n, p3 is tolerance (i= f 0, will act like 0.001). ; SLOW. Use predefined hrynewtnnn_j where possible. z =3D pixel, a =3D p1, n =3D p2, n1 =3D n - 1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D z^n1 zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + n*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt_m { ; p2 is exponent n, p3 is tolerance (if 0, will act like 0.00= 1). ; SLOW. Use predefined hrynewtnnn_m where possible. z =3D 0, a =3D pixel, n =3D p2, n1 =3D n - 1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D z^n1 zn =3D z*zn1 zno =3D (zn - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + n*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt2_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 2. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z2 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 2*z*zzz z =3D z - pz/ppz, |pz| > r } hrynewt2_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 2. a =3D pixel, ap1 =3D a + 1, IF((real(ap1) >=3D 0) || ((abs(real(ap1))*(3^(0.5))) < abs(imag(ap1)))) t =3D ((0.1 + (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3), ELSE t =3D ((0.1 - (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3), ENDIF z =3D t + 0.1*ap1/t, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z2 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 2*z*zzz z =3D z - pz/ppz, |pz| > r } hrynewt3_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 3. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z3 - 1) zzz =3D zno - a*z tz2 =3D 3*z2 pz =3D zno*zzz ppz =3D zno*(tz2 - a) + tz2*zzz z =3D z - pz/ppz, |pz| > r } hrynewt3_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 3. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z3 - 1) zzz =3D zno - a*z tz2 =3D 3*z2 pz =3D zno*zzz ppz =3D zno*(tz2 - a) + tz2*zzz z =3D z - pz/ppz, |pz| > r } hrynewt4_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 4. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z*z3 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 4*z3*zzz z =3D z - pz/ppz, |pz| > r } hrynewt4_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 4. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z*z3 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 4*z3*zzz z =3D z - pz/ppz, |pz| > r } hrynewt5_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 5. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(z2) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 5*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt5_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 5. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(z2) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 5*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt17_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act li= ke 0.001). ; n =3D 17. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(z2))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 17*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt17_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 17. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(z2))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 17*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt33_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act li= ke 0.001). ; n =3D 33. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(sqr(z2)))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 33*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt33_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 33. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(sqr(z2)))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 33*zn1*zzz z =3D z - pz/ppz, |pz| > r }
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Yesterday even= ing I set out to find an interesting family of Newton-based Mandelbrot ma= ppings. The result was the collection of formulae below.
 = ;
You'll note the lack of 3D disease -- false alarm. All my po= sts to another listserv had it, but I now think it must be that listserv = rather than MSN Exploder. Or rather, some interaction between the two (si= nce some posts from that listserv don't have it), like the weird "em= ail laser" that happened with this listserv last week (lots of peoples' m= essages were being duplicated, but for some reason mine were showing up i= n sets of five or six!)...Anyone who's an expert on listservs care to spe= culate further on what might be going on? One thing is clear: recent mail= software and listservs have unnecessary complexity, and while we like co= mplexity in our fractals, we could do without chaos erupting in the mail = system we depend on to communicate here...
 
T= he hrynewt_j and hrynewt_m formulae iterate Newton's method for p(z) =3D = (z^n - 1)(z^3 - az - 1). Both n and a are parameters, as is the tolerance= , an inverse bailout radius about the roots of p. The Mandelbrot variant = has a vary over the screen while initial z is zero; this is a critical po= int but not a root of p for n real and greater than 2. You can = plug in other values of n -- arbitrary negative or even complex values --= but won't generally be able to find minibrots unless n has a positive re= al part greater than two. If n is not an integer, there will be branch cu= ts in both the Mandelbrot and the Julia variants.
 
=
The hrynewtnnn_j and _m formulae are optimized versions with specif= ic values for n, mostly small positive integers. They avoid a slow arbitr= ary exponentiation, and for the smaller values of n re-use powers that ar= e used on both sides of the polynomial or its derivative. The hrynewt2_m = formula also has the feature of using a critical point for initial z, ins= tead of zero (which is *not* a critical point for n =3D 2). The result is= a proper Mandelbrot view, but it has a branch cut due to a square root i= n the calculation of the critical point, which is a-dependent. The branch= cut has been intentionally manipulated to put it in a fairly unobtrusive= place, but can't be eliminated; the full Mandelbrot for this one lives o= n a two-layer Riemann sheet like that of the square root function.
=
 
The hrynewtnnn_m formulae also use an (XAXIS) symm= etry declaration. (The generic hrynewt_m can't use this without trashing = the output for non-real values of n.)
 
Observ= ations:
* Certain choices of n produce three-fold-symmetric Ma= ndelbrot sets. Find out which!
* Mangled and occasionally also= intact Mandelbrots can be extracted when n is "strange" but has a real p= art greater than 2.
* You get radial petals with n real, conce= ntric patterns with n imaginary, and logarithmic spirals with complex n; = the ratio of
   real to imaginary parts determines w= hether the spiral is steep (n close to real) or shallow (n close to imagi= nary).
* The Mandelbrots are always quadratic -- for real n &g= t; 2, the critical point at zero is nondegenerate, and the critical point=
   pair for hrynewt2_m is degenerate only at o= ne specific value of a.
 
The formula file beg= ins with an extensive comment that details the mathematical constructions= that informed their design.
 
comment {
&n= bsp; We want a Newton's method with a large number of basins, most of whi= ch are fixed and predictable.
  This is accomplished by choosing = a polynomial function to solve composed of two factors, one with many fix= ed roots,
  the other with a few mobile ones:
  p(z) =3D = (z^n-1)(z^3-az-1).
  The Newton iteration is:
  z -> r= (z)
  where
  r(z) =3D z - p(z)/p'(z)
  &nbs= p;    =3D (zp'(z) - p(z))/p'(z)
  We easily discov= er p'(z) to be
  p'(z) =3D (z^n-1)(3z^2-a) + (nz^(n-1))(z^3-az-1)=
        =3D (3+n)z^(n+2) - a(n+1)z= ^n - nz^(n-1) - 3z^2 + a
  so
  (zp'(z) - p(z)) =3D (z^n-= 1)(3z^3-az)+(nz^n)(z^3-az-1) - (z^n-1)(z^3-az-1)
   &nb= sp;           &nbs= p;  =3D (z^n-1)(2z^3+1)+(nz^n)(z^3-az-1)
    =             &= nbsp; =3D (2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1
  and  r(z) =3D ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)/((3+n)z= ^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a)
  Using the quotient ru= le the numerator of r'(z) is
  ((3+n)z^(n+2) - a(n+1)z^n - nz^(n-= 1) - 3z^2 + a)((n+2)(n+3)z^(n+2) - an(n+1)z^n - n(n-1)z^(n-1) - 6z^2) -    ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)((n+= 2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z)
  which fac= tors into
  ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) -= 6z)
  and
  ((3+n)z^(n+3) - a(n+1)z^(n+1) - nz^n - 3z^3 = + az) - ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)
  which = simplifies to
  z^(n+3) - az^(n+1) - z^n - z^3 + az + 1
 = Note that p(z) =3D z^(n+3) - az^(n+1) - z^n - z^3 + az + 1.
  Th= us the critical points of r(z) are the roots of p(z) and the roots of
=   q(z) :=3D ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z= )
  These latter are the "interesting" critical points, as the ot= her critical points of r(z) are all superattracting.
  Note that = q(z) is divisible by z, so 0 is an "interesting" critical point of r(z), = for n not one of 2, 1, or -1.
  This is the critical point used i= n the below hrynewt_m formulas except for hrynewt2_m. For n =3D 2,
&nb= sp; q(z)/2 =3D 10z^3 - 3(a+1)z - 1
  Put z =3D y + (a+1)/10y to g= et
  q(z)/2 =3D y^6 - y^3/10 + (a+1)^3/1000
  so
 = ; 2y^3 =3D 1/10 +/-sqrt(1/100 - 4(a+1)^3/1000),
  y =3D ((1/10 +/= -sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3)
  and
  z =3D ((1/= 10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) + (a+1)/((1/10 +/-sqrt(1/100 = - 4(a+1)^3)/1000)/2)^(1/3)
}
 
hrynewt_j { = ; p1 is Julia parameter, p2 is exponent n, p3 is tolerance (if 0, will ac= t like 0.001).
         &= nbsp;  ; SLOW. Use predefined hrynewtnnn_j where possible.
 = z =3D pixel, a =3D p1, n =3D p2, n1 =3D n - 1, r =3D p3
  IF(r =3D= =3D 0)
    r =3D 0.001
  ENDIF
  :
&= nbsp; z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D z^n1
 = ; zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D z= no*zzz
  ppz =3D zno*(3*z2 - a) + n*zn1*zzz
  z =3D z - p= z/ppz,
  |pz| > r
}
 
hrynewt_m = { ; p2 is exponent n, p3 is tolerance (if 0, will act like 0.001).
&nb= sp;           ; SLOW. U= se predefined hrynewtnnn_m where possible.
  z =3D 0, a =3D pixel= , n =3D p2, n1 =3D n - 1, r =3D p3
  IF(r =3D=3D 0)
 &nbs= p;  r =3D 0.001
  ENDIF
  :
  z2 =3D sqr(z)<= BR>  z3 =3D z*z2
  zn1 =3D z^n1
  zn =3D z*zn1
&n= bsp; zno =3D (zn - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D z= no*zzz
  ppz =3D zno*(3*z2 - a) + n*zn1*zzz
  z =3D z - p= z/ppz,
  |pz| > r
}
 
hrynewt2_j= { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.001).<= BR>           &nbs= p; ; n =3D 2.
  z =3D pixel, a =3D p1, r =3D p3
  IF(r =3D= =3D 0)
    r =3D 0.001
  ENDIF
  :
&= nbsp; z2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z2 - 1)
&= nbsp; zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  ppz =3D z= no*(3*z2 - a) + 2*z*zzz
  z =3D z - pz/ppz,
  |pz| > r=
}
 
hrynewt2_m (XAXIS) { ; p3 is tolerance= (if 0, will act like 0.001).
      &nbs= p;      ; n =3D 2.
  a =3D pixel,
&nb= sp; ap1 =3D a + 1,
  IF((real(ap1) >=3D 0) || ((abs(real(ap1))= *(3^(0.5))) < abs(imag(ap1))))
    t =3D ((0.1 + (0.= 01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3),
  ELSE
  &= nbsp; t =3D ((0.1 - (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3),
 = ; ENDIF
  z =3D t + 0.1*ap1/t, r =3D p3
  IF(r =3D=3D 0)<= BR>    r =3D 0.001
  ENDIF
  :
  z= 2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z2 - 1)
  z= zz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  ppz =3D zno*(3*z= 2 - a) + 2*z*zzz
  z =3D z - pz/ppz,
  |pz| > r
}
 
hrynewt3_j { ; p1 is Julia parameter, p3 is t= olerance (if 0, will act like 0.001).
     &n= bsp;       ; n =3D 3.
  z =3D pixel= , a =3D p1, r =3D p3
  IF(r =3D=3D 0)
    r =3D= 0.001
  ENDIF
  :
  z2 =3D sqr(z)
  z3 =3D= z*z2
  zno =3D (z3 - 1)
  zzz =3D zno - a*z
  tz= 2 =3D 3*z2
  pz =3D zno*zzz
  ppz =3D zno*(tz2 - a) + tz2= *zzz
  z =3D z - pz/ppz,
  |pz| > r
}
&n= bsp;
hrynewt3_m (XAXIS) { ; p3 is tolerance (if 0, will act li= ke 0.001).
          = ;   ; n =3D 3.
  z =3D 0, a =3D pixel, r =3D p3
&nbs= p; IF(r =3D=3D 0)
    r =3D 0.001
  ENDIF
&n= bsp; :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z= 3 - 1)
  zzz =3D zno - a*z
  tz2 =3D 3*z2
  pz =3D= zno*zzz
  ppz =3D zno*(tz2 - a) + tz2*zzz
  z =3D z - pz= /ppz,
  |pz| > r
}
 
hrynewt4_j = { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.001).            = ; ; n =3D 4.
  z =3D pixel, a =3D p1, r =3D p3
  IF(r =3D= =3D 0)
    r =3D 0.001
  ENDIF
  :
&= nbsp; z2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z*z3 - 1)  zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  ppz =3D= zno*(3*z2 - a) + 4*z3*zzz
  z =3D z - pz/ppz,
  |pz| >= ; r
}
 
hrynewt4_m (XAXIS) { ; p3 is tolera= nce (if 0, will act like 0.001).
      &= nbsp;      ; n =3D 4.
  z =3D 0, a =3D p= ixel, r =3D p3
  IF(r =3D=3D 0)
    r =3D 0.001=
  ENDIF
  :
  z2 =3D sqr(z)
  z3 =3D z*z= 2
  zno =3D (z*z3 - 1)
  zzz =3D (z3 - a*z - 1)
 = pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 4*z3*zzz
  z =3D= z - pz/ppz,
  |pz| > r
}
 
hryn= ewt5_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.= 001).
          &nbs= p;  ; n =3D 5.
  z =3D pixel, a =3D p1, r =3D p3
  I= F(r =3D=3D 0)
    r =3D 0.001
  ENDIF
 = :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D sqr(z2= )
  zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
 = ; pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 5*zn1*zzz
  z= =3D z - pz/ppz,
  |pz| > r
}
 
= hrynewt5_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001).
&n= bsp;            ; = n =3D 5.
  z =3D 0, a =3D pixel, r =3D p3
  IF(r =3D=3D 0= )
    r =3D 0.001
  ENDIF
  :
 = z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D sqr(z2)
  = zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D zno= *zzz
  ppz =3D zno*(3*z2 - a) + 5*zn1*zzz
  z =3D z - pz/= ppz,
  |pz| > r
}
 
hrynewt17_j = { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.001).            = ;  ; n =3D 17.
  z =3D pixel, a =3D p1, r =3D p3
  I= F(r =3D=3D 0)
    r =3D 0.001
  ENDIF
 = :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D sqr(sq= r(sqr(z2)))
  zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1= )
  pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 17*zn1*zzz<= BR>  z =3D z - pz/ppz,
  |pz| > r
}
 <= /DIV>
hrynewt17_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0= .001).
          &nb= sp;   ; n =3D 17.
  z =3D 0, a =3D pixel, r =3D p3
&= nbsp; IF(r =3D=3D 0)
    r =3D 0.001
  ENDIF  :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D= sqr(sqr(sqr(z2)))
  zno =3D (z*zn1 - 1)
  zzz =3D (z3 - = a*z - 1)
  pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 17*z= n1*zzz
  z =3D z - pz/ppz,
  |pz| > r
}
=  
hrynewt33_j { ; p1 is Julia parameter, p3 is tolerance = (if 0, will act like 0.001).
       = ;       ; n =3D 33.
  z =3D pixel, = a =3D p1, r =3D p3
  IF(r =3D=3D 0)
    r =3D 0= .001
  ENDIF
  :
  z2 =3D sqr(z)
  z3 =3D= z*z2
  zn1 =3D sqr(sqr(sqr(sqr(z2))))
  zno =3D (z*zn1 -= 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  p= pz =3D zno*(3*z2 - a) + 33*zn1*zzz
  z =3D z - pz/ppz,
  = |pz| > r
}
 
hrynewt33_m (XAXIS) { ; p3 = is tolerance (if 0, will act like 0.001).
    &nbs= p;         ; n =3D 33.
  = z =3D 0, a =3D pixel, r =3D p3
  IF(r =3D=3D 0)
  &n= bsp; r =3D 0.001
  ENDIF
  :
  z2 =3D sqr(z)
&= nbsp; z3 =3D z*z2
  zn1 =3D sqr(sqr(sqr(sqr(z2))))
  zno = =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz=
  ppz =3D zno*(3*z2 - a) + 33*zn1*zzz
  z =3D z - pz/ppz= ,
  |pz| > r
}


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------=_NextPart_001_0000_01C0F062.2AD5F4A0-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: "Multiple Bogeys" Subject: (fractint) Hairy Newton Date: 08 Jun 2001 21:29:56 -0400 ------=_NextPart_001_0000_01C0F062.2AD5F4A0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Yesterday evening I set out to find an interesting family of Newton-based= Mandelbrot mappings. The result was the collection of formulae below. You'll note the lack of 3D disease -- false alarm. All my posts to anothe= r listserv had it, but I now think it must be that listserv rather than M= SN Exploder. Or rather, some interaction between the two (since some post= s from that listserv don't have it), like the weird "email laser" that ha= ppened with this listserv last week (lots of peoples' messages were being= duplicated, but for some reason mine were showing up in sets of five or = six!)...Anyone who's an expert on listservs care to speculate further on = what might be going on? One thing is clear: recent mail software and list= servs have unnecessary complexity, and while we like complexity in our fr= actals, we could do without chaos erupting in the mail system we depend o= n to communicate here... The hrynewt_j and hrynewt_m formulae iterate Newton's method for p(z) =3D= (z^n - 1)(z^3 - az - 1). Both n and a are parameters, as is the toleranc= e, an inverse bailout radius about the roots of p. The Mandelbrot variant= has a vary over the screen while initial z is zero; this is a critical p= oint but not a root of p for n real and greater than 2. You can plug in o= ther values of n -- arbitrary negative or even complex values -- but won'= t generally be able to find minibrots unless n has a positive real part g= reater than two. If n is not an integer, there will be branch cuts in bot= h the Mandelbrot and the Julia variants. The hrynewtnnn_j and _m formulae are optimized versions with specific val= ues for n, mostly small positive integers. They avoid a slow arbitrary ex= ponentiation, and for the smaller values of n re-use powers that are used= on both sides of the polynomial or its derivative. The hrynewt2_m formul= a also has the feature of using a critical point for initial z, instead o= f zero (which is *not* a critical point for n =3D 2). The result is a pro= per Mandelbrot view, but it has a branch cut due to a square root in the = calculation of the critical point, which is a-dependent. The branch cut h= as been intentionally manipulated to put it in a fairly unobtrusive place= , but can't be eliminated; the full Mandelbrot for this one lives on a tw= o-layer Riemann sheet like that of the square root function. The hrynewtnnn_m formulae also use an (XAXIS) symmetry declaration. (The = generic hrynewt_m can't use this without trashing the output for non-real= values of n.) Observations: * Certain choices of n produce three-fold-symmetric Mandelbrot sets. Find= out which! * Mangled and occasionally also intact Mandelbrots can be extracted when = n is "strange" but has a real part greater than 2. * You get radial petals with n real, concentric patterns with n imaginary= , and logarithmic spirals with complex n; the ratio of real to imaginary parts determines whether the spiral is steep (n clos= e to real) or shallow (n close to imaginary). * The Mandelbrots are always quadratic -- for real n > 2, the critical po= int at zero is nondegenerate, and the critical point pair for hrynewt2_m is degenerate only at one specific value of a. The formula file begins with an extensive comment that details the mathem= atical constructions that informed their design. comment { We want a Newton's method with a large number of basins, most of which = are fixed and predictable. This is accomplished by choosing a polynomial function to solve compose= d of two factors, one with many fixed roots, the other with a few mobile ones: p(z) =3D (z^n-1)(z^3-az-1). The Newton iteration is: z -> r(z) where r(z) =3D z - p(z)/p'(z) =3D (zp'(z) - p(z))/p'(z) We easily discover p'(z) to be p'(z) =3D (z^n-1)(3z^2-a) + (nz^(n-1))(z^3-az-1) =3D (3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a so (zp'(z) - p(z)) =3D (z^n-1)(3z^3-az)+(nz^n)(z^3-az-1) - (z^n-1)(z^3-az-= 1) =3D (z^n-1)(2z^3+1)+(nz^n)(z^3-az-1) =3D (2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1 and r(z) =3D ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)/((3+n)z^(n+2)= - a(n+1)z^n - nz^(n-1) - 3z^2 + a) Using the quotient rule the numerator of r'(z) is ((3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a)((n+2)(n+3)z^(n+2) - an= (n+1)z^n - n(n-1)z^(n-1) - 6z^2) - ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)((n+2)(n+3)z^(n+1) - = an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) which factors into ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) and ((3+n)z^(n+3) - a(n+1)z^(n+1) - nz^n - 3z^3 + az) - ((2+n)z^(n+3) - anz= ^(n+1) - (n-1)z^n - 2z^3 - 1) which simplifies to z^(n+3) - az^(n+1) - z^n - z^3 + az + 1 Note that p(z) =3D z^(n+3) - az^(n+1) - z^n - z^3 + az + 1. Thus the critical points of r(z) are the roots of p(z) and the roots of q(z) :=3D ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) These latter are the "interesting" critical points, as the other critic= al points of r(z) are all superattracting. Note that q(z) is divisible by z, so 0 is an "interesting" critical poi= nt of r(z), for n not one of 2, 1, or -1. This is the critical point used in the below hrynewt_m formulas except = for hrynewt2_m. For n =3D 2, q(z)/2 =3D 10z^3 - 3(a+1)z - 1 Put z =3D y + (a+1)/10y to get q(z)/2 =3D y^6 - y^3/10 + (a+1)^3/1000 so 2y^3 =3D 1/10 +/-sqrt(1/100 - 4(a+1)^3/1000), y =3D ((1/10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) and z =3D ((1/10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) + (a+1)/((1/10 +/= -sqrt(1/100 - 4(a+1)^3)/1000)/2)^(1/3) } hrynewt_j { ; p1 is Julia parameter, p2 is exponent n, p3 is tolerance (i= f 0, will act like 0.001). ; SLOW. Use predefined hrynewtnnn_j where possible. z =3D pixel, a =3D p1, n =3D p2, n1 =3D n - 1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D z^n1 zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + n*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt_m { ; p2 is exponent n, p3 is tolerance (if 0, will act like 0.00= 1). ; SLOW. Use predefined hrynewtnnn_m where possible. z =3D 0, a =3D pixel, n =3D p2, n1 =3D n - 1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D z^n1 zn =3D z*zn1 zno =3D (zn - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + n*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt2_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 2. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z2 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 2*z*zzz z =3D z - pz/ppz, |pz| > r } hrynewt2_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 2. a =3D pixel, ap1 =3D a + 1, IF((real(ap1) >=3D 0) || ((abs(real(ap1))*(3^(0.5))) < abs(imag(ap1)))) t =3D ((0.1 + (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3), ELSE t =3D ((0.1 - (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3), ENDIF z =3D t + 0.1*ap1/t, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z2 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 2*z*zzz z =3D z - pz/ppz, |pz| > r } hrynewt3_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 3. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z3 - 1) zzz =3D zno - a*z tz2 =3D 3*z2 pz =3D zno*zzz ppz =3D zno*(tz2 - a) + tz2*zzz z =3D z - pz/ppz, |pz| > r } hrynewt3_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 3. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z3 - 1) zzz =3D zno - a*z tz2 =3D 3*z2 pz =3D zno*zzz ppz =3D zno*(tz2 - a) + tz2*zzz z =3D z - pz/ppz, |pz| > r } hrynewt4_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 4. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z*z3 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 4*z3*zzz z =3D z - pz/ppz, |pz| > r } hrynewt4_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 4. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zno =3D (z*z3 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 4*z3*zzz z =3D z - pz/ppz, |pz| > r } hrynewt5_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act lik= e 0.001). ; n =3D 5. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(z2) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 5*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt5_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 5. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(z2) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 5*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt17_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act li= ke 0.001). ; n =3D 17. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(z2))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 17*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt17_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 17. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(z2))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 17*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt33_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act li= ke 0.001). ; n =3D 33. z =3D pixel, a =3D p1, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(sqr(z2)))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 33*zn1*zzz z =3D z - pz/ppz, |pz| > r } hrynewt33_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). ; n =3D 33. z =3D 0, a =3D pixel, r =3D p3 IF(r =3D=3D 0) r =3D 0.001 ENDIF : z2 =3D sqr(z) z3 =3D z*z2 zn1 =3D sqr(sqr(sqr(sqr(z2)))) zno =3D (z*zn1 - 1) zzz =3D (z3 - a*z - 1) pz =3D zno*zzz ppz =3D zno*(3*z2 - a) + 33*zn1*zzz z =3D z - pz/ppz, |pz| > r }
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------=_NextPart_001_0000_01C0F062.2AD5F4A0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Yesterday even= ing I set out to find an interesting family of Newton-based Mandelbrot ma= ppings. The result was the collection of formulae below.
 = ;
You'll note the lack of 3D disease -- false alarm. All my po= sts to another listserv had it, but I now think it must be that listserv = rather than MSN Exploder. Or rather, some interaction between the two (si= nce some posts from that listserv don't have it), like the weird "em= ail laser" that happened with this listserv last week (lots of peoples' m= essages were being duplicated, but for some reason mine were showing up i= n sets of five or six!)...Anyone who's an expert on listservs care to spe= culate further on what might be going on? One thing is clear: recent mail= software and listservs have unnecessary complexity, and while we like co= mplexity in our fractals, we could do without chaos erupting in the mail = system we depend on to communicate here...
 
T= he hrynewt_j and hrynewt_m formulae iterate Newton's method for p(z) =3D = (z^n - 1)(z^3 - az - 1). Both n and a are parameters, as is the tolerance= , an inverse bailout radius about the roots of p. The Mandelbrot variant = has a vary over the screen while initial z is zero; this is a critical po= int but not a root of p for n real and greater than 2. You can = plug in other values of n -- arbitrary negative or even complex values --= but won't generally be able to find minibrots unless n has a positive re= al part greater than two. If n is not an integer, there will be branch cu= ts in both the Mandelbrot and the Julia variants.
 
=
The hrynewtnnn_j and _m formulae are optimized versions with specif= ic values for n, mostly small positive integers. They avoid a slow arbitr= ary exponentiation, and for the smaller values of n re-use powers that ar= e used on both sides of the polynomial or its derivative. The hrynewt2_m = formula also has the feature of using a critical point for initial z, ins= tead of zero (which is *not* a critical point for n =3D 2). The result is= a proper Mandelbrot view, but it has a branch cut due to a square root i= n the calculation of the critical point, which is a-dependent. The branch= cut has been intentionally manipulated to put it in a fairly unobtrusive= place, but can't be eliminated; the full Mandelbrot for this one lives o= n a two-layer Riemann sheet like that of the square root function.
=
 
The hrynewtnnn_m formulae also use an (XAXIS) symm= etry declaration. (The generic hrynewt_m can't use this without trashing = the output for non-real values of n.)
 
Observ= ations:
* Certain choices of n produce three-fold-symmetric Ma= ndelbrot sets. Find out which!
* Mangled and occasionally also= intact Mandelbrots can be extracted when n is "strange" but has a real p= art greater than 2.
* You get radial petals with n real, conce= ntric patterns with n imaginary, and logarithmic spirals with complex n; = the ratio of
   real to imaginary parts determines w= hether the spiral is steep (n close to real) or shallow (n close to imagi= nary).
* The Mandelbrots are always quadratic -- for real n &g= t; 2, the critical point at zero is nondegenerate, and the critical point=
   pair for hrynewt2_m is degenerate only at o= ne specific value of a.
 
The formula file beg= ins with an extensive comment that details the mathematical constructions= that informed their design.
 
comment {
&n= bsp; We want a Newton's method with a large number of basins, most of whi= ch are fixed and predictable.
  This is accomplished by choosing = a polynomial function to solve composed of two factors, one with many fix= ed roots,
  the other with a few mobile ones:
  p(z) =3D = (z^n-1)(z^3-az-1).
  The Newton iteration is:
  z -> r= (z)
  where
  r(z) =3D z - p(z)/p'(z)
  &nbs= p;    =3D (zp'(z) - p(z))/p'(z)
  We easily discov= er p'(z) to be
  p'(z) =3D (z^n-1)(3z^2-a) + (nz^(n-1))(z^3-az-1)=
        =3D (3+n)z^(n+2) - a(n+1)z= ^n - nz^(n-1) - 3z^2 + a
  so
  (zp'(z) - p(z)) =3D (z^n-= 1)(3z^3-az)+(nz^n)(z^3-az-1) - (z^n-1)(z^3-az-1)
   &nb= sp;           &nbs= p;  =3D (z^n-1)(2z^3+1)+(nz^n)(z^3-az-1)
    =             &= nbsp; =3D (2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1
  and  r(z) =3D ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)/((3+n)z= ^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a)
  Using the quotient ru= le the numerator of r'(z) is
  ((3+n)z^(n+2) - a(n+1)z^n - nz^(n-= 1) - 3z^2 + a)((n+2)(n+3)z^(n+2) - an(n+1)z^n - n(n-1)z^(n-1) - 6z^2) -    ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)((n+= 2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z)
  which fac= tors into
  ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) -= 6z)
  and
  ((3+n)z^(n+3) - a(n+1)z^(n+1) - nz^n - 3z^3 = + az) - ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)
  which = simplifies to
  z^(n+3) - az^(n+1) - z^n - z^3 + az + 1
 = Note that p(z) =3D z^(n+3) - az^(n+1) - z^n - z^3 + az + 1.
  Th= us the critical points of r(z) are the roots of p(z) and the roots of
=   q(z) :=3D ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z= )
  These latter are the "interesting" critical points, as the ot= her critical points of r(z) are all superattracting.
  Note that = q(z) is divisible by z, so 0 is an "interesting" critical point of r(z), = for n not one of 2, 1, or -1.
  This is the critical point used i= n the below hrynewt_m formulas except for hrynewt2_m. For n =3D 2,
&nb= sp; q(z)/2 =3D 10z^3 - 3(a+1)z - 1
  Put z =3D y + (a+1)/10y to g= et
  q(z)/2 =3D y^6 - y^3/10 + (a+1)^3/1000
  so
 = ; 2y^3 =3D 1/10 +/-sqrt(1/100 - 4(a+1)^3/1000),
  y =3D ((1/10 +/= -sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3)
  and
  z =3D ((1/= 10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) + (a+1)/((1/10 +/-sqrt(1/100 = - 4(a+1)^3)/1000)/2)^(1/3)
}
 
hrynewt_j { = ; p1 is Julia parameter, p2 is exponent n, p3 is tolerance (if 0, will ac= t like 0.001).
         &= nbsp;  ; SLOW. Use predefined hrynewtnnn_j where possible.
 = z =3D pixel, a =3D p1, n =3D p2, n1 =3D n - 1, r =3D p3
  IF(r =3D= =3D 0)
    r =3D 0.001
  ENDIF
  :
&= nbsp; z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D z^n1
 = ; zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D z= no*zzz
  ppz =3D zno*(3*z2 - a) + n*zn1*zzz
  z =3D z - p= z/ppz,
  |pz| > r
}
 
hrynewt_m = { ; p2 is exponent n, p3 is tolerance (if 0, will act like 0.001).
&nb= sp;           ; SLOW. U= se predefined hrynewtnnn_m where possible.
  z =3D 0, a =3D pixel= , n =3D p2, n1 =3D n - 1, r =3D p3
  IF(r =3D=3D 0)
 &nbs= p;  r =3D 0.001
  ENDIF
  :
  z2 =3D sqr(z)<= BR>  z3 =3D z*z2
  zn1 =3D z^n1
  zn =3D z*zn1
&n= bsp; zno =3D (zn - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D z= no*zzz
  ppz =3D zno*(3*z2 - a) + n*zn1*zzz
  z =3D z - p= z/ppz,
  |pz| > r
}
 
hrynewt2_j= { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.001).<= BR>           &nbs= p; ; n =3D 2.
  z =3D pixel, a =3D p1, r =3D p3
  IF(r =3D= =3D 0)
    r =3D 0.001
  ENDIF
  :
&= nbsp; z2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z2 - 1)
&= nbsp; zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  ppz =3D z= no*(3*z2 - a) + 2*z*zzz
  z =3D z - pz/ppz,
  |pz| > r=
}
 
hrynewt2_m (XAXIS) { ; p3 is tolerance= (if 0, will act like 0.001).
      &nbs= p;      ; n =3D 2.
  a =3D pixel,
&nb= sp; ap1 =3D a + 1,
  IF((real(ap1) >=3D 0) || ((abs(real(ap1))= *(3^(0.5))) < abs(imag(ap1))))
    t =3D ((0.1 + (0.= 01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3),
  ELSE
  &= nbsp; t =3D ((0.1 - (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3),
 = ; ENDIF
  z =3D t + 0.1*ap1/t, r =3D p3
  IF(r =3D=3D 0)<= BR>    r =3D 0.001
  ENDIF
  :
  z= 2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z2 - 1)
  z= zz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  ppz =3D zno*(3*z= 2 - a) + 2*z*zzz
  z =3D z - pz/ppz,
  |pz| > r
}
 
hrynewt3_j { ; p1 is Julia parameter, p3 is t= olerance (if 0, will act like 0.001).
     &n= bsp;       ; n =3D 3.
  z =3D pixel= , a =3D p1, r =3D p3
  IF(r =3D=3D 0)
    r =3D= 0.001
  ENDIF
  :
  z2 =3D sqr(z)
  z3 =3D= z*z2
  zno =3D (z3 - 1)
  zzz =3D zno - a*z
  tz= 2 =3D 3*z2
  pz =3D zno*zzz
  ppz =3D zno*(tz2 - a) + tz2= *zzz
  z =3D z - pz/ppz,
  |pz| > r
}
&n= bsp;
hrynewt3_m (XAXIS) { ; p3 is tolerance (if 0, will act li= ke 0.001).
          = ;   ; n =3D 3.
  z =3D 0, a =3D pixel, r =3D p3
&nbs= p; IF(r =3D=3D 0)
    r =3D 0.001
  ENDIF
&n= bsp; :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z= 3 - 1)
  zzz =3D zno - a*z
  tz2 =3D 3*z2
  pz =3D= zno*zzz
  ppz =3D zno*(tz2 - a) + tz2*zzz
  z =3D z - pz= /ppz,
  |pz| > r
}
 
hrynewt4_j = { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.001).            = ; ; n =3D 4.
  z =3D pixel, a =3D p1, r =3D p3
  IF(r =3D= =3D 0)
    r =3D 0.001
  ENDIF
  :
&= nbsp; z2 =3D sqr(z)
  z3 =3D z*z2
  zno =3D (z*z3 - 1)  zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  ppz =3D= zno*(3*z2 - a) + 4*z3*zzz
  z =3D z - pz/ppz,
  |pz| >= ; r
}
 
hrynewt4_m (XAXIS) { ; p3 is tolera= nce (if 0, will act like 0.001).
      &= nbsp;      ; n =3D 4.
  z =3D 0, a =3D p= ixel, r =3D p3
  IF(r =3D=3D 0)
    r =3D 0.001=
  ENDIF
  :
  z2 =3D sqr(z)
  z3 =3D z*z= 2
  zno =3D (z*z3 - 1)
  zzz =3D (z3 - a*z - 1)
 = pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 4*z3*zzz
  z =3D= z - pz/ppz,
  |pz| > r
}
 
hryn= ewt5_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.= 001).
          &nbs= p;  ; n =3D 5.
  z =3D pixel, a =3D p1, r =3D p3
  I= F(r =3D=3D 0)
    r =3D 0.001
  ENDIF
 = :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D sqr(z2= )
  zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
 = ; pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 5*zn1*zzz
  z= =3D z - pz/ppz,
  |pz| > r
}
 
= hrynewt5_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001).
&n= bsp;            ; = n =3D 5.
  z =3D 0, a =3D pixel, r =3D p3
  IF(r =3D=3D 0= )
    r =3D 0.001
  ENDIF
  :
 = z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D sqr(z2)
  = zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D zno= *zzz
  ppz =3D zno*(3*z2 - a) + 5*zn1*zzz
  z =3D z - pz/= ppz,
  |pz| > r
}
 
hrynewt17_j = { ; p1 is Julia parameter, p3 is tolerance (if 0, will act like 0.001).            = ;  ; n =3D 17.
  z =3D pixel, a =3D p1, r =3D p3
  I= F(r =3D=3D 0)
    r =3D 0.001
  ENDIF
 = :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D sqr(sq= r(sqr(z2)))
  zno =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1= )
  pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 17*zn1*zzz<= BR>  z =3D z - pz/ppz,
  |pz| > r
}
 <= /DIV>
hrynewt17_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0= .001).
          &nb= sp;   ; n =3D 17.
  z =3D 0, a =3D pixel, r =3D p3
&= nbsp; IF(r =3D=3D 0)
    r =3D 0.001
  ENDIF  :
  z2 =3D sqr(z)
  z3 =3D z*z2
  zn1 =3D= sqr(sqr(sqr(z2)))
  zno =3D (z*zn1 - 1)
  zzz =3D (z3 - = a*z - 1)
  pz =3D zno*zzz
  ppz =3D zno*(3*z2 - a) + 17*z= n1*zzz
  z =3D z - pz/ppz,
  |pz| > r
}
=  
hrynewt33_j { ; p1 is Julia parameter, p3 is tolerance = (if 0, will act like 0.001).
       = ;       ; n =3D 33.
  z =3D pixel, = a =3D p1, r =3D p3
  IF(r =3D=3D 0)
    r =3D 0= .001
  ENDIF
  :
  z2 =3D sqr(z)
  z3 =3D= z*z2
  zn1 =3D sqr(sqr(sqr(sqr(z2))))
  zno =3D (z*zn1 -= 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz
  p= pz =3D zno*(3*z2 - a) + 33*zn1*zzz
  z =3D z - pz/ppz,
  = |pz| > r
}
 
hrynewt33_m (XAXIS) { ; p3 = is tolerance (if 0, will act like 0.001).
    &nbs= p;         ; n =3D 33.
  = z =3D 0, a =3D pixel, r =3D p3
  IF(r =3D=3D 0)
  &n= bsp; r =3D 0.001
  ENDIF
  :
  z2 =3D sqr(z)
&= nbsp; z3 =3D z*z2
  zn1 =3D sqr(sqr(sqr(sqr(z2))))
  zno = =3D (z*zn1 - 1)
  zzz =3D (z3 - a*z - 1)
  pz =3D zno*zzz=
  ppz =3D zno*(3*z2 - a) + 33*zn1*zzz
  z =3D z - pz/ppz= ,
  |pz| > r
}


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------=_NextPart_001_0000_01C0F062.2AD5F4A0-- Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: harry Subject: Re: (fractint) Hairy Newton Date: 09 Jun 2001 00:27:37 -0400 Hairy Newton ??? Brother of "Fig" by any chance ??? BTW you are coming through in two's tonight.... Perhaps your posts are bifurcating ??? H^) Harry (not Newton) Multiple Bogeys wrote: > Yesterday evening I set out to find an interesting family of > Newton-based Mandelbrot mappings. The result was the collection of > formulae below. You'll note the lack of 3D disease -- false alarm. All > my posts to another listserv had it, but I now think it must be that > listserv rather than MSN Exploder. Or rather, some interaction between > the two (since some posts from that listserv don't have it), like the > weird "email laser" that happened with this listserv last week (lots > of peoples' messages were being duplicated, but for some reason mine > were showing up in sets of five or six!)...Anyone who's an expert on > listservs care to speculate further on what might be going on? One > thing is clear: recent mail software and listservs have unnecessary > complexity, and while we like complexity in our fractals, we could do > without chaos erupting in the mail system we depend on to communicate > here... The hrynewt_j and hrynewt_m formulae iterate Newton's method > for p(z) = (z^n - 1)(z^3 - az - 1). Both n and a are parameters, as is > the tolerance, an inverse bailout radius about the roots of p. The > Mandelbrot variant has a vary over the screen while initial z is zero; > this is a critical point but not a root of p for n real and greater > than 2. You can plug in other values of n -- arbitrary negative or > even complex values -- but won't generally be able to find minibrots > unless n has a positive real part greater than two. If n is not an > integer, there will be branch cuts in both the Mandelbrot and the > Julia variants. The hrynewtnnn_j and _m formulae are optimized > versions with specific values for n, mostly small positive integers. > They avoid a slow arbitrary exponentiation, and for the smaller values > of n re-use powers that are used on both sides of the polynomial or > its derivative. The hrynewt2_m formula also has the feature of using a > critical point for initial z, instead of zero (which is *not* a > critical point for n = 2). The result is a proper Mandelbrot view, but > it has a branch cut due to a square root in the calculation of the > critical point, which is a-dependent. The branch cut has been > intentionally manipulated to put it in a fairly unobtrusive place, but > can't be eliminated; the full Mandelbrot for this one lives on a > two-layer Riemann sheet like that of the square root function. The > hrynewtnnn_m formulae also use an (XAXIS) symmetry declaration. (The > generic hrynewt_m can't use this without trashing the output for > non-real values of n.) Observations:* Certain choices of n produce > three-fold-symmetric Mandelbrot sets. Find out which!* Mangled and > occasionally also intact Mandelbrots can be extracted when n is > "strange" but has a real part greater than 2.* You get radial petals > with n real, concentric patterns with n imaginary, and logarithmic > spirals with complex n; the ratio of real to imaginary parts > determines whether the spiral is steep (n close to real) or shallow (n > close to imaginary).* The Mandelbrots are always quadratic -- for real > n > 2, the critical point at zero is nondegenerate, and the critical > point pair for hrynewt2_m is degenerate only at one specific value > of a. The formula file begins with an extensive comment that details > the mathematical constructions that informed their design. comment { > We want a Newton's method with a large number of basins, most of > which are fixed and predictable. > This is accomplished by choosing a polynomial function to solve > composed of two factors, one with many fixed roots, > the other with a few mobile ones: > p(z) = (z^n-1)(z^3-az-1). > The Newton iteration is: > z -> r(z) > where > r(z) = z - p(z)/p'(z) > = (zp'(z) - p(z))/p'(z) > We easily discover p'(z) to be > p'(z) = (z^n-1)(3z^2-a) + (nz^(n-1))(z^3-az-1) > = (3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a > so > (zp'(z) - p(z)) = (z^n-1)(3z^3-az)+(nz^n)(z^3-az-1) - > (z^n-1)(z^3-az-1) > = (z^n-1)(2z^3+1)+(nz^n)(z^3-az-1) > = (2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1 > and > r(z) = ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - > 1)/((3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a) > Using the quotient rule the numerator of r'(z) is > ((3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a)((n+2)(n+3)z^(n+2) - > an(n+1)z^n - n(n-1)z^(n-1) - 6z^2) - > ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)((n+2)(n+3)z^(n+1) > - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) > which factors into > ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) > and > ((3+n)z^(n+3) - a(n+1)z^(n+1) - nz^n - 3z^3 + az) - ((2+n)z^(n+3) - > anz^(n+1) - (n-1)z^n - 2z^3 - 1) > which simplifies to > z^(n+3) - az^(n+1) - z^n - z^3 + az + 1 > Note that p(z) = z^(n+3) - az^(n+1) - z^n - z^3 + az + 1. > Thus the critical points of r(z) are the roots of p(z) and the roots > of > q(z) := ((n+2)(n+3)z^(n+1) - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) > These latter are the "interesting" critical points, as the other > critical points of r(z) are all superattracting. > Note that q(z) is divisible by z, so 0 is an "interesting" critical > point of r(z), for n not one of 2, 1, or -1. > This is the critical point used in the below hrynewt_m formulas > except for hrynewt2_m. For n = 2, > q(z)/2 = 10z^3 - 3(a+1)z - 1 > Put z = y + (a+1)/10y to get > q(z)/2 = y^6 - y^3/10 + (a+1)^3/1000 > so > 2y^3 = 1/10 +/-sqrt(1/100 - 4(a+1)^3/1000), > y = ((1/10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) > and > z = ((1/10 +/-sqrt(1/100 - 4(a+1)^3/1000))/2)^(1/3) + (a+1)/((1/10 > +/-sqrt(1/100 - 4(a+1)^3)/1000)/2)^(1/3) > } hrynewt_j { ; p1 is Julia parameter, p2 is exponent n, p3 is > tolerance (if 0, will act like 0.001). > ; SLOW. Use predefined hrynewtnnn_j where possible. > z = pixel, a = p1, n = p2, n1 = n - 1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = z^n1 > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + n*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt_m { ; p2 is exponent n, p3 is tolerance (if 0, will act like > 0.001). > ; SLOW. Use predefined hrynewtnnn_m where possible. > z = 0, a = pixel, n = p2, n1 = n - 1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = z^n1 > zn = z*zn1 > zno = (zn - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + n*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt2_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will > act like 0.001). > ; n = 2. > z = pixel, a = p1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zno = (z2 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 2*z*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt2_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). > ; n = 2. > a = pixel, > ap1 = a + 1, > IF((real(ap1) >= 0) || ((abs(real(ap1))*(3^(0.5))) < > abs(imag(ap1)))) > t = ((0.1 + (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3), > ELSE > t = ((0.1 - (0.01 - 0.004*sqr(ap1)*ap1)^(0.5))/2)^(1/3), > ENDIF > z = t + 0.1*ap1/t, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zno = (z2 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 2*z*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt3_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will > act like 0.001). > ; n = 3. > z = pixel, a = p1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zno = (z3 - 1) > zzz = zno - a*z > tz2 = 3*z2 > pz = zno*zzz > ppz = zno*(tz2 - a) + tz2*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt3_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). > ; n = 3. > z = 0, a = pixel, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zno = (z3 - 1) > zzz = zno - a*z > tz2 = 3*z2 > pz = zno*zzz > ppz = zno*(tz2 - a) + tz2*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt4_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will > act like 0.001). > ; n = 4. > z = pixel, a = p1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zno = (z*z3 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 4*z3*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt4_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). > ; n = 4. > z = 0, a = pixel, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zno = (z*z3 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 4*z3*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt5_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will > act like 0.001). > ; n = 5. > z = pixel, a = p1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = sqr(z2) > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 5*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt5_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). > ; n = 5. > z = 0, a = pixel, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = sqr(z2) > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 5*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt17_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will > act like 0.001). > ; n = 17. > z = pixel, a = p1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = sqr(sqr(sqr(z2))) > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 17*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt17_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). > > ; n = 17. > z = 0, a = pixel, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = sqr(sqr(sqr(z2))) > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 17*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt33_j { ; p1 is Julia parameter, p3 is tolerance (if 0, will > act like 0.001). > ; n = 33. > z = pixel, a = p1, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = sqr(sqr(sqr(sqr(z2)))) > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 33*zn1*zzz > z = z - pz/ppz, > |pz| > r > } hrynewt33_m (XAXIS) { ; p3 is tolerance (if 0, will act like 0.001). > > ; n = 33. > z = 0, a = pixel, r = p3 > IF(r == 0) > r = 0.001 > ENDIF > : > z2 = sqr(z) > z3 = z*z2 > zn1 = sqr(sqr(sqr(sqr(z2)))) > zno = (z*zn1 - 1) > zzz = (z3 - a*z - 1) > pz = zno*zzz > ppz = zno*(3*z2 - a) + 33*zn1*zzz > z = z - pz/ppz, > |pz| > r > } > > ----------------------------------------------------------------------- > Get Your Private, Free E-mail from MSN Hotmail at > http://www.hotmail.com. Thanks for using Fractint, The Fractals and Fractint Discussion List Post Message: fractint@lists.xmission.com Get Commands: majordomo@lists.xmission.com "help" Administrator: twegner@fractint.org Unsubscribe: majordomo@lists.xmission.com "unsubscribe fractint" ------------------------------------------------------------------------------- From: bmc1@airmail.net Subject: Re: (fractint) Hairy Newton Date: 09 Jun 2001 01:26:59 -0500 I'm getting Multiple Bogeys in Multiples tonight , too. D. Freed harry wrote: > Hairy Newton ??? > > Brother of "Fig" by any chance ??? > > BTW you are coming through in two's tonight.... Perhaps your > posts are bifurcating ??? > > H^) Harry (not Newton) > > Multiple Bogeys wrote: > > > Yesterday evening I set out to find an interesting family of > > Newton-based Mandelbrot mappings. The result was the collection of > > formulae below. You'll note the lack of 3D disease -- false alarm. All > > my posts to another listserv had it, but I now think it must be that > > listserv rather than MSN Exploder. Or rather, some interaction between > > the two (since some posts from that listserv don't have it), like the > > weird "email laser" that happened with this listserv last week (lots > > of peoples' messages were being duplicated, but for some reason mine > > were showing up in sets of five or six!)...Anyone who's an expert on > > listservs care to speculate further on what might be going on? One > > thing is clear: recent mail software and listservs have unnecessary > > complexity, and while we like complexity in our fractals, we could do > > without chaos erupting in the mail system we depend on to communicate > > here... The hrynewt_j and hrynewt_m formulae iterate Newton's method > > for p(z) = (z^n - 1)(z^3 - az - 1). Both n and a are parameters, as is > > the tolerance, an inverse bailout radius about the roots of p. The > > Mandelbrot variant has a vary over the screen while initial z is zero; > > this is a critical point but not a root of p for n real and greater > > than 2. You can plug in other values of n -- arbitrary negative or > > even complex values -- but won't generally be able to find minibrots > > unless n has a positive real part greater than two. If n is not an > > integer, there will be branch cuts in both the Mandelbrot and the > > Julia variants. The hrynewtnnn_j and _m formulae are optimized > > versions with specific values for n, mostly small positive integers. > > They avoid a slow arbitrary exponentiation, and for the smaller values > > of n re-use powers that are used on both sides of the polynomial or > > its derivative. The hrynewt2_m formula also has the feature of using a > > critical point for initial z, instead of zero (which is *not* a > > critical point for n = 2). The result is a proper Mandelbrot view, but > > it has a branch cut due to a square root in the calculation of the > > critical point, which is a-dependent. The branch cut has been > > intentionally manipulated to put it in a fairly unobtrusive place, but > > can't be eliminated; the full Mandelbrot for this one lives on a > > two-layer Riemann sheet like that of the square root function. The > > hrynewtnnn_m formulae also use an (XAXIS) symmetry declaration. (The > > generic hrynewt_m can't use this without trashing the output for > > non-real values of n.) Observations:* Certain choices of n produce > > three-fold-symmetric Mandelbrot sets. Find out which!* Mangled and > > occasionally also intact Mandelbrots can be extracted when n is > > "strange" but has a real part greater than 2.* You get radial petals > > with n real, concentric patterns with n imaginary, and logarithmic > > spirals with complex n; the ratio of real to imaginary parts > > determines whether the spiral is steep (n close to real) or shallow (n > > close to imaginary).* The Mandelbrots are always quadratic -- for real > > n > 2, the critical point at zero is nondegenerate, and the critical > > point pair for hrynewt2_m is degenerate only at one specific value > > of a. The formula file begins with an extensive comment that details > > the mathematical constructions that informed their design. comment { > > We want a Newton's method with a large number of basins, most of > > which are fixed and predictable. > > This is accomplished by choosing a polynomial function to solve > > composed of two factors, one with many fixed roots, > > the other with a few mobile ones: > > p(z) = (z^n-1)(z^3-az-1). > > The Newton iteration is: > > z -> r(z) > > where > > r(z) = z - p(z)/p'(z) > > = (zp'(z) - p(z))/p'(z) > > We easily discover p'(z) to be > > p'(z) = (z^n-1)(3z^2-a) + (nz^(n-1))(z^3-az-1) > > = (3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a > > so > > (zp'(z) - p(z)) = (z^n-1)(3z^3-az)+(nz^n)(z^3-az-1) - > > (z^n-1)(z^3-az-1) > > = (z^n-1)(2z^3+1)+(nz^n)(z^3-az-1) > > = (2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1 > > and > > r(z) = ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - > > 1)/((3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a) > > Using the quotient rule the numerator of r'(z) is > > ((3+n)z^(n+2) - a(n+1)z^n - nz^(n-1) - 3z^2 + a)((n+2)(n+3)z^(n+2) - > > an(n+1)z^n - n(n-1)z^(n-1) - 6z^2) - > > ((2+n)z^(n+3) - anz^(n+1) - (n-1)z^n - 2z^3 - 1)((n+2)(n+3)z^(n+1) > > - an(n+1)z^(n-1) - n(n-1)z^(n-2) - 6z) > > which factors into > > ((n+2)(n+3)z^(n+1) - an(n+1)z^