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Triternions
by Russell Walsmith 22 Jan '03

22 Jan '03

Hola, Fellow Phractophiles, There was a brief murmur of interest in triternions when Tim Wegner broached the subject on 12/9/02, which was about as much as the concept, then embryonic, warranted at the time. Its evolved some since then, and the Tsets now offer more opportunity for exploration; so I thought I put the latest formulas out there and see what happens next. TMan { c1=real(pixel),c2=imag(pixel),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2-c2*c2,z3=t3+c3 z=z1+z2+z3 |z| < 8 } There is, I believe, a rather nice rendition of this set on page 12 of http://fibonacci-arrays.com/Triternions.pdf The formula is a 2-D slice (in the X,Y plane) of a 3-D object at whatever value is chosen for p1. In appendix #2 of the paper, some variations are explored. Ive also noticed that sign changes may have nice effects. E.g., z1=-t1+c1,z2=t2+c2*c2,z3=t3+c3, t3=-z2*z2+2*z3*z1, and so forth. It seems a bit challenging to define a Julia set version of the TMan; Maybe someone can see a way to go with that. I'm looking forward, eventually, to seeing these objects in full-on 3-D. Ciao, Russell Russ Walsmith russw(a)lycos.com _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus

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Running fractint
by Fr. Jim Lacasse 22 Jan '03

22 Jan '03

This may be a question answered many time, however, when Jim Muth and/or others put the long set of parameters for the given fractal at the bottom of their posting, Where or how is that to be entered into Fractint so I can make and or explore the same fractal. Thanks Jim LaCasse

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Re: [Fractint] Triternions
by Russell Walsmith 22 Jan '03

22 Jan '03

-- On Wed, 22 Jan 2003 10:27:11 Lee H. Skinner wrote: >Hi Russell, Morgan, et al, I would like to request whenever anyone makes revisions or improvements to a formula, that a revision number be included in the formula name. Otherwise there may be confusion among the list members as to which version to use in any posted pars. Sounds like a reasonable course of action, Lee. If you are to send me the colormap or whatever other info I need in order to behold 'The Cave of the Fire Pit' in all of its blazing glory (wait... throw in the 'Fractal Forest too...) then I guess I will be persuaded to your cause. Thanks, Russ russw(a)lycos.com _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus

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FOTD 22-01-03 (Antibrot [6])
by JimMuth＠aol.com 22 Jan '03

22 Jan '03

FOTD -- January 22, 2003 (Rating 6) Fractal visionaries and enthusiasts: Today's image is named "Antibrot". I named it when I noticed the almost-Mandelbrot shapes surrounding the obvious Mandelbrot shape at the center. It is another of my evaporated scenes consisting totally of trapped inside points made visible by the < fmod > inside option. The scene was created by combining Z^(-0.7), Z^(-6.4) and (1/C). The rating of a 6 might be a bit generous for an image that uses so few colors of the palette, but those colors create a good impression of fractal objects floating in space, so the rating could be considered appropriate. The render time of 59 minutes is quite slow. The download from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> A fresh dusting of snow and a temperature of 26F -3C kept the dynamic fractal cats confined to the indoors all day Tuesday. But they endured their hardship intrepidly, and slept most of the time by the heat. Today is starting sunny but even colder. I hope they can maintain their intrepidness. As for me, I'm ready to have an intrepid day at the work desk. When I finish, I'll take a trip to Fractal Land. The best thing I find there will appear as tomorrow's FOTD. Until then, take care, and when drowning, strap on a fractal. They weigh almost nothing and will keep you afloat indefinitely. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Anti_Mandelbrot { ; time=0:59:10.93--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=t center-mag=+1.7148895301837/-0.0375140875302/29702\ .9/1/57.5/-2.1386011017643369e-010 params=0.6/-0.7\ /0.7/-6.4/0/1e+100 float=y maxiter=1800 inside=fmod proximity=0.45 logmap=yes periodicity=10 colors=000Kgj`UWCjbXVUF7FKyoWidfVV4GITGKhJJkIKmHKo\ GLQySXmQcbOjRMRl0dWBk9maLhcSfdZcfeahlZisXai`V`dOSh\ GIl99p20s43p66m88jABgCEdEGaFJ_HLXJOULRRNTOPWLQYJZS\ KhMLS28d9F9NLnOooMgpK_qISoQRpNPpLOqIMUmhJjj8hkBmjE\ riHwhAV`lmhqn_uoRypJCyIDoWEeiFXvpaNhOIqUMqNLfRlJvz\ ``efXViSSlOQoKNmaYoUTpNP7VGMhnU`faU_iNSD0wVHQfGNcZ\ ZeQEgujl`Y6EsIFjUFaeGTXOS`9aiDTbPbjKUT3tIz9RnC_bFh\ RIlw`nlXoaTpRPQkrSDqdF`7TeJP_VMVfJQebQpKlNqubZbDo8\ RcDdSHneLoZLpTLqMLMHwaGcKkX`WR7tGNfIbTKC8yNAnXCdgE\ VW`jfQYaCaEFbjShlPbnMXpJRKpa`YT2k3SWC0y5YL3bJ8gICl\ HH`9QeBOiDNmFMOH6YGBgGGUcDeSHLi0aVBNzwYjjgVYehbiZX\ mPRY8YgCRzmUz_RqMOA7A`ADpCGhEJcxaglXjaTnRPMX58f_qQ\ e3oRky_miVoVQ1k3Fc8SWCdOH1lxSWd7kLVWL_avS6idBXQyx_\ ijhVYptiqf`qTT0IbRHUm_geZGiSImMKNpZVfVbYSjPO4gyMSr\ aMawe_uXVsOQSUk`PbiKUtzerbVcHGE13S69dBFgzZ7ozknQnX\ NfejjXanOT4epJ8G`CJpnh3us } 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==================================

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Re: [fractint] FOTD 19-01-03 (Maximum Spirals [5])
by Lee H. Skinner 21 Jan '03

21 Jan '03

The TMan1 formula can really produce some incredible images. Here's one deep in the parent of Jim's FOTD of Maximum Spirals. I think it's one of the most beautiful images that I've ever created, but well worth the wait to calculate at the highest resolution you can. It reminds me of some of the images I created with the Carr2821 PHC formula, but the TMan1 formula is not a PHC formula. 60119015 { ; Fractal Forest t= 0:40:10.96 ; on a P4 2Gh at 1600x1200 Jan 21, 2003 22:06:37 ; Image Copyright 2003 by Lee H. Skinner ver=2002 ; Version 2002 Patchlevel 3 reset=2002 type=formula formulafile=fractint.frm formulaname=tman1 passes=1 center-mag=-0.00595484452179773/+0.74507406570897650/2.889005e+008/1/-44\ .9999986561148546/-1.7144370605542969e-006 params=0/0 float=y maxiter=4200 inside=0 symmetry=none periodicity=0 colors=000ah0<3>`g0`f0`f0<31>DSCDRDCRDBQEAQE9PF<5>CNBDMBDMA<3>GK9HK9JJ9<\ 22>j73l63m63<2>q42r42r72<3>tM1uP1uT1vW1v_1wc0<4>xmPxoUxqZ<2>ywmzzsyzs<12\ >_pvYpvWov<2>QmwOlwOlu<17>MiGMiEMiC<2>Mi5Lh2Mh3<24>prkqrmssn<2>vttxuvxuv\ yvv<6>ypSyoNynJ<2>yk5zj0xh0<8>ZP1XN2UK2<3>IB3<38>`g1 } frm:TMan1 { c1=flip(real(pixel)),c2=imag(pixel),c3=p1 z1=z2=z3=0: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+c1,z2=t2-c2*c2,z3=t3+c3 z=z1+z2+z3 |z| < 8 }

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Re: Fractint digest, Vol 1 #368 - 7 msgs
by JackOTradez 21 Jan '03

21 Jan '03

In recent fractint mailing lists there were postings about the TMan1 (Russ Wallsmith, tweaked by Jim Muth), and Sylvie Gallet's "rising lake" effect. I have figured how to animate them both; they are currently rendering right now and should be posted on my site in a few days or so. They look 2B seriously cool. Lee Skinner also posted 4 PAR formulas in the last mailing, one of them, "Cave of the Fire Pit" is awesome! If you didn't check it out, do it! I rendered it at 1600x1200, printed out a hardcopy, and hung it up on my wall - I don't often do that. I really see a view from the air of a shallow Carribean sea dotted with little islands. In the middle is a gaping black hole, as if an asteroid had just impacted, with a definite 3D appearance and a wall/cliff (tidal wave?) next to it. And fire spewing out of it like there would be. Now if I could animate the fire w/o affecting the rest of it ... ...

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FOTD 21-01-03 (Sub-Marine Scene [6])
by Jim Muth 21 Jan '03

21 Jan '03

FOTD -- January 21, 2003 (Rating 6) Fractal visionaries and enthusiasts: As the extended fractional value of the real p1 and p2 para- meters of today's image shows, the coefficients were found by Fractint's evolver feature, the robot fractal explorer that can go intrepidly where men would dare not venture. The exponents however were chosen by the man, who had no time to dare to venture intrepidly. The image itself is another of those ghostly ones created by finding an interesting scene in a fractal which contains negative exponents, then raising the bailout until every point is trapped and the image consists of nothing but the infinite fractal ocean. At this point, if an inside fill such as bof60 is applied, a reflection of the original image almost magically reappears, sometimes looking even better than the original itself. I named the image "Sub-Marine Scene", though it has little resem- blance to the underwater world. I rated it at a 6, which might be a bit optimistic considering the hasty and somewhat muddy coloring. The render time of 12 minutes (on my tired machine) can be avoided by downloading the completed image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> With a warm sun, the temperature reached 37F 3C here at Fractal Central on Monday, but the winds were over 40mph 64kph, which made the intrepid duo quite skittish, and cut their outdoor time to a few minutes. They must be getting used to being house- bound, because they were actually quite happy with themselves through the long afternoon. At dinner time I rewarded their patience with a treat of tuna. But today is much colder, and light snow is falling. Their patience will be tried far more severely. Looking at the clock on the wall over my desk, I see that it is time to get busy on the unexpectedly tall pile of tasks on the shelf beside me. Until next time, take care, and be sure to try the fractal diet. Eat only fractals for a month and watch those pounds and ounces (kilograms and grams) vanish like magic. Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ Sub-Marine_Scene { ; time=0:12:16.60--SF5 on a p200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+2.27593538850584600/+0.293765268933583\ 40/3.698853e+008/1/57.5/-8.49740486379493842e-005 params=-11.49790215765862/-1.3/-10.61464473403119/\ -9.4/0/1e+100 float=y maxiter=1200 inside=bof61 symmetry=none periodicity=10 colors=000zlJzmJznKzoKzpKzqLzrMzsOztQzuSzvUNBWQCZR\ C`SCbAcaAc`9b_8bZ8bY7bX6aW6aV5aU5aT6bV6bX7cY0007da\ 8db8ed9ef9fg9fiUYVSXXQVZOU`MTcKReIQgGPiENlCMnALp5G\ q9KrDOrGSrKWrNZrRbrVfrYjranr`wsdqrhlqkgqoapsXoxQpv\ SotUnrWmqYlo_km`jkbijdhhfgfhfeiedjdckcclcbmbambana\ `oa_p`_p`Zq_Zr_YsZXtYXtYWuXVvX000UwWTxVTyVSzUSzUUw\ WWuXYsY_pZan_cl`ejaggbieckcdmaeoZfqXgsVhvRguTitVjs\ XkrZmq`nt`pqaoq`pp_qpZroYsoXtnWumUvuWDrYGo_JlbNidQ\ ffUbiX_k_XmcUpfRrjOtmKwqKwpKwpLwnMwmNwkOwiOwhPwfQw\ eRwcSwaTw`TwZUwY000WwUXwTYwRYwQZwO_wM`wLawJbwHdrHg\ lGjgFmaEpXDsRCvLBrSEqTGpUIoVKnWMmXOlYQkZSj_Ui`Wgxz\ fzzevzduzctzbszarz`qy_pxZowYnvWmuYjqYioXhmXgkWfjWe\ hWdfVcdVbbU`aU__TZYTXWTVUSTTSRRRPPRNNQKLQLLROMRPMS\ SNSTPRWRQYTP`VObXNdZMf`LibKkdJnfIpiOndTobYo`bpZgpX\ lqVrrTsmQthNufNvdNwbNx`NyZMzXMzYMzZMz_Mz`LzaLzbLzc\ LzdLzeKzfKzgKzhKziKzjJzkJ } 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==================================

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Re: [Fractint] Re: Fractint digest, Vol 1 #366 - 3 msgs
by Franktal Gallery 21 Jan '03

21 Jan '03

Check out my Mandelbrot SunSet (animated GIF) on : http://www.geocities.com/franktal2002/gal2.html it's a bit crude but uses the cool "lake" fractint feature. regards, Miguel Fliguer Franktal Gallery - SHUT UP 'N' DRAW YER FRACTALS http://www.geocities.com/franktal2002 (I know, no updates on 2002....) >From: "JackOTradez" <JackOTradez(a)email.msn.com> >Reply-To: fractint(a)mailman.xmission.com >To: <fractint(a)mailman.xmission.com> >Subject: [Fractint] Re: Fractint digest, Vol 1 #366 - 3 msgs >Date: Sun, 19 Jan 2003 12:25:23 -0500 > >Most Intriguing. The Russ-Dude strikes again! The rising Mandelbrot is way >cool. Now I must figure out a way to animate it and make a viddie - like a >Mandelbrot "sunrise" from an ocean horizon....... > _________________________________________________________________ The new MSN 8 is here: Try it free* for 2 months http://join.msn.com/?page=dept/dialup

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An afternoon visiting an old friend.
by Morgan L. Owens 21 Jan '03

21 Jan '03

Guess I should have seen it coming... Finished an article earlier than expected today. Had a few email inquiries to get through and just finished the last one (someone wanting to know what exactly Hilbert spaces are), made some posts of my own, found myself with a whole afternoon with (just for a moment) no idea what to do in it. Brief moment of panic: what had I forgotten? Nope, just no active contracts right now. Nothing needing writing, nothing needing coding, nothing needing expediting, and my friend's nephew is still on holiday, so he won't be in for his tutorial. Halfway through January and things are still quiet. Thought of a few things to do, but I'd already planned trips to the art gallery to see the new Chinese exhibit and the museum for a bit of digging through its archives for Thursday ... wasn't keen on the beach at all ... seen the movies I've been planning to see ... read a book? (glance at the shelves - nope, nothing saying "READ me! Read ME!") ... take a nap? ... could go to the pub ... the usual suspects seemed strangely boring. The proveribal drawing board was as metaphorically quiet as the literal bookshelves. Ended up spending the afternoon messing 'round on Fractint. Well, you should have been able to guess that - otherwise this post about my day and psychological state would have been entirely off-topic; it's not as though any of you would actually be interested in my psychological state while I munched lunch. It's just that I don't play with Fractint as much as I did ... good thing I didn't pay for it, I suppose (Would I use it more if I _had_ paid for it? Judging from past form, no. It would just sit there with its cost/benefit ratio steadily climbing.) I figured twenty pars worth keeping would be a convenient figure; to make things more interesting for myself, I resolved to use only those formulae that I'd coded myself. WFA, here are the final score: Morgan L. 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Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=chby5.frm formulaname=sc06-05 passes=d center-mag=6.19973/9.57075/2.446735/1/10 params=1/0/1/0 float=y maxiter=1023 inside=255 outside=atan invert=0.5/6.2038774233896/9.8444537114261 periodicity=0 colors=653<9>cb`fecjig<3>xwu<9>qYXpVVoSS<2>mKLlHIjII<8>QRLOSLMTL<3>CXN<9\ >ekKhmKknK<3>wuI<6>caM`_NYXN<3>MLRJISGFSDCT<9>YCS_CSaCS<3>jCR<5>XIZVJ_TK\ `<3>KOfIPhFRiDSkATl<9>8EL8DI7BF<3>653<9>cb`fecjig<3>xwu<9>qYXpVVoSS<2>mK\ LlHIjII<8>QRLOSLMTL<3>CXN<9>ekKhmKknK<3>wuI<6>caM`_NYXN<3>MLRJISGFSDCT<9\ >YCS_CSaCS<3>jCR<5>XIZVJ_TK`<3>KOfIPhFRiDSkATl<14>AA9 } Not_Your_Job { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=chby5.frm formulaname=Ca10-05 passes=1 center-mag=+0.07662876962040530/+0.09346251373957647/6.137269e+012/1/-14\ .997/-0.001 params=2/0/1/0 float=y maxiter=1023 inside=255 periodicity=0 sound=beep/fm polyphony=3 wavetype=1 colors=00032t40t50s60s<36>ps`qt`rv_sw_uyZvzZwzYxzYzyXzwXzvW<30>OCHNBGLAG\ K8FJ6F<14>I5EI5EI5E<3>I5EIzE<4>IzE<3>IzEIzEH7F<33>CDLCELCEL<2>CELBFMDEM<\ 3>I8OJ6PK5PL3QN2QO0RP0RQ0S<22>rZbs_cuac<2>xeezgezhfzjf<5>rsiqtjpvjnwkmyk\ lzlkzlizm<21>JUxITxGRy<3>BLzAKz9Iz<5>19w08v06v05u13u } Mandelbrot_Forgery { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=more.frm formulaname=Mandelbar passes=1 center-mag=-0.62872052566293170/+0.11121047612571350/23956.75/0.1483/-12\ .868/-50.835 float=y maxiter=512 inside=255 logmap=-43 periodicity=0 colors=000Xtu<8>XvuXvuXvuXvuXwv<23>eeOfdNfdL<2>gaHh`Fg`F<25>FCSEBTDATC9U\ B8U97V<158>8RD7SC7SC<10>9SE9SE9SE<3>ASFXtu } Parallel_Transport { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=ifs.frm formulaname=A4Page passes=1 center-mag=1/-0.666667/1.5 float=y maxiter=16 inside=startrail outside=0 periodicity=0 colors=@default } Two_Flows { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=chby24.frm formulaname=P03-24 function=sin/sqr/sinh passes=t center-mag=-1.19245/26.7544/0.2931875/17.119/180 params=0.1/0 float=y maxiter=1023 inside=255 periodicity=0 sound=off polyphony=3 wavetype=1 colors=000LLL<9>LLjLLlLLo<2>LLwLLzLNx<8>Lj`LlZLoW<3>LzL<9>LzjLzlLzo<2>Lz\ wLzzNxx<8>j``lZZoWW<3>zLL<9>zLjzLlzLo<2>zLwzLzzNx<8>zj`zlZzoW<3>zzL<9>zz\ jzzlzzo<3>zzz<9>OOPKKMGGI<3>002<8>00Q00S00V<2>00b00e02c<8>0QG0SE0VB<3>0e\ 0<9>0eQ0eS0eV<2>0eb0ee2cc<8>QGGSEEVBB<3>e00<9>e0Qe0Se0V<3>e0e<9>eDGeEEeF\ B<3>eL0<9>eYQeZSe_V<3>eee<14>NNN } Altern_AntiSpiral { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=more.frm formulaname=PokornyJul center-mag=0.200125/0.776177/2.506266 params=0.25/0.9 float=y maxiter=1023 inside=255 logmap=yes periodicity=0 colors=@altern } Puffer_Train { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=evenmore.frm formulaname=Rossler2Man center-mag=+0.87099255681842290/+0.45740250376426910/5.001221e+012/1.000\ 1 params=0/0/12/0 float=y maxiter=1023 inside=255 logmap=57 periodicity=0 colors=653<9>cb`fecjig<3>xwu<9>qYXpVVoSS<2>mKLlHIjII<8>QRLOSLMTL<3>CXN<9\ >ekKhmKknK<3>wuI<9>VUOSRPPOQ<3>DCT<9>YCS_CSaCS<3>jCR<9>OMcMNeKOf<3>ATl<9\ >8EL8DI7BF<3>653<9>cb`fecjig<3>xwu<9>qYXpVVoSS<2>mKLlHIjII<8>QRLOSLMTL<3\ >CXN<9>ekKhmKknK<3>wuI<9>VUOSRPPOQ<3>DCT<9>YCS_CSaCS<3>jCR<9>OMcMNeKOf<3\ >ATl<14>AA9 } FriendlyBiological { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=evenmore.frm formulaname=Rossler2Man passes=1 center-mag=+0.80768598348989470/+1.07767747322206200/622.4066/1/28 params=0/0/12/0 float=y maxiter=1023 inside=255 logmap=9 periodicity=0 colors=tuu<3>tvvtvvtvvtvvuww<6>lwwkwwjwwiwwgww<3>ctvbtvasu<3>XosVnsUmr<1\ 8>AXh9Wg8Vg7Uf6Te<4>5Pc5Oc5Ob4Na<18>17R17Q16Q<2>13O02N14O02N<8>16Q27Q27Q\ <3>3AT3BU3CV<6>4Ka4Lb5Mc<5>4Si4Tj4Uk3Vl3Wm<4>2_o1_p1`p<2>0cr0cr0br<3>1_q\ 1Zp1Yp<8>3Qg3Pf3Oe<2>4La4K`3IZ<3>4GS4FQ5FO<3>6CG7BD7BD<70>oqqpqqqrrqrrrs\ s<8>suu } Corrosion { ; Kerry Mitchell colour map ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=chby17.frm formulaname=H04-17 function=ident/log passes=1 center-mag=-0.405748/-0.631308/5.253812/1/-34.999 params=0/0/1/0 float=y maxiter=1023 inside=255 periodicity=0 colors=222qU6yaEqaMyiUeYQYQIaaaiiiqiayqiqqqyyyemuYemQeuIYmAYu2QmAQe2IYMU\ aUaiQQQIIIAIQ2AIAAA642IC6UKAYK6iSAqS2u_Eq_IugUi_QYSMY_aigeqgYuoiqomuwyio\ uYgqUguI_qA_y6SmASi6KYISaU_eUSQIKMAKU6CIACEA62IEAUI6YMAiQ6uU2qYEuaIqeUiY\ MYUQYaeieauiYqmiuqmquyimqYiuUeqIau6YyAUm6QiAMYIUeUYaUQMIMQ6IUAEI6AEA86IG\ EQG6YOEeO6yW2qWAycIqcQiWIaWQYciecaykYqkeysmqsuikmakuUcmMcu2WyAWq2OiAOaIW\ iQWaUOIMOQ2GUAGM28EEA6MIEQE2aQEeM2yY6mUAyeMmaQeUIaYUaeieaYymamieyuqmquei\ mamyQamMey2UuEYq2MeEQaMYiQUYQMIMQU2EQEIM26AE82MGAUG2aOAiO2uW6mWEucMmcUeW\ MYWUaceicYukamkiusqmsyekqYkyQcqIcy6WuEWm6OeEOYMWeUWYQOMIOU6GQEGI68AIA2QI\ AYI2eQAmQ2uYAmYIueQiaUaUMYYYeeemeYumemmmuuuiqyaiqUiyMaqEay6UqEUi6MaIQYQY\ eUUUMMMEMU6EMEEE666MC2QKEaK2eSEmS6y_Am_MygQe_UaSIa_Yegimgayoemoqywueoyag\ mQgyM_mE_u2SqESe2KaMSYQ_iQSUMKIEKQ2CMECA246ME6UMEaM6iUE } Mitrocoma { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=chby26.frm formulaname=T03-26 function=abs/ident/ident passes=1 center-mag=-0.833021/0.838198/7.751938 params=2/0.2/0/0/8/1 float=y maxiter=1023 inside=255 periodicity=0 colors=(a)blues.map } Six_OClock { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=twodne~1.frm formulaname=twodnewton_o passes=t center-mag=+17.61898205646764000/+11.36059478637080000/109.2896/1/-22.49\ 9 params=0.1/-1 float=y maxiter=1023 inside=255 periodicity=0 colors=000e0_e0be0e<9>eDGeEEeFB<3>eL0<9>eYQeZSe_V<3>eee<9>TTTSSSRRR<2>NN\ NLLLLLN<8>LLjLLlLLo<2>LLwLLzLNx<3>LYm<21>MruMsvMtv<3>Nxx<8>j``lZZoWW<3>z\ LLzLNzLQzLS<18>zgVzhVzjV<2>zmVzoWzrTztRzwOzzL<9>zzjzzlzzo<3>zzz<3>kkkggg\ ccd__`<31>__`<4>__`0_60b30e0<9>0eQ0eS0eV<2>0eb0ee2cc<8>QGGSEEVBB<3>e00<1\ 2>e0Y } frm:distortpendulumMG{ bailout=4 dt=real(p1) a=imag(p1) b=real(pixel) c=imag(pixel) time=0 z=pixel: x=real(z) y=imag(z) z=x+dt*y+flip(y+(dt*fn1(b*x-c*time)-x)*a) time=time+dt |z|<=bailout} frm:HenonQuadmapJulGen{ bailout=real(p2)+4 c=p1 d=1-c z=pixel: x=real(z) nx=c*x-imag(z)-d*fn1(x) z=nx+flip(x-c*nx+d*fn2(nx)) |z|<=bailout} frm:Ca03-19 {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = (2(a+2)xCa(2)-(2a+1)Ca(1))/3 a=p2, b=(a+a)*(a+1), c=(a+a+4)/3 t=p1, bailout=4, z=pixel: x=real(z), y=imag(z) y=y+t*(b*x*(c*x*x-1)) x=x-t*(b*y*(c*y*y-1)) z=x+flip(y) |z|<=bailout } frm:C03-15 {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = zC02-C01 r=p1, bailout=real(p2)/10000, z=pixel: zz=z*z f=z*(zz-3) fd=3*zz-3 oz=z z=z-r*f/fd bailout<=sqrt(|(|z|)-(|oz|)|) } frm:P07-07 (XYAxis) {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = (13zP06-6P05)/7 bailout=real(p2)/10000, z=pixel: zz=z*z f=z*(3*zz*(11*zz*(13*zz-21)+105)-35)/16 fd=(21*zz*(11*zz*(13*zz-15)+45)-35)/16 oz=z w=z-f/fd ww=w*w fw=w*(3*ww*(11*ww*(13*ww-21)+105)-35)/16 z=w-fw/fd bailout<=|z-oz| } frm:quarterblend_iJJJJ{ z=pixel: if(imag(z)>0) if(real(z)<0) c=p1 else c=p2 endif else if(real(z)>0) c=p3 else c=p4 endif endif z=sqr(z)+c |z|<=4} frm:threecircles2 { ; three circles located at the cube roots of 1 ; real(p1) specifies their common radius - note that they're tangent to ; each other when real(p1)=sqrt(3)/2. ; ; p2 specifies a point (the "radiant") from which test particles are ; launched. More precisely, every point in the complex plane emits a ; test particle, which travels directly away from p2 ; ; First, we need to establish the centres of the three circles. centre1=1 centre2=exp(flip(2*pi/3)) centre3=sqr(centre2) ; Reckon this is faster than another exp()? rho=real(p1) direction=pixel-p2 z = pixel trapped=(|z-centre1|<sqr(rho) || |z-centre2|<sqr(rho) || |z-centre3|<sqr(rho)) ; For efficiency, we see if z is _inside_ a circle. If it is, we can ; predict how long it will take to escape :-) ; Note also that since Fractint's || operator is nonstandard, we use ; sqr(rho) instead of rho. : ; End of initialisation section. Now beginning the loop. if(trapped) escaped=0 ; We'll continue to iterate, though - so that the insides of ; the circles really are classified as "inside" else ; Let's get dangerous ; Right, we have a ray that passes through the point z in the direction ; theta. Finding the intersections (if any) of this ray with a circle ; involves finding the solutions of a certain quadratic with real ; coefficients. If the quadratic has two real roots, then the ray cuts the ; circle. If it has one real root (necessarily of multiplicity two) then ; the ray is tangent to the circle. And if both roots are complex, then ; the ray misses the circle completely. escaped=1 ; As far as we know at the moment. hit=9999 ; Meaningless when escaped==1 ;Examining circle 1 coefA = |direction| coefB = real(z-centre1)*real(direction)+imag(z-centre1)*imag(direction) coefC = |z-centre1|-sqr(rho) discriminant=coefB*coefB-coefA*coefC ; If discriminant==0, then the ray is tangent to circle 1 - we won't ; count this as a "reflection" ; If discriminant<0, then the ray misses circle 1 completely. if(discriminant>0) ; Two solutions t1=(-coefB-sqrt(discriminant))/coefA t2=(-coefB+sqrt(discriminant))/coefA if(t1>0) ; If t1<0 then this hit is actually behind us escaped=0 ; Sorry, not this time. hit=t1 target=1 ; The first circle endif if(t2>0 && t2<t1) ; Hits t2 before hitting t1 escaped=0 hit=t2 target=1 endif endif ; Circle 2 ; coefA = |direction| is already known coefB = real(z-centre2)*real(direction)+imag(z-centre2)*imag(direction) coefC = |z-centre2|-sqr(rho) discriminant=coefB*coefB-coefA*coefC if(discriminant>0) t1=(-coefB-sqrt(discriminant))/coefA t2=(-coefB+sqrt(discriminant))/coefA if(t1>0 && t1<hit) escaped=0 hit=t1 target=2 endif if(t2>0 && t2<t1 && t2<hit) escaped=0 hit=t2 target=2 endif endif ; Circle 3 coefB = real(z-centre3)*real(direction)+imag(z-centre3)*imag(direction) coefC = |z-centre3|-sqr(rho) discriminant=coefB*coefB-coefA*coefC if(discriminant>0) t1=(-coefB-sqrt(discriminant))/coefA t2=(-coefB+sqrt(discriminant))/coefA if(t1>0 && t1<hit) escaped=0 hit=t1 target=3 endif if(t2>0 && t2<t1 && t2<hit) escaped=0 hit=t2 target=3 endif endif ; We now know when we hit a circle (hit) and which circle it is (target) ; The hit point is z+direction*hit, the normal at this point is ; z-centre?, which we normalise to a unit vector (un). ; ; We set z to the hit point, and the reflected vector is ; direction-2(direction.un)un if(target==1) un=(z-centre1)/cabs(z-centre1) elseif(target==2) un=(z-centre2)/cabs(z-centre2) else un=(z-centre3)/cabs(z-centre3) endif r=direction-2*(real(direction)*real(un)+imag(direction)*imag(un))*un z=z+direction*hit direction=r endif ;bailout continue until we escape escaped==0 } frm:newton_elliptics_4{ x=real(pixel) y=imag(pixel) A=real(p1) B=imag(p1) phi=tan(real(p2)) t=imag(p2) z=x+flip(y) : oz=z x=real(z) y=imag(z) d=2*y*phi-(3*sqr(x)+A) f=(sqr(x)+A)*x+B-sqr(y) g=phi*x+t-y nx=x-f+2*y*g ny=y-phi*f+(3*x^2+A)*g z=nx+flip(ny) |z-oz|>0.0001 } frm:Mandelbar(Xaxis){ z=pixel: z=sqr(conj(z))+pixel |z|<=4} Julibar(Origin){ z=pixel: z=sqr(conj(z))+p1 |z|<=4} frm:A4Page(XYAxis){ s=-sqrt(2) z=pixel: x=real(z) y=imag(z) if(x>0) z=s*(flip(conj(z)-1)) else z=s*(flip(conj(z)+1)) endif |z|<=300} frm:Sc06-05 {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = Sc[6]=2(zO[5]-cos(5*pi/2)^2)/5 r=p1, bailout=real(p2)/10000, z=pixel: z2=z*z, z4=z2*z2 f=((((((72-8*z)*z-192)*z+384)*z-704)*z+768)/(z4*z)) fd=((((384-72*z)*z-1152)*z+2816)*z-3840)/(z4*z2) oz=z z=z-r*f/fd bailout<=|z-oz| } frm:twodnewton_o{ h = real(p1) m = imag(p1) z=pixel: oz=z x=real(z),y=imag(z) f=2*x-y-m*(x-x*x) g=2*y-x-m*(y-y*y) fx=2-m*(1-2*x) gy=2-m*(1-2*y) delta =(-g-f*gy)/(fx*gy-1) epsilon=(-f-g*fx)/(fx*gy-1) nx=x+h*delta ny=y+h*epsilon z=nx+flip(ny) |oz-z|>.001 } frm:Rossler2Man(XAxis){ bailout=real(p2)+4 z=p=pixel: x=real(z) y=imag(z) z=abs(sqr(x)-sqr(y))-real(p)+flip((x+x)*y-imag(p)) |z|<=bailout} frm:PokornyJul(Origin){ bailout=4 c=p1 z=pixel: z=1/(sqr(z)+c) |z|<=bailout} frm:P03-24 {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = ( 5zP02-2P01)/3 t=p1, bailout=4, z=fn3(pixel): x=real(z), y=imag(z) y=y+t*(fn1(x*(5*x*x-3)/2)) x=x-t*(fn2(y*(5*y*y-3)/2)) z=x+flip(y) |z|<=bailout } frm:T03-26 {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = 2zT02-T01 t=p1, bailout=real(p3), z=pixel, v=imag(p3): x=real(z), y=imag(z) Tx=(fn1(x*(4*x*x-3))) Ty=(fn1(y*(4*y*y-3))) x=x-t*Ty, y=y+t*Tx w=fn1(x+flip(y)) z=fn3(v/fn2(w*w))+p2 |z|<=bailout } frm:H04-17 {; V.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = 2(xH(3)-3H(2)) z=pixel, oz=0 fz=12 zt=fn1(z)+p1: foz=fz zz=z*z fz=4*(4*zz*(zz-3)+3) z=z-p2*fz*(z-oz)/(fz-foz) oz=zt zt=fn2(z) |fz|>=0.0001 } frm:Ca10-05 {; V.1.1.1 - earlier versions may be discarded ; Copyright (c)1998,1999 Morgan L. Owens ; = ((a+9)xCa(9)-(a+4)Ca(8))/5 a=p1 b1=a*(a*(a*(a*(a+10)+35)+50)+24)/382860 c1=15*(a+5) d1=2*(a+6) e1=a*(a+15)+56 f1=2*(a+9)/45 g1=14*(2*a+13)/5 b2=a*(a*(a*(a*(a*(a+15)+85)+225)+274)+120)/12762 e2=(a+9)/9 f2=21*(2*a+13)/5 bailout=real(p2)/10000, z=pixel: zz=z*z f=b1*(c1*zz*(d1*zz*(81*zz*(e1*zz*(f1*zz-1)+g1)-1418)+2127)-31903) fd=b2*z*(d1*zz*(162*zz*(e1*zz*(e2*zz-2)+f2)-2881)+2127) oz=z z=z-f/fd bailout<=|z-oz| }

3 2

Oops, belay that earlier Tjul formula
by Russell Walsmith 20 Jan '03

20 Jan '03

I've noticed that cutting and pasting doesn't always save time. TJul { z1=real(pixel),z2=imag(pixel)*(0,1),z3=p3: t1=z1*z1+2*z2*z3 t2=z3*z3+2*z1*z2 t3=z2*z2+2*z3*z1 z1=t1+p1,z2=t2+p2,z3=t3+p3 z=z1*z2+z2*z3+z3*z1 z < 64 } _____________________________________________________________ Get 25MB, POP3, Spam Filtering with LYCOS MAIL PLUS for $19.95/year. http://login.mail.lycos.com/brandPage.shtml?pageId=plus&ref=lmtplus

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