Path: unixg.ubc.ca!news.bc.net!news.mic.ucla.edu!library.ucla.edu!europa.eng.gtefsd.com!newsxfer.itd.umich.edu!gatech!howland.reston.ans.net!swiss.ans.net!prodigy.com!usenet From: XKBR53C@prodigy.com (CHRIS BEHRENS ) Newsgroups: sci.fractals Subject: Re: Fractal Formulas Date: 28 Nov 1994 23:58:24 GMT Organization: Prodigy Services Company 1-800-PRODIGY Lines: 24 Distribution: world Message-ID: <3bdqr0$n9i@usenetw1.news.prodigy.com> NNTP-Posting-Host: inugap2.news.prodigy.com X-Newsreader: Version 1.2 Here are the formulas, then: Flattop { c = z = pixel: z = sin(z+2) + (c*z)/(z-2); |z| <= 4; } Rabadon { z = pixel/2; q = z^sin(pixel); g = pixel: z = z^q/g; z = (log(q) + sqr(g)) / z; z = 4^z / q^g*2; |real(z)| <=8; } The only caution is that Rabadon sometimes won't work- I have occasionally gotten a division by zero error. Have fun. - Chris Behrens xkbr53c@prodigy.com Path: unixg.ubc.ca!unixg.ubc.ca!news.mic.ucla.edu!nntp.club.cc.cmu.edu!godot.cc.duq.edu!news.duke.edu!news.mathworks.com!uunet!news.sprintlink.net!cs.utexas.edu!news.cs.utah.edu!news.cc.utah.edu!u.cc.utah.edu!kb9727 From: kb9727@u.cc.utah.edu (Kim Best) Newsgroups: sci.fractals Subject: Fractint Par Date: 10 Feb 1995 21:33:02 GMT Organization: University Of Utah Computer Center Lines: 106 Message-ID: <3hgm2e$9c7@news.cc.utah.edu> NNTP-Posting-Host: u.cc.utah.edu X-Newsreader: TIN [version 1.2 PL2] A few more formulas and parameter files, I have had interesting results with. mfn2fn = { z=pixel: power = fn2(z); z = fn1(z)^power + pixel, |z| < p2 } jfn2fn = { z=pixel: power = fn2(z); z = fn1(z)^power + p1, |z| < p2 } jtofn = { z=pixel: power = fn2(z); z = z^power + p1, |z| < p2 } mtofn = { z=pixel: power = fn2(z); z = z^power + pixel, |z| < p2 } Now some parameters to use with these functions... log2sqr { ; A Section of the Mandelbrot set for the formula ; z = log(z) ^ sqr(z) reset=1821 type=formula formulafile=kim.frm formulaname=mfn2fn function=log/sqr corners=2.6266707454/2.6273926625/0.35959516412/0.35935878535/2.62704409\ 74/0.35909736147 params=0/0/100 float=y maxiter=500 inside=0 colors=mU0a06<27>c00c00b00<26>20K00L00M<28>00u00w00w<45>T0VU0UU0UU0U<114\ >a06 } sin2sqr { ; z1 = sin(z0) ^ sqr(z0) reset=1821 type=formula formulafile=kim.frm formulaname=mtofn function=sin/sqr corners=1.336035534/1.346864405/0.7162926354/0.7085313265/1.343659361/0.\ 7061275462 params=0/0/100 float=y maxiter=1000 inside=0 colors=000z0z<7>z00<5>zz0<7>0z0<7>0zz<7>00z<7>m0m<7>m00<7>mm0<7>0m0<7>0m\ m<7>00m<7>c0c<7>c00<7>cc0<7>0c0<7>0cc<7>00c<7>U0U<7>U00<7>UU0<7>0U0<7>0U\ U<6>04U00U20T<6>K0K<6>K03K00K20<5>KH0KK0IK0<6>0K0<6>0KH0KK0IK<6>00K<6>80\ CA0AD0D<13>w0wjWr } sin2sqr2 { ; z1 = sin(z0) ^ sqr(z0) reset=1821 type=formula formulafile=kim.frm formulaname=mtofn function=sin/sqr corners=1.3414287342/1.343201919/0.7147275786/0.71510027391/1.3418881868\ /0.71411497597 params=0/0/100 float=y maxiter=1000 inside=0 colors=CCCcK0<44>rrXssYssW<8>ssEssCsqC<19>sECsCCqCC<21>ACCCCC<21>sCs<20>\ ECsCCsCEs<19>CqsCssCsq<8>Cs_CsYECE<20>sCs<20>ECsCCsCEs<19>CqsCssCsq<6>Cs\ c } Maelstrom { ; z = sin(z) ^ sinh(z) ; Look at the wierd junk entering the whirlpool reset=1821 type=formula formulafile=kim.frm formulaname=mfn2fn function=sin/sinh corners=0.542924405/0.5484006094/3.774413923/3.775816907/0.5442224063/3.\ 772683255 params=0/0/100 float=y maxiter=500 inside=0 colors=CCC00z<7>z0z<6>z00z00<15>zz0<7>0z0<7>0zz<15>00z<7>z0z<6>z00z00<15\ >zz0<7>0z0<7>0zz<15>00z<7>z0z<6>z00z00<15>zz0<7>0z0<7>0zz<15>00z<7>z0z<6\ >z00<15>zz0<7>0z0<6>0zz<15>04z } Splat! { ; ; reset=1821 type=formula formulafile=kim.frm formulaname=mfn2fn function=sin/cotan passes=1 corners=5.6526622/5.7742505/-8.55473708/-8.46354582 params=0/0/100 float=y maxiter=500 inside=0 outside=real colors=000z0z<7>z00<5>zz0<7>0z0<7>0zz<7>00z<7>m0m<7>m00<7>mm0<7>0m0<7>0m\ m<7>00m<7>c0c<7>c00<7>cc0<7>0c0<7>0cc<7>00c<7>U0U<7>U00<7>UU0<7>0U0<7>0U\ U<6>04U00U20T<6>K0K<6>K03K00K20<5>KH0KK0IK0<6>0K0<6>0KH0KK0IK<6>00K<6>80\ CA0AD0D<13>w0wjWr } Vines { ; Bizarre Alien Plant life from a ; B rate Sci-Fi horror flick reset=1821 type=formula formulafile=kim.frm formulaname=mfn2fn function=sin/log passes=1 corners=-10.032862/-9.1796235/0.5694353/-0.5682166/-9.1796235/-0.5682166 params=0/0/100 float=y maxiter=500 inside=253 outside=mult colors=0000U0<13>0z0<46>02u00w00w<46>U0UU0UT0V<44>10v00w01v<45>0x20z00z0\ <43>0U0m00Y3uZ2t } Hope you enjoy these, please send in your own. -- Kim Best ************************* * Origamist * Rocky Mountain Cancer Data System * Are good with thier * 420 Chipeta Way #120 * Hands * Salt Lake City, Utah 84108 ************************* Path: unixg.ubc.ca!unixg.ubc.ca!news.bc.net!news.mic.ucla.edu!library.ucla.edu!agate!newsxfer.itd.umich.edu!news.itd.umich.edu!PM012-00.dialip.mich.net!asdalton From: asdalton@umich.edu (Andrew Dalton) Newsgroups: sci.fractals Subject: organic structures-- .par file Date: Sun, 12 Feb 1995 11:28:46 Organization: University of Michigan Lines: 86 Message-ID: NNTP-Posting-Host: pm012-00.dialip.mich.net X-Newsreader: Trumpet for Windows [Version 1.0 Rev A] I am interested in the ways that fractals relate to living organisms, and I have found some surprising structures in the Mandelbrot set (and its accompanying Julia sets). As far as I know, all of these are original except for the octopus. All of the flower structures have realistic phyllotaxis. For those of you who do not know, phyllotaxis involves the ways that plants arrange their leaves and petals. These tend to be arranged in spiral bands whose numbers are adjacent numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, ...) which is the most efficient arrangement, as one might expect in living organisms. "Cauliflower" looks just like the real thing; the real plant has not only 8,5 phyllotaxis but also repeated self-similarity. Most of these fractals can be generated fairly quickly, except for the sunflowers. "Sunflower2" took 5 hours to complete on a 486SX/25. Octopus { ; Mandelbrot zoom reset type=mandel passes=2 corners=-0.746975139/-0.746948462/0.098474076/0.0984940771 maxiter=2000 inside=0 logmap=yes colors=CCCICsGCsECsCCs<20>CqsCssCsq<8>Cs_CsYECE<20>sCs<20>ECsCCsCEs<19>CqsCs\ sCsq<19>CsECsCEsE<19>qsqsssssq<19>ssEssCsqC<19>sECsCCqCC<21>ACCCCC<21>sCs<17\ >KCs } Nautilus { ; Mandelbrot zoom reset type=mandel passes=2 corners=-0.76762367/-0.77014899/0.115081733/0.115788504/-0.76887231/0.116747\ 128 maxiter=1000 inside=0 logmap=yes colors=000Boz<25>zzz<46>2zz0zz0yz<45>02z00z00y<59>002000000000<29>00k00m01m<\ 29>0ky0mz1mz<3>9oz } Cauliflower { ; 8,5 phyllotaxis reset=1821 type=julia passes=2 corners=-0.718398/-0.067584753/0.10392352/0.59223706 params=-0.40820767616000003/0.6551210359716666 inside=0 colors=@BLUES.MAP } Sunflower1 { ; 89,55 phyllotaxis reset=1701 type=julia passes=2 corners=-0.79349196/0.067583824/-0.005007999/0.64106843 params=-0.39072456565125002/0.58730403248999996 maxiter=20000 inside=0 logmap=1000 colors=@volcano.map } Sunflower2 { ; 89,55 phyllotaxis reset=1701 type=julia passes=2 corners=-0.47305561/-0.26295291/0.25774024/0.41538307 params=-0.39072456565125002/0.58730403248999996 maxiter=20000 inside=0 logmap=2500 colors=@volcano.map } daisy { ; julia set--21,13 phyllotaxis reset type=julia passes=2 corners=-1.389237/1.384229/-1.042988/1.03798 params=-0.3949223450975/0.595677474375 maxiter=1000 inside=0 logmap=yes colors=000x11z00<30>zx0zz0zz1<29>zzxzzzzzz<61>zV1zU0zU0zT0<28>z10z00z00y00<3\ 0>c00b11a11`22_22<23>GEEFFFFFFFFF<28>v11 } daisy2 { ; julia set--21,13 phyllotaxis reset type=julia passes=2 corners=-0.826907/0.214445/-0.051141/0.730525 params=-0.3949223450975/0.595677474375 maxiter=1000 inside=0 logmap=yes colors=@grey.map } Shell1 { ; Mandelbrot zoom reset type=mandel passes=2 corners=-0.078649875/-0.077912094/0.658022877/0.657816993/-0.078285569/0.657\ 536978 maxiter=1000 inside=0 logmap=yes colors=@blues.map } - Andrew Dalton - asdalton@umich.edu -- --- "Faith, n. Belief without evidence in what is told by one who ----- speaks without knowledge, of things without parallel." -------- --Ambrose Bierce ------------- --------------------- ---------------------------------- -------------------------------------------------------