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Re: [Fractint] Volumetric Fractals, Julia and Mandelbrot mysteries (and a few railways)
by Morgan L. Owens 03 Aug '02

03 Aug '02

> At 02:39 02/08/2002, geoff stanton wrote: > >> I'm mostly interested in 3D graphics, surround sound and model >railways. > > >Fractal railways!! Now there's a project I'd love. Yes! With their dimensionality determined by boxcar counting! Morgan L. Owens "And with plenty of critical points."

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Re: [Fractint] Volumetric Fractals, Julia and Mandelbrot mysteries
by Franktal Gallery 02 Aug '02

02 Aug '02

>>>http://mschreier.tripod.com/ Marc, very interesting page... >From your page : "The Mandelbrot set is an index for Julia sets, but now it was demonstrated the Mandelbrot set is in fact a picture made of Julia sets. I swear, I didn't take a mosaic application to use Julias matching the Mandelbrot, all of it was done buy pure calculation using Fractint!!!" Yeah, we believe you ! ;-) >>>http://mschreier.tripod.com/english/jam/pic/julia_band2d.png This can be also achieved with Fractint, by using the evolver feature. I don't have it installed right now, but I remember a post a while ago with the specific instructions to get a picture just like that one. Somebody in the list will probably find out the required evolver settings, in a matter of microseconds. Best regards, Miguel Fliguer - Buenos Aires, Argentina Franktal Gallery - http://www.geocities.com/franktal2002 >From: "Marc Schreier" <marc.schreier(a)gmx.de> >Reply-To: fractint(a)mailman.xmission.com >To: <fractint(a)mailman.xmission.com> >Subject: [Fractint] Volumetric Fractals, Julia and Mandelbrot mysteries >Date: Wed, 31 Jul 2002 22:41:41 +0200 > >Dear Fractint users, > >Ever wanted to see fractals in 3D? Then use Fractint's built in stereo >support. >But there is an interesting alternative: volume rendering!!! >A friend of mine is working with volumes in his PhD thesis and I did some >nice datasets just for fun - and to explore fractals in a very different >way. >While my computer calculated lots of Julia sets I also discovered a >surprising >phenomena, but see yourself :) >You are welcome to visit my latest work, the "Julia and Mandelbrot" page, >at http://mschreier.tripod.com/ > >To my person: > >Actually I'm living in south of Germany, I'm working for a local >car supplier compagny, and just got the MSc in Distributed Computing >Systems Engineering at the Brunel University (Uxbridge, West London). >I was exploring fractals since 20 years, first on the good old C64 >(remember the Mandelbrot renderer that used also the 1541 floppy CPU?), >then with Fractx86 and later with Fractint. >I'm mostly interested in 3D graphics, surround sound and model railways. > >Best regards > >Marc > >--------------- >JX-8P Resource Center >http://www.synthzone.com/~mschreier/ >--------------- > > >_______________________________________________ >Fractint mailing list >Fractint(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint _________________________________________________________________ MSN Photos is the easiest way to share and print your photos: http://photos.msn.com/support/worldwide.aspx

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FOTD 02-08-02 (Twirling Banners [6])
by Jim Muth 02 Aug '02

02 Aug '02

FOTD -- August 02, 2002 (Rating 6) Fractal visionaries and enthusiasts: At first glance, today's image looks like an exotic midget created by the MandelbrotMix4 formula, but it is not. The image, with it's almost-disconnected appearance, is actually a scene in our faithful old friend, the Mandelbrot set. The scene is located deep on a filament in the northeast suburbs of the large midget on the negative stem of the M-set. The exact location can be found in a few seconds by outzooming. As one backs out, the magnitude soon falls within the range of conventional math, so the trip back to the surface is a fast one. Today's scene almost appears disconnected, but it is not. All parts of the Mandelbrot set, including the midgets and the elements around them, are connected to the main body of the set. The shoreline of every midget and the main body as well is a single unit. True, the connection is often made by an infinitesimally thin filament, but the connection is there. By comparison, many fractals created by the MandelbrotMix4 formula are not connected. In these fractals, there frequently is no main body, but only a cloud of fractal debris scattered around the area like ejecta around an explosion. And like at the scene of an explosion, isolated bits of debris are often found far from the center, totally disconnected from the rest of the fractal. I named today's definitely connected fractal "Twirling Banners" when I was reminded of synchronized gymnasts twirling their banners. If it were not part of the M-set, today's image might have rated a 7. As it is, a rating of a slightly-above-average 6 will have to suffice. Since the image is in the magnitude range where arbitrary precision math is required, it takes over 22 minutes to render on my P200 machine. A download of the completed GIF image from: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> will be considerably faster. The weather Thursday here at Fractal Central was of the typical high-summer variety -- a temperature of 97F 36C, blazing sun, high humidity, and light winds. In the evening, thunder clouds developed on the horizon, but never moved in. These conditions kept the fractal cats confined to the cool indoors through the afternoon. In the evening, when the sun sank low and they finally did venture outdoors, the local catbird began annoying them, apparently aware that their bird-catching days are nearly finished. And now it's Friday morning -- another hot day -- and I've got a pile of work to finish. It's time to reconfigure the fractal shoppe into a workroom and get busy. Until next FOTD on the 4th of August, take care, and fractals are too complex to be natural, yet they are not artifacts. So what are they? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START PARAMETER FILE======================================= Twirling_Banners { ; time=0:22:51.60--SF5 on a P200 reset=2002 type=mandel passes=1 center-mag=-1.74644456175032895642/0.0051806197286\ 35784144/3.752443e+015/1/-122.5/3.8857805861880478\ 9e-016 params=0/0 float=y maxiter=6000 bailout=8 inside=0 symmetry=none periodicity=10 sound=off colors=000ho_eoXcoU`nQZnNZqNXnKWdHVVFUTFUSFURFUQFU\ PFUOFTNFTMFTKFTJFTIFTHFTGFSFFSEFSDFSCFSAFS9FR8FR7F\ R6FR5FR4FR3FR2FV5KY8O00000000000000000000000000000\ 0000000J0IJHFIEBIB8I84I51K66M7BN8GP8KQ9PSAUUBZVBdX\ CiYDn_Er`Et`Hr`Ko`Nl`Pj`Sg`Vd`Xa`__`bX`dU`gR`jP`lM\ `oJ`rG`tEVfDQTDKFDF2DH4CI6CJ8CKABLCBNEBOGBPIAQKARM\ ATO9UQ9VS9WU9XW8ZY8__8`a7ac7be7dj5cg7cd8cbAc_BcYDc\ VEbSGbQHbNJbLKbIMcENbGNaINaJN`LN`MN_ON_PNZRNZSNYUN\ YVNXXNXYNW_NW`NYbN_cNaeNcfNfhNiiNmkNplNpnNuoNzqNzr\ NzpMpoLjnKdmKalJajIbiIbhHcgGdfGfdFgcEibEjaDl`Cm_Cl\ aGkcKjdOifShhWhi_gkcfmgenkdpocrscswanp`iiZdcY_XWVR\ VQKXIBULESNGQQIOSLMUNKXPIZSGaUEcWCeZAh`8jb6ld8jcAi\ bChaDg`Ff_HeZIdYKcXMbWNaVP`UR_TSZSUYRWXQXWPZVO`UNb\ TMcSLeRKgQJhPIjOHlNGmMFoLEqKDrJCtIBvHAwG9vFBuFDtEE\ sEGrDHqDJpCLoCMnBOmBPlBRkATjAUi9Wh9Xg8Zf8`e7ad7cc7\ dd8cd9be9aeA`fA_fBZgBZgCY } END PARAMETER FILE=========================================

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Volumetric Fractals, Julia and Mandelbrot mysteries
by Marc Schreier 01 Aug '02

01 Aug '02

Dear Fractint users, Ever wanted to see fractals in 3D? Then use Fractint's built in stereo support. But there is an interesting alternative: volume rendering!!! A friend of mine is working with volumes in his PhD thesis and I did some nice datasets just for fun - and to explore fractals in a very different way. While my computer calculated lots of Julia sets I also discovered a surprising phenomena, but see yourself :) You are welcome to visit my latest work, the "Julia and Mandelbrot" page, at http://mschreier.tripod.com/ To my person: Actually I'm living in south of Germany, I'm working for a local car supplier compagny, and just got the MSc in Distributed Computing Systems Engineering at the Brunel University (Uxbridge, West London). I was exploring fractals since 20 years, first on the good old C64 (remember the Mandelbrot renderer that used also the 1541 floppy CPU?), then with Fractx86 and later with Fractint. I'm mostly interested in 3D graphics, surround sound and model railways. Best regards Marc --------------- JX-8P Resource Center http://www.synthzone.com/~mschreier/ ---------------

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FOTD 30-07-02 (Aquatic Life [8])
by Jim Muth 30 Jul '02

30 Jul '02

FOTD -- July 30, 2002 (Rating 8) Fractal visionaries and enthusiasts: Today's fractal reminds me of another real-world object. In this case, I see the graceful swaying of seaweed, a few meters under the surface near some tropical shore. The scene is also filled with schools of small, colorful tropical fish, though these are more difficult to pick out. With all this underwater activity in mind, I named the image "Aquatic Life". Then I stepped back to evaluate the result. After much mental wrestling, I rated the image a much-above- average 8. The formula of the aquatic scene is Z^(-1.1)+Z^(-11)+(1/C), a simple expression that does more than it should. But as I stated recently, it is not the formula that works the magic; it is the 280 iterations. The parent fractal of today's scene appears as a large open bay with many buds and valleys around its perimeter. Today's scene is located in the center of a blunt valley on the south shore of the main bay. The aquatic effect was achieved by rendering the scene with the outside set to the <tdis> option, followed by careful coloring. The render time of just over 1 minute makes running the attached parameter file the quickest way of seeing the seaweed. But downloading the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> might be a lot more fun. The weather Monday here at the (IMO) center of the fractal world was hot, hot, hot. A breeze and slightly lower humidity made it feel less oppressive than it otherwise might have been, but 100F 38C is still hot -- there's no escaping it. The heat was so severe that it buckled the rails of a local rail line, causing a train to derail. The dynamic fractal cats avoided the excessive heat by passing the day in the air-cooled indoors, though they did check several times just to be sure it really was that hot outside. Today is Tuesday, and it's another scorcher -- a perfect excuse for me to stay indoors and get a little ahead on the upcoming convention program that needs to be finished within a week. Until next time, which will come in 3 days on August 2, take care, and why are tropical shores with all their humidity and heat considered paradise on earth?. Jim Muth jamth(a)mindspring.com START 20.0 PAR-FORMULA FILE================================ Aquatic_Life { ; time=0:01:27.77--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+0.856206964880088/-1.054799381804275/2\ 124528/1/-160/-3.37734701108538538e-008 params=1/-1.1/1/-11/0/525 float=y maxiter=280 inside=0 outside=tdis periodicity=10 colors=000z01z03y04u04o06j06f07`07X09S09M0AI0AD0C7\ 0C30D00D13I47M7CQAGTDLYGPaJTfLYiOamQfrTjvXoy_szaxz\ czzUzzKzz7zz6vz4lz3`z1Qv0Gs6FuCDvIDxOCyTAzYAz`CzcF\ zfGziIzlLzoMzrOzuQzxSzyTzxVzxXvxYrvYmv_iv`dv``uaXu\ cSudOsdJsfFsgAsg6mc9g`CcYFYVGTSJOOMJLPDIQ9FT3CX09Y\ 6C_DD_LF_SG_TI_TPcPXgLaiGikCmaJqSPu0Mz6LuAJoGGjLFd\ QD_VAT`9Pd7Jj4Do37u13y00z00z00z10z60zC0xG0sM1oQ4iX\ 7d`A_fDVjGQpJLuMGzPAzUUzc1zmYzuVzzSzzQzzOzyMzxJzvI\ zuFzsDzrAzp9zo6zm4yl1yj0xl3vl6vl9umCumFsmGroJroMpo\ PppSopVopXmrUlsUluUjuUjvUitUimUmcUqUPvCdz0sv4osCjo\ JflQaiYYddValQ_sMVzISzDPzATzMXsX`jfc`pgQzjIzgMzdPz\ cSx`Vs__oXajVdfSgaQjYYXcdIgl4lgAicGg_MfVSdQYaMc`Ii\ _DoY9uV4zT0zS0zQ0zV0z_0za0yf0vj0sm0pr0mv0jy0gz0dz0\ az0_z0Yz0Vz0Tz0Qz0Pz0Mz0Ly0JxIiTJfOJdILcDLa7M`1M_0\ OY0OX0ld3sXJzQYy6ls0yo0xl0xi0xf0va0v_3vX6uT9uPCuMF\ sJIsGLsCi09i07i0`_7zPTz9F } 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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FOTD 28-07-02 (The Helm [7])
by Jim Muth 28 Jul '02

28 Jul '02

FOTD -- July 28, 2002 (Rating 7) Fractal visionaries and enthusiasts: I named today's 7-rated image "The Helm" because of the resemblance it bears to a ship's wheel. Almost all my FOTD fractals are named after a resemblance they bear to a real-world object. This tendency to find the familiar in the unfamiliar is a universal human trait, perhaps best illustrated by the "faces in the clouds" illusion. No ship's wheel exists in today's image, which is merely a mapping to the screen of the number of iterations required before the iterated points of the image pass beyond the bailout radius. The coloring does not exist in the formula, but was consciously added by myself to enhance the resemblance to a wheel. Today's image is in fact nothing but one of the many ways in which the basic underlying mathematical abstraction could have been made manifest. The same formula could just as easily have created a piece of fractal 'music'. (Though I have yet to hear a piece of fractal 'music' that was worth being listened to more than once. [Or even once, for that matter.]) Today's generating formula -- 1.5(Z^1.1)+1.5(Z^11)+(1/C) -- is unusually simple. This is not surprising. When working with fractals, long convoluted formulae are unnecessary, and in fact often not worth the effort of writing them. Complex formulae are not needed, because the iterations work the magic. Great images come from skillful searching, not necessarily from 200- line-long formulae. The above statements are my personal opinion of course, especially the statement about fractal 'music' being worthless. Since beauty exists in the mind of the beholder, it may be that my mind is not tuned to the mental frequency of the appreciation of fractal 'music' and all the beauty in the music passes right through me like light through the invisible man. Forgetting the worth or non-worth of fractal music, the render time of today's image is a fast 3 minutes on my brave little P200 fractal machine. Approximately the same time will be required to download the completed GIF image from Paul's FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's FOTD site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The route to enjoyment of the image is left to the viewer. The weather Saturday here at Fractal Central was not perfect, but it could have been a lot worse. The temperature was held to a muggy 83F 28C by heavy clouds, which threatened all day but produced only a few drops of rain. The cats, who dislike high humidity, briefly checked the outdoor conditions, then decided to spend the day indoors. And I'm about ready to spend today relaxing. Until next time, take care, and where is that "Theory of Everything for Dummies" book? Jim Muth jamth(a)mindspring.com jimmuth(a)aol.com START 20.0 PAR-FORMULA FILE================================ The_Helm { ; time=0:03:11.41--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-6.14601973153377700/-1.981123197536113\ 00/1.483931e+010/1/-28.746056810576011/-0.02765394\ 21627924401 params=1.5/1.1/1.5/11/0/0 float=y maxiter=900 inside=0 logmap=63 periodicity=10 colors=0006AI7AJ8AK9APAJUBMXCNcvl4kc8UNCzu8faBTODm\ A3`B7KBBk_ocTbWNVOHNb`dVSWNJNzfvbR`labcVXWOROHLu9x\ gAhUBUOZ9KNCDwBEcDFPErCkeB`TBQg_t`TiUN_NHP80PB4LE8\ I2Y_z8cj9X`ARQBLfl1FwzGfjGRVvrt1GQ6EMBCI6Y8AQBDIDq\ 1lnkYbZRRNLkwd`fWQRNLThIM6HI9GECHtkGYVY1dP6S9iPDSK\ RPJNKHJFG9W6CP9EIC1_p5Tf9NYDHOA3HD7G4q0jFk_D`QCQWr\ 8SgAOXCKMEFYGGMFKWYILOE6zF9bGDMmYVXMNv9H6HSBELsoZi\ dU_VPQLK9YikQ3WI9T1YjhRVSL24e77XC9O0eV8QNR_rOThLNZ\ IHPKZeIRXHJOrUOkpT`bOQPJYv_TjUPZPKNK__JTRHMJG5wdBZ\ SCQJDMIEIHFEG4pd7eYAWSDLLJOCfEM69P9AMBAKEBHTaSOTNK\ KJavGUfFNRFNboA5NC7KE9HdwuYjjSZ_MNP6JxBFa9nFDVFONZ\ Vy4Uba_h4TX8MMCP4MM6KK8IIAG5wK8jIBZHENGjwlvfomiddl\ UWoJNq8OuRPxhMflJdpGesHfjIgaIhUUiWejXqkYXl`CmbroaZ\ qQ4szAub1wKzyjnzbbzVRzNqz5gz8ZzAPzDzzPbzKQz83zK7zI\ AzHDzGYzBSzDMzEDzNszH_zG2zm7zaCzQxzflz_azTRzMOzELz\ FIzFQz2Nz6Lz9IzC6ztBz_OzW } 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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FOTD 26-07-02 (Quadrapod Minibrot [8])
by Jim Muth 26 Jul '02

26 Jul '02

FOTD -- July 26, 2002 (Rating 8) Fractal visionaries and enthusiasts: After a long fractal drought, we finally have a fractal that rates an 8, a rating that equals much above average on my conservative 0-to-10 scale of fractal worth. Actually, the fractal itself is quite humdrum in the mathematical sense. Most of the high rating is based on the coloring, which in a moment of excessive fractal zeal, I devoted an entire half-hour to. The parent fractal of today's scene consists of a Mandelbrot figure with several large irregular open holes at the end of some of its filaments, and strings of smaller holes at the ends of others. There is also a string of holes extending outward from the center of East Valley. Today's image lies in an isolated, cut-off feature near this string of holes. The string-of-holes nature of the area in which today's image lies is reflected in the four strings of holes radiating from the brilliant red flower around the central midget. Since there are four of these 'arms', I named the image "Quadrapus Minibrot". (I have no idea what a real 'quadrapus' might be, though the fractal cats might fit the description.) The formula of the image uses the golden ratio, 1.618..., which often appears in the designs of nature. The ratio also appears at places in the Julibrot, where least expected, but that's another story for another day. The render time of 3 minutes from the parameter file is close to the time required to go online and download the completed GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> or from Scott's site at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> The fractal weather Thursday here at Fractal Central featured a pleasant break from the oppressive heat and humidity that usually plagues this part of the world at this time of year. Lots of clouds and a cool breeze off the ocean held the temperature to 79F 26C. The grass, still wet from an early morning shower, was a bit too uncomfortable for the dynamic duo, who prefer their paws dry, but they did spend a few relaxing hours on the dry porch, watching the local birds. And I'm also going to relax -- but not until I finish the work that needs to be done. Until next time, on Monday the 28th, take care, and keep the ratio golden. Jim Muth jamth(a)mindspring.com START 20.0 PAR-FORMULA FILE================================ Quadrapus_Minibrot { ; time=0:03:39.53--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=-2.85097707624026200/-3.007861364773440\ 00/5821280/1/-115/1.56494616987412627e-008 params=1.5/1.618/1.5/16.18/0/0 float=y maxiter=400 inside=0 logmap=71 periodicity=10 colors=00020K10K00K40MBAQI9WPMcXcadt_lzZtzXlzUdvRX\ qOPlLKgIPcFU_C_W9eR7lM4sI1yE0z90zd0zZ0yS0sB0nB2gC5\ cE7XF9TFCMHEIIHCKK8KL7PH5WCBTEHRFLOFRMHXLHaIIgHIlE\ KsCKyBLz8Lz7Mz5Mz8Pz9RzCTwEUtFWoIXlKZgMadOc_PdXTeT\ UgPXiLZjI_lIZlIWlBTg4Pc0PZ0PU0tU0PW0PW0PW0PWIIa09W\ 0BW0BX0BX0CX0CX1EX1EZ1EZ1FZ2FZ2FZ2HZ2H_4I_4I_4I_5K\ _5Ka5La5La7La7Ma7Ma7Mc8Oc8Oc8Pc8Pc9Pd9Rd9Rd9RdBTdB\ TdBUeBUeCUeCWeCWeCWgEXgEXgEZgEZgFZgF_iF_iF_iHaiHai\ HdjHggIjdLnaOqZRtWUtTUtPUtITtBTt4Tt1Rt2Rt5Rt7Pt9Pt\ BPtEOtFOqIOnKMjMMgOMdRLaTLZWLWXKW_KXaK_dPacXcadd_l\ dZtdXzcWzcUycTwcTtcUscWqcWncXlcZicZgd_dd_cda_dcZdc\ XddUdeTdePddXaXdZPlWItTBwR4vT0vU0tW0tX0sZ0s_0qa0qc\ 0od0ne0ng0li0lj1jl1jn2io4iq5gs7gt8et9dt9dtBctCctEa\ tFatH_tI_tKZtKXtLXtMWtOWtPUtRUtTTtTTtURtWRtXPt_Otc\ OteMtiMtlLtoLtsLtvKtyItzKtzItzItzItzItzItzItzItzIt\ zItzItzItzItzItzItz5tq4to } 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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xfractint now available for Gentoo Linux
by James S. Martin 26 Jul '02

26 Jul '02

Folks, I took it upon myself to write an ebuild for xfractint(development version 20.02.4, no extra map files) for Gentoo Linux . After obtaining the latest updates by doing an "emerge rsync" you should be able to type "emerge xfractint" and watch it download, compile, and install xfractint for you, with your CPU optimizations. James

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Re: [Fractint] Yet another trivial...
by Jim Muth 25 Jul '02

25 Jul '02

While toying with the Lyapunov Mandelbrot formula recently posted by Morgan Owens, I noticed that the formula produces images virtually identical to those drawn by the Fractint type=hypercomplex Mandelbrot formula. I have included two parameter files, one of which uses the Lyapunov Mandelbrot formula, while the other uses the hypercomplex Mandelbrot formula, yet the two files draw identical images. Perhaps one of the math experts on the list can explain what is happening here. . . . Jim Muth jamth(a)mindspring.com START PARAMETER FILE======================================= Lyapunov { reset=2002 type=formula formulafile=allinone.frm formulaname=LyapunovMandel center-mag=-1.07668/7.77156e-016/1.344086 params=0/0/0/1/16/0 float=y maxiter=500 inside=0 logmap=yes periodicity=0 sound=off colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\ zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\ 00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\ VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\ lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\ lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\ E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\ OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\ OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\ SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\ G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\ G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\ G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\ GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\ G000000000000000000000000 } Hypercomplex { reset=2002 type=hypercomplex function=sqr center-mag=-0.5759/7.77156e-016/1.344086 params=0/0/0/0.5 float=y maxiter=500 bailout=7 inside=0 logmap=yes periodicity=0 sound=off colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\ zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\ 00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\ VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\ lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\ lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\ E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\ OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\ OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\ SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\ G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\ G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\ G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\ GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\ G000000000000000000000000 } frm:LyapunovMandel { narg=real(p2) nmag=imag(p2) bailout=4*(real(p3)==0)+real(p3)*(real(p3)!=0) z0=c0=pixel z1=c1=pixel+nmag*(cos(narg)+(0,1)*sin(narg)) : z0=z0*z0+c0 z1=z1*z1+c1 cabs(z0-z1)<bailout } END PARAMETER FILE=========================================

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Yet another trivial messing around with the Mandelbrot set accompanied by a long-winded (yet on-topic) intro.
by Morgan L. Owens 25 Jul '02

25 Jul '02

At least there are only two formulae this time, rather than thirty-two. As you are no doubt all aware, one of the defining characteristics of chaotic systems is that they have a property known as "sensitive dependence on initial conditions". Two different starting states for a given system, however close to each other they may be initially, would eventually diverge and produce entirely different histories for the system's evolution. Equivalently, any inaccuracy in measuring or specifying the initial state of the system would eventually be magnified to the point that the inaccuracy dominates and the system could be in virtually any possible state. Usually, the rate of divergence is exponential - i.e. the difference between the two states is multiplied by some factor every iteration or unit of time. Measuring the rate of divergence hence provides a way of helping identify chaotic systems. This is the origin of the Lyapunov exponent, developed by Aleksandr Lyapunoc (or Liapunov or Ljapunov or whatever other spellings can result from a Cyrillic->English transliteration.) in the late nineteenth century. To avoid describing Jacobian matrices and characteristic values, I'll just give an explicit definition in the case of a one-dimensional system, where the state of the system at any instant is described by a single number: n L = lim (1/n)SUM log|df(x )/dx| n->inf i=1 i Where f is the dynamical rule in question; x_0 is the initial state; and x_1, x_2, ... are subsequent states. L is the Lyapunov exponent. Roughly it says we measure by how much each step magnifies (or shrinks) the divergence at the previous step and average these measurements over larger and larger numbers of iterations, continuing until the average settles down to a steady value. If L is less than zero, then the divergence becomes a _con_vergence and the system has a periodic cycle. If it's equal to zero, the system is described as Hamiltonian; an example would be the system x = x+1 - everything just cruises along. If L is _greater_ than zero - for many, the most interesting situation - then one has a chaotic situation. Higher-dimensional systems have several Lyapunov exponents, and determining them and seeing which are larger and which are smaller, which are positive and which are negative, one can learn quite a few things about the system's sensitivity to initial conditions. Interesting side note: When f is one-dimensional which depends on a given parameter a, then if there is a particular value for the parameter - call it a_inf - at which the behaviour of f becomes chaotic (i.e., where L becomes positive), it can be shown that log|2|/log|delta| L approx. |a - a | inf Where delta = 4.6692016.... Now that I've gone into all that detail about what the Lyapunov exponent is, you can safely ignore it, because most of it is purely background justification for my formulae. Someone else might like to implement the Lyapunov-calculation code in a Fractint formula, so that, e.g., one could have the pixel value specify the parameter to a function, and have Fractint colour that pixel according to the value of L; but all I do here is the perturbation thing. For each pixel, I start the usual quadratic map iteration (z->z^2+c) both for the pixel value itself and a perturbed version, the perturbation being specified in polar coordinates by p2. The bailout condition is simply whenever the two orbits diverge by a sufficient minimum amount, specified in p3 (which defaults to 4 if the user doesn't say otherwise). And as I found with my initial random explorations, they can be _very_ sensitive to different perturbations. Gee, all I really needed to write was that last paragraph; but that wouldn't have been anywhere _near_ as informative. Morgan L. Owens "As an aside, it's tempting to speculate that the generic quadratic behaviour of smooth functions ensured by Morse's Theorem accounts for the fact that almost all the basic laws of physics (conservation of energy, Fermat's principle, and the principle of least action, for example) are expressed as quadratic forms. Since it's both stable and 'typical' to be locally quadratic, it's reasonable to expect that the local laws of classical physics will also be expressible as quadratics, as indeed they are." Lyamand_1 { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=lyamand.frm formulaname=LyapunovMandel passes=1 center-mag=-0.162602/0.700584/5.464481 params=0/0/1/0.1/0/0 float=y maxiter=255 inside=255 logmap=5 periodicity=0 colors=000zzz<42>zzBzz9zz8<3>yy4yy4yx4<72>yX0yX0yW0<127>000 } LyapunovJulia_1 { ; Version 2001 Patchlevel 8 reset=2001 type=formula formulafile=lyamand.frm formulaname=LyapunovJulia center-mag=+0.46966854283928040/-0.00187656380316929/57.97101 params=-1/0/3.14/0.4/0/0 float=y maxiter=1023 inside=255 logmap=36 periodicity=0 colors=000<42>00s00t00v00w00x00z<41>0qz0rz0sz<3>0yz<9>bymfyljyk<3>zye<9>\ zyGzyEzyB<3>zy0<41>z80z70z50<2>z20z00y00<24>T00S00R00<3>L00<9>jaDleEoiF<\ 3>zyL<9>z_8zY7zV6<3>zK0<11>D50930520000 } frm:LyapunovMandel { narg=real(p2) nmag=imag(p2) bailout=4*(real(p3)==0)+real(p3)*(real(p3)!=0) z0=c0=pixel z1=c1=pixel+nmag*(cos(narg)+(0,1)*sin(narg)) : z0=z0*z0+c0 z1=z1*z1+c1 cabs(z0-z1)<bailout } frm:LyapunovJulia { narg=real(p2) nmag=imag(p2) perturb=nmag*(cos(narg)+(0,1)*sin(narg)) bailout=4*(real(p3)==0)+real(p3)*(real(p3)!=0) z0=pixel c0=p1 z1=z0+perturb c1=c0+perturb : z0=z0*z0+c0 z1=z1*z1+c1 cabs(z0-z1)<bailout }

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