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Path: unixg.ubc.ca!vanbc.wimsey.com!scipio.cyberstore.ca!math.ohio-state.edu!uwm.edu!hookup!olivea!uunet!mr.net!dawn.mmm.com!newsdist.tc.umn.edu!news.d.umn.edu!ub.d.umn.edu!dfintonFrom: dfinton@ub.d.umn.edu (david finton)Newsgroups: sci.fractalsSubject: A deep-zoom fractal contest?Date: 15 Mar 1995 03:49:19 GMTOrganization: University of Minnesota, DuluthLines: 29Message-ID: NNTP-Posting-Host: 131.212.134.2X-Newsreader: TIN [version 1.2 PL2]I've been fooling around with the deep zooming mode available on Fractintv 19.0 and I love it. I've been able to view pictures that I couldn'thave possibly seen on my computer before I got it. While it takes alittle while longer to calculate the images, they are well worth it.Tim Wagner proposed a deep zooming fractal contest. I want to submitthe following .par file to the contest. It's not a very deep picture (Ittook only two hours to calculate on a Pentium), but I like it and I wantto share it anyway. Here it is:---- Cut Here ---------------------------------------------------------

Flower { ; Swirls within swirls, ad infinitum; (When is anything in the Mandelbrot set NOT; ad infinitum?)

reset=1900 type=mandel passes=tcorners=-1.74731064714761698304/-1.74731064714761683396/-5.5936523e-17/5\.5936523e-17 params=0/0 float=y maxiter=20000 inside=0 logmap=-1050symmetry=xaxis

colors=000I00L00O20fE0jG0kI0zz0W00C00z0000000z\W51X60X60X70yxFyyFzzGzzHzzz0z0z00

}----- Cut Here -----------------------------------------------------------Enjoy!

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Path: unixg.ubc.ca!nntp.cs.ubc.ca!newsxfer.itd.umich.edu!agate!tcsi.tcs.com!uunet!in1.uu.net!ankh.iia.org!birdsehFrom: birdseh@iia.org (Henry Birdseye)Newsgroups: sci.fractalsSubject: 10^233 magnificationDate: 5 Jul 1995 00:26:50 GMTOrganization: International Internet Association.Lines: 29Message-ID: NNTP-Posting-Host: iia.orgX-Newsreader: TIN [version 1.2 PL2]Here is a huge magnification .par file I've been zooming for a while.Cut here ----------------------233 { ; mag = 10^233

; This takes a while; Henry Birdseye - birdseh@iia.org; Mail me your deep .par files

reset=1920 type=mandel passes=2center-mag=-0.7320299298281361984552930152710573055695167223049229392827\649564115771810297845444840812404043441296278932140339697145134072822677\086003318252081469770916042159178755967032000445188081331280125432690897\

715003022308347206188135908325909287/0.362254943051205664189634244517662\428237917023435025634249637350403140930892080222768288921485422473252016\083128169244094674454003420702120965466655270667981376175896195517699326\5252623488386153939460345614941145789560489349509360628096195/1.306558e+\233 params=0/0 float=y maxiter=999990colors=000zbOzv4wz07z00z30zs3wzw3zz2xzlEzt6zy1qz02z00z\90zx7szkFzs7zz0zzeLzjGznCzt6zy1qz02z00z90zx7szkFzs7zz0\zzeLzjGznCzt6zy1qz02z00z90zn60tz00z60zy0tz0Hz0Ex50g\Z0Sh0Jn08u20xv04z20zm0zu0xz0Tz0Lz0Fy22jW0bb0Sh0Jn08u20xv0\4z20zm0zu0xz0Tz0Lz0Fy22jW0bb0Sh0Eq03x60tz00z60zy0tz0Hz\0Ex50gZ0Ye0Nk0Eq03x60t3wzw3zz2xzYT}

------------------------- END

Let's see some more deep zoomers!-- Henry B.

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Path: unixg.ubc.ca!vanbc.wimsey.com!news.cyberstore.ca!math.ohio-state.edu!howland.reston.ans.net!news.sprintlink.net!gryphon.phoenix.net!newsFrom: Twegner@phoenix.net (Tim Wegner)Newsgroups: sci.fractalsSubject: Re: 10^233 magnificationDate: Thu, 06 Jul 1995 15:08:29 GMTOrganization: Phoenix Data SystemsLines: 86Message-ID: References: Reply-To: twegner@phoenix.netNNTP-Posting-Host: dial56.phoenix.netX-Newsreader: Forte Free Agent v0.55birdseh@iia.org (Henry Birdseye) wrote:>Here is a huge magnification .par file I've been zooming for a while.This is an impressive accomplishment, and I commend you for your diligence.However, you too have now joined the "Deep Zoom Whirlpools of SimilarityVictims Club", of which David Chapman and I are charter members,The problem with relatively blind deep zooming is that there is a danger of

creating deep zoom images that are essentially similar to much shallowerzooms. The appeal of deep zooming is at least partly the hope of discoveringsome new images not visible at shallower depths. If this is your motivation,then you must strive to avoid the dreaded "whirlpools of self-similarity"that are legion in the Mandelbrot. The only technique I know for avoidingthis horrible fate is to search for baby Mandelbrots and leap from babyMandelbrot to baby Mandelbrot while zooming. This technique is unfortunatelymuch more time consuming than the easier (but more dangerous for thosefearing whirlpools) technique of zooming into any "fractally" looking area.My procedure for testing the condition of being trapped in such a whirlpoolis as follows. First, I generate the deep zoomed image at as low aresolution as possible to save time, requiring only enough detail to see the

main structure. Then I make a copy of the deep zoom PAR entry in center-magform, and edit the magnification to bring it within range of normal doubleprecision (usually about e+14 or so). I generate the much-shallower-zoomedimage, and look for structures resembling the deep zoom. After framing thestructure as accurately as possible, I then rotate the colors to get anapproximate match. This whole procedure usually takes only a few minutesafter generating the deep zoom, which is the sloweest step.

Such quick success at finding similar structures at such vastly differentmagnifications (greatly exceeding the ratio of the size of the visibleUniverse to tiny sub-atomic quantum effects which is a mere 1.0e+61) ispretty good evidence of self-similarity.

The results for the "233" zoom are as follows. The PAR below includes a copyof your deep zoom as well as my shallower zoom. I have only tried to matchthe framing and colors enough to convince myself of self-similarity; I'm surea better match could be achieved with a little care.I challenge Henry and other sci.fractals readers to produce a zoom this deep(e+233) that is *not* a self-similar copy of a much shallower image at thesame location.Keep on deep zoomin'!

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Tim

233 { ; mag = 10^233; This takes a while; Henry Birdseye - birdseh@iia.org; Mail me your deep .par files

reset=1920 type=mandel passes=2center-mag=-0.7320299298281361984552930152710573055695167223049229392827\649564115771810297845444840812404043441296278932140339697145134072822677\086003318252081469770916042159178755967032000445188081331280125432690897\715003022308347206188135908325909287/0.362254943051205664189634244517662\428237917023435025634249637350403140930892080222768288921485422473252016\083128169244094674454003420702120965466655270667981376175896195517699326\5252623488386153939460345614941145789560489349509360628096195/1.306558e+\233 params=0/0 float=y maxiter=999990colors=000zbOzv4wz07z00z30zs3wzw3zz2xzlEzt6zy1qz02z00z\90zx7szkFzs7zz0zzeLzjGznCzt6zy1qz02z00z90zx7szkFzs7zz0\zzeLzjGznCzt6zy1qz02z00z90zn60tz00z60zy0tz0Hz0Ex50g\Z0Sh0Jn08u20xv04z20zm0zu0xz0Tz0Lz0Fy22jW0bb0Sh0Jn08u20xv0\4z20zm0zu0xz0Tz0Lz0Fy22jW0bb0Sh0Eq03x60tz00z60zy0tz0Hz\0Ex50gZ0Ye0Nk0Eq03x60t3wzw3zz2xzYT}

233-shallow { ; Arrgggh! A much shallower copy of Henry Birdseye's; "233" image. Self-similarity strikes again.; Tim Wegner

reset=1920 type=mandelcenter-mag=-0.7320299298280675908/0.3622549430511640843/1.1816e+014params=0/0 float=y maxiter=999990colors=000Q0`v04z20zm0zu0xz0Tz0Lz0Fy22jW0bb0Sh0Jn08u20xv0\4z20zm0zu0xz0Tz0Lz0Fy22jW0bb0Sh0Eq03x60tz00z60zy0tz0Hz\0Ex50gZ0Ye0Nk0Eq03x60t3wzw3zz2xzYTzbOzgJzlEzq9zv4wz07z00z30zs3wzw3zz2xzlEzt6zy1qz02z00z90zx7szkFzs7zz0zzeLzjGzn\Czt6zy1qz02z00z90zx7szkFzs7zz0zzeLzjGznCzt6zy1qz02z00z90zn60tz00z60zy0tz0Hz0Ex50gZ0Sh0Jn08u20xA0pI0h

}

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Path: unixg.ubc.ca!news.bc.net!info.ucla.edu!library.ucla.edu!agate!news.duke.edu!godot.cc.duq.edu!newsfeed.pitt.edu!dsinc!netnews.upenn.edu!mipg.upenn.edu!deweyFrom: dewey@mipg.upenn.edu (Dewey Odhner)Newsgroups: sci.fractalsSubject: Re: 10^233 magnificationDate: 10 Jul 1995 13:08:41 GMTOrganization: University of PennsylvaniaLines: 15Distribution: worldMessage-ID: References: NNTP-Posting-Host: mipgsun.mipg.upenn.edu

In article , Twegner@phoenix.net (Tim Wegner) writes:|> I challenge Henry and other sci.fractals readers to produce a zoom this deep|> (e+233) that is *not* a self-similar copy of a much shallower image at the|> same location.Horiz.Hold { ; by Dewey Odhner. Public domain.

reset=1920 type=mandelcenter-mag=-1.9999999999999999999999999999999999999999999999999999999999\999999999999999999999999999999999999999999999999999999999999996416785918\148134530430458843562012392471211940405947277366380516671161895619483871\9021305389817240974903756614325041160443974/0/3.3e+242params=0/0 float=y maxiter=3000 inside=0}

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Path: unixg.ubc.ca!info.ucla.edu!library.ucla.edu!agate!howland.reston.ans.net!news.sprintlink.net!gryphon.phoenix.net!newsFrom: Twegner@phoenix.net (Tim Wegner)Newsgroups: sci.fractalsSubject: Re: 10^233 magnificationDate: Tue, 11 Jul 1995 04:35:33 GMTOrganization: Phoenix Data SystemsLines: 24Message-ID: References: Reply-To: twegner@phoenix.netNNTP-Posting-Host: 199.3.234.129X-Newsreader: Forte Free Agent 1.0.82dewey@mipg.upenn.edu (Dewey Odhner) wrote:

>In article , Twegner@phoenix.net (Tim Wegner) writes:>|> I challenge Henry and other sci.fractals readers to produce a zoom this deep>|> (e+233) that is *not* a self-similar copy of a much shallower image at the

>|> same location.>Horiz.Hold { ; by Dewey Odhner. Public domain.> reset=1920 type=mandel> center-mag=-1.9999999999999999999999999999999999999999999999999999999999\> 999999999999999999999999999999999999999999999999999999999999996416785918\> 148134530430458843562012392471211940405947277366380516671161895619483871\> 9021305389817240974903756614325041160443974/0/3.3e+242> params=0/0 float=y maxiter=3000 inside=0> }This sure is weird. It looks like a bug, but I have investigated a little andcan't see what it is. Maybe it is for real! If anyone has the time andcomputer power to make a 640x480 image of this I'd like to see it.

Tim

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Path: unixg.ubc.ca!news.bc.net!news.uoregon.edu!newsfeed.internetmci.com!news.mathworks.com!zombie.ncsc.mil!simtel!daffy!uwvax!newssinet!news.u-tokyo.ac.jp!wnoc-tyo-news!wnoc-sfc-news!wnoc-kyo-news!kuis-news!kudpc!sakura.kudpc.kyoto-u.ac.jp!a51511From: a51511@sakura.kudpc.kyoto-u.ac.jp (unknown)Newsgroups: sci.fractalsSubject: Re:10^233 magnificationDate: 21 Jul 95 09:36:44Organization: Fukui-nct,Fukui Pref.,Sabae,JAPAN.Lines: 56Distribution: worldMessage-ID: NNTP-Posting-Host: sakura.kudpc.kyoto-u.ac.jpADewey Odhner writes>> Horiz.Hold {: by Dewey Odhner. Public domain>> reset=1920 type=mandelMy theory for the series converging to c=-2.0 can explainDewey Odhner's picture.< Series of small M ( with period m=3,4,5,...) converging

to c=-2.0 >

For small M with period m ( m>>1), the location c(m) and the sized(m) are given byc(m)=-2.0+eps(m)=-2.0+(1.5*pi^2/4^m)+...d(m)=(6*pi^2/16^m)+...

respectively.The value of c(m) for large m can easily be calculated by takingthe above principal value as a initial value.Dewey Odhner's picture locates at relative Xctr=0.68 position,i.e. near the casp point of the small M. The period of this smallM is m=202.The magnification, which makes the small M to the size of FullM, is given by 0.5/d(m).The relative branch length L(m,k) of the branch, which has the

direction of (pi/2^k)*odd ( which is distorted by the cardioidlike shape very near M ) and is normalized by d(m), is given byL(m,0)=eps(m)/d(m)=4^(m-1)L(m,k)=sqrt(L(m,k-1)), k=1,2,... .

Therefore, the every relative branch length become longer as mincreases.From these results, my theory can say: " The number of (additional)branches at the same coressponding area near small M is proportionalto m."This can explain the reason that many horizontal branches appear nearthe casp point of small M for large m.I calculated many pictures near the small M with period m=25,50,100,200, ( and 400). These pictures support my theory.

For the case of M=400, per file is made, but it is not calculatedyet, because it takes very long time.The following is the par file of small M with period m=25.Zoom areas you want. I hope you can enjoy pictures and my theory.Figures can be calculated in shoter time , though number of branches isabout 1/8 compared with the case of m=202.m25a1 { : by Minoru Morikawa. small M with m=25.reset=1900 type=mandelcenter-mag=-1.99999999999998685104553996190604824145973215854934\

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/0.0/1.07e+028/1 params=0/0 float=y maxiter=10000 inside=0}

m25horz { : by Minoru Morikawa. Horiz. Linesreset=1920 type=mandel passes=tcenter-mag=-1.999999999999986851045539961874934/2.250452e-45/3.032787e+0\29 params=0/0 float=y maxiter=10000 inside=0}

i

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Path: unixg.ubc.ca!info.ucla.edu!csulb.edu!nic-nac.CSU.net!usc!cs.utexas.edu!swrinde!tank.news.pipex.net!pipex!news.mathworks.com!zombie.ncsc.mil!simtel!daffy!uwvax!newssinet!news.u-tokyo.ac.jp!wnoc-tyo-news!wnoc-sfc-news!wnoc-kyo-news!kuis-news!kudpc!sakura.kudpc.kyoto-u.ac.jp!a51511From: a51511@sakura.kudpc.kyoto-u.ac.jp (unknown)Newsgroups: sci.fractalsSubject: Re:10^233 magnification and 10^1000Date: 22 Jul 95 09:28:02Organization: Fukui-nct,Fukui Pref.,Sabae,JAPAN.Lines: 55Distribution: worldMessage-ID: NNTP-Posting-Host: sakura.kudpc.kyoto-u.ac.jp

Last time, I mentioned my theory for the series converging to c=-2.0,and as examples gave only par files for m=25 and forgot ones for m=50.I give here par files m25a1,m25horz(correction),m50a1,and m50horz.m25a1 { : by Minoru Morikawa. small M with m=25.reset=1920 type=mandelcenter-mag=-1.99999999999998685104553996190604824145973215854934\/0.0/1.07e+028/1 params=0/0 float=y maxiter=10000 inside=0

}m25horz { : by Minoru Morikawa. Horiz. Linesreset=1920 type=mandel passes=tcenter-mag=-1.999999999999986851045539961874934/0.0/3.0e+029params=0/0 float=y maxiter=10000 inside=0}

m50a1 { : by Minoru Morikawa. small M with m=50.reset=1920 type=mandelcenter-mag=-1.9999999999999999999999999999883213824069750497300385583139\2087677/0.0/1.36e+058/1 params=0/0 float=y maxiter=10000 inside=0

}

m50horz { : by Minoru Morikawa. Horiz. Linesreset=1920 type=mandel passes=tcenter-mag=-1.9999999999999999999999999999883213824069750497300385583138\956/0.0/3.85e+059/1 params=0/0 float=y maxiter=10000 inside=0}

You can see that the number of branches becomes double as m becomesdouble.The following is an example of very deep zoom. This has no meaning except

one for a test of Fractint 19.2.As a test to show that a 10^1000 magnification picture can be calculatedcorrectly by Fractint 19.2, the well known Xctr=0.0 and Yctr=1.0self similar case is treated.The par file is very simple and is given as follows:

test1000 { : by Minoru Morikawa 10^10000reset=1920 type=mandelcenter-mag = 0.0/1.0/1.0e+1000/1.0params=0/0 float=y maxiter=100000 inside=0}

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The result is nice. As expected, the pictures for for mag=1.0e+1000are similar with ones for mag=1.0e+50 , except rotation. Time was 60hours 24min for videomode F3 by Pentium 90.This shows Fractint 19.2 can calculate deep zoom accurately.----

morikawa@fukui-nct.ac.jp