Print Page - Smallest midget?

Title: Smallest midget?
Post by: lkmitch on September 26, 2011, 10:47:30 PM

I'm playing around with some deep zooms and I was wondering what's the smallest midget (deepest magnification) that anyone's ever seen? Especially if they zoomed in to find it, as opposed to knowing where if was and just setting the coordinates there. My deepest is to a magnification of 2e358, but I knew where I was going before I got there.

Title: Re: Smallest midget?
Post by: David Makin on November 18, 2011, 09:36:52 PM

Have never really tried doing really-really deep-zooming, but if I was looking for deep-zoom midgets (either brot or juli) I'd use my "magic formula" with the settings that produce the disconnected midgets within each iteration band - I realise that the formula's not exactly optimum for this, being a tad more complicated than z^2+c but it can't be beaten when it comes to the choice of midgets to play with....

Code:

Fractal1 {
fractal:
  title="Fractal1" width=944 height=708 layers=1
  credits="Dave Makin;11/18/2011"
layer:
  caption="Background" opacity=100 method=multipass
mapping:
  center=-0.00035267065657380982670697635/-0.0001865049272874642352037\
  912 magn=3.8239644E14
formula:
  maxiter=256 percheck=off filename="mmf3.ufm" entry="MMF3d-Magic"
  p_search=None p_searchdepth=12 p_startscale=1E20 p_searchreal=yes
  p_searchimag=no p_autooff=no p_value=0/0 p_zstart=0/0 p_c=4/0
  p_c1=1/0 p_c2=20000/0 p_c3=1/0 p_c4=0.5/0 p_c5=1/0 p_c6=1/0
  p_power=2/0 p_power1=2/0 p_power2=0.5/0 p_showmap=yes
  p_usevalue="Unused (ie. nothing)" p_usepixel="Constant c"
  p_usecentre=0/0 p_usescale=1.0 p_swvalue="Constant c"
  p_swpixel=Zstart p_swcentre=0/0 p_swscale=1.0 p_showmod=yes
  p_sigma=no p_product=no p_selfrot=no p_ang=0.0 p_t=0/0 p_m=1/0
  p_n=0/1 f_fn1=ident p_fixfn=no p_flip=no p_smallbail=divergent
  p_bailout=128 p_bailout1=1E-5 p_smooth=Frac+Atan
inside:
  transfer=none
outside:
  transfer=linear filename="mmf3.ucl" entry="MMF3f-ExtCilia"
  p_method="Hinrich's Div." p_bailout=128 p_dauto=yes p_power=2/0
  p_usedfudge=no p_fudge=1.0 p_fudge1=1.0 p_ciliamode="Smooth saw"
  p_split=2 p_olditer=No p_iterval=0 p_iterweight=60
  p_logiter="Log(Log())" p_skew=0 p_rot=0.5 p_fixrot=no p_rpwr=0.1
  p_fixskew=no p_smallbail=1E-5 p_cauto=yes p_cpower=2/0
  p_usecfudge=no p_fudgec=1.0 p_fudge1c=1.0 p_cfixed=no
  p_cfixedval=1/0 p_cciliamode=Sawtooth p_csplit=2.0 p_fixconv=yes
  p_colditer="1 older" p_citerval=0.0 p_iterweightc=50.0
  p_logiterc=Ident p_cskew=0.0 p_crot=0.0 p_cfixrot=no p_crpwr=5.0
  p_cfixskew=no p_convoffset=0.0 p_zbasin=Off p_zscale=0.1 p_ip=1.0
gradient:
  smooth=yes rotation=1 index=235 color=1792 index=399 color=16777215
opacity:
  smooth=no index=0 opacity=255
}

In that render there is another mini-Julia *and* I think a mini-brot at every point where two arms of the colouring meet in a point, I think that goes for *all* of them !!
You can of course start at normal magnification looking for midgets rather than here at mag 1e15 already ;)

Title: Re: Smallest midget?
Post by: Pauldelbrot on May 03, 2012, 04:05:31 AM

Quote from: lkmitch on September 26, 2011, 10:47:30 PM

I just found a minibrot in Elephant Valley that is around 2-3x10^-325 in length. That's the big elephant valley, not one of the minibrot ones on the spike! Iterations are only in the high six figures around the perimeter of a square a few times the minibrot's size, so it's fairly low iter for something that deep in that part of the set. It's also not at the end of a really boring zoom, like 150 factor-of-10 zooms at the center of an elephant trunk spiral and then find a double spiral and go for the minibrot at its center. It's at the end of a fairly interesting zoom that adds several layers of baroqueness instead. And yes, I zoomed in to find it.

It took a few hours of cumulative work to find it -- the last few with extreme tunnel vision, with a few large colored blocks with a vague bullseye pattern to guide me toward the minibrot I knew was down there. Eventually I was having to leave it for 10 minutes, come back and find just enough information to center the next zoom, and then zoom it by a factor of 2-3 and wait again, etc.

Was your "2e358" referring to a 2x10^-358 or so large image? If so it's a little bit deeper than what I just found. Where was it?

Title: Re: Smallest midget?
Post by: lkmitch on May 04, 2012, 05:16:50 PM

Actually, the magnification is 2 x 10^359; not sure how I missed that extra order of magnitude. Here's the parameter set:

three00 {
; Similar to:
; http://classes.yale.edu/fractals/MandelSet/MandelScalings/
; HRScaling/Midget300.html
fractal:
title="three00" width=320 height=240 layers=1
credits="Kerry Mitchell;9/12/2003"
layer:
caption="Layer 1" opacity=100 method=multipass
mapping:
center=-1.9999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999999999999999999999999999999999999999999\
99999999999999999999999999999999999999999999999999999643225664396618\
52224658242034900926942265581617082564244406089532095341856110462695\
59415490628286794516178700972122986354853480254194026614869173154340\
22372718984082957654492895386803516338/0 magn=2E359
formula:
maxiter=10000 percheck=off filename="Standard.ufm"
entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
transfer=none solid=4283451440 filename="lkm.ucl" entry="basic"
p_colorby=iteration
outside:
transfer=linear filename="lkm.ucl" entry="basic" p_colorby=iteration
gradient:
smooth=yes index=0 color=16777215 index=190 color=3098447 index=199
color=0 index=210 color=3098447
opacity:
smooth=no index=0 opacity=255
}

Title: Re: Smallest midget?
Post by: Pauldelbrot on May 05, 2012, 01:09:16 AM

If I recall correctly, in UF a magnification of 1 indicates an image with a side length of 4, so mine would have a UF magnification of around 1e324.

Your coordinates are very very close to the spike tip; you indicated you chose them algorithmically? How, solving for roots of (...(((c² + c) ² + c) ² + c) ... ) ² + c = 0 on the real axis with an initial guess near -2?

The minibrot I found is near 0.275337647746737993588667124824627881566714069895426285916274363067437510130230301309671975356653639860582884204637353 84997362663584446169657773339617717365950286959762265485804783047336923365261060963100721927003791989610861331863571141 065592841226995797739723012374298589823921181693139824190379745910243872940870200527114596661654505 + 0.006759649405327850670181700456194929502189750234614304846357269137106731032582471677573582008294494705826194131450773 10704967071714678595763311924422571027117886784050420240236249129631789483532106497151867377563025274513529470021667381 5790733343134984120108524001799351076577642283751627469315124883962453013093853471898311683555782404i. That exact point should be in its interior somewhere.

There's an Images Showcase thread (http://www.fractalforums.com/index.php?topic=11776.0 (http://www.fractalforums.com/index.php?topic=11776.0)) that contains some shallow zooms centered on that point.

Quote from: lkmitch on May 04, 2012, 05:16:50 PM

; http://classes.yale.edu/fractals/MandelSet/MandelScalings/

404.

Title: Re: Smallest midget?
Post by: lkmitch on May 08, 2012, 05:49:33 AM

Quote from: Pauldelbrot on May 05, 2012, 01:09:16 AM

Your coordinates are very very close to the spike tip; you indicated you chose them algorithmically? How, solving for roots of (...(((c² + c) ² + c) ² + c) ... ) ² + c = 0 on the real axis with an initial guess near -2?

Yep, that's exactly what I did. If I remember correctly, the cardioid has a period of 300.

Title: Re: Smallest midget?
Post by: Pauldelbrot on May 08, 2012, 10:13:03 AM

That's interesting. The period 3 one is close to 10² magnification, the period 4 one near 10³, and the period 5 one near 10⁴. It seems like the relationship between period and magnification exponent, for these particular minibrots, is nearly the identity. There's a slight excess accumulation of magnification, enough to add another 80 or so over 300. Perhaps it's close to x + 15 ln x?

On the original topic, what's the deepest minibrot anyone's found by zooming and exploring, rather than algorithmically?

Title: Re: Smallest midget?
Post by: simon.snake on May 08, 2012, 03:33:58 PM

Hi Guys

Don't know if this is the deepest I've ever seen (it might be close), but I zoomed in using Fractal eXtreme into the leftmost tip and kept going for quite some time. Although a low resolution zoom into quite a boring area, it still took some time to create the zoom movie.

http://www.youtube.com/watch?v=jSnvMzteXFY

Simon

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 14, 2012, 10:11:05 PM

There's one at -1.74876509068805410962773875505207110634576035706311800327228481694592778810422550891956899419
07808399086853342299293798922482382262740521452822297984675731844990632736969879149894781646055529967892379
49925418146358395989577130421112084098971448722810917137355091650980758907978785827621252596328851665213669
65735633984329449119432534175531446380151420418202769599220904270088453901458432389143676124792875892664599
029406171857676759343115184449384887247981685788 + 0.0000000000000000000000000000000000001577971993849821132
38416389070162724691179727746352796520127459180522183933737640599922647121179429947859622784957954529685812
85516668040614678910161140753194431629264574612632541501773808517848069693547543189850509604973885793653540
78882759548442740163890299926329691830694718497962083502704697888578417268864044963249060842425014979778694
8948580632318758553922008084138651913479453906009414805276702037582919661222910246289i that's about 10^-459 times the size of the full M-set.

Yeah, it's near the spike, but it's not right on it and it's not near the tip. Indeed, you might recognize from the real coordinate being just above -1.75 and the imaginary coordinate much closer to zero that it's somewhere in those tree shapes dotting the spike inside the largest minibrot's Elephant Valley.

A square image centered on the minibrot and with it about a quarter the image width has iterations in the low seven figures at the edges -- 2 million or so.

Needless to say, rendering any decent-resolution image of this beast would take a ridiculously long time.

Title: Re: Smallest midget?
Post by: Dinkydau on June 15, 2012, 05:13:16 AM

I filled in the coordinates in fractal extreme but it doesn't work. At a magnification of about 2.51 * 10^58 it starts zooming into nothing.

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 15, 2012, 05:47:06 AM

Could be a bug in FX, but I don't think so. The numbers I posted have line breaks, to fit better in the forum post without horizontal scrolling or other irritations. I paused my current Mandelbrot Safari render and temporarily plugged in the coordinates you get if you stop at the first line break in each coordinate I posted and dialed up the magnification and it went off the rails right where you said your copy of FX did. So, you probably just didn't remove the line breaks, and FX probably didn't ignore them and instead cut off each number after the first break.

Try pasting the numbers into notepad, hitting del at the end of each line, and then copying and pasting into FX. It should work then. Though you might have to wait an hour to get even a very low-res (as in, like 20x20 pixel) preview of the minibrot. :)

Title: Re: Smallest midget?
Post by: Dinkydau on June 15, 2012, 02:23:16 PM

Oh yeah, I forgot about that. Now it should work.

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 15, 2012, 05:29:49 PM

Quote from: Dinkydau on June 15, 2012, 02:23:16 PM

Oh yeah, I forgot about that. Now it should work.

Did it?

Now you have me wondering if coordinates that long or magnifications that big break any of the popular fractal programs.

Though I've now had the chance to confirm that it works in Ultra Fractal. Very, very slowly mind you even on fast hardware.

Title: Re: Smallest midget?
Post by: Dinkydau on June 16, 2012, 12:41:56 AM

It works well. I stopped the rendering after 9 hours:
(http://i538.photobucket.com/albums/ff342/formule/pauldelbrot2.png)

max 6 500 000 iterations

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 16, 2012, 12:48:21 AM

And we've still only just scratched the M-set's surface...

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 16, 2012, 04:23:08 AM

I am the fractelunker that zooms in the night! I am the sticking GPU fan that ruins your Fragmentarium render. And there is a minibrot about 10^-485 times the size of the mother set at:

-1.7857627143661715948507568154269450949853004359999760273065188303818231643695567509108342190268802692293728460766907234851902462853124268226079443808297404646245096070658294594783386448692680630339091818722707001152538073105983018846867987645355993804327742890388021629163570112150776691822262034335782952091901363826907927783044812545066760960400038155694491979977068076151471951680434553872242585523941147631795674846595017343381607796237295518640552995702019715364251758710541413719386

+

-0.0000000000000000000000000344294684012881190016060448033825310665822326187533946781957054651339700665075294851677962711973348852576596640727857176279146654610793524484054867761809323164364342970320304285933492000113727746555199564551269537686628963030016052689742943735599355883216179515029152168317211594619589348509850403669264515211373220244104590242193243349441805446592352529745562332461413013945560603249534402052597060753080367183975379421297219541773020116137369587001536763487043i.

Title: Re: Smallest midget?
Post by: Dinkydau on June 22, 2012, 07:04:15 AM

(http://i538.photobucket.com/albums/ff342/formule/Mandelbrot_2965_zooms4.png)

Magnification:
2^2965
3.5804461285038769039428991186338 * 10^892

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 22, 2012, 08:45:57 AM

where

whut

how the fuhhh

Title: Re: Smallest midget?
Post by: Dinkydau on June 22, 2012, 06:53:29 PM

The coordinates are:

Re:
-1.999 998 012 788 599 994 431 399 136 931 166 563 233 407 924 358 905 461 394 663 515 090 233 821 242 166 119 137 547 883 952 336 047 036 411 645 849 813 525 405 462 193 062 169 011 343 570 931 897 485 843 865 924 983 806 303 294 536 412 498 514 087 769 454 942 433 176 932 962 405 768 590 012 052 098 708 676 428 826 695 424 473 516 454 216 129 321 345 695 964 314 017 500 422 836 308 439 525 222 962 387 529 724 377 377 610 604 215 025 393 875 848 655 955 925 002 459 289 897 451 947 878 099 541 083 464 390 776 517 531 603 221 470 323 535 061 330 656 295 811 032 140 090 818 834 447 359 422 844 144 652 651 800 314 338 420 143 933 711 256 782 771 739 745 565 626 065 496 837 230 277 293 815 108 462 821 748 960 408 893 460 952 067 517 602 222 865 007 872 698 662 810 716 255 313 366 055 070 820 891 309 274 776 735 626 105 557 974 928 100 995 173 992 857 752 325 854 671 352 651 178 903 616 592 612 538 354 138 045 862 061 413 789 935 809 170 119 018 491 726 143 401 523 213 099 734 963 525 748 394 221 672 678 501 636 396 489 904 637 015 220 959 201 103 834 783 303 644 372 851 975 216 658 620 454 491 248 609 511 938 673 943 287 349 960 265 315 485 007 434 101 226 394 286 576 249 570 989 8

Im:
+0.000 000 000 000 000 000 000 003 637 989 561 919 930 210 258 131 359 782 508 367 221 047 678 465 360 184 672 656 163 222 014 467 166 679 824 776 709 801 254 414 108 957 215 248 431 263 478 267 748 864 866 931 277 081 543 409 791 104 057 622 205 768 475 352 308 136 467 399 903 705 566 076 771 118 520 264 321 164 431 087 124 551 129 394 204 278 743 413 557 317 978 711 322 635 465 116 237 356 626 295 474 453 506 473 598 393 593 101 109 149 211 136 206 991 528 112 684 931 728 467 353 829 121 702 642 851 582 870 911 565 253 369 158 404 853 021 350 394 295 067 263 214 707 576 670 434 176 651 080 977 948 705 749 015 465 095 739 299 878 999 213 615 961 880 117 711 152 103 913 346 706 799 540 489 510 780 137 761 069 884 742 137 317 102 456 703 468 467 203 007 998 595 332 445 111 958 105 870 625 066 107 899 189 688 349 497 929 335 191 068 662 089 234 319 475 848 706 451 531 919 613 171 217 725 423 380 588 641 953 448 511 352 573 438 159 632 270 690 965 658 525 813 379 939 239 782 739 092 576 399 492 708 236 466 732 745 831 107 124 224 891 765 919 955 637 557 264 428 260 870 408 232 201 330 027 363 645 417 350 999 074 913 299 466 384 957 350 699 212 368 089 116 170 486 782 642 753 121 3

This minibrot is in the middle of the julia set that I posted here:
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/the-complexity-of-a-line/
It's not the first minibrot from there, the second one.

I want to render a zoom movie to this thing as well, but don't expect it soon. I may cancel it if it takes too long and end it at for example a julia set instead.

Oh yeah: only 1 400 000 iterations here, otherwise it would have been pretty undoable to get there.

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 23, 2012, 12:06:04 AM

-1.999998? Wuss. Try finding one that deep near -1.75. I dare you. ;)

Title: Re: Smallest midget?
Post by: Dinkydau on June 23, 2012, 02:52:31 AM

Easy: I can just avoid everything until 2^1500 and then go straight for the minibrot.

Re:
-1.749 957 021 188 409 705 030 333 940 495 588 916 169 404 405 442 996 252 662 369 010 324 927 042 991 938 840 620 122 837 148 377 567 225 919 286 402 410 661 802 566 493 401 899 870 882 300 173 626 958 344 545 346 763 058 551 742 804 292 626 217 249 883 237 102 043 671 591 034 799 487 751 725 206 073 216 851 641 954 933 541 712 857 560 583 355 632 675 157 396 726 892 542 370 683 776 285 225 847 444 523 074 630 796 907 859 095 099 467 499 994 351 021 611 181 441 527 551 007 568 385 332 361 355 498 983 496 163 393 762 242 481 528 465 721 770 620 772 839 890 309 575 080 321 539 208 465 910 738 961 132 132 815 320 440 636 907 464 847 700 643 408 244 870 653 496 609 922 150 316 172 823 118 130 774 518 350 225 122 951 064 858 945 852 673 778 642 326 779 031 974 264 149 378 725 593 828 024 816 169 897 611 154 275 087 086 260 863 661 127 130 837 412 669 704 467 510 689 045 504 989 603 371 194 278 305 676 369 357 072 997 806 318 558 233 782 195 935 417 781 654 635 137 026 333 917 555 900 538 683 678 428 203 585 289 246 083 555 842 694 087 885 582 585 920 111 453 904 028 168 733 256 558 555 439 581 916 098 920 772 795 609 900 711 686 605 438 238 305 359 694 000 298 096 759 258 886 820 193 866 428 060 200

Im:
+0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 097

Magnification:
2^3010
1.2597574882929839891779162247954 * 10^906

Much less heavy to render than the other one: 250 000 iterations seems more than enough for a smooth minibrot already.

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 23, 2012, 02:57:24 AM

BOOOO-ring! Mine followed all kinds of branches and layered on various structures. ;)

Title: Re: Smallest midget?
Post by: Dinkydau on June 23, 2012, 04:55:56 AM

You didn't challenge me to make it in any way interesting. xD

Title: Re: Smallest midget?
Post by: cKleinhuis on June 23, 2012, 05:06:04 AM

lol, how you find those coordinates ? searching for periods with length > 500 ?!?!??!?!

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 23, 2012, 05:07:35 AM

Quote from: Dinkydau on June 23, 2012, 04:55:56 AM

You didn't challenge me to make it in any way interesting. xD

Making my own interesting didn't qualify? ;)

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 23, 2012, 05:08:31 AM

Quote from: cKleinhuis on June 23, 2012, 05:06:04 AM

lol, how you find those coordinates ? searching for periods with length > 500 ?!?!??!?!

lkmitch's 10³⁵⁰ish one was found that way, but Dinkydau and I have been manual zooming to find deep minibrots.

Title: Re: Smallest midget?
Post by: Pauldelbrot on June 28, 2013, 09:11:16 AM

Quote from: Pauldelbrot on May 03, 2012, 04:05:31 AM

Anyone who was interested in this, the minibrot discussed above is about to be posted in the images showcase in the next few days ... as it's the final image in the Mandelbrot Safari sequence. :)

Welcome to Fractal Forums

Fractal Math, Chaos Theory & Research => Mandelbrot & Julia Set => Topic started by: lkmitch on September 26, 2011, 10:47:30 PM