Smallest midget?

[lkmitch]

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.

[David Makin]

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

[Pauldelbrot]

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.

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?

[lkmitch]

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
}

[Pauldelbrot]

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) that contains some shallow zooms centered on that point.

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

404.

[lkmitch]

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.

[Pauldelbrot]

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?

[simon.snake]

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.

<a href="http://www.youtube.com/v/jSnvMzteXFY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/jSnvMzteXFY&rel=1&fs=1&hd=1</a>

Simon

[Pauldelbrot]

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.

[Dinkydau]

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.

[Pauldelbrot]

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.

[Dinkydau]

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

[Pauldelbrot]

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.

[Dinkydau]

It works well. I stopped the rendering after 9 hours:


max 6 500 000 iterations

[Pauldelbrot]

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