Pictures of the Mandelbrot set in Black and White

[youhn]

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-Warped space, no time-
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See also http://jeroensnake.deviantart.com/#/art/Warped-space-no-time-449610230 for comments, more of the same or download of non-AA original 6400x3600 image.

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[youhn]

Ragged S-cross black edition



source : http://jeroensnake.deviantart.com/art/Ragged-S-cross-black-edition-450258313

Original location file accidentally overwritten, but it should be zooming out from:

Re: -1.40115516384389530792701723770285364315981214502170119286520288557123933115806853
Im: -0.00000001858364976811376541766798861781602333777142651643267904531480007674790182
Zoom: 3.00000000000E59
Iterations: 570141

[SeryZone]

Wow, super works! Very well matched colors! I love it! Not deep, notso hard, but beautiful!

[Dinkydau]

Nice work

[youhn]

Thank you both Dinkydau and SeryZone!

I'm planning to explore more in the neighbourhood of this location, all in black-and-white to focus on the shapes. Creativity through constrains.

More focus on the straightness of the spokes in this small one:

[Kalles Fraktaler]

 

[youhn]

Flowing gears, greyish flames



from : http://jeroensnake.deviantart.com/art/Flowing-gears-greyish-flames-450492517

Params for Kalles Fraktaler 2:
Re: -1.4011584382489438118120621342707175518581508137348123883954740927973377046960618155008687234396776486263048195010826199222946131453976628
Im: -0.0002503257975558324067372149433433108180876134328018133589226101492143552313613654053314626687113203840442179651064285118990895031806688
Zoom: 1.20000000000E107
Iterations: 103092

[youhn]

-= Bugs, insects and swirly things in a cross =-



source : http://jeroensnake.deviantart.com/art/Bugs-insects-and-swirly-things-in-a-cross-450895171

Another one in the small study serie. Location in the Mandelbrot fractal somewhere around the border of the onset of chaos. At least when the bifurcation diagram of the logistics maps is projected to the Mandelbrot set. All in black and white to focus on the shapes. Software used was Kalles Fraktaler 2.

Params:
Re: -1.36960127406136382428632167628285595355095049806292764554
Im: -0.00641878040272599517536848233872963303268614180232362463
Zoom: 8.20000000000E39
Iterations: 43582

[aleph0]

One from me...



Ultra Fractal, custom distance estimator formula, 100x (10x10) anti-aliased.

Centre co-ordinates:
-1.748768589747256568351
9.3572166111081456160205e-6
Magnification:
5.5124492E12

[Kalles Fraktaler]

Location:

Code:

Re: -1.9405270059781348712173985652907944854290533438728392433081576487211965598459503599645158527157595940620508693479138199370347954626237684037840499594072869715663858841052693187303192870210338074564657214281711083070542723663721255250292232751628531944696257641455533230418963094074665458255774903771272853922456500381323837718367342759896623739547324694650017604772049377808750423050564535548209554607109580287941779740858403807218744632912983988726146601138351065142557862355296864043789131553002293611783096496329464858553843439724350872493279971530275963011386676571411135523762267816551071263235978947824350934573177973519438561694144899
Im: 0.0000000129592632119114886870809736636234580629726367599512104765303892057424926875389161038390241308678612188393441144619211960050994065367926631043421816185869047382588089490216130471700588430914710094396183012215489645707281564671771610383518849445752905758606570438479136290791950806676690803964274253864812787110369328048642170882144515210072246136068133708230252034083736006227904179425910228691613365099646086368385795934913324175867495175065996871250838986777483648164954577394285183080801805874635025024750268857083565930616389002484641987427931636451939382462913217406413084716076296453024541752478986897351246715568682880433773868
Zoom: 1.58977852651E605

[youhn]

aleph0, your method really brings more clarity to the basic shapes of the set. Wonderful.

[aleph0]

aleph0, your method really brings more clarity to the basic shapes of the set. Wonderful.

Thanks. The heavy anti-aliasing is a big part of this image, as needed to bring out the really fine filigree smoothly. This zoom was within Ultra Fractal's extended precision range, so render time was tolerable. Even so it took a while to generate the whopping 19200x12000 image file.

[tit_toinou]

The last image is awesome.. What do you mean by "custom" distance formula ?

[aleph0]

What do you mean by "custom" distance formula ?

It's a distance estimator formula that smooths the filaments, rather than using a hard black and white rendering (black for areas near or in the set, white for areas far from the set). It uses eight grey scale levels mapped to distance. Distance threshold is adjustable to fine-tune filament width for a given image. Grey-scale gradient as you approach the set is also adjustable from 'fast as you cross the distance threshold/slowing as you approach the set' through linear to 'slow as you cross the distance threshold/accelerating as you approach the seat'.

The filament and outside set colours are actually freely adjustable (as is the inside set colour as a standard UF feature), so black and white renderings are just part of what the formula can do. It can also do a hard rendering too if you want to produce an image with a little 'grit' in the filaments, but I tend not to use that very often, as the smoothed approach works so well; it delivers most of the benefit of anti-aliasing with little computational overhead. It's still not sufficient to generate something like the posted image though. To achieve that level of fine detail, generating an oversized image then downsampling is essential.

The formula combines DE implementation techniques previously used by:
- Robert Munafo in his MANDELZOOM programs for Mac. See http://mrob.com/pub/muency/color.html (section following "Convert  the  scaled  distance  estimate  to  a  brightness" - the filament smoothing technique).
- Duncan Champney in his program Fractal Works for Mac. See http://www.fractalforums.com/mandelbrot-and-julia-set/how-are-bandw-images-from-'beauty-of-fractals'-created/msg4290/#msg4290 (colour gradient handling and ability to select filament colour).

I should also mention Paul Derbyshire's (Pauldelbrot's) awesome DE images and techniques.

There may well be other software/formula authors that have used similar DE techniques, but these are the sources that inspired my own exploration of DE (plus of course the wonderful books written by Heinz-Otto Peitgen and his colleagues).

[xenodreambuie]

Well done Aleph0, a custom gradient is the way to go, along with adjustable boundary thickness. The program I'm working on also does that. Here's my version with a simple smooth gradient with midpoint 0.25, and 25 samples per pixel.