"A simple formula produces strange and unpredictable forms ⁠— wispy, chaotic ghosts composed of 10 million points. "

https://fronkonstin.com/2017/11/07/drawing-10-million-points-with-ggplot-clifford-attractors/

and the formulas are incredibly simple

I follow Pickover on twitter, and he made this suggestion today &quot;A simple formula produces strange and unpredictable forms ⁠&mdash; wispy, chaotic ghosts composed of 10 million points. &quot; here&#039;s the link he provided https://fronkonstin.com/2017/11/07/drawing-10-million-points-with-ggplot-clifford-attractors/ and the formulas are incredibly simple

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Among many, many books by Pickover (check on Amazon!), you might like this one with many more examples of chaos and fractal images:

Computers, Pattern, Chaos and Beauty
Clifford A. Pickover
St. Martin's Press
1990

Inexpensive used copies are available online.

Among many, many books by Pickover (check on Amazon!), you might like this one with many more examples of chaos and fractal images: _Computers, Pattern, Chaos and Beauty_ Clifford A. Pickover St. Martin&#039;s Press 1990 Inexpensive used copies are available online.

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Some similar chaotic images can be made using a pixel formula and Latööcarfian Attractors in mt.ucl. Here's an image:

and the upr:

`lato {::LT2D2gn2ViVTvNOOM07Fo/HM89map8ZnF6Qn2pFYw09yOFY39SgitcsmRWyQSppJ/6HaZ70m  GJ70bOiPSR+EFFZy10ULV8lLvIKyytCGJWQtq4ot8MbBZRSSUBjvuwSmBfKo7YaDBVjOVzy4  WDJ+78yovK2w+L81zvGngHHf5FOgOjmSrsclkE/j6limEHpqopc7OCCMYJzWoyIlbEWeF1Yu  8iSaVFXuuRXm0y0kkrBg01y69NXpBwUn4S6r8a5ooKmOtgl+biKPPKnLYSaJEJ/Ssa0m8y4I  wO6dk4K+rMxvqA/jLN8MWTUrpSTOYGBXyo6IjSwzITwzXgHPuOofzel2RbSFx1a1axS7V10l  Klqz5UZcU1yyNWqLi/HLVmR1Z1rRflMOB+gSuC82lrIjrNS1yUSyo5ThPy6+QLzISF8hwWoZ  mCY9xwvMMWGBhHPZ6s5Lu8C1G7hAIjBBDwnjQfmYpz7+YE9ITy080H10MOs0dKhSDnHxt+b7  PvTAHWtEsY1X++P+6y75Gwkps+0Y0LLX3Yf3p/pS7+dt4A636YDiESUrPFw4kFJoTE3wM7YG  f6uPbPZ/JrSlrhrHn615A49+MDlggM38Ps6DEDXW4D/DPMA/4A0cGDXDCF5OUbMM9Zh8Naa+  Yv7Yv+bFBftXTn/8PPO94JFbFfj5Zwv+ZnfFHQzjphwgGMAbgdWcRDUfeJBHWD4s/lTOhdCq  ZtQaRTc108KCVn0EURsLlynoVJ9p4KEU4Jkm4Qaa3qygb1N1jOWEwQPxlPB11kKfqmlVylEk  v1BdCcGYOcp+JVmXrW7NHX15f/v/POE0hzjaR1V7igRoE0NhR++iK34HjdXFrP/fkxHf6kwl  wGYDxoOjLUr9p+BmDFarlIfnGgNRfCCFdeMK68pU05wpoBJVUYWFNMti64VvPGc/5n05wOMH  1B7M4oWo9zRNg6jjaQ4ljaE1LH1a/hy9wBz9wBy9wfgXvjrTFsgpf4zL9Df+pf4jeAE3jJds  bj/FAB8WYY1NQ7yM5y2OaD03RD0311rfA6+2oCYl9Ec4o12n266WXDdY11BUY15ylQnpkvxt  FQv+BoJ12mBI8LOTt0kShtJ81V8no4O+TWiH7rQ/Ag7u41HGzwxzglpwD8GgqktkKEk4obFC  vXiAKA2hJnsehabJVuL47y1q57OcreY/HobdEjHPtarFGrZq34seUQS2J9U9orL0TgvKUrLp  u9O+WNkKCDZyTjuFYX6a/FXurJlzXP3mfzhx2eJ4rDvvllTbviLpCyfrkMf6C3Eg75neVdFl  LqvGh+m/CSaWKvy5PdDD1MtsqsE+GGL7eWOFoxonFQbvRP0MifUH4RxRmSlCmuHKDFpVtjre  F2dgwlZsXJwoou9kgmhwoFTn2uO6ggJzmeDqdV8bwnO7NjAWsd5FzRzwLqnbtZ2fn/26EQ+R  D8kD/zA4pgr8H0si0WK=}`

{Anti-aliasing is not useful for chaotic images.)

Some similar chaotic images can be made using a pixel formula and Lat&ouml;&ouml;carfian Attractors in mt.ucl. Here&#039;s an image: ![63e30764257a1.jpg](serve/attachment&amp;path=63e30764257a1.jpg) and the upr: lato { ::LT2D2gn2ViVTvNOOM07Fo/HM89map8ZnF6Qn2pFYw09yOFY39SgitcsmRWyQSppJ/6HaZ70m GJ70bOiPSR+EFFZy10ULV8lLvIKyytCGJWQtq4ot8MbBZRSSUBjvuwSmBfKo7YaDBVjOVzy4 WDJ+78yovK2w+L81zvGngHHf5FOgOjmSrsclkE/j6limEHpqopc7OCCMYJzWoyIlbEWeF1Yu 8iSaVFXuuRXm0y0kkrBg01y69NXpBwUn4S6r8a5ooKmOtgl+biKPPKnLYSaJEJ/Ssa0m8y4I wO6dk4K+rMxvqA/jLN8MWTUrpSTOYGBXyo6IjSwzITwzXgHPuOofzel2RbSFx1a1axS7V10l Klqz5UZcU1yyNWqLi/HLVmR1Z1rRflMOB+gSuC82lrIjrNS1yUSyo5ThPy6+QLzISF8hwWoZ mCY9xwvMMWGBhHPZ6s5Lu8C1G7hAIjBBDwnjQfmYpz7+YE9ITy080H10MOs0dKhSDnHxt+b7 PvTAHWtEsY1X++P+6y75Gwkps+0Y0LLX3Yf3p/pS7+dt4A636YDiESUrPFw4kFJoTE3wM7YG f6uPbPZ/JrSlrhrHn615A49+MDlggM38Ps6DEDXW4D/DPMA/4A0cGDXDCF5OUbMM9Zh8Naa+ Yv7Yv+bFBftXTn/8PPO94JFbFfj5Zwv+ZnfFHQzjphwgGMAbgdWcRDUfeJBHWD4s/lTOhdCq ZtQaRTc108KCVn0EURsLlynoVJ9p4KEU4Jkm4Qaa3qygb1N1jOWEwQPxlPB11kKfqmlVylEk v1BdCcGYOcp+JVmXrW7NHX15f/v/POE0hzjaR1V7igRoE0NhR++iK34HjdXFrP/fkxHf6kwl wGYDxoOjLUr9p+BmDFarlIfnGgNRfCCFdeMK68pU05wpoBJVUYWFNMti64VvPGc/5n05wOMH 1B7M4oWo9zRNg6jjaQ4ljaE1LH1a/hy9wBz9wBy9wfgXvjrTFsgpf4zL9Df+pf4jeAE3jJds bj/FAB8WYY1NQ7yM5y2OaD03RD0311rfA6+2oCYl9Ec4o12n266WXDdY11BUY15ylQnpkvxt FQv+BoJ12mBI8LOTt0kShtJ81V8no4O+TWiH7rQ/Ag7u41HGzwxzglpwD8GgqktkKEk4obFC vXiAKA2hJnsehabJVuL47y1q57OcreY/HobdEjHPtarFGrZq34seUQS2J9U9orL0TgvKUrLp u9O+WNkKCDZyTjuFYX6a/FXurJlzXP3mfzhx2eJ4rDvvllTbviLpCyfrkMf6C3Eg75neVdFl LqvGh+m/CSaWKvy5PdDD1MtsqsE+GGL7eWOFoxonFQbvRP0MifUH4RxRmSlCmuHKDFpVtjre F2dgwlZsXJwoou9kgmhwoFTn2uO6ggJzmeDqdV8bwnO7NjAWsd5FzRzwLqnbtZ2fn/26EQ+R D8kD/zA4pgr8H0si0WK= } {Anti-aliasing is not useful for chaotic images.)
edited Feb 8 at 2:27 am

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Thanks to you both. Cool stuff. But when I paste Jim's parameters into UF I get this message (even after updating my formulas):

Ultra Fractal was unable to find ‘pixeljp’ in ‘jlb.ufm’

Thanks to you both. Cool stuff. But when I paste Jim&#039;s parameters into UF I get this message (even after updating my formulas): &gt; Ultra Fractal was unable to find &lsquo;pixeljp&rsquo; in &lsquo;jlb.ufm&rsquo;

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Thanks to you both. Cool stuff. But when I paste Jim's parameters into UF I get this message (even after updating my formulas):

Ultra Fractal was unable to find ‘pixeljp’ in ‘jlb.ufm’

I get this error too but you just need a plain pixel formula for this to work. UF suggested "pixeljp" in jp.ufm which works just fine.

&gt;Thanks to you both. Cool stuff. But when I paste Jim&#039;s parameters into UF I get this message (even after updating my formulas): &gt;&gt; Ultra Fractal was unable to find &lsquo;pixeljp&rsquo; in &lsquo;jlb.ufm&rsquo; I get this error too but you just need a plain pixel formula for this to work. UF suggested &quot;pixeljp&quot; in **jp.ufm** which works just fine.

Chris Martin
Gallery: Velvet--Glove.deviantart.com

Currently using UF6.05 on Windows 11 Professional 64-bit

edited Feb 8 at 6:57 pm

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Thanks Chris. UF Mac didn’t suggest a replacement formula but that one does the job.

Thanks Chris. UF Mac didn&rsquo;t suggest a replacement formula but that one does the job.

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I had copied that pixel formula into jlb.ufm, just for convenience, but must have forgotten to upload the file.

I got inspired, so more chaotic attractors coming soon.

I had copied that pixel formula into jlb.ufm, just for convenience, but must have forgotten to upload the file. I got inspired, so more chaotic attractors coming soon.
edited Feb 9 at 12:46 pm

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