Inky Nebula  
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Description: Dark and swirling clots of gas dot this spacescape. Triple matchmaker Julia set.
Stats: Views: 119 Total Favorities: 0 View Who Favorited Filesize: 468.99kB Height: 1080 Width: 1920 Posted by: pauldelbrot January 20, 2018, 01:48:45 AM Rating: by 3 members. Total Likes: 0 Image Linking Codes


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KRAFTWERK  January 20, 2018, 10:22:58 AM Great stuff Paul! 
pauldelbrot  January 20, 2018, 10:32:03 AM Thanks! 
quaz0r  January 21, 2018, 12:27:14 AM this matchmaker stuff, have you ever explained how you do it? is it your own secret sauce or has it been implemented by others also? 
pauldelbrot  January 21, 2018, 01:52:01 AM Neither. It's not secret, but it's not to my knowledge implemented by anyone else either. Triple matchmaker is this rational map: \( \frac{z + \frac{a}{\sqrt{3}}}{b(z^3  \sqrt{3}az^2 + cz + \frac{ac}{\sqrt{3}} + d \) As you can see, it has four parameters, and the degree of the denominator is higher  fruitful ground for finding finite attractors and also wholeRiemannsphere Julia sets, not so fruitful if you want infinity to superattract. To see structure in wholeRiemannsphere Julia sets (and in parameter space regions rife with parameters for such Julia sets) requires colorings based on the whole of an orbit, and typically to iterate every point to a fixed, sometimes large, number of iterations. UF has an Elliptic Harlequin coloring (found in aklmmath.ucl) that does a nice job, when tweaked appropriately, of bringing out the structure in these fractals. A few other methods can give interesting results, notably Triangle Inequality Average and similar methods, various other orbitweightedsum approaches (Harlequin seems to use this in particular), and potentially orbit trap colorings that cumulate something (e.g. based on distance from the trap) on every iteration, rather than only the final iteration or only the first that lands in a trap. 
pauldelbrot  January 21, 2018, 01:54:29 AM Disregard the above comment. Edit Comment is not working correctly and Delete Comment is missing completely for some reason. Neither. It's not secret, but it's not to my knowledge implemented by anyone else either. Triple matchmaker is this rational map: \( \frac{z + \frac{a}{\sqrt{3}}}{b(z^3  \sqrt{3}az^2 + cz + \frac{ac}{\sqrt{3}}} + d \) As you can see, it has four parameters, and the degree of the denominator is higher  fruitful ground for finding finite attractors and also wholeRiemannsphere Julia sets, not so fruitful if you want infinity to superattract. To see structure in wholeRiemannsphere Julia sets (and in parameter space regions rife with parameters for such Julia sets) requires colorings based on the whole of an orbit, and typically to iterate every point to a fixed, sometimes large, number of iterations. UF has an Elliptic Harlequin coloring (found in aklmmath.ucl) that does a nice job, when tweaked appropriately, of bringing out the structure in these fractals. A few other methods can give interesting results, notably Triangle Inequality Average and similar methods, various other orbitweightedsum approaches (Harlequin seems to use this in particular), and potentially orbit trap colorings that cumulate something (e.g. based on distance from the trap) on every iteration, rather than only the final iteration or only the first that lands in a trap. 
quaz0r  January 23, 2018, 12:14:44 AM cool! maybe you can post the explanation in its own forum thread sometime so people can find it 