Deep zooming to interesting areas

[youhn]

How can this be? I thought the Mandelbrot set border was connected as a whole? Nevertheless I can follow your example. It does ring a bell. But I think only the shapes within one iteration-band can look disconnected, but still are connected some iterations further down. Is this a mistake?



The near-white parts are disconnected only by the mapping of colors to the different iteration bands.

[aleph0]

How can this be? I thought the Mandelbrot set border was connected as a whole?

Yes, the M-Set itself is connected. When Robert refers to the embedded Julia being disconnected, he means the swirling arcs that comprise the Julia rather than the M-Set atoms (the miniature M-Sets). Look at the fifth image down this page:
- http://www.mrob.com/pub/muency/2foldembeddedjuliaset.html

The arcs appear to be continuously connected structures. But that's an illusion. The arcs are merely a by-product of the colouring method (DE in his example for the dark areas), which reveals the presence of an infinite array of tiny M-Set atoms within the structure. The atoms themselves are indeed connected, but they connect via a branching, dendritic structure of infinitely thin tendrils. In the image, the connections enter from centre right (the direction in which the main M-Set lies), then branch outwards from the centre of the image. If you follow the sequence of branches up and to the right and clockwise, you can see they form a 'swirl' inside the swirl of the Julia arc in that quadrant of the image. That inner swirl is the locus of interconnection, not the Julia arc itself.

Hope that makes sense. I too found this very confusing when I first read that page and tried to relate it to what I knew about the connectedness of the set.

[aleph0]

Getting a little off the main topic with the connectedness point, but I think worth just a couple more clarifications.

See this Ultra Fractal Wiki page for an example embedded Julia Set and counterpart Julia Set:
- http://ultrafractalwiki.fractalforums.com/Embedded_Julia_Set
- The example Julia is a Cantor dust.
- The embedded Julia also contains the filaments which connect the M-Set atoms.

See gallery image http://www.fractalforums.com/index.php?action=gallery;sa=view;id=15883 for a re-rendering of the fifth image in the Mu-Ency web page, using UltraFractal and black and white DE. The blue lines outline the Julia arcs. Red lines roughly outline one connection track. Orange lines show some branches of the connection track. The actual connection track is of course infinitely more convoluted and branched, as becomes quickly apparent if one zooms into the connection track down in those spirals.

[knighty]

That's a nice thread. Thank you all for all the valuable informations -and thanks to moderators for making it sticky-.

[Botond Kósa]

One fascinating aspect of the embedded Julias, as pointed out by Robert Munafo, is that although they appear connected they are actually Cantor dusts, i.e. disconnected. This becomes very apparent if you spend any significant time zooming with a DE-based rendering. Structures which appear connected at a certain zoom level continually 'melt away' as you zoom ever deeper. Again, this can be very difficult to spot when using a complex colouring scheme.

Au contraire! Julias for the outer points of the Mandelbrot sets are Cantor dusts, but embedded Julias in the M-set are actually connected by very thin filaments. Those filaments are only slightly deeper than their surroundings and you may need careful coloring to even recognize them. See this comparison on Wikipedia: http://en.wikipedia.org/wiki/Mandelbrot_set#mediaviewer/File:Relationship_between_Mandelbrot_sets_and_Julia_sets.PNG