Entanglement

[Pauldelbrot]

Entanglement



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=15445

Matchmaker Mandelbrot. Spacefilling in this region, except in some tiny minibrots.

The location is in the intersection of a Scepter Valley (contributes the green shapes) and a Double Scepter Valley (yellow). It's deeper in the latter valley and closer to its walls, producing stronger and more tightly-wound gold spirals among the less spirally, more just wiggly green forms. The main gold area is connected with embedded minibrots, as it's part of a dendrite directly attached to the valley wall, to wit, one of the scepters.

The whole thing seems to crackle with energy.

[simon.snake]

These are particularly stunning.

What is the code like that is generating these?  I'd like to recreate something like this using FractInt, but I don't know how you're doing it.

Thanks.

[lkmitch]

I like the eyes popping out of the lower center.   

[Pauldelbrot]

Thanks.

Matchmaker is z -> (z + a)/b(z² + 1) but the attractorless images are tricky to visualize. One needs to use a coloring method that uses the entire orbit in some manner. UF ones that can work include exponential smoothing, triangle inequality average, and curvature average, but the one that seems to often give the best results (least noisy, good contrast to structures) seems to be the Elliptic Harlequin coloring in akl-m-math.ucl (named elliptic_module_transform in the actual code). Finding and/or porting that for Fractint might be nontrivial. It has a side effect of its own: preimages of infinity become bullseyes, magnetic fields, or other similar shapes of bunched-together color bands. Some of the transform function choices (e.g. atan, log) can reduce the banding in those areas but they may still be washed out by a bright glow or similarly.

Using one of the W-Sum options for "Dist. Mode" and playing with a) which one and b) the bottommost two parameters, Start Function and Additional Function, seems to give the best results so far.

The advantage over many of the other methods is that those tend to have trouble with the intricate fractals: high iterations results in outer areas being noisy, while low iterations result in inner spirals being smooth and un-detailed. This also happens using Exp. Smoothing on ducks-type fractals with high detail.

Many of the most interesting maps have a repelling fixed point at infinity so the disadvantage of the elliptic module transform can become significant, consistently hitting a particular structure in the Julia sets. An ideal modification would allow it to treat each point as having had a more flexible set of transforms applied to it before sampling, than just what the bottom two parameters enable. Inversions about arbitrary points, say, or something that works natively with the Riemann sphere rather than a plane topology.