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Spring Garden XIa



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=12313

A zoom in the center of Spring Garden Xa, showing an apparent doubling zone in the pinkish filament structures.

Thanks! I think.

Thanks. It is a jungle in there. Have you been following the whole series?

Thanks!


It's a c-plane visualization of z -> e^2πiαz²(z − c)/(1 − zc) + b, which I call Supernova. In this instance, b = 0 and it reduces to the old Herman Ring formula; α = 0.51803398. Values of α near 1/2 produce seahorse designs and values of α far from ratios of small integers produce (approximately) the potential for Herman rings. (Technically, α must be irrational and not well-approximated by successive ratios of small integers in a complicated sense of "well-approximated". In practice, α can't be irrational as input, but on the other hand the calculated value of e^2πiα is going to be forced to have rational real and imaginary components by the limitations of precision and the value of α that corresponds to the actual complex number used is then irrational if it isn't an integer multiple of 1/4.)

The dynamics of the above map, when b = 0, admits a superattracting fixed point at infinity and another at zero, plus two wandering critical points that can be captured by either or by as many as two additional, α- and c-dependent attractors. When b is near zero, but not exactly zero, the superattracting fixed point at zero moves and becomes merely attracting. It becomes c-dependent as well, and for large |c| it bifurcates or vanishes.

In this instance, c varies while a fixed choice of critical point is iterated. Zero attracts the chosen critical point for blue c values, while infinity captures it in the pink areas. The purple bulbs are regions where the chosen critical point finds an attracting cycle someplace else; the fractured, mirror-image bulbs become whole if the other critical point is chosen instead. Separate multiwave color gradients were assigned to each of these three region types.

It is a peculiar feature of Supernova that the seahorses and similar structures have their central "eyes" replaced by lakes in which zero and infinity exchange roles with regard to which captures the critical point. The effect can be something of a yin-and-yang pattern.

Spring Garden I is one. http://www.fractalforums.com/index.php?action=gallery;sa=view;id=10831

The fractal extends outside the image, though, because it extends all the way to infinity. To "fully unzoom" it requires a Mollweide projection or similar trick.

Thanks!


I've attached three files here. Put Supernova.txt and MultiplexedMultiwave.ucl in "My Formulas", rename Supernova.txt to Supernova.ufm, and put AutumnForest.upr in "Parameters", then open AutumnForest.upr using the UF browser. You should be able to recreate (and explore inside of) the new Autumn Forest 1 image I posted this morning. There's a lot of tweakable parameters (but leave the "Draw What" set to "Multiplex All" if you want sane coloring). Note that this gets you Supernova, multiwave coloring that works with Supernova, and some other goodies (check out the angle-sensitive gradients in the buds in Autumn Forest 1). If you have any questions or problems (this port looks somewhat fragile to me), let me know.

Spring Garden can be done by setting b (Re) to 0.000001 and b (Im) to 0, setting Draw What Period to 1, setting different color parameters (if you want to replicate the Spring Garden series's colors and not just the shapes), and then zooming around.

I also have a Spring Garden Mollweide projection at http://www.fractalforums.com/index.php?action=gallery;sa=view;id=12672 if you want a "fully unzoomed" image to look at.

Thanks. Good luck.


There is, though someone would have to program it into M3D, Fragmentarium, or one of them.


Sorry -- haven't a clue.