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Talispirals

(http://nocache-nocookies.digitalgott.com/gallery/10/3869_30_03_12_10_29_01.jpeg)

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=10860

Ultra Fractal

Julia of neg-scaled Talis formula:

Z=(Z+1/Z+C)*-0.50

Coloring method: Magnitude average

Nice!

This seems to have Mandelbrot features to it -- doublings, so the spiral contains double spirals contain quadruples. I haven't seen this in Kaliset-esque fractals before.

Well, this formula is more a "mandelbrot-esque" one indeed. The original formula Z=Z+1/Z+C, setting initial Z value to 1, gives a strange Mandelbrot, without even using any power function (Ok, 1/Z is equal to Z^-1, but...) My finding is the results of adding a negative scale factor. It's all-inside, it doesn't diverge, but there's no folding so no Kaliset-esque patterns here. You can add abs function for that. But I think we shall call the patterns "Duck-esque" as it was Sam Monnier's formula which made them popular first ;)

Have you tried zooming up one of the doubling-zones? I bet there's minibrots.

The formula you state is a rational map with critical points. With no higher powers of z infinity won't attract so there's just finite attractors. Some quick calculus shows the derivative proportional to 1 - 1/z², which has zeros at 1 and -1, so there are two critical points. So there can be up to two attractors. The doubling-zones suggests that at least one of them bifurcates in response to changes to c. If you start with z=1 (or -1, but not 0 or anything else) and zoom up a doubling-zone in the c-plane minibrots seem to be a mathematical certainty.

If the shapes are being forced to space-filling, which it looks like, then likely for large chunks of the parameter plane there are zero attractors and the Julia set is the whole complex plane. So there's no omnipresent basin (like A(infinity) in the classic Mandelbrot) to "separate" the dendrites from one another; the dendrites are forced to fill all of the space not inside of minibrots.

Awesome fractal, very nice

Nice one. I haven't had any sucsess with talis formula, hadn't found how to use.

IMHO this is not exactly talis, I found z=z+1/z+c under the name of "tails". I believe "talis" are from "tails" at it looks like sheep tails;) Talis is alsou very rare man name, but tails sounds as beeing first.

A bit of experimentation got me the attached image in short order. There's a minibrot inside a spiral, colored by calculating the Lyapunov exponent of the moduli of the orbit values attained in the first 1000 iterations. The critical point iterated was 1.

Notable mathematical features: First, in the dynamic plane, zero is the only finite preimage of infinity. Infinity is a fixed point which is repelling if |scale| < 1, indifferent if |scale| = 1, and attracting if |scale| > 1. So, there are escaping points if the scale is positive. It's never superattracting, and the shape of its basin can be selected by varying the argument of scale.

Further, if |scale| < 1, there are generally chunks of the c-plane near c = 0 where there are no attractors at all -- all periodic points are unstable and the Julia set is the whole complex plane. In the rest of the c-plane one or both critical points find finite attractors, sometimes finding two separate ones. A coloring method like exponential smoothing or Lyapunov can reveal the dendrite structures. In the attached image, the spirals are the dendrite structures, which are largely space-filling here except in the minibrot, an island of stability where the critical point 1 is associated with a bifurcating attractor.

The normal Mandelbrot set has a "default". Infinity is always attracting, and is itself a critical point that is captured by itself. The other critical point, zero, can go there or to a finite attractor. A rule of rational maps of the complex plane is that every attractor must capture at least one critical point; so there is no attractor other than infinity unless zero finds it. So, the fate of zero tells you everything about the dynamics. Moreover, though, when zero isn't on the Mandelbrot figure (going to a finite attractor) or on its boundary, including on a dendrite, then it must go to infinity. If infinity disappeared as an attractor, the only choices for zero would be finite attractor or dendrite. So the space between the M-set's dendrites, where zero goes to infinity, would have to disappear and the dendrites would have to expand to fill all of that space. That is what has happened here: whereas a normal seahorse spiral has a clear space between the spiral arms, here the spiral arms are crammed up against one another without any room between them, because every point not in a minibrot must be on a dendrite here.