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Author Topic: Spring Garden IIa  (Read 502 times)
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Pauldelbrot
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pderbyshire2
« on: April 01, 2012, 07:35:28 PM »

Spring Garden IIa



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=10889

A strange reflection of Seahorse Valley, in bright flowery colors.
« Last Edit: April 05, 2012, 03:11:33 AM by Pauldelbrot » Logged

klixon
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« Reply #1 on: April 02, 2012, 12:40:38 AM »

What the... That's awesome. What is that?

(see what you did? you made me use capitals!)
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Pauldelbrot
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Posts: 2592



pderbyshire2
« Reply #2 on: April 02, 2012, 01:56:48 AM »

What the... That's awesome.

Thanks!

Quote
What is that?

It's a c-plane visualization of z -> eiαz2(zc)/(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 eiα 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.
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