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Siegels and Seahorses Forever
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Description: Introducing Triskelion: a rational map with fixed points at the cube root of unity and several complex parameters, three of which exactly determine the derivatives at said fixed points. This means that pretty much any three desired Julia sets can be forced to intertwine -- just dial a Julia by punching in an internal angle and a stability value for each.

Here's the Fibonacci Siegel disk Julia set at 1, a Siegel disk near Elephant Valley at $-\frac{1}{2} + \frac{\sqrt{3}}{2}i$, and a Julia set from inside a bud in Seahorse Valley at $-\frac{1}{2} - \frac{\sqrt{3}}{2}i$. The latter results from setting the stability value greater than 1 rather than equal to it, so the fixed point becomes repelling. It's the center of the biggest radiation of white petals a bit down and left from center and directly beneath one of the biggest red Siegel disks.

In this case, we've actually chosen three Fatou sets and the Julia set is technically their shared boundary, but this same system can combine three disconnected quadratic Julia sets to create a spacefilling fractal with the disconnected quadratic Julia sets identifiable inside it, copies of them endlessly intertwined with each other to fractally tile the plane.
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Keywords: Triskelion
Posted by: Pauldelbrot February 14, 2014, 09:26:05 PM

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