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Author Topic: Seahorse Valley, With a Twist  (Read 824 times)
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Pauldelbrot
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pderbyshire2
« on: October 16, 2011, 10:56:41 AM »

Seahorse Valley, With a Twist



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=8985

A coiled exponential map around a nanobrot in Seahorse Valley. This one doesn't go very deep.

An interesting fact about the exponential map: it shows us the outer shapes "as the target point sees them", and the target point often sees things in the shapes that later adorn the nanobrots one finds. In the Forks case, the outermost "fork" in the exponential map doesn't even look like a fork in the normal view, but like a propeller with two blades. But its tightly-packed cousins close to the microbrot next reached do look like forks. From the microbrot's perspective, according to the exponential map, so does the original.

In this one, we get a linear-ish embedded Julia from near the spike of a microbrot. This embedded Julia has sections that look like fishhooks, and the whole thing looks like two fishhooks in the exponential map. Not only that, though, the microbrot itself looks like a fishhook from the perspective of the deeper nanobrot! Most of which is its spike, but the longer rising arm of it is bulked out by the body of the microbrot.

We also see that from the perspective of a seahorse spiral, the next seahorse up the valley looks curved into the same convex-backed form seen alternating with ones that have concave backs along seahorse spirals. In a sense, the exponential map is telling us why that form occurs there, and nowhere else.
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