Spring GrowthIce OctopodesWheels Within Wheels
Ice Octopodes
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Description: Adapted to the northern seas.

Found deep in elephant valley (over 10200).
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Posted by: pauldelbrot March 09, 2019, 09:13:05 PM

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marcm200
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March 10, 2019, 10:41:49 AM
Nice! Do you discover these enclosed structures by an automatic process?
pauldelbrot
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March 10, 2019, 10:54:42 AM
Thanks! And what do you mean by your question? "Enclosed structures"?
marcm200
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March 10, 2019, 11:26:08 AM
The image revolves around the central point, symmetrically. And finding that spot, is what I meant. Do you use an automatic survey that finds spots so that the resultung image looks that symmetrical? Or maybe, the question would've been better suited under your minibrot images, refering to them as "enclosed structures".
pauldelbrot
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March 11, 2019, 12:31:21 AM
For this one I likely did it manually, zooming at the center of symmetry repeatedly. It is possible to do it automatically, as follows. The orbit of a point in a cardioid or bulb center passes through 0 periodically. This means that (((c2 + c)2 + c)2 + c) ... 2 + c is zero, for some number p of repetitions of ... 2 + c. Now let's say you have an image. Somewhere in that image is a mini (if the boundary of the set is in the image). Pick a point near a filament and calculate its orbit. The first point of that orbit is 0; ignore that and find the minimum modulus among the rest. Count how many iterations to get from the initial 0 to this minimal-magnitude point, and that should be the p for a mini that's on that filament and near the chosen point on the filament. Next apply Newton's method to the polynomial in c above with p repetitions of its structure, using the filament point as initial guess, and it will typically in a few to a few dozen iterations have converged to the point in the center of the cardioid of the mini. Now you have the coordinates of a mini and can just center an image there and use successively higher zooms to find structures like the above that surround the mini, as well as the mini itself.

The Kalles Fraktaler program implements this method to locate minibrots.
marcm200
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March 12, 2019, 09:18:57 AM
Thanks a lot! I think I'll add minibrots to my to-do-list. It'd be very cool to code a program, start it, go away - and then it beeps after having automatically found a minibrot!
claude
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March 12, 2019, 10:09:56 PM
Quote from: pauldelbrot
The Kalles Fraktaler program implements this method to locate minibrots.


KF uses a slightly different method for finding the period - it iterates the corners of a triangle until it surrounds the origin. This tends to work when zoomed "farther out" than the "atom domain" method you describe.  Next version (due May/June) will have the option to use an alternative Jacobian-based method, which may be more robust in the presence of skewed views (or not, will see when testing...).  https://www.mrob.com/pub/muency/period.html for the current method

KF also has a size estimate, so it can automatically zoom to the mini (or embedded Julia set, or in between, as desired).
Last modified by: claude March 12, 2019, 10:11:02 PM

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