Escape paths' properties, namely magnitude, length, displacement, and sinuosity

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Offline sjhalayka

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« on: October 26, 2019, 02:45:18 AM »
Traditionally, the escape path's magnitude (squared) is used to generate the fractal set. The escape paths also have length, displacement, and sinuosity. Using these alternative criteria generates fractals that are practically equivalent in shape, despite their inherent differences.

Do you know of anyone who has done this before? I'm looking for references.

The paper is at: http://vixra.org/abs/1906.0407

Thanks for your time.

Linkback: https://fractalforums.org/share-a-fractal/22/-escape-paths-properties-namely-magnitude-length-displacement-and-sinuosity/3160/


Offline marcm200

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« Reply #2 on: October 26, 2019, 11:23:38 AM »
Traditionally, the escape path's magnitude (squared) is used to generate the fractal set. The escape paths also have length, displacement, and sinuosity. Using these alternative criteria generates fractals that are practically equivalent in shape, despite their inherent differences.
Afaik polynomial iterations like z²+c grow exponentially for escaping points at some |z| when the influence of c is negligable. The step from one iterate to the next can then get arbitrarily large (displacement), so does the length of the orbit. So every escaping point exceeds every finite threshold for magnitude, length and displacement at some iteration.

But wouldn't that mean that in the mathematical limit the same points escape or stay bounded independent of the escape criteria as long as it somehow reflects distance to the origin? So the described fractals are actually the same mathematical object and only look different like e.g. when using too low of an escape radius.




Offline sjhalayka

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« Reply #3 on: October 26, 2019, 08:41:34 PM »
Thanks for the insight!

Offline sjhalayka

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« Reply #4 on: October 26, 2019, 11:28:56 PM »
But wouldn't that mean that in the mathematical limit the same points escape or stay bounded independent of the escape criteria as long as it somehow reflects distance to the origin?

Perhaps I'm not understanding your question, but I do believe that the origin is only used in the magnitude criterion.

Thanks for your time and interest!

Offline marcm200

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« Reply #5 on: October 27, 2019, 09:51:08 AM »
What I was trying to say, is: If arbitrarily large displacements occur for points not in the classically defined Mset (magnitude), those points are not in the Mset defined by displacement either - if using a displacement value of 4 (diameter of the 2-disc where all Mset points are contained within) or larger.

Using smaller ones is a nice way of selecting points according to a path property and could be used for coloring, but to describe the object one has to use the mathematical limit, hence a displacement value so that the set of points not exceeding that value does not change by increasing it further.

And my guess was that this limit set and the classical Mset are identical: { z | z²+c, escaped if magnitude > 2} = { z | z²+c, escaped if displacement > 4}.


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