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Author Topic: Search for an agreeable definition of fractals in nature  (Read 5550 times)
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kram1032
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Posts: 1863


« Reply #45 on: May 08, 2014, 12:01:29 PM »

I've only encountered these deformed minibrots at low magnifications surrounding the mainbrot.

Quasi-Self-Similarity which isn't strict Self-Similarity doesn't require this to be the case at all scales. It suffices to find few examples.

And I can think of two reasons why you might not have encountered a distorted Minibrot at deeper zooms.
The first is the one you gave yourself: you never actively searched.
The second, however, would mean that you actually saw numberous examples at all scales:
Attractors have the property that things not *quite* but nearly in the attractor rapidly converge towards the attractor but - and this is the catch - never actually quite land directly on the attractor.
This might mean that, as you zoom deeper and reveal Minibrots which require more iterations to form, the distortions do indeed diminish - and quickly so - but they never quite actually go away. So while they visually do not seem like they are distorted anymore, they still most definitely are.
If I had to guess, the only Minibrots that are not distorted at all are probably those on the real axis.
Oh and those which require infinite zooming.

Given how the M-Set seems to tend to work, though, to find maximally distorted Minibrots, you probably need to find the maximally distorted Minibrot at a given scale and then "follow the same path" you got to here from the full Mandelbrot. That, I'd think (but I don't know) would lead to Minibrots which still are distorted. And if they are, that could be used as a good first test (but not a proof) for my hypothesis: If each time you "follow that path" your Minibrots becomes less and less distorted (but they keep being slightly distorted in the beginning) what I guessed above just got some evidence.
It can't be considered perfectly disproven if you don't find any distorted Minibrots right after the next scale jump though: It might be that those distortions go away really quickly.
In that case, to really prove it (really the only way to be dead sure about this) is to go a more mathematical route: The resolutions and contrasts of our screens are finite and if the distortions drop off too quickly, there might no longer be a visual difference while there actually is one if you compare precise values.
However, if you actually find Minibrots that are more and more distorted as you go down, both of our thoughts on the matter can be considered disproven.
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binjiang
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Posts: 14


« Reply #46 on: July 10, 2017, 10:57:59 PM »

Hi, I have a very simple definition of fractal as follows:

A set or pattern is fractal if there are far more small things than large ones in it. I called it the third definition of fractal https://www.researchgate.net/publication/309428627_The_third_definition_of_fractal

Let me know if it makes sense to you. Thanks!
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0xbeefc0ffee
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Posts: 17



« Reply #47 on: July 13, 2017, 01:00:24 AM »

I think a fractal is completely characterized by its high complexity, and must be generated by some kind of algorithm.
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