Is a fractal always a predictable pattern?

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Discussion Overview

The discussion revolves around the nature of fractals, specifically whether they are always predictable patterns or if they can exhibit unpredictable characteristics. Participants explore the definitions and properties of fractals, including examples like coastlines and the Mandelbrot set, while questioning the implications of self-similarity and periodicity.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants assert that fractals do not exhibit unpredictable patterns, suggesting a level of determinism in their structure.
  • Others argue that coastlines, often cited as fractal examples, are not deterministic and can be considered pseudo-non-deterministic.
  • A question is raised about whether a fractal can be irrational and non-repeating, prompting discussion on the nature of self-similarity and periodic components in fractals.
  • One participant notes that while the Mandelbrot fractal reveals increasingly elaborate patterns upon zooming in, it does not repeat in a conventional sense, highlighting its complexity.
  • Another participant mentions that a fractal's Hausdorff dimension exceeds its topological dimension, suggesting a deeper mathematical property that may influence its predictability.

Areas of Agreement / Disagreement

Participants express differing views on the predictability of fractals, with some suggesting inherent determinism while others highlight examples of non-determinism. The discussion remains unresolved regarding the implications of self-similarity and periodicity in defining fractals.

Contextual Notes

There are limitations in the definitions and assumptions regarding what constitutes a fractal, particularly concerning periodicity and the nature of self-similarity. The discussion does not resolve these complexities.

OMGMathPLS
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It never goes crazy and starts doing it's own unpredictable number pattern or sequence, right? Thanks in advance.
 
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OMGMathPLS said:
It never goes crazy and starts doing it's own unpredictable number pattern or sequence, right? Thanks in advance.

Coastlines are often considered fractal, in fact they are one of the earliest known structures to be considered fractal and they are not deterministic. Same with fractal landscaped used in CGI (pseudo-non-deterministic).

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zzephod said:
Coastlines are often considered fractal, in fact they are one of the earliest known structures to be considered fractal and they are not deterministic. Same with fractal landscaped used in CGI (pseudo-non-deterministic).

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Can a fractal be irrational and non repeating?
 
OMGMathPLS said:
Can a fractal be irrational and non repeating?

Strictly speaking a fractal is any structure which exhibits self-similarly at any scale. So in some sense it kind of has to have a periodic component. But it can still be quite rich in features. For instance, take a look at the Mandelbrot fractal, if you zoom in on it at various levels of detail you tend to see more and more elaborate patterns, but ultimately they are just the same few types of symmetry repeated over and over again in slightly different ways.
 
Bacterius said:
For instance, take a look at the Mandelbrot fractal, if you zoom in on it at various levels of detail you tend to see more and more elaborate patterns, but ultimately they are just the same few types of symmetry repeated over and over again in slightly different ways.

Interestingly, I believe that the Mandelbrot fractal does not repeat.
Except for its symmetry, its shape is really different everywhere.
It's kind of fun to zoom in somewhere, and discover that it does not show the repeatable pattern that you would sort of expect.
 
Bacterius said:
Strictly speaking a fractal is any structure which exhibits self-similarly at any scale. So in some sense it kind of has to have a periodic component. But it can still be quite rich in features. For instance, take a look at the Mandelbrot fractal, if you zoom in on it at various levels of detail you tend to see more and more elaborate patterns, but ultimately they are just the same few types of symmetry repeated over and over again in slightly different ways.

Fractal is a structure where its Hausdorff dimension (strictly) exceeds its topological dimension.

See here

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