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Diagram: Fractals

Fractals

While Fractals are visually striking on their own, such topological regularities in complex systems are significant not only due to their inherent beauty , but because of the dynamics at work which generate such patterns.

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The Sierpinski Fractal shown here is a nice example of a scale-free/power law system. At each level of zoom we encounter the same pattern. Such self-similar structures follow power-law distributions of their components:  that is to say, a few elements within the overall pattern are very large, but there are many more elements within the structure that become vanishingly small. 


Let us take a look at another well known fractal, the Koch Curve. The Fractal demonstrates how multiple, {{timeiterative}} have the power of generating something highlighly unexpected:

First four iterations of the Koch Curve

We begin with a line and a simple three step rule protocol:

  • for every line segment break it into three equal segments;
  • Form an equilateral triangle wherever the center segment falls (removing the base)
  • Repeat this process for each of the newly created line segments.

As seen in the image above, a repeated simple rule set is all that is required to  generate a high level of complexity.  This is the basic principle at work in creating all Fractals - where basic instructions  generate an output, whose new properties (going from one line segment to four line segments), become the new context upon which to re-apply the rule.

While we can obtain a lot of richness from such phenomena, it would be incorrect to state that the resulting figure 'adapts' or 'learns' - instead it is more accurate to state that it 'unfolds' over multiple iterations.



 


Cite this page:

Wohl, S. (2022, 1 June). Fractals. Retrieved from https://kapalicarsi.wittmeyer.io/definition/fractals-1

Fractals was updated June 1st, 2022.

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