157 – On Fractals & The Mandelbrot Set …

Fractal, from the Latin fractus meaning “broken” or “fractured”, is the recurrence of a pattern or shape which is replicated at decreasing or increasing scales and used to conceptualize fractional dimensions in Nature that look similar to the whole (small sections of clouds, mountain ranges, lightning bolts, blood vessels, snowflakes, cauliflowers). The recursive nature of fractals is also called Expanding Symmetry or Evolving Symmetry and is obvious, for instance, in a branch from a tree, which is a miniature replica of the whole, not identical, but similar in nature. If the replication of the pattern or shape is the same at every scale, it is called a Self-Similar Pattern and is best illustrated in the magnifications of the Mandelbrot set.

Mandelbrot Set

Mandelbrot Set

The elaborate boundary of the Mandelbrot set reveals ever-finer detail at increasing magnifications, with each magnification incorporating smaller repetitions of the set. So, the fractal property of self-similarity applies to the entire set and not just to its parts (the link at the bottom is a 2.14 Minutes YouTube video showing the recurrence of the set at increasing magnifications).

The Mandelbrot set has become popular outside mathematics, both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization. [1]

Along with Holography (see post 154), the Mandelbrot set is one of the best analogies of Self-similarity I can find in the natural sciences: Everything in Nature, including you and I, is similar to everything else in the way we all unfold according to a common blueprint: Become, Self-organize, Bond, Self-generate, Self-regulate, Adapt, Self-perpetuate, Transform. This common blueprint endows every universal component – every one of us, every molecule, every galaxy – with the capacity to perpetuate the recursive cycle that begins with Becoming and always lead to Transformation. It is thus I believe that the perpetuation of this common blueprint is the universal foundation for building cohesive structures that can adapt to changing environments and engender hierarchical orders of complexity.

The Mandelbrot set is a computer-generated program in which we find the recurrence of identical repetitions, but ‘naturally evolving systems’ [2] are diverse, highly malleable, and completely reliant on a creative symbiosis with the evolutionary processes in which we exist; therefore, self-similarity in Nature is not identical as in the computer-generated Mandelbrot set, but is instead a supple and evolving symmetry towards greater and greater complexity.

Self-similarity in Nature can be ascending or descending, recurring from the whole Universe to atomic particles and vice-versa. Everything begins by becoming, then on to self-organizing, bonding, self-generating, self-regulating, adapting, self-perpetuating, and ultimately, transforming. This kind of cohesion can only be achieved with a level of communication we are not yet able to comprehend even though it is constantly occurring in us. How else can we communicate our own blueprint (DNA) to every cell we create in our bodies? How else can they in return, as a combined whole, communicate to us what we are, what we must do as the captain of our vessel, so we can follow primal instincts yet are able to choose?

It is obvious to me that the common knowledge of self-similarity in Nature can help us understand a bit better our intimate relationship with Nature, what is not obvious is why this knowledge is not in every curriculum of our educational systems?


[1] The information above has been abstracted from Wikipedia

[2] By ‘natural evolving system’ I refer to ‘open systems’ which are generated by natural processes to interact freely with environments, as opposed to ‘closed systems’ which are man-made to fit specific expectations.

Revised September 2020   

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