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 (small sections of clouds, mountain ranges, lightning bolts, blood vessels, snowflakes, cauliflowers, that look similar to the whole). The recursive nature of fractals is also called expanding or evolving symmetry and is obvious, for instance, in a piece from a cauliflower, which is a miniature replica of the whole, not identical, but similar in nature. If the replication of the pattern or shape is exactly the same at every scale, it is called a self-similar pattern and is best illustrated in the magnifications of the Mandelbrot set pictured below.

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-minute 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 and the Universe itself, is similar to everything else in the way we all unfold according to a common blueprint: become, self-organize, bond, self-generate, adapt, self-regulate, self-perpetuate, and transform with a degree of freedom to choose. This common blueprint endows everything in Nature with the capacity to find ingenious ways to reiterate its mandate, thus perpetuating the recurrence of progressive cycles that begin with becoming and lead to transformation. Therefore, I believe that this common blueprint is the foundation for building cohesive structures that can adapt to changing environments and engender hierarchical orders of progressive 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 they exist; therefore, self-similarity in Nature is not identical like in the Mandelbrot set but is instead an evolving symmetry.

Self-similarity in Nature goes both ways, recurring from the whole to its atomic particles, and vice versa. Whether it is an atom or the entire Universe, everything begins by becoming, then on to self-organizing, bonding, self-generating, adapting, self-regulating, self-perpetuating, and ultimately, transforming. This kind of cohesion and interconnectedness can only be achieved with a degree of communication that, even though it is constantly occurring in us, we cannot yet comprehend. How else can we communicate our own blueprint, our DNA, to every cell we create in our bodies? How else can they in return communicate to us what we are: the captain of our vessel following primal instincts yet able to choose?

Obviously, the knowledge of self-similarity in Nature can help us understand 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 ‘naturally evolving systems’ I refer to ‘open systems’ which are generated by natural processes to interact freely with environments, as opposed to ‘closed systems’ which are made by man to fit specific expectations.

Revised December 2020   

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