Fractal Geometry: A Journey from Chaos to Order

 

Seeing a world in a grain of sand
and a sky in a wildflower
is to have infinity in the palm of the hand
and eternity in an hour.

In this poem, the famous English author and artist William Blake was able to capture something essential about the natural world that surrounds us and that exists inside us. It highlights the connection between the "big world" and the "small world," and notes that similar patterns occur on vastly different scales across space and time. Without realizing it, Blake managed to capture the essence of a concept that was only fully described 200 years after his poem - fractals.

A fractal is a geometric pattern that repeats itself at different scales of observation, both spatial and temporal, in both a micro and a macro level. To better understand this idea, imagine an image or a shape in which, as you zoom in or zoom out, you end up finding similar shapes to the ones you previously saw. Fractals have a feature called "self-similarity" - this gives us the feeling that, as we keep zooming in on the image, it appears as though we are exactly where we were before... but we are not. Does it seem confusing? Let's go step by step then.


 
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Going back to Ancient Greece, the Greek philosophers began to notice some interesting associations. Structures as distinct as the spiral of the shell of a snail, the ramifications of a tree or the ornament of a peacock's tail seemed to reproduce the same periodicities and patterns. They realized that the number 1.6180 seemed to rationally explain these repetitions - and this proportion later came to be known as the "golden ratio".

Hermes Trimegisto, the father of Alchemy, also had an interesting and similar point of view. In his titled "Principle of Correspondence," he tells us: “That which is below is like that which is above, and that which is above is like that which is below, to perform the miracles of one only thing.”  This is a quote from one of his works, the "Emerald Tablet", that gave rise to Alchemy, the goal of which was to achieve wisdom through the understanding of the mysterious relationship between the micro and the macrocosm. 

Throughout human history, there have been numerous references to this phenomenon in the most diverse areas, from art to philosophy, through architecture, geography, and biology. But the ultimate understanding and grasp of these fascinating pieces of the infinite only came up in 1979 through mathematics, regarded by many as the mother of all sciences. Math was considered "the music of reason" by James Joseph Sylvester and "the poetry of logical ideas" by Albert Einstein. Through the realm of numbers, Benoit Mandelbrot was able to describe the fractals in a surprisingly simple expression, resulting in a structure of immense complexity called “The Mandelbrot Set”.

 
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As Einstein's theories solved the limitations of Newton's laws, this new Mandelbrot geometry, called fractal geometry, succeeded in explaining phenomena that Euclid's geometry could not, thus interweaving the apparent randomness of parts to the whole, and giving an order to that which was once thought of as chaos.

In Breathe Portugal, we find fractals truly fascinating because they reveal a hidden yet decipherable pattern present throughout numerous processes in the cosmos. How crazy is it, to find incredibly similar shapes in completely different scales? Just look at the following images and let yourself be amazed by the mesmerizing fractality present in them:

 
 

As we said before, fractals can be found in drastically different scales. Recently, scientists found a beautiful fractal structure in the quantum realm. The image below represents a type of fractal figure called Sierpinski triangle and, in this case, it shows the density of electrons on the surface of copper, as you can see in this video.

 
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But not only physical objects present fractal shapes… When we try to represent certain phenomena on a diagram, these reveal patterns that show the same fractal quality present on aerial photos of rivers or the on lightning that branches out. An evolutionary tree is a good example of that kind, as you can see in the images below.

 
 
 
 

A very interesting platform that might leave you inspired, it’s the One Zoom Fractal Tree of Life. It is a software that allows you to explore the entire tree of life condensed into a single page. This was only possible because it was done using fractal geometry techniques.

 
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Beyond their magical aesthetic qualities, did you know that fractals are actually soothing for our eyes and minds and help restore our cognitive abilities too? There is a reason, as Agnes van den Berg, Yannick Joye and Sander Koole said in the title of their study “why viewing nature is more fascinating and restorative than viewing buildings”. In particular, according to Stephen Kaplan’s Attention Restoration Theory, natural scenes and landscapes “capture our attention in a pleasant and effortless manner, allowing the mind to rest and wander freely while the capacity for directing attention is replenished”.

 
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This pleasant way of paying attention is described as “soft fascination” (such as looking at the flames in a fireplace or contemplating a forest) and is very different from the way we, in our daily lives, are used to paying attention, such as at work for example. What we usually do within society and in urban areas is a much harsher way of concentration that can not only exhaust us, but also cause depletion of our ability to concentrate. No wonder you feel as though you should take a break from studying or being on the computer, but you don’t really feel as though you should take a break from looking at a natural landscape. Quite the contrary, after contemplating nature, you most probably feel fresher, relaxed and rejuvenated.

Back to van den Berg, Joye and Koole’s study, the reason why looking at nature is soothing and restorative, is most probably because of the fractal quality of nature’s structures, from landscapes to details. Natural fractals (even more than artificial fractals) have just the right amount of visual complexity to replenish our cognitive abilities and not leave us depleted of them.

 
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Because of their importance and contribution to the many beneficial qualities of nature, even our logo in Breathe Portugal is a simple representation of a fractal: a branch of a tree (or even a tree trunk) dividing into smaller, identical branches… you could even keep dividing them infinitely.

 
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The beauty of all of this is that, not only fractals are fascinating and seem miraculous in their own way, they are all around you.


You just need to go outside.

Article by: João Gonçalo Dias & Maria Branco