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30 July 2020

Fractals: Nature’s Geometric Code

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Pythagoras’ theorem, the formulas for calculating the surface area and volume of geometric shapes, the number pi…These are all concepts of classical or Euclidean geometry taught in schools, alongside analytic geometry (which translates these figures into algebraic expressions such as functions or equations), and accurately reflect the world we humans have created

But what if there were a “raw” geometry behind nature’s behavioral patterns? A geometry that does not reflect the world humans have created so much as everything that was here before we existed, and even the functioning of our own bodies. A new perspective to decipher the natural processes that occur around us: fractal geometry arrived (and came to stay) at the end of the last century.

The discovery of fractal geometry barely 50 years ago allowed us to mathematically explore the “anomalies” of nature in their many forms. What logic shapes the growth of tree-branches? Or mountain peaks or even the pathways of lightning in a storm, the growth cycle of microbes or the formation of stars in a galaxy. All these natural phenomena can be decrypted thanks to fractal geometry.

Image: a Romanesco broccoli is highly illustrative of the principle of self-similarity of fractals / Source: Unsplash

According to the mathematical principle of self-similarity, the same form is repeated on a gradually decreasing scale indefinitely, i.e., an identical form within the previous one and so forth. To infinity. Forms, rhythms, sounds or trajectories, because all these phenomena can be broken down into self-replicating structures. Self-replication is the key feature of fractals.

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.” This was the statement made in the late 1970s by Benoit Mandelbrot, the mathematician responsible for coining the term “fractal” (from the Latin fractus, broken or fractured) in 1975. At the time Mandelbrot was working for IBM at the Thomas Watson Research Institute in New York after teaching at several American universities. His task was to find out why there was white noise interference in the telecommunication system he was working on. Benoit Mandelbrot (1924-2010), born Polish and naturalized French and American in the context of World War II, had an exceptionally visual mind. This allowed him to find the mathematical basis of fractals, even though these figures appeared irregular to the human eye. 

Mathematics: a more powerful lens than a microscope

Following his instinct to interpret the problems in visual terms, Mandelbrot analyzed the graph representing white noise turbulence and discovered that, regardless of the scale of the graph, the data for a day, an hour or a second always had the same pattern. That’s when he turned to the work of the mathematicians Pierre Fatou (1878-1929) and Gaston Maurice Julia (1893-1978), who had studied the iteration of functions (the basis of the principle of self-similarity in mathematics). Using powerful computers, Mandelbrot was able to replicate this equation infinitely to obtain one of the most iconic images of science, the Mandelbrot set. This strangely organic and irregular shape follows the mathematical principle of fractal self-similarity and is infinitely expandable: the pattern of the edges is constantly repeated as one looks deeper into the image.

Image: visualization of the Mandelbrot set built from an equation replicated infinitely by computer. / Source: Wikimedia, author: Lars H. Rohwedder

Years later, Mandelbrot published Fractal Geometry of Nature (1982), a work that attracted attention and acclaim worthy of the creator of a new field of knowledge. As a mathematician hitherto regarded by academia as unorthodox, Mandelbrot argued that fractals are more natural and intuitive than objects based on Euclidean geometry, artificially generated and regularized by humankind.  While Mandelbrot was not the only scientist responsible for the birth of fractal geometry, he literally shaped previous knowledge using the power of computers. In science, as in fractals, there is always an intellectual form within a larger one, although in this case the principle of self-similarity is not fulfilled. Thanks to the discovery of fractals, for the first time a simple equation can explain forms of great complexity that, moreover, over time have been shown to arise in the larger processes of nature.

Dory Gascueña para OpenMind

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