The infinite, hidden beauty and chaos of fractals
By Andy Han ‘26
The Mandelbrot Set Courtesy of Wikipedia
Infinity seems daunting, yet it surrounds us constantly, whether we’re aware of it or not. It appears in the blood vessels that run through our bodies, in the peculiar patterns of plants and vegetation, and in the shapes of rivers, mountains, and perhaps even the universe itself. Known as fractals, this pattern of infinity is a universal constant, observable from the infinitely vast to the infinitely small.
Dictated by the mysterious laws of nature, our comprehension of these infinite structures is only restrained by our finite understanding. So where do these fractal shapes come from, and why are they here?
Benoit Mandelbrot at TED2010, Courtesy of Wikipedia
Fractals were first popularized and brought to light by Polish-born French-American mathematician Benoit Mandelbrot, dubbed by many as the “father of fractals.”
In a TEDTalk presentation in 2010, Mandelbrot began his lecture by stating that “roughness is part of human life, forever and forever," the “roughness” referring to the perpetual chaos of life, complexity, accidents, and randomness. He devoted his studies to exploring the juxtaposition of order within chaos, dubbing the term “fractal” as we know it today.
The term fractal originated from the Latin word fractus, meaning fragmented or broken. According to the Fractal Foundation, a fractal is defined as “an infinitely complex pattern that repeats itself across different scales.”
Upper school math teacher Kathy Sloan says that fractals demonstrate "self-similar behavior," in that fragmented parts resemble the whole when magnified or vice versa and can continue to do so indefinitely.
Ferns are her favorite example of fractals in nature, as the smaller leaflets represent the larger plant.
The Mandelbrot set, as seen above, is a mathematical function graphed on a complex plane.
The function above determines the intricate and infinitely repeating pattern of the Mandelbrot fractal, which is governed by simple rules.
Starting with a complex number represented by variable c, it is added to the squared value of variable z, which starts at 0. The output of the initial formula will be inputted as the new value for z into the next iteration, repeating as the function grows in a self-similar manner within a finite boundary.
The z value in the Mandelbrot set remains bounded in finite numbers while the number of iterations is infinite, thus causing the infinitely small patterns to repeat indefinitely within its domain. Simply put, the Mandelbrot set is a visual representation of Mandelbrot’s work of “finding order within roughness.”
Sure, the Mandelbrot set looks cool, but what are its practical uses, and what is the significance of its existence? For one, research into fractals provided valuable insights into other branches of mathematics and science, such as chaos theory.
A cloud in three scales, Courtesy of Akseli Palén
Imagine a butterfly fluttering its wings in a flower garden. Would you believe the flapping of its wings could eventually cause a tornado? While this specific scenario, otherwise known as the “butterfly effect,” is mostly speculation and has been heavily scrutinized by critics as unrealistic, it acts as an effective allegory for chaos systems and the impact of initial conditions.
Like Mandelbrot’s set, the outcomes of chaos systems are highly interconnected with their initial conditions. Small, seemingly insignificant changes can have far-reaching effects on the larger system, much like how clouds form in the atmosphere.
The formation of clouds is affected by many factors, such as temperature, humidity, and altitude, thereby making cloud formation an inherently chaotic process. A small change in temperature or air pressure in one area of the atmosphere can result in overcast skies, storm clouds, or clear weather.
However, when looking at clouds, they don’t seem to be very similar at all, but when we zoom in, we realize that the magnified portion of the cloud is nearly indistinguishable from the whole.
Natural fractals can be hard to identify and seldom express exact self-similarity; rather, a more intriguing factor is at play.
Tree, courtesy of ScienceNewsExplores
Consider this image of a tree. At its base is the trunk, which splits off into a few large branches as it grows, eventually dividing into progressively smaller branches and twigs. If you cut off a branch of this tree, it'll likely look similar to the whole tree, demonstrating self-similarity.
In fact, there are actually more fractal elements in this tree. If we look below the ground, we’ll find roots that extend and branch out into the dirt, similar to how the tree branches are structured. The leaves of the tree contain one large vein that splits off into progressively smaller veins in order to provide nutrients to the entire leaf.
The existence of fractals is not accidental. In nature, fractal structures are often found due to a legitimate evolutionary benefit. For example, the self-similar structure of branches and leaves maximizes sunlight absorption by spreading leaves across a large surface area, also aiding in structural durability by distributing forces applied by winds or harsh weather.
However, unlike mathematical and geometric fractals such as the Mandelbrot set, Menger sponge, and Sierpinski triangle, natural fractals are harder to spot and rarely perfect due to external factors that induce “chaos” into the structure.
For example, in human bodies, fractal structures play a crucial role in maintaining the functionality of our circulatory, nervous, and respiratory systems.
Diagrams of the human circulatory, respiratory, and nervous systems, Courtesy of Cleveland Clinic
According to upper school science teacher Joey Grissom, to generate blood flow across the entire body with maximum efficiency, the heart pumps blood through large arteries, which branch into smaller arteries, then into thinner arterioles, and finally into tiny capillaries, in accordance with traditional fractal structure.
The nervous system is similar, with large nerve bundles split into smaller nerves, which branch out much like a tree, ultimately reaching specific organs, muscles, and skin.
Both the circulatory and nervous systems are structured similarly, with the arteries, veins, nerves, and even the nerve cells themselves possessing a branching, fractal shape. Through evolution, these fractal structures allow these body systems to fulfill their function efficiently while maintaining the compactness necessary within the body.
Aerial shot of coastline, courtesy of Chris Moran
However, there is an exception when it comes to naturally occurring fractals with evolutionary benefits: coastlines.
Coastlines are non-living, so it’s impossible for them to gain any evolutionary benefit from fractal structures. Nevertheless, a multitude of factors, such as currents, waves, and tides, lead to random and chaotic coastline erosion that somehow always looks like a fractal.
French physicist Bernard Sapoval suggested that since a self-similar fractal coastline has a very large perimeter, larger than any non-fractal shape, perhaps “the coastline takes on a fractal shape that is the most stable, since it is most effective at absorbing the energy.”
Strangely, factors of evolution couldn’t influence the coastline, yet these fractal coastlines are perfectly designed to avoid excess erosion. However, according to chaos theory, there must be a root to this phenomenon. So what natural, or even divine, power dictates it to be so?
Romanesco Broccoli, Courtesy of Wikipedia
The puzzling and common occurrence of fractals spurred the proliferation of theories, with Mandelbrot himself arguing for the existence of the “fractal multiverse," and some even claiming that these structures prove the existence of God. Although the mystery surrounding fractals is yet to be fully uncovered, we know one thing for sure: it makes for some cool-looking broccoli.