Is God a Mathematician? Fractal Geometry of Nature.
What do galaxies, cloud formations, our nervous system, coastlines, and a snowflake have in common? They all contain never-ending patterns known as fractals. A fractal pattern finds constancy in randomness. By establishing a connection between the very large and the very small and observing that similar patterns can emerge at vastly different scales, spanning across time and space. This phenomenon is evident all around us, from the majestic spiral galaxies’ shapes to the intricate weather patterns of hurricanes, even down to the tiny eddies in a stream. Whenever we encounter a single object that exhibits these recurring patterns across various scales, and where each small component resembles the whole, we refer to it as a fractal.
But fractals are not just external to us. Our lungs are fractals. That’s how they manage to pack the surface area of a racquetball court, folded down into our ribcage. Our circulatory system is a fractal as well, which is how 60,000 miles of branching blood vessels and capillaries fit inside every human being. That’s more than twice the distance around the world.
History of Fractals
Computers made it possible to usher in a new e-era in mathematics. The development of computer science introduced a whole new discipline of mathematics, with one of the most famous examples being fractal geometry.
Simple geometry failed to explain the properties of these complex shapes. However, Benoit Mandelbrot found himself studying a very simple equation. With this simple equation, he unlocked the mysteries of these complex shapes which led to the development of an entirely new field of chaos theory and fractal geometry.
The Koch Snowflake: A shape with infinite perimeter but finite area.
There are many shapes in nature that look like fractals. We’ve already seen some examples at the beginning of this article. Other great examples are snowflakes and ice crystals:
To create our own fractal snowflake, we once again have to find a simple procedure we can apply over and over again.
The first step involves drawing an equilateral triangle. Each side of this triangle can be divided into three equal sections, and the middle section can be further divided into another equilateral triangle. We can repeat this process infinitely. As we continue, the shape will gradually transform into a snowflake-like pattern.
This intriguing creation is known as the Koch snowflake, and it possesses a remarkable property. Regardless of where we look or how closely we zoom in, the same intricate pattern repeats. We can actually generate a Koch snowflake on a computer by instructing it to graph a specific mathematical equation. With each iteration, the triangle’s sides multiply. After the first repetition, we’ll have 3 x 4¹ = 12 sides. Following the second repetition, we’ll have 3 x 4² = 48 sides. After n repetitions, we’ll have 3 x 4^n sides. If we continue this process infinitely, we’ll have an infinite number of sides. Consequently, the perimeter of the Koch snowflake becomes infinite, but its area remains finite. We can always draw a circle with a finite area around the snowflake, and it will fit, no matter how many times we increase the number of sides. This is what makes the Koch fractal’s perimeter infinite while its area remains finite.
The Coastline Paradox
One common trait shared by all the fractals we’ve explored is the ability to keep zooming in and discovering new patterns. In the early 1920s, a British mathematician named Lewis Fry Richardson, known for his work in weather forecasting and inventions like iceberg-detecting acoustic devices, tackled a unique challenge. He tried to figure out the likelihood of two countries going to war based on the length of their shared borders. However, he faced a significant hurdle — the lack of accurate border measurements. But this problem led to an interesting revelation: boundaries and coastlines have fractal-like characteristics.
He realized that the same idea applied to the borders or coastlines of many countries. Starting with the country’s basic shape, as you zoom in, you encounter more details, such as river inlets, bays, cliffs, rocks, pebbles, and so on:
Calculating a country’s border length presents a challenge. How do you decide how much to zoom in and which details to include? One way to measure the length of Britain’s coastline, for instance, is to take a long ruler, walk along its beaches, and add up the distances. If the ruler is 100 kilometers long, you’d use it 16 times, resulting in a total coastline length of 1600 kilometers. You could keep going with smaller rulers, and each time the coastline measurement would get a bit longer. Just like the Koch snowflake, Britain’s coastline seems to be endlessly long. This fascinating idea is often called the ‘coastline paradox’.
Generating Fractals
At its core, the Mandelbrot Set emerges from the straightforward concept of iteration. Let’s understand this idea with the aid of an example, such as taking the number 7 and squaring it, resulting in 49. If we continue to square it repeatedly, at some point, this number will expand into infinity. However, this isn’t the case with the number 1; when we square 1, it remains 1, providing a fixed value.
Now, consider what happens when we square a number smaller than one, like 0.5. In the first iteration, it becomes 0.25, and then 0.0625, gradually diminishing in value.
So, sometimes, numbers blow up, which occurs when they are greater than one. However, the number 1 behaves differently and does not explode, maintaining its stability. Numbers less than one, such as ½, transform into a quarter when squared, and further iterations yield even smaller fractions. Yet, for numbers significantly less than one, like negative numbers, squaring them turns them into positive values that eventually escalate into infinity.
This entire idea can be compressed into a formula that Mandelbrot came up with i.e
Here in the above examples, the value of c was taken 0, we were just squaring the value to get the next iteration. Now let us visualize this in a complex plane.
Julia Set
The letter ‘c’ indicates a number that can be added with each iteration. There is a reason why there is a circle on the screen in the 2D complex plane. Anytime we move inside this circle, it provides us with a value that can be deemed stable. This circle of stability is defined by the equation x² + y² = 1. However, when we introduce a constant after squaring the number, the boundary of stability undergoes disruption. Inside the circle, we can find points that tend towards infinity, while outside the circle, we encounter values that remain stable. Hence, it becomes evident that the circle itself is not of paramount importance, as the boundary of stability is subject to change.
When the value of ‘c’ is zero, the boundary of stability is a circle. However, when ‘c’ is not zero, we get different shapes. These regions, which aren’t necessarily circles, are called Julia sets. In other words, they represent the boundary of stability for different values of ‘c’.
It’s worth noting that the discovery of Julia sets owes itself to the efforts of two French mathematicians, with Gaston Julia being one of the pioneers who co-discovered the Julia set with Pierre Fatou in 1918.
Back in their era, there were no computers to help visualize what Julia's sets actually looked like. Mathematicians like Julia and Fatou were able to reason about them mathematically, but they only ever saw rough, hand-drawn sketches of what they might look like.
Fast forward to today, and our perspective has dramatically evolved. The images below lay bare a multitude of diverse Julia sets. Their distinctive colors serve as guides, revealing the pace at which the sequences at each point spiral into divergence.
Mandelbrot Set
When we created various Julia sets, we noticed some values of ‘c’ where every sequence goes wild, and the entire complex plane stays white. Decades after Julia and Fatou’s time, a new generation of mathematicians wanted to explore these mysterious areas.
This unique fractal is known as the Mandelbrot set. If you rotate it by 90°, it almost resembles a person with a head, body, and two arms. In 1978, mathematicians Robert Brooks and Peter Matelski first defined and drew it in a research paper.
A few years later, Benoit Mandelbrot, while working at IBM, harnessed early computers to create visual representations of fractals. In 1980, he discovered the famous Mandelbrot set. Mandelbrot used IBM’s powerful computers to craft a highly detailed visualization of the fractal, which was later named after him. At first, the printouts appeared different from what he expected. It turned out that the technicians operating the printers were erasing the “fuzziness” around its edges, assuming it was due to dust or printer errors rather than an essential feature of fractals!
Like all fractals, we can continuously “zoom into” the Mandelbrot set, revealing new patterns at every scale. Here, you can zoom into a part of the Mandelbrot set known as the Seahorse Valley. Black points reside within the Mandelbrot set, where the sequence remains bounded. Colored points lie outside the Mandelbrot set, where the sequence goes wild, and the various colors show how quickly it heads towards infinity.
Mandelbrot’s lifelong dedication to fractals, roughness, and self-similarity has left a significant mark on various disciplines. His work extends its influence into physics, weather science, neurology, economics, geology, engineering, computer science, and beyond.
Fractals in Various Fields
Fractals have found practical applications in diverse fields, enhancing our understanding and capabilities in each.
Computer Science
One of the most significant uses of fractals in computer science is in fractal image compression. By leveraging the principles of fractal geometry, this compression technique achieves superior results compared to traditional methods like JPEG or GIF formats. One remarkable advantage of fractal compression is that when an image is enlarged, it remains free from pixelation. This means that the picture often looks even better when its size is increased.
Fluid Mechanics
Fractals play a crucial role in the study of turbulence in fluid flows. Turbulent flows are inherently chaotic and complex to model accurately. A fractal representation of these flows has provided engineers and physicists with a valuable tool to comprehend intricate flow patterns better. Furthermore, fractals are instrumental in simulating flames and modeling porous media, particularly in the realm of petroleum science.
Telecommunications
Telecommunications has benefited from the development of fractal-shaped antennae. Companies like Fractenna specialize in these antennae, significantly reducing the size and weight of antennas. The specific advantages depending on the type of fractal used, the frequency of interest, and other factors. In general, fractal components introduce “fractal loading,” allowing antennas to be smaller for a given frequency of operation. This results in practical size reductions of 2–4 times while maintaining acceptable performance, often exceeding expectations.
Surface Physics
Fractals have become indispensable in characterizing the roughness of surfaces. In surface physics, rough surfaces are described as a combination of two distinct fractals, providing a comprehensive understanding of their intricate topography.
Medicine
In the field of medicine, fractals have opened doors for studying biosensor interactions. This innovative approach provides valuable insights into the intricacies of biological systems. Additionally, fractal analysis aids in the detection of abnormalities in lung CT scans, potentially helping diagnose lung diseases by examining global fractal dimensions.
“To See a World in a Grain of Sand And a Heaven in a Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour” — William Blake
In these lines, William Blake captured the essence of seeing grandeur in the minutiae, infinity in the finite. Normally, when we gaze at something or someone, like a face, we don’t consciously take in every tiny detail all at once. Our brain is hard at work, simplifying and filtering what we see to make it manageable. However individuals who’ve had psychedelic experiences often describe a feeling as though these filters have been lifted, and the full spectrum of information floods in all at once. It’s like a sensory overload.
When we’re looking at intricate fractals like these, the same sort of thing might be happening. They’re so incredibly complex that our brain can’t figure out how to simplify them, so the filters fail, and all the information rushes in. This creates an overwhelmingly sensory experience. What’s intriguing is that recent studies suggest that controlled doses of substances like psilocybin might have measurable positive effects on patients dealing with PTSD or depression. If these fractal images activate similar areas in the brain, they could potentially have some of those same benefits.
So, if fractals can impact the brain to such an extent, it raises another fascinating question. We know that the lungs and circulatory system are fractals. Could the brain be one as well? To explore this, we can look at artificial intelligence and machine learning. These technologies often involve processing information repeatedly while periodically feeding back the original data to continue the process. It’s somewhat reminiscent of how Mandelbrot’s fractal equation operates. This suggests that, whether or not the brain is a fractal, it’s something even more remarkable. It’s an engine for generating levels of complexity akin to fractals.
So, it might not be that the brain is a fractal, but the mind is. This could explain why these fractal images resonate with us so profoundly. When we gaze at them, it’s almost as though we’re seeing a reflection of ourselves. These images offer a practical purpose too, shedding light on the mysteries of human perception and consciousness. These fractals have so much to teach us. They continuously inspire us with the wonder of science and math. They remind us that when we’re confronted with something exceptionally intricate, the key to understanding might just be surprisingly simple.
As we look at these images, much like William Blake imagined, we do indeed glimpse eternity in an hour and the world in a grain of sand.
