Fractals: Mapping the Patterns of Chaos
How can we predict the unpredictable? Fractals hold the key to harnessing the chaos of our universe. Fractals are beautifully mysterious shapes. They walk the line that separates chaos and order and exist between the three dimensions we experience. These geometric shapes have repeating patterns, meaning that the shape of the fractal is repeated infinitely throughout itself. This creates captivating models such as the Mandelbrot set and Sierpinki’s triangle (Britannica, 2024). But fractals are not just pretty shapes. They exist everywhere in nature, from snowflakes to the layout of dark matter in the universe, and understanding them can help scientists predict the unpredictable.
Mathematically, fractals are described as geometric shapes that can be infinitely broken into pieces, each of which is a scaled version of the original. This property is called self-similarity. Additionally, their dimensions are fractional. Humans exist in three dimensions, the shapes analyzed in geometry are two-dimensional, and the lines represented on graphs are one-dimensional. Fractals however, are none of these. Their dimensions are not whole numbers but rather are between whole numbers (Fractal Foundation, 2018). But perhaps the most enticing property of mathematical fractals is their ability to model chaos. Chaos in math is a sequence where a slight change in initial position results in a completely different final position (Britannica, 2024). As an example, take two adjacent air particles. While they start out only a few nanometers apart, after minutes on a windy day they can end up miles apart (Britannica, 2024). Understanding chaos allows scientists to predict chaotic systems like the stock market, weather, or star clusters (Britannica, 2024). Most fractals have a connection to a chaotic system and are thus very intriguing to scientists (Fractal Foundation, 2018).
For an example of a fractal, draw an equilateral triangle. Then, draw another one with vertices at the midpoints of the first one. This should create four congruent triangles. Repeat this process as many times as possible with the three outermost triangles, leaving the middle one empty. The resulting shape should look like this:
This is Sierpinki’s triangle, named after the mathematician Wacław Sierpiński who discovered it. It is a straightforward and comprehensive example of a fractal. Each part of the larger triangle is a scaled-down version of the original.
Sierpinski's triangle can also be derived through a chaotic system. The vertices of a triangle are drawn and a random point p, chosen inside of that shape, is also marked. Then, one of the vertices is selected randomly, and a new point, q, is drawn halfway between p and the vertex. The process is repeated with point q. The resulting shape is Sierpinski’s triangle.
Another famous fractal that models chaos is the Mandelbrot set. The Mandelbrot set is famous for being an aesthetic masterpiece despite its humble mathematical formula. The formula is defined as Zn+1 = Zn2 + C where Z is a set of numbers, Zn is the nth term in the sequence, Zn+1 is the following number in that sequence, and C is a complex number. The way the graph is created is that Z1 (the first number in the sequence) starts as 0, and C is any point on the complex plane (the plane that contains real and complex numbers (a+bi) rather than (x,y) coordinates). The output Z2 (the second number in the sequence) is determined by adding Z12 and C. Then, Z3 is found by squaring Z2 and adding C, and so on. This will result in one of two tendencies: either Zn will expand outwards approaching infinity or it will collapse and approach 0. By graphing only the complex numbers C which, when put in this equation, approach 0, the Mandelbrot set (named after Benoit Mandelbrot, its creator) is defined (Fractal Foundation, 2018). This shape is a graph of a chaotic system since small differences in the C value can result in completely different outcomes. While the Mandelbrot set and its many variations generally remain only for mathematical analysis, there are many fractals that have practical applications.
Fractals can be found nearly everywhere in nature and arise in all different areas of science. In fact, many things in our everyday lives – trees, mountains, the circulatory system, snowflakes, leaves, rivers – are fractalesque (Montana State University, 2011). While nothing in nature can be a true fractal (infiniteness, going against the fundamental principles of nature, requires that it remains confined to the mathematical realm), we come across many systems that are fractal-like, meaning that they are self-similar, just not infinitely so. Here are three principles of science that have their roots in fractals:
Chemistry
Snowflakes are great examples of fractals and a chaotic system. Snowflakes form in layers which are created all throughout their journey to the ground. Each layer forms according to factors in its environment such as temperature, humidity, and speed. As the snowflake falls, it passes through different environments, driving each layer of the arms to look different from the previous one, yet the same as the other arms on that layer. At the end of its journey, the snowflake has picked up fractal-like properties. Each arm is similar to the others, and each branch of the arms are similar to the arms themselves, and so on. These fractals also model a chaotic system. Two adjacent raindrops, on their path to the ground, will experience vastly different environments and usually reach the ground far apart. This is the reason why each snowflake is unique (NOAA, 2022).
Physics
Lichtenberg figures occur when high voltage charges pass through an insulator (a non-conductive material). The electricity spreads out through the insulator in a lighting-like shape. These figures, first discovered by German scientist Georg Lichtenberg, often appear as scars on lightning strike victims since the human body acts as an insulator and lighting is a natural occurrence of very high voltage (Stone Ridge Engineering, 2024). The figures look like this:
The high voltage forces electrons into the insulator. Once they enter, wanting to minimize the potential energy of the electrons, they spread out to cover the maximum area in order to decrease the voltage drop. The shape they make is the Lichtenberg figure. These can also model chaos since any given electron has hundreds or thousands of different paths it can take as it spreads through the insulator and any slight difference in the initial position of an electron can significantly change its path (Stone Ridge Engineering, 2024).
Biology
The body is a masterpiece of elegant functionality. The paradigmatic example of this is the circulatory system. The job of the circulatory system is to provide oxygenated blood to the cells. To bring blood to each of the 36 trillion cells in the human body, the circulatory system utilizes a fractal branching structure, similar to that of the Lichtenberg figure. The main blood vessels, the ones that directly connect to the heart, split off, creating smaller vessels that in turn split into even smaller ones. This self-similar pattern creates a network of blood vessels that reach the farthest corners of the body. In fact, as a result of the fractal-shaped network, there is thought to be about 60,000 miles of blood vessels in the human body, which is enough to circle the Earth twice (Fractal Foundation, 2018).This feat of evolution is designed to maximize the reach of the blood while minimizing the number of vessels required (Montana State University, 2011).
Observing fractals helps scientists understand the world. In a system that seems chaotic and unpredictable, fractals can be used to predict them. This allows scientists to learn about complex ideas like brain waves and bacterial growth, and develop new technologies that depend on fractals, like cancer detection and antennas (University of Waterloo, 2001) (Obert, et al., 1990)(Montana State University, 2011). While Normal cells usually reproduce in a fractal pattern, cancerous cells are much less organized. To find abnormalities, scientists measured the dimensions of the growth pattern of healthy cells and cells at different stages of cancer. By comparing these dimensions to the ones detected in the patient, they were able to isolate cancerous regions (Elkington, et al., 2022). This has major implications for the future of cancer detection and treatment.
In the field of bacteriology, scientists also found fractal structures. They discovered that some bacteria grow in fractal patterns also with fractional dimensions (Obert, et al., 1990). This is important because it can help biologists to predict how bacteria will grow which could disrupt many biological industries. For the technology sector, fractals can help build antennas. These are not the antennas that stick up from old TVs, but rather the small electronic ones that communicate with other devices through frequencies such as the ones in your phone or wireless earbuds. These are very useful but they have their limitations. Most antennas can only receive frequencies within a small range. To expand the range of antennas, fractals can be used. Fractals are self-similar on different scales. This means that each iteration can detect a different frequency. With only a few iterations of fractal geometry in their construction, the upper and lower limits of the antenna can be stretched significantly (Yale, n.d.). These are just a few ways that demonstrate how fractals have and will continue to have fascinating impacts on our life and the future of science. These beautiful and mysterious figures deserve more recognition. They are the key to discovering the order in the chaos that is our universe.
References
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NOAA. (2022, December 21). How do snowflakes form? Get the science behind snow. noaa.gov. Retrieved November 5, 2024, from https://www.noaa.gov/stories/how-do-snowflakes-form-science-behind-snow
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