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Fractal Geometry: Nature Explained
It is the artful and boisterous side of mathematics; it is fractal geometry. It helps mathematics explain the beauty and irregularity of the natural world. While mathematicians did not always know what they were, fractals existed and were used by artists, architects and others throughout history. Overtime, the Cantor set helped relate fractals to noise, and later, the Mandelbrot set laid the foundation for a new branch of mathematics, fractal geometry, which was used to explain all sorts of shapes and relationships found. If one looks a little closer, fractals can be seen all around in nature. The human body, coastlines and the structure of trees, and, through trees, entire forests, are excellent examples of natural fractals. Historically, fractal geometry was unknown to mathematicians, but today its many different sets allow mathematicians to explain numerous aspects of nature, such as the length of coastlines and the structure of trees. Fractal geometry enables geometry to leave the world of rigid conformity and enter the world of originality. Benoit Mandlebrot, founder of fractal geometry, explains why geometry is considered barren and trite, “Clouds are not spheres, mountains are not cones, and bark is not smooth, nor does lightning travel in a straight line” (1). Fractal geometry, however, is not lifeless; it uses complicated equations to create or otherwise explain crazy shapes. J.R. Murdock, author of “Fractal,” defines a fractal as “a geometric figure, often characterized as being self-similar; that is, irregular, fractured, fragmented, or loosely connected in appearance” (1). Through their irregularity, fractals create their own dimensions, and these dimensions indicate how crinkled or rough something is. Fractal dimensions were discovered when mathematicians were able to create a line that wound within itself and, theoretically, was able to touch all the points on a plane without crossing over itself. Using the fractal dimension, mathematicians describe different figures that occur in between the basic dimensions. A fractal dimension of a figure could be a decimal such as 1.26, between the first and second dimension, or 2.79, between the second and third dimension. A good example that shows this is a piece of paper. A flat piece of paper is a plane with two dimensions, length and width, but if one crumples the piece of paper into a ball, it is neither a plane nor a sphere but somewhere in between the second and third dimensions (Briggs 69-70). This ability to describe complex figures with multiple dimensions allows fractals to be applied to many different aspects of nature. The founding of the branch of mathematics known as fractal geometry was itself irregular, loosely connected, and fractured. As Michael Frame and Nial Negar, professors at Yale University, point out, “The intricate design of Gothic, Renaissance, and Baroque architecture, especially as expressed in cathedrals, frequently exhibited scaling over several levels” (1I). Many cathedrals would have arches that consisted of smaller arches that consisted of even smaller arches and so on. As far back as the seventeenth century, a German mathematician named Gottfried Wilhelm Leibniz thought of concepts that later helped Mandelbrot define fractals. In 1872, Karl Weierstrass, another German mathematician, discovered a function that would later be called a fractal (Murdock 1). Around the same time, a third German mathematician, Gayor Cantor, created what he knew to be the first fractal, initially called, a monster. Cantor took a line, divided it into thirds and took the middle third out. He then repeated this action over and over again. This fractal was the first fractal dust (Hunting). Over a quarter of a century later, in 1904, Helge von Koch, a Swedish mathematician, refined Weierstrass’s work and created another function which is known today as the Koch curve, or the Koch snowflake (Murdock 2). Gaston Julia, a French mathematician, then further expanded the theory of fractals when he thought that if one put a number in an equation, and then put the answer back into the equation, one could get a monster. This idea was called the Julia set. Finally, Benoit Mandelbrot, in the 1960s, was able to build on all this prior work and combine the many different pictures of the Julia set into a relatively simple function: f(z) = zn + c (Hunting). He called this function a fractal, deriving fractal from the Latin word, fractus, meaning broken or irregular (Murdock 2). Until Mandelbrot was finally able to explain and name them, artist, architects, and mathematicians used fractals without knowing what they were. In addition to revealing itself in art and architecture, fractal geometry, and in particular, the work of Cantor and Mandelbrot, gave mathematicians the foundation for using fractals to describe figures in nature. While one of the simplest fractals, the Cantor set, describes noise. More specifically, the Cantor set is known as a dust because it creates a set of points that is somewhere in between the dimensions of zero and one (Mandelbrot 74). The Cantor set may sound simpler than other, more popular, sets, but it is, at the very least, as extensive as the Koch curve (Mandelbrot 79). Also, many mathematicians use the Cantor set to create fractal dusts which resemble the graphs of random noise (Mandelbrot 75). With this explanation, the Cantor set proves that random noise may not be as random as it appears. Benoit Mandelbrot created the more complicated Mandelbrot set, which is used as the basis for much of fractal geometry and is the full embodiment of fractal geometry. The Mandelbrot set uses an iterative equation, meaning it uses the final answer of the equation to recalculate the equation. This allows for an infinite number of results (Briggs 75). All the small copies of the Mandelbrot set are connected to each other, and bigger copies, through a trail of even smaller copies. Unlike the Koch curve, these small copies are slightly distorted, but it is this distortion that, according to Frame and Negar, “makes the Mandelbrot set much more interesting” (3C). The edge of the Mandelbrot set is so intricate that it is considered a line with a dimension of one, while it has a fractal dimension of two (Briggs 70). In the equation for the Mandelbrot set, f(z) = zn + c, n commonly equals two, but if one substitutes a number that is closer to infinity for n, the set starts to resemble a disc. John Briggs, author of Fractals: The Patterns of Chaos, says, “Largely because of its haunting beauty, the Mandelbrot set has become the most famous object in modern mathematics” (74). The more simple Cantor set laid the ground work for explaining random noise. This combined with the more complicated Mandelbrot set, the most popular fractal and the foundation of fractal geometry, opened the floodgates for explaining nature through fractals. Fractals can be used to explain virtually every form in the world around us, whether mother nature or humans created it. According to Ralph Abraham, a professor at the University of California Santa Cruz, “… fractal geometry has given us a much larger vocabulary and with a larger vocabulary, we can read more of the book of nature” (qtd. in Hunting). Most natural objects are not just made up of one fractal, but multiple different fractals wound within one another (Briggs 71). Further, not only shape plays an important role in the development of organisms, but fractals can be used to explain efficiency as well. James Brown, a professor at the University of New Mexico, explains, “It would be incredibly inefficient to have a set of blueprints for every single stage of increasing size. But if you have a fractal code… then a very simple genetic code can produce what looks like a complicated organism” (qtd. in Hunting). Finally, fractals are not limited to just explaining the natural world. They also present themselves in superficial worlds created by people. For example, fractal equations are used in films to create the appearance of chaos. Both an explosion of lava in Star Wars and an entire separate world in Star Trek: Wrath of Kahn were created by simple fractal codes reiterated over and over again (Hunting). Fractals are found naturally, and are being created all over the place; one just has to know where to look. Most would agree that the human body is one of the most complicated organisms in nature; as such, it is a prime example of fractals in nature. Blood vessels are an ever shrinking branching structure transporting blood from the heart to the rest of the body (Briggs 124). Also, the heart beat itself is a fractal structure. While many may think of the heartbeat as a periodic rhythm, that is not so. When a special kind of plot is created, the heartbeat shows a kind of irregular fractal web instead of the circular pattern that would be created by a periodic rhythm (Briggs 126). Briggs explains why nature uses fractals in organisms saying, “Because each of the body’s fractal-shaped structures is redundant and irregular, parts of fractal systems can be injured or lost with relatively minor consequences” (124).Finally, Richard Taylor, a professor at the University of Oregon, is experimenting with the eye to find out how it is able to absorb so much information. The eye could be yet another fractal creation within the body (Hunting). Blood vessels, the heartbeat, and the eye are only a few of the many examples of fractal structure with in the human body. Another natural example of fractals is coastlines which can be explained by more complex versions of the Koch curve. Helge von Koch created the Koch curve when he took an equilateral triangle and substituted a part of its side with two new lines creating a triangular bump. If this is repeated infinitely, it creates a curve that is infinitely long. Koch curves are lines that fulfilled the requirements of a curve, but are not considered actual curves (Hunting). Because Koch curves are too systematic, they are not considered the best representation for coastlines (Mandelbrot 35). Still, coastlines are considered a natural fractal because first, they are irregular; and second, their irregularity is the same at all levels of scaling (Murdock 3). In addition to this, when one accepts more minute details into the length of a coastline, the coastline’s length becomes longer. This is shown when one would measure a coastline using only the details visible from space versus the details visible from a helicopter ride along the coastline (Mandelbrot 25). Loren Carpenter, the co-founder and chief scientist for Pixar Animation Studios, explains this bizarre phenomenon well: “If you measure the coastline of Britain with a one mile yardstick, you would get so many yardsticks, which gives you so many miles. If you measure it with a one foot yardstick, it turns out that it is longer. And every time you use a shorter yardstick; you get a longer number” (qtd. in Hunting). Although it is impossible to determine the exact length of a coastline, different coastlines can be compared to one another by comparing their fractal dimensions. Finally, trees are a note worthy example of natural fractals. Entire forests can be represented by the fractal structure of a single tree. In addition, the single tree is a fractal and is itself made up of numerous fractals. For example, its bark and its veins are fractal structures (Briggs 36). Geoffrey West, a Santa Fe Institute professor, James Brown, a University of New Mexico professor, and Brian Enquist, a University of Arizona professor, performed an experiment intending to prove their theory that fractals hold the key to how organisms and ecosystems work. Enquist believed that by knowing the fractal basis of a tree, one could mathematically predict the structure of a whole forest. In their experiment they planned to find first the fractal ratio of the branches within a tree, then, the fractal ratio of a sample of the forest. After this, they would plot the data. Their results were staggering. The narrator of Hunting the Hidden Dimension explains, “The relative number of big and small trees closely matches the relative number of big and small branches.” Using the information from a tree’s fractal structure, one can easily estimate the fractal structure of the entire forest. For centuries, fractals were present in art and architecture but were unknown to mathematicians. In time, patterns were discovered that were initially called monsters. These monsters were the outcasts of mathematics; yet, they held enough interest in the minds of mathematicians to create even more complicated sets. Ultimately, Mandelbrot named and explained fractals, and with that, fractals shed their role as outcasts and became the darling of mathematics. Ever since mathematicians took up the use of fractals, they permitted mathematics to be in relationship with the wonders of nature. This relationship allows for the explanation of the real nature people live in and the creation of synthetic natures.