frac·tal
/ˈfraktəl/
MATHEMATICS
noun
a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation.
Preface:
In our universe, fractals and patterns are in abundant existence, whether it be in the scale of microcosms or macrocosms. The scale of self-similarity between the micro and macro level is outstanding, certain patterns and fractals form throughout all scales of the universe in the most effective and efficient ways. The concepts of fractal self-similarity, repetition, and iteration help us as people form visions and principles of ecology and interconnection. Complementing the understanding of an atom, fractals provide humans and a deeper comprehension and further understanding of the universe itself. In ancient Mesopotamia, astrology was derived from the thought people expressed that mathematics and the placement of the universe and the stars can tell us a deeper understanding of ourselves, and is rooted in self-similarity.
Based on the concept of the Law of Correspondence, the smaller processes happening in the universe at the micro level are almost equivalent to those processes happening at the macro level. A prime example of this is that the depiction of the electromagnetic field of a human heart is identical to that of Earth's and other galaxies' electromagnetic fields, just on a smaller scale.
When envisioning science and mathematics, there is the often popular misperception that reality is smooth, while in reality both nature and the texture of the world is rough. For example, in nature a leafs outermost layer is rough in order to retain as much surface tension as possible in order to hold water and nutrients. Nature in itself can be described as rough where fecundity and excess of life shows us the fragility and brutality of the world's order. The mentors selected to provide further understanding of this concept are two transcendentalists Annie Dillard and Henry David Thoreau, and a mathematician, Benoit Mandelbrot, who is famous for the Mandelbrot Set. It can be argued that all three of these mentors through ecology, philosophy, mathematics, and immersion try to perceive and understand the border between chaos and order. From Dillard focussing on the fractals of a leaf and the complexity, as well as simplicity, of its efficient ways to grow using repetition of nature, to Thoreau scaling out and observing nebulae within eyes and ponds. Throughout this essay, I will be proving the relationship between fractal geometry and the theory of interconnectedness between the macro and micro scales. The two mentors from this course I have chosen are Dillard and Thoreau, as they both have interesting perspectives through immersion on the natural world. It’s necessary to include the mathematician Mandelbrot as a reading mentor since he developed the key concept of fractal geometry.
This term fractal was coined by a Polish-born mathematician named Benoit Mandelbrot. The term derived from the Latin word "fractus," meaning "broken" or “fragmented” refers to the rough texture in the natural world. Mandelbrot rose as a fractal pioneer in the 1980s, publishing the seminal work The Fractal Geometry of Nature, which laid the foundation for mathematical and philosophical fractal theories. This significant foundation has since been adopted by analysts, scientists, and theoreticians from a wide range of disciplines, from art to astrophysics, economics to natural sciences, social media to medicine, and climatology to human psychology. This key idea of fractal patterns helped shape ecology into the concept of interconnection. Interconnection is the concept describing the mutual feeling of relativity between one or multiple entities. As science and math appears to show nature as this smooth order, because of numbers being entirely certain and calculated, Mandelbrot helps show that nature is anything but that, and is in fact rough and fractured. As perception down the ladder turns into creation, our own perceptions have the power to manipulate the world around us. An example of this is Mandelbrot's idea of a coastline being measured in various ways depending on who is conducting the measurements. Perception is the key to understanding how something is defined and ultimately created. Mandelbrot describes the world as the border between chaos and order. Fractals and patterns may be found in all kinds of things, including animals, vegetation, and minerals. One very interesting example of this is the formation of human lungs compared to that of the structure of a tree. Another prime example of these patterns found amid the chaos of nature, could perhaps be the shape of lightning paths and their similarity to the way a river fragments the land and rips through the earth. These patterns found throughout nature are indeed fractals. A fractal is a pattern that can be found in both nature and art. When you notice a series of patterns repeating again and over at many different scales, and any small component mimics the whole, that's a fractal. Fractals are fascinating not only because of their mathematical or conceptual depiction, but also because you can see the remarkable arithmetic geometry in action.
Dillard and Thoreau alike had deep perceptions on nature, viewing the natural world from a neutral stance, also known as a full immersion into nature, physically and mentally. Dillard’s recognition of the repetition and iteration taking place throughout nature helps us better understand the idea of nature constantly repeating itself. In the chapter titled Intricacy in The Pilgrim at Tinker Creek, Dillard is able to shape her ecological vision by expanding on her fractalized view of the world. She focuses on the frayed and fringed textures of the world. This has a deep correlation to Mandelbrot, who claimed the world is fractured and resembles that of broken glass. This texture of roughness in the world also plays into Dillard's writing as a whole, and more specifically, in the Fecundity chapter. The world is not a particularly happy, smooth place; rather it’s a rough harsh one that doesn't relent. The roughness and repetition of Dillard’s writing fits within the bigger theme at play, relating back to the natural world around us. Her writing, like the world, nature, and organisms, is rough for many different purposes. Texture makes a sizable impact within our very own bodies as the human lungs are not perfectly smooth. This is vital to our lungs as this rough texture gives leeway for many small cavities that are designed to capture more oxygen, just as a leaf’s rough texture allows it to hold more water on its surface. While her writing is rough in order to symbolize the natural world around us, it is also repetitive for this very same reason. She has observed the repetition of nature first hand and this style of writing is a very perceptive and poignant way of illustrating the repetition within the universe. This repetition is what Dillard describes as the loop. She sees a shedded snake skin and ultimately relates that discarded piece of epidermis to a cycle that is constantly in motion, making the same steps in the future that happened similarly in the past. I perceive her writing as a fringed tangle, emulating the texture she discovers in the movement between her thinking and her perceiving, similar to the movement between creek water and creek bank. This could be described as fractal texture. The book as a whole is a fractal. One way to imagine the concept of fractal texture in Dillard's writing is that if you keep magnifying an image of a border or a coastline, say, or the edge of a cloud, each larger or closer image will have a pattern similar to the original. So, to reiterate without exact repetition, the spiraling loop Dillard describes is the constant intricacy based on a beautiful and incomprehensible simplicity. This loop articulated by Dillard is the constant repetition in nature referred to by Mandelbrot's conceptualized idea of fractals. Repetition occurs all throughout nature, and efficiency and iteration keep fecundity in excess. Chaos and order occur within the same universe. When Dillard is trying to make sense of this she ultimately comes to the conclusion that the world is neither chaos nor order, but rather a balance in the middle. As nature's growth is orderly fashioned and often repeated uniformly, whether it’s tributaries, mountain ranges, lightning, waterfalls, trees, wood burning, or coral, all growth follows their respective, exact efficient structures. This is the order side of the universe. Then chaos comes into play with the opposite of growth. Where there is a yin, there is a yang. With growth comes disease, death, parasites, shrinking, and withering that occurs sporadically and without reason. One could argue the reason for the chaos is a counter balance to the order.
Thoreau, like Dillard, immerses himself within nature in order to have the balanced perspective necessary to create a valid and legitimate opinion about the natural world. In Walden’s chapter titled Spring, there is a key moment where Thoreau looks at the bank.
“The Maker of this earth but patented a leaf. What Champollion will decipher this hieroglyphic for us, that we may turn over a new leaf at last? This phenomenon is more exhilarating to me than the luxuriance and fertility of vineyards. True, it is somewhat excrementitious in its character, and there is no end to the heaps of liver lights and bowels, as if the globe were turned wrong side outward; but this suggests at least that Nature has some bowels, and there again is mother of humanity.”(310)
This is a key moment in the chapter as Thoreau is talking about the cycle taking place of the eroding bank and the tree losing its leaves. This very cycle is the same cycle, or loop, Dillard examines with the snake skin. Thoreau gives nature and the bank human-type qualities, and gives a sort of interconnected vision to his ecological principles. For example, within the text he exclaims that nature has bowels and describes the mother of humanity. The scale at which Thoreau takes is similar to Dillard, however, I would argue it’s more of a macro level, whereas Dillard focuses more on the micro scale. Thoreau compares nebulae in space to the eyes of a person, and even states that Walden pond is the eye of the earth. This comparison made by Thoreau highlights another perspective of interconnection between the human eye to that of the unknown nebulae in space and the consciousness of the earth. This process, if thoroughly explored, might be a way of understanding natural phenomena that occur and are usually classified as random or chaotic. These patterns illustrated by Thoreau give a certain meaning of unity and harmony to what is otherwise labeled as a random or chaotic universe, with a human connection to it all. Chaos theory, sometimes known as the butterfly effect, is an area of mathematics that studies systems which appear to be ordered, but actually exhibit chaotic behavior. Paradoxically, it also deals with systems that appear chaotic, but are actually ordered beneath the surface. This theory, for me, is heavily connected to the ideas of both Thoreau and Dillard, as we see them observe what appears to be at first natural chaos, but it is actually ordered efficiently and consistently throughout nature. This study conducted by Thoreau and Dillard of non-linear, dynamic systems are highly sensitive to initial conditions. This study is popularly referred to as the butterfly effect. The reason that Thoreua and Dillard are able to grasp and understand these perceptions is because they are not just thinking all the time. A person who thinks all the time has nothing to think about except thoughts, leading to a loss of touch with reality, resulting in living in a world full of illusions. Dillard and Thoreau are perceiving the natural environment around them and then posing thoughts relating to that which is happening. This is a crucial way to develop perception and understand the reality in a world filled with illusions.
The Law of Correspondence as Hermes Trismegistus wrote in the Emerald Tablet states, "That which is Below corresponds to that which is Above, and that which is Above corresponds to that which is Below, to achieve the miracles of the One Thing." As above, so below. This means that by understanding ourselves or even subatomic atoms, we can start to begin to understand the cosmic universe and vice versa. Dillard, Thoreau, and Mandelbrot each brought about diverse perspectives on the fractalized structure of the universe and its connection to human beings. This interconnectedness, through the concept of fractals, helps manipulate visions in a very harmonious and unifying way, creating a sense of togetherness with the environment and natural world. A sense of meaning coming from beyond consciousness and coming from a higher power, whether that be the aura of the cosmic universe or something else, fractals help shape our understanding, vision, and perceptions of one another and the physical and metaphysical world around us.
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