Fractal Utility: The Reduction of Angst

I have felt that fractals have a somewhat mystical nature. I have been seeing fractal like images since childhood and have engaged in fractal like artwork since that time as well. Of course, seeing fractal art was a whole new experience in its’ own right. In fractals and fractal images a person can somewhat lose themselves.

This has led to some interesting literature and the deeper I dug the more interesting the literature became. In this paper I intend to explore some areas in which fractals have been shown to be useful and where they have contributed to advances in the fields of mathematics, technology, and art.

Fractals emerged in the field of mathematics. Fractals are algebraic non-linear equations. The simplest form of a fractal is znew = zold2 + c. Fractals form beautiful and bizarre geometric shapes. They exhibit a property called self-similarity. At different scales they look the same in that you can take a small sample of the shape and it looks the same as the entire original shape. Mathematical fractals are also incredibly complex. You can zoom into them forever and more detail keeps emerging.

Fractals have served as a metaphor for many of realities processes. They are examples of how structures can be formulated from the simple to the complex through repetition, iteration, feedback, and the principle of wholeness. In Shipshape Earth and other Fractal Holding Patterns, Jackson writes, “Fractal images are like diagrams or mosaic mandalas of the systemic complications held together by an underlying simplicity of principle. They are visible and useful in imaging the structures of networks and the dynamics of mutual interpenetration and co-belonging of all living organisms in their communities and environments” (Jackson, 2004, p. 61).

Since their discovery and the coinage of the term ‘fractal’ by Benoit Mandelbrot, mathematician and engineer, in the 1980’s, fractals have taken part in a scientific revolution where human ingenuity and adaptability are directly evident. In “Fractal Art: Closer to Heaven? Modern Mathematics, the Art of Nature, and the Nature of Art” Saitis writes,

"Computers became an integral part of this scientific revolution, being capable of calculating non-linear dynamical systems and visualizing the results overtime; scientists were able to watch the chaotic evolution of a naturally occurring human phenomenon on a computer’s screen. Eccentric shapes, irregular geometric objects and extraordinary figures emerged, revealing a world where the unreal (graphic representations of algorithmic processes) implied the real (nature). Benoit Mandelbrot was the first mathematician to shape this new area into an individual self-standing theory, which instantly became the most-popular of all. He introduced the neologism fractal to unite all these strange objects under one term” (Saitis, 2007, p. 155).

Fractals are everywhere in nature. Branching is a fractal quality. We find this pattern repeated again and again. Inside our own bodies blood vessels and neurons display the branching process. In nature lightning bolts, trees, river networks, and ferns are examples of the branching process. Another fractal design commonly found in nature and our bodies (which are of course an extension of nature) is the spiral. Spirals are found in fossils, hurricanes, DNA, galaxies, plants, and the turbulent forces of fluids. Other naturally occurring fractals are clouds, coastlines, snowflakes, shells, flowers, the heartbeat, and the human brain. In After the Clockwork Universe: The Emerging Science and Culture of Integral Society Goerner writes, “Fractals are important scientifically, however, because they are everywhere. Any thing that is rough---with zigs, zags, scatter, bumps, or wobble---is likely to be a fractal and when you look closely, virtually everything is rough. Thus, more and more members of the new science are standing up and saying that fractal are nature’s geometry” (Goerner, 1999, p. 123).

Given the ubiquitous nature of fractals and the omnipresence of fractal processes one can begin to imagine the ingenious ways in which fractals and their principles have been employed. Fractals have been used in some interesting and novel ways. I will now explore some of the ways in which fractal have been put to use.

Fractals have been very influential in the field of Mathematics itself. It has served in a place of promotion and advertising for that field. Instead of regarding math with dread, a young student may be enticed by fractals into the field. The Fractal Foundation in “Fractal Pack 1” writes “The great value of fractals for education is that they make abstract math visual. When people see the beautiful and intricate patterns produced by equations, they lose their fear and instead become curious” (Fractal Foundation, 2009, p. 8). In the same pack they provide visual depictions of mathematical abstractions such as 3-fold symmetry as opposed to 4-fold symmetry, and Bifurcation (branching in these fractals): (Fractal Foundation, 2009, p. 9-10).

“Fractal Pack 1”, a treasure trove of information on fractals, also provides fractal applications in twentieth century technology and medicine (Fractal Foundation, 2009, p.11) It provides the examples of a computer chip cooling circuit etched in a fractal branching pattern, fractal antennas for cell phones, and fractal analysis used in the field of medicine. Very simple processes using fractals are also used to generate computer images of natural elements like water, mountains, and clouds. It also has promise for providing graphic texturing. Reverse processes offer hope in the area of image compression (Mathigon, p. 11). In “Fractals,” it is stated, regarding shapes in nature that are ‘fractal like,’ “These shapes appear to be completely random, but as with fractals-there is an underlying pattern that determines how the shapes are formed and what they will look like. Mathematics can help us understand the shapes better, and this has applications in medicine, biology, geology, and meteorology” (Mathigon, p. 11).

Speaking of the graphic potential of fractals, this brings us to a topic where fractal depictions are the heart of the matter—fractal art. Saitis writes, “Artists were instantly attracted to fractals. Artistic interest burst out, resulting in a new form of digital art, which rapidly became popular both inside and outside the artistic and scientific communities” (Saitis, 2007, p. 154). He also gives us a simple definition of fractal images. He states simply “Fractal images are digital images that either consist of a single fractal or are composed of several fractal and non-fractal objects.

How are fractal images produced? According to Saitis there are three ways in which fractals can be generated each corresponding to a classification of self-symmetry. The first way in which fractals may be generated is as “Escape-time fractals: defined by a recurrence relation at each point in space.” The Second way is as “Iterated function systems (IFS): a fixed geometric replacement rule exists.” The third way is as “Random fractals: these are generated by stochastic rather than deterministic processes” (Saitis, 2007, p. 155). Simion also addressed the topic of generating fractals. She writes, “Three key elements are required to generate fractals: the initiator the whole from which it begins, the generator- the rule to make a new shape from the whole and iteration- the repetition of the rule infinitely” (Simion, 2015, p. 1).

This is all very technical but if you go to the Fractal Foundation website, they have a wonderful listing of fractal generating software programs. Some of these are offered for free. This is where you can begin generating fractal images and learn to develop fractal images into dynamic expression. This list can be found at http://fractalfoundation.org/resources/fractal-software/. Saitis states, “Software tools have become widely available, whereby the artist is able to algorithmically generate a fractal, to apply color patterns, and to compose digital images.” He lists three of the most used fractal generating programs: Mojo world; Apophysis; and Ultra Fractal. If you are seriously interested in producing works of fractal images, one of these programs might be worth the investment. I would do a little experimenting and researching before I made that decision. Some programs offer free trial memberships during which you can experiment to see if a particular program is right for you.

This concludes our preliminary investigation on fractal utility. This utility, however, also extends to the soft sciences and spirituality where they have not only incorporated fractal images into their research (Simion, 2015) but have postulated fractal structuring of the human self, consciousness, and the human brain. According to Simion, “Other studies conducted at Harvard Medical School showed that electric signals of the nervous system follow a fractal pattern (model). It seems that our mind has an innate property to recognize and resonate with fractals” (Simion, 2015, p.2). It all starts with the human eye. Jackson states,

“When we look at something we are actually also examining the nature of our own eyeballs with their photoreceptive rods and cones, and the nature of light, and the nature of the brain, as well as the thing seen: all these examinations are simultaneous and usually about two-thirds conscious. Hence, we are a partially submerged mutual feedback system (and this has implications for beauty)--we echo, project, and replicate and appreciate—we are part of a rhythm, participating in mutually reflecting and generating patterns” (Jackson, 2004, p. 67).

We are nested and embedded in larger systems of complexity and systems of complexity are nested and embedded in us. Simion writes, “It can easily be said that humans are attracted and respond to those stimuli which resembles their internal structure. By doing so, they connect their inner world with the outside objects that are one way or another self-similar with what’s inside their own body (a fractal relation)” (Simion, 2015, p.2). Jackson further states, “They illustrate our own embeddedness in a larger interrelated system around us. We can observe how fractal processes involved in the yin and yang interfaces of “out there” and “in here” become second nature to us” (Jackson, 2004, p. 67). This explains why study after study has shown that viewing fractal images reduces stress and promotes wellbeing (Simion, 2015, p. 2). So, get yourself a book on fractal illustrations (there are many) or google search “fractals” and just lose yourself amongst the curves and depths, you may just lose your angst.

REFERENCES

Fractal Foundation. (2009). “The Fractal Pack 1 Educators’ Guide.” Retrieved from:

Fractal foundation.org/fractivities/FractalPacks-Educator’sGuide.pdf.

Goerner, S.J. (1999). After the Clockwork Universe: The Emerging Science and Culture of Integral Society. Edinburgh, Scotland: Floris Books.

Jackson, W. (2014). Heaven’s Fractal Net. Bloomington IN: Indiana University Press.

Mathigon, date unknown, “Fractals” retrieved from: https://mathigon.org/word/Fractals.

Saitis, C. (2007). “Fractal Art: Closer to Heaven? Modern Mathematics, the Art of Nature, and the Nature of Art.” retrieved from: researchgate.net/publication/233760832_Fractal _Art_closer_to_Heaven_Modern_mathematics_the_art_of_Nature_and_the_nature _of_Art.

Simion, R.M. (2015) “The Fractal Technique- An Experiential Approach of Fractal Images In Reducing Perceived Stress Through Therapy of Unification” in Journal of Experiential Psychotherapy., vol. 18, N.2 (70), June 2015.