A Panorama of Fractals and their Uses

Until recently, all of nature was thought to follow Euclidean geometric principles, where simple shapes make up all shapes in the natural world and simple statistics describe the occurrences of natural phenomena. However, the study of fractal geometry, spearheaded by Benoit Mandelbrot, shows that fractals – rather than linearity – may be the predominant tool in designing the universe. Fractals occur in nature both organically and inorganically, both tangibly and intangibly. Tangible inorganic fractals, like mountains, are those that can be touched and studied without extensive equipment, whereas intangible inorganic fractals, like lightning, cannot. There are some scientific contentions over the fractal nature of nature and some links between organic and inorganic fractals.

Organic fractals occur in living beings, such as ammonites, trees, broccoli, and even humans. These fractals manifest in external structure, internal structure, and in outward design. In ferns and leaves, the external structure of the plant is self-similar at multiple scales. One can describe the construction of ferns and their design using Barnsby’s fern formula (McNally, 2010).  Zebra stripes and leopard spots can be simulated by using elliptical neighborhoods for cellular automata that determine the coloring of fur. Changing the inhibitor concentration pattern changes the size, density and location of the spots or stripes; this same mechanism works for both spots and stripes (Panorama of Fractals and their Uses).  Fractals are also useful in the internal structure of living organisms. In ammonites, sutures connecting different parts of the shell are organized in a fractal pattern. In humans, both the lungs and the cardiovascular system are built using fractals. Fractals in lungs allow for maximum efficiency in getting oxygen into the blood supply. According to West and Goldberger in their article on fractals in physiology, “lung structure appears roughly similar from one generation of tubes to the next.” The heart is a particularly violent and powerful pump, and the fractal structure of the cardiovascular system surrounding it damps the force of each beat (Panorama). The fractal structure of organic parts of nature is reflected in inorganic parts of nature.

Inorganic fractals manifest in both tangible and intangible ways. Tangible inorganic fractals include mountains, salt flats, coastlines, and rivers, among others. When measured with area-perimeter relations, we find that mountains have dimensions between digits, making them fractal (Panorama). Mountains can be most effectively forged in computer simulations when using fractals. Salt flats crack in fractal formations. Salt flats have large cracks that are then iteratively broken down into smaller and smaller generations of capillaries. Coastlines are self-similar at almost every scale, and are generally only seen as non-rough curves when physically on the beach. Sapoval theorized that coastlines damp waves to help minimize coastal erosion, and fractals help stabilize the coast as fractals are especially effective at damping the force of waves. Rivers and waterfalls are constantly moving. Rivers help irrigate land and move water from one body of water to another; to maximize water distribution, many river deltas are organized in fractal designs. Gravity and irregular rock surfaces cause waterfalls, creating fractals (waterfalls) within fractals (rivers). Besides these tangible forms, fractals also exist in intangible forms.

Nontangible inorganic natural fractals occur in clouds, lightning, earthquakes, and galaxies. Fractals in clouds could be caused by turbulence, and cause disturbances in the shapes of cumulus clouds. Clouds, like coastlines, can be measured with area-perimeter relations. The equation for these relations is  (Mandelbrot, Avnir, et al., 1998; Panorama). Benoit Mandelbrot, in the blurb of his book “The Fractal Geometry of Nature,” mentions that lightning is fractal (Mandelbrot, 1982). When lightning strikes, the bolt ends up in a self-similar, branching design. This is because its path to the ground is formed by a step-by-step, iterative journey. Earthquakes are more thoroughly fractal, in that several aspects of earthquakes are fractal. The magnitude-frequency distribution, spatial distribution of hypocenters, and frequency of aftershock occurrences are all fractal (Matsuzaki, 1994). Fractals in galaxies are a point of contention within the scientific community, as some scientists believe galaxies are distributed fractally and others do not (Panorama). The distribution of galaxies could be fractal, and there is general agreement about fractal structures out to about fifty million light years away from us. Galaxies include self-similar fractal distributions at all, or nearly all, scales (Panorama). The cosmological principle states that the physical laws of the universe are constant across all of the universe; every part of the universe is at least somewhat similar to every other part of the universe. Scaling every part of the universe to be similar is self-similarity, and therefore fractal-like.

As stated before, there is some scientific contention over the fractal nature of the universe. Where before the contention was over galaxies, there is also contention over the span of fractals in nature. After Avnir et al. wrote a paper on the fractal nature of nature, Benoit Mandelbrot wrote a letter criticizing the paper (Mandelbrot et al. 1998). Mandelbrot wrote that nature is not purely made of fractals, and that there are always upper or lower cutoffs for scaling, making nature not a true fractal. Mandelbrot points out that seeing fractals in everything in nature is a “side effect of enthusiasm” (Mandelbrot, 1998, p. 783). The authors of the original paper responded to this letter, attesting that their research shows that “the limited-range empirical fractals are the dominant justification for the fractal geometry of nature” (Avnir et al., 1998). This response cites Mandelbrot’s own work with power laws and talks about Lovejoy’s report on the fractality of clouds (clouds can be measured with area-perimeter relations).

The “Forgeries of Nature” page in the Panorama of Fractals and their Uses shows how computer models can make lifelike recreations of mountains using fractals. Simulations that use fractals are more lifelike than those that use Euclidean geometry. Simulations are supposed to emulate how things really work; this could suggest that fractals are an inherent part of nature. Using iterative equations that employ fractals can produce more than just lifelike mountains; they can also produce simulations of both organic and inorganic natural structures. The fractal nature of both lungs and coastlines function as a damping tool to minimize the effects of a powerful incoming force. Lungs stabilize the amount of blood going into the cardiovascular system, which is necessary as a surge in the amount of blood going into the bloodstream would cause permanent damage and perhaps even death. Coastlines use fractals to dampen the force of waves upon the land behind the coast; fractals are extremely efficient at dissipating this force so that strong waves do not seriously erode land masses. Rivers spread out in fractals to maximize efficiency in irrigating land; cardiovascular systems do so to maximize efficiency in spreading oxygen throughout the body.

Fractals are widespread throughout nature, as the Panorama of Fractals and their Uses points out. Fractals exist both in living beings and in inorganic surroundings such as mountains, rivers, and clouds. Organic fractals feature external structure, internal structure, and design. While there is some debate over the span of fractals in nature, simulations of and similarities between fractal structures show that fractals are an inherent part of nature, even if they’re not the only way the universe is organized. This may be because recreating a structure iteratively over multiple scales takes less energy or planning than individually manipulating each iteration. This may also be because fractals exist between dimensions, making them more efficient at spreading things around than structures that exist in two or three dimensions. The prevalence of fractals in nature allows for important biological functions to exist, as well as the existence of planetary structures and weather conditions.



Works Cited

A Panorama of Fractals and Their Uses. Yale University. Retrieved from http://users.math.yale.edu/public_html/People/frame/Fractals/Panorama/Welcome.html

Mandelbrot, B., Pfeifer, P., Biham, O., Malcai, O., Lidar, D., & Avnir, D. (1998). Is Nature Fractal? Science, 279(5352), 783-786. Retrieved from http://www.jstor.org/stable/2894997

Mandelbrot, B. B. (1982). The fractal geometry of nature. Freeman, Oxford.

Matsuzaki, M. (1994). Fractals in Earthquakes. Philosophical Transactions: Physical Sciences and Engineering,348(1688), 449-457. Retrieved from http://www.jstor.org/stable/54221

McNally, J. (2010, September 10). Earth’s Most Stunning Natural Fractal Patterns. Wired Magazine. Retrieved from https://www.wired.com/2010/09/fractal-patterns-in-nature/

West, B., & Goldberger, A. (1987). Physiology in Fractal Dimensions. American Scientist, 75(4), 354-365. Retrieved from http://www.jstor.org/stable/27854715