# Fractals Fractals Everywhere!

What is a fractal?

After centuries of mathematical thought and mystery, Benoit Mandelbrot solidified fractal geometry in the 1970s. Unlike the geometry you learned in high school, which consists of smooth shapes and straight lines (circles, squares, triangles, etc.), fractal geometry describes “rough” complex patterns and shapes that are seen all over nature. These seemingly complex shapes are made by reiterating simple rules over and over again at different sizes. Fractal patterns are self-similar, so they look the same when you zoom in or zoom out.

Fractals are everywhere!

Why?

They are created by simple iteration. Over time, replicating processes produce these patterns. Also, fractal patterns are thought to be an efficient and optimal way to fill space and transfer resources1. For example, fractal patterns in blood vessels allow for space filling of tissue and transfer of nutrients. Branching veins and arteries disperse all throughout our bodies, connecting back to central pathways. This is much more effective for distributing nutrients to the farthest corners of our fingertips than say, straight lines all connecting to the heart or a square grid pattern.

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Fractals can even be found in behavior patterns.

Although the examples above show fractal space-filling patterns, a behavior can have fractal patterns across time. Instead of looking at how often or how long a specific behavior occurs, researchers consider the pattern in which a behavior occurs. A behavior can occur in a repetitive pattern, a more complex fractal-like pattern, or completely randomly. Researchers have found fractal patterns in social behavior of chimpanzees2, foraging behavior of chicks3, locomotion and foraging behavior of Japanese Macaques4, surfacing behavior of dolphins5, and dive sequences in penguins6.

Like other fractal patterns in biology, fractal behavior patterns are thought to be a good way to interact with the environment. Complexity in the pattern of which an animal acts theoretically allows them to interact optimally with resources. For example, two animals could spend the same amount of time searching for food, but one may do so in a more complex pattern that is sensitive to what has previously happened, as opposed to a repetitive pre-determined pattern that would limit their opportunities.

Studies have shown that when an animal is exposed to a stressful state such as parasitism, their behavior patterns become less complex and more repetitive. Similarly, virus-infected pepper plants have lowered complexity in branch structure compared to healthy plants7. Illnesses or stress use up energy, so less energy is available to uphold a complex behavior pattern. This instead produces what is thought to be a sub-optimal pattern.

Fractal patterns have emerged over and over again in nature as efficient solutions for transferring information and resources. From plant growth, to neuron networks, to the branching of our lungs, to the patterns of animal behaviors, fractals are everywhere!

Citations:

(1) West, B. J. (1990). Physiology in Fractal Dimensions: Error Tolerance. Annuals of Biomedical Engineering, 18, 135-149

(2) Alados, C. L. & Huffman, M. A. (2000). Ethology, 106, 105-116

(3) María, G. A. Escós, J. M., & Alados, C. L. (2004). Applied Animal Behavior Science, 86, 93-104

(4) MacIntosh, A. J. J., Alados, C. L, & Huffman, M. A. (2011). Journal of the Royal Society Interface, 8, 1497-1509

(5) Seuront, L., & Cribb, N. (2011). Physica A, 390, 2333-2339

(6) MacIntosh, A. J. J., Pelletier, L., Chiaradia, A., Kato, A., Robert-Coudert, Y. (2013) Scientific Reports, 3: 1884, 1-10

(7) Escós, J. M., Alados, C. L., & Emlem, J. M. (1995). Oikos, 74, 310-314

(8) Goldberger, A. L., Amaral, L. A., Hausdorff, J. M., Ivanov, P. C., Peng, C. K., & Stanley, H. E. (2002). Proceedings of the National Academy of Sciences of the United States of America, 99, 2466-2472