History, Development, and Applications of Fractal Geometry

Fractal [frak-tl], noun

  1. A geometric or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.[i]

It was the work of philosopher and mathematician Gottfried Leibniz in the 17th century that first provided groundwork for the early development of fractal geometry.[ii] However, the nature of recursive self-similarity (a core tenet in the study of many fractal-like objects) and the mathematical “monsters”[1] to which this emerging concept gave birth, consequently delayed meaningful research for roughly two centuries.[iii] Nearly two hundred years elapsed before Karl Weierstrass, in 1872, presented the first definition of a function whose graph by today’s standards would be considered fractal. That is, Weierstrass had shown it was possible to define a function with the non-intuitive property that it could be both everywhere continuous and nowhere differentiable.[iv] Subsequent works including those of Georg Cantor (specifically “Cantor sets”), Felix Klein, and Henri Poincaré were crucial in laying a foundation on which much of the modern mathematical investigation of fractal geometry unfolded. Without the aid of modern computational and graphical tools, however, much of the early research in fractal geometry was severely limited. It would be more than a half-century until, in the 1960’s, equipped with the work of his predecessors (i.e., Helge von Koch, Wacław Sierpiński, Pierre Fatou, Gaston Julia, Felix Hausdorff, and Paul Lévy to name a few) did the French-American mathematician Benoit Mandelbrot succeed in uniting hundreds of years of mathematical research by coining the word “fractal”[2] and illustrating his mathematical definition with the aid of remarkable computer-generated visualizations.[v] Most notable among Mandelbrot’s demonstrations was his use of infinite recursion to define what is known today as the Mandelbrot set. Mathematically speaking, the Mandelbrot set is defined to be the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex polynomial zn+1=zn2+c remains bounded. Equivalently, a complex number c is part of the Mandelbrot set if, when starting with 0=0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.[vi] This seemingly innocuous procedure is responsible for generating the beautiful, infinitely detailed Mandelbrot set pictured below.

Mandelbrot set. Image: Wolfgangbeyer, via Wikimedia Commons.

This figure, and others like it, is generated with the aid of modern computational tools capable of carrying out a myriad of successive iterations far greater in number than the computational limitations of “by-hand” human calculation. The study of infinite descent is also seen in other areas of mathematics such as algebraic geometry (specifically Fermat’s Last Theorem). In these fields, infinite descent is a tool by which the existence of certain special triangles may be determined. For example, infinite descent may be used to prove that no right-angled triangle having integral side lengths will have a square area. The mysterious nature of objects when examined at their infinite limits is, indeed, quite peculiar.

In addition to producing stunning artistic images, fractal geometry has also found diverse applications in fields such as structural engineering, medicine, telecommunications, urban planning, and more.[viii] Iterative methods have been used to create high-strength construction cables by interweaving a series of thinner wires into thicker ones that, in turn, are used in the next iteration to make even larger and stronger cables than before. This simple procedure, usually no more than a few iterations for most industrial purposes, bears a resemblance to a fractal pattern and allows us to make use of the special properties that fractal geometry offers.

Iterative construction techniques used to create high-strength construction cable. Image: Bernard S. Jansen and Jonathan Wolfe, via Fractal Foundation.

As medical knowledge continues to improve, the usefulness of fractals in curing and identifying health concerns becomes increasingly more apparent. Biomimicry, the concept of deriving inspiration for human designs from the natural world, is currently being employed in an attempt to solve the problem of fluid transport by mimicking the fractal patterns of our blood vessels and lungs.viii

Fractal heat exchanger etched in silicon and designed by Deb Pence at Oregon State University, via Fractal Foundation.

According to researchers at Oregon State University, the above figure can be etched into silicon chips, allowing for a cooling fluid (such as liquid nitrogen) to uniformly flow across the surface of the chip, keeping it cool. Researchers say this fractal pattern was derived from human blood vessels and provides a simple low-pressure system to easily cool sensitive computer chips.viii In this case, the relationship between biomimicry and fractals becomes quite clear. The natural question to ask, then, is to what extent may biomimicry assist researchers in addressing biological and natural phenomena? The future is certainly promising.

While classical Euclidian geometry and other related areas of mathematics are often used to understand and predict natural phenomena, these traditional modes of thinking may prove insufficient in answering some of the more complex questions that arise in nature (some of which have been discussed here). After all, as Benoit Mandelbrot himself once said, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”[ix] It is the heart of fractal geometry that attempts to analyze and understand many of these more complex phenomena.

[1] These “monsters” were mathematical problems of immense complexity that, according to Leibniz and others, were believed to be beyond the scope of contemporary geometric knowledge.

[2] From the Latin word frāctus meaning “broken” or “fractured.”

[i] “fractal.” Dictionary.com Unabridged. Random House, Inc. 07 Feb. 2015. <Dictionary.com http://dictionary.reference.com/browse/fractal>.

[ii] Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. p. 310. ISBN 978-1-4027-5796-9. Retrieved 2011-02-05.

[iii] Gordon, Nigel (2000). Introducing fractal geometry. Duxford, UK: Icon. p. 71. ISBN 978-1-84046-123-7.

[iv]Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8.

[v]Albers, Donald J.; Alexanderson, Gerald L. (2008). “Benoît Mandelbrot: In his own words”. Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0.

[vi]“Mandelbrot Set Explorer: Mathematical Glossary”. Retrieved 2007-10-07.

[vii] Mandelbrot Set. Digital image. Mandelbrot Set. Wikipedia, 30 Jan. 2015. Web. 7 Feb. 2015.

[viii] “Fractal Applications.” Fractal Foundation Online Course. Fractal Foundation, 30 Apr. 2003. Web. 07 Feb. 2015.

[ix] Mandelbrot, Benoit. “THE FRACTAL GEOMETRY OF NATURE – Introduction.” THE FRACTAL GEOMETRY OF NATURE – Introduction. Cut.the.knot.org, 11 June 2001. Web. 07 Feb. 2015.

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