This post contributed by Nadine Lymn, ESA Director of Public Affairs
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.
—Mandelbrot, The Fractal Geometry of Nature
As the year 2010 draws to a close and we find ourselves in the midst of winter’s icy grip, we might pause to marvel at the geometric beauty of a snowflake. Earlier this year, the “father of fractal geometry,” Benoit Mandelbrot, passed away. The New York Times obituary described how this “maverick mathematician” proposed a simple but radical way to quantify crookedness by assigning it a “fractal dimension.” Mandelbrot defined a fractal as “a shape made of parts similar to the whole in some way.” This method allowed for a way to quantitatively measure such complex outlines as clouds (or snowflakes) or coastlines and contributed to fields as varied as geology, engineering, medicine, and ecology.
A recent PBS program explored the world of fractals and its corresponding website offers numerous ways to better understand the concept and its applications. Described on the website as “irregular, repeating shapes found in cloud formations and tree limbs, in stalks of broccoli and craggy mountain ranges, even in the rhythm of the human heart,” the use of fractals enable the prediction of patterns at different scales.
As described on the PBS website, the so-called Mandelbrot set (see above video), which is the “breeding ground for the world’s most famous fractals,” is an “odd-shaped infinite swarm of points clustered on what is known as the ‘complex number plane.’” To visualize it, the website suggests imaging real numbers such as 1, 2, 3…as spaced out along a number line. Making complex numbers tangible requires two lines (axes) to create a plane and to accommodate the complex numbers’ “real” and “imaginary” parts. The advent of computers and Mandelbrot’s exploratory work at IBM made visualizing fractals possible. This “zooming in” on the Mandelbrot set’s boundary reveals its details by “magnifying it” and allowing people to discover patterns. If you want to try it yourself, the website lets users design their own fractal.
Applications in ecology include mapping patterns of soils at multiple spatial scales, comparing different landscapes and measuring magnitudes of fluctuations in populations.
Ecologists Monica Turner, R.H. Gardner, and Robert V. O’Neill’s 2003 book on landscape ecology offers this description of fractals:
“The essence of fractals is the recognition that, for many phenomena, the amount of resolvable detail is a function of scale. An important corollary is that increasing the resolution does not result in an absolute increase in precision, but rather it reveals variation that passed unnoticed before.”
They go on to note that:
“Fractals have two important characteristics: (1) they embody the idea of self-similarity, the manner in which variations at one scale are repeated at another; and (2) their dimension is not an integer, but rather a fraction, hence the fractal dimension, from which these objects acquired the name.”
A Nature article describes how in the mid-1990s, Brown and fellow ecologist Brian Enquist teamed up with physicist Geoffrey West. The trio scrapped Euclidian geometry in favor of fractals to develop a theory to explain how organisms use resources. The branching structures of resource distribution networks, such as the xylem that transports water through plants, lends itself well to fractal geometry.
A later paper by Brown and colleagues, “The fractal nature of nature: power laws, ecological complexity and biodiversity” states that:
“Now we are faced with the challenges of understanding the structures and dynamics of the complex systems themselves. We know this cannot be done simply by assembling the parts in ever larger subsystems. There are just too many possibilities. Power laws and other emergent general features of these systems offer invaluable clues to the universal mechanisms that constrain the diversity of life and the complexity of nature.”
Photo credits: Trees, Matthew Venn; leaf close-up, kvd