## Fractals

Fractals are awesome
Figure 1. Fractal courteous of http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/mandelbrot-safari/

A fractal is a function that when you graph it appears the same at all scales,infinitely. You can zoom in forever on the tiniest detail and find yourself amid geometry that closely if not exactly mirrors the scale you started at. Check out the below video for an amazing example of this!
http://vimeo.com/12185093

A French-American by the name of Benoit Mandelbrot first described fractals lucidly (others had noted the presence of them before but hadn’t realized the mathematics or meaning behind them).

Figure 2. Benoit Mandelbrot, the man himself. Image courtesy of http://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrot

This a geoblog though, not a math blog and I can’t justify showing you fractals just because they are beautiful.  However, I can show you fractals because they are beautiful and fundamental parts of geomorphology!

Figure 3. Malaysian river system with a fractal form. Courtesy of http://www.mymodernmet.com/profiles/blogs/paul-bourke-google-earth-fractals

Figure 4. River system showing fractal form in Mexico. Courtesy of http://photography.nationalgeographic.com/photography/photo-of-the-day/baja-california-rivers/

Figure 5. Greenland coastline showing fractal form. Courtesy of http://artsonearth.com/2008/08/11-phenomenal-images-of-earth.html

Fractals aren’t just a cool thing we can observe qualitatively in landscapes around us either.  An important question is, at what scale are these fractal landforms most variable? If you wanted to measure the length of Britain’s coast line, what length ruler would you use? Is most of the erosional work in a river system done by a few big, wide rivers or by the net effect of many smaller tributary streams? Quite a few papers have been published around the quantification of fractals in landforms. One of the more particular interesting ones called “Fractal properties of landforms in the Ordos Block and surrounding areas, China” takes a series of cross sections through topography in such a way that the ground surface is represented as a wave on a graph (figure 6a).
Figure 6. a represents the cross-section through topography. b represents the complex topographical wave broken down into a set of sine waves.  c is the wavelengths of these sine waves graphed with respect to amplitude. This process is called taking a fourier transform.

A mathematical function called a Fourier transform converts the topographical wave (figure 6a) into a set of sine waves (figure 6b) that added together recreate the topographical wave. The sine wave with the biggest vertical distance between the peak and trough (known as the amplitude) is the dominant scale at which that stretch of topography changes over. A flat desert with lots of dunes would have a dominant wave on the scale of the dunes whereas a mountainous area with smooth exposed bedrock would have a dominant wave on the scale of the mountains. These results might seem to be something that you could just easily observe by looking at the landscape, but they provide a quantified representation that would be difficult to obtain otherwise. This allows scientists to define fundamental aspects about landscapes in a way that could allow a deeper understanding of erosion, geomorphology and geology as a whole.

Pretty sweet I say! -TB

If you would like to see more of these fractals yourself, or you don’t believe me check out this link that includes kmz files that bring you to places in google earth that have fractals!

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