POTD: Little Landscapes #1

Little Landscapes #1
Cathedral Gorge, Nevada
2019

Hiking around Cathedral Gorge, I was reminded of how often closeups of desert terrain can look just like large-scale landscapes that one might see from the air. I’m sure at least a passing explanation for this can be found by appealing to the recursive quality of the Mandelbrot set (a special set of complex numbers) and more generally, fractals. Per Wikipedia:

…a fractal is a subset of a Euclidean space for which the fractal dimension strictly exceeds the topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set; because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales called self similarity,[5] also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar.

(The relevant part of this quote is in the second sentence. I threw in the rest for the math majors out there–and just to spread the confusion around that I experience every time I try to delve into this level of theoretical mathematics or physics!)

3 thoughts on “POTD: Little Landscapes #1”

  1. Not being a math type guy, I struggled often with fractals in my GIS duties early in the 90’s. Fireman, friend and math type, the late Paul Gleason would devil me in discussions about results of early forest service gis systems. His argument was that ecologic habitats and communities could hide in the space of fractals when our systems would analyze areas based on non-fractal containing polygons. Later on, fractal geometry was incorporated in our systems. I did some work at the Remote Sensing Labs in Salt Lake where we used “supervised” learning classification to refine results from infrared remote sensed plots, and topographic and other over-layed data sets that we had robust data sets on to try to tease out potential rare communities using fractal geometry.

    1. Interesting application of fractals, and an indication that you know more about them than I do! About the only thing I’ve seen them used for is making pretty pictures of the sets that are recursive.

      1. I was introduced to the concept but just one member of a team whose skills did not include higher math! At my level of understanding, sometimes I thought we were arguing over how many angels could dance on the head of a pin.

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