December 29, 2019

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!)