You should look up Mandelbrot’s 1967 paper “How Long is the Coastline of Britain?”.
Very interesting discussion of how our measurements are dependent on the scale of our rulers. For fractal objects, as the length of the ruler is reduced towards a limit of zero, the length of the object grows without limits.
That is exactly right. It is called fractal dimension.
Thank you Dr. Swamidass!
Fractal analysis is a deep and powerful tool in the study of many natural processes, including hydrology and the modeling of river basins (one of my geohats).
When I get a bit more time I will write a topic on this, with a focus on the natural, complex and functional nature of river basins, as well as their very large unique topological forms.
This is a map of Lechiguilla Cave, near the Carlsbad Caverns in SE New Mexico. Although the pattern of fracturing in the host rock is apparent, the secondary dissolution patterns exhibit a clear fractal pattern.
In other fractal news:
Check out today’s Google Doodle. Today is Beniot Mandlebrot’s 96th birthday.
I look forward to it! I had to learn a bit of hydrology for my my graduate school research, and still find it fascinating (but I always did like mucking about in water and mud!). 
A bit over 30 years ago Joan and I were in France talking to a mathematician at a meeting that was the reason for the trip. Joan said that her older son had been drawing the Mandelbrot Set on his computer, and asked the mathematician if he knew of the Mandelbrot Set. (I cringed). The mathematician said “I not only know of the Mandelbrot Set, for a time I shared an apartment with Benoit Mandelbrot!”
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