How can different scales generate different interpretations of geographic observations?

A famous example explained by Benoit Mandelbrot is how scale affects measuring a country's coastline. If you ignore all detail below a "ruler" one mile long applied to the coast, the length of the coastline will be entirely different than if you use a "ruler" six inches long. Mandelbrot (who developed fractal geometry) explained that only fractal geometry can come close to measuring a coastline with some accuracy, because it accounts for the small redundancies that linear measurement doesn't.