It reminds me of how we use measures like the VIX in finance; not because markets are actually log-normal, but because having a standardized way to compare "choppiness" across different periods is incredibly useful. I like your fractal dimension example too. Even if real coastlines aren't truly fractal, being able to say "this coastline has dimension 1.3 vs 1.7" gives you meaningful information about erosion patterns, wave energy, and rock composition. The cube metric could work similarly for forensic geology.