Problems with applying standard measurement to coastlines

Even a moment's reflection reveals a problem with the standard approach: a smaller measuring scale is sensitive to more details.
This is a real issue because many geological features exhibit similar structure at finer detail. That is, they are scale invariant, at least for some range of scales.
This is reasonable because the forces that sculpt coastlines operate in approximately the same way over a wide range of scales.
Nevertheless, surveyors measure the length of a coastline by
selecting a measuring scale d,
approximating the coastline by N line segments of length d, and
deducing the length of the coastline is L(d) = N⋅d.
Here is a picture from NASA's website. Click the picture to see two polygonal approximations of the coastline.
If this picture does not convince you, click here for another NASA photograph.
Imagine the difficulty of measuring the length of this coastline, using smaller and smaller scales.
Evidently, a smaller measuring scale will detect more detail of the coastline, hence give a greater length.
Self-similarity of coastlines casts doubt on the hope that these measurements will converge as smaller scales are used. In fact, these doubts are justified.

Return to Coastlines.