Young Brunette Girl On The Rocky Coast

The coastline problem
At some time in the years immediately preceding 1951, Lewis Fry Richardson in researching the possible effect of border lengths on the probability of war noticed that the Portuguese reported their measured border with Spain to be 987 km, but the Spanish reported it to be 1214 km. This was the beginning of the coastline problem, which is how to arrive at an estimate of a boundary that is infinite.
The prevailing method of estimating a border (or coastline) was to lay off n equal straight-line segments of length ℓ with dividers on a map or aerial photograph. Each end of the segment must be on the boundary. Investigating the discrepancies in border estimation Richardson discovered what is now termed the Richardson Effect: the sum of the segments is inversely proportional to the common length of the segments. In effect, the shorter the ruler, the longer the measured border; thus, the Spanish and Portuguese geographers were using different-length rulers.
The result most astounding to Richardson is that, as ℓ approaches zero, the length of the coastline approaches infinity. Richardson had believed, based on Euclidean geometry, that a coastline would approach a fixed length, as do similar estimations of regular geometric figures. For example, the perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides (and decrease in the length of one side). In Geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.