Lab / 03
How long is Denmark?
In 2014 Denmark's coastline grew by about 1,400 km overnight. Nothing moved. The Danish Geodata Agency just measured it again on a more detailed map. That is the coastline paradox: the shorter your ruler, the longer the coast. Drag the ruler below and watch Denmark shrink much faster than South Africa.
The divider method
Open a pair of dividers to a fixed width, put one leg on the coast, swing the other until it touches the coast again, and repeat. Count the steps and multiply by the width. Lewis Fry Richardson did this by hand in the 1950s and noticed the answer never settles. It keeps growing as the dividers close.
Plot it on log-log axes and you get a roughly straight line. The slope gives D, the fractal dimension. A perfectly smooth line has D = 1. The rougher the coast, the higher D. Richardson's own estimate for South Africa was about 1.02, one of the smoothest he measured. With the same map data, Denmark is much rougher.
Why Denmark falls off a cliff
South Africa is one long, fairly straight coast from the Orange River to Kosi Bay. Big ruler or small, it mostly follows the same line. Denmark is fjords, peninsulas and islands. Long rulers jump straight across Limfjorden, and once the ruler is longer than an island, that island stops counting at all. At 80 km most of Denmark has fallen through the gaps.
The official numbers tell the same story. The old figure for Denmark was 7,314 km, measured at 1:100,000. Remeasured at 1:10,000 it became 8,750 km. The map used here is 1:10,000,000, so it bottoms out around 4,300 km - the map itself is a ruler, and a long one.
Coastlines: Natural Earth 1:10m (public domain), land borders removed. Official figures: The Local, 2014. Source on GitHub