The figure of 245 kilometres appears identically from one travel guide to the next and from one encyclopaedia to the next, with none of them saying what ruler produced it. The calculation below, run directly on open coordinates for Romania's coastline, reproduces that number almost exactly with a ruler a few kilometres long. With a ruler a hundred and fifty metres long, the same coast measures nearly 379 kilometres, a mathematical property described in the 1950s by the meteorologist Lewis Fry Richardson and worked out mathematically by Benoît Mandelbrot in 1967. Drag the ruler below and watch the compass step along Romania's real coastline.
Current ruler: —. The thin line shows the coastline at the open data's resolution; the orange one, the compass's steps at the chosen ruler.
The method is called the “compass method” and starts from an observation by the British meteorologist Lewis Fry Richardson: take a cartographer's divider, open it to a fixed distance, the ruler, and walk it along the coast, hopping from point to point. The number of steps times the ruler length gives the measured length. On a straight line, the result stays the same whatever ruler you use. On a real coast, full of bays, headlands and sandbars that repeat at every scale, a smaller ruler catches details a larger one skips over entirely, so the measured length grows as the ruler shrinks, in principle without limit. Richardson's data went largely uncited for decades, by Benoît Mandelbrot's own account; Mandelbrot formalised it mathematically in 1967, in a Science article bluntly titled “How Long Is the Coast of Britain?” The exponent describing how fast that length grows has been called the fractal dimension ever since.
The chart below plots all sixteen measurements, from a ruler a hundred and fifty metres long to one a hundred and fifty kilometres long, on a logarithmic scale on both axes. On a coast with no repeating structure, the points would fall on a flat horizontal line: length wouldn't depend on the ruler at all. Here they fall on a constant slope instead, a sign that small bays statistically resemble large ones at every scale tested. That slope gives the fractal dimension directly: for Romania's Black Sea coast, the fit comes out at D ≈ 1.12, over the range between a 250-metre and a 40-kilometre ruler. The dotted line marks the 245-kilometre threshold, the figure repeated by guides and encyclopaedias: its value falls exactly between the two neighbouring readings in the table, 269 kilometres at a ruler of roughly four kilometres and 230 kilometres at one of six-and-a-half kilometres.
| Ruler | Length | Steps |
|---|---|---|
| 150 m | 378.9 km | 2,526 |
| 250 m | 362.4 km | 1,449 |
| 400 m | 336.0 km | 840 |
| 650 m | 324.2 km | 498 |
| 1 km | 309.8 km | 309 |
| 1.6 km | 295.8 km | 184 |
| 2.5 km | 272.7 km | 109 |
| 4 km | 269.1 km | 67 |
| 6.5 km | 230.1 km | 35 |
| 10 km | 215.8 km | 21 |
| 16 km | 215.3 km | 13 |
| 25 km | 209.7 km | 8 |
| 40 km | 206.7 km | 5 |
| 65 km | 202.1 km | 3 |
| 100 km | 197.0 km | 1 |
| 150 km | 189.7 km | 1 |
Richardson didn't only measure Britain. Other researchers have applied the same method to coastlines with very different shapes, and the results line up in an order that confirms the intuition: the more a coastline is cut by fjords and inlets, the higher its fractal dimension, closer to 2, a shape that would fill almost the whole plane, than to 1, a straight line.
Romania's coastline, at D ≈ 1.12, comes out almost as smooth as Australia's and far gentler than Norway's fjords.
The coordinates used here come from OpenStreetMap and include the harbour breakwaters at Constanța and Mangalia, built structures rather than natural shoreline, which lengthens the result slightly compared with an entirely unmodified coast.
Data resolution sets a floor of its own: below roughly a hundred metres, the open map no longer has nodes dense enough for the ruler to find new detail, so values below that threshold show the practical limit of the available public data rather than a real ceiling on the phenomenon.
The figures for South Africa, Australia, Great Britain and Norway come from separate studies, with their own source maps and methods; their relative order is comparable with Romania's coastline, but not down to the decimal.
Coastline coordinates come from OpenStreetMap (natural=coastline elements, in the range 43.70–45.25 N / 28.50–29.75 E), fetched through the Overpass interface on 29 August 2026 and assembled into a single outline from Vama Veche to near the Sulina mouth. The compass method and the fractal dimension were calculated directly from these coordinates, with an independent arithmetic check on every published value. The 245-kilometre figure comes from the Wikipedia article “Geography of Romania” and from tourism sources that repeat it without specifying a measurement method. The comparison figures for South Africa, Australia and the west coast of Great Britain come from Mandelbrot, B. B. (1967), “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,” Science, 156(3775), 636–638; the figure for southern Norway comes from Feder, J. (1988), Fractals, Plenum Press.