How winding is the coast of Britain ? Conformal invariance of rocky shorelines

We show that rocky shorelines with fractal dimension 4/3 are conformally invariant curves by measuring the statistics of their winding angles from global high-resolution data. Such coastlines are thus statistically equivalent to the outer boundary of the random walk and of percolation clusters. A simple model of coastal erosion gives an explanation for these results. Conformal invariance allows also to predict the highly intermittent spatial distribution of the flux of pollutant diffusing ashore.