The Coastline Paradox

By one measurement, the Earth has 217,490 miles of coastline.

For comparison, the average distance between the Earth and the Moon is roughly the same (238,854 miles). 217,490 miles is a large, but comforting measurement.

Humans have an affinity for naming and measuring. From measuring grows understanding. Yet measurement also gives us a deceptive sense of mastery.

There's more.

The edge of the sea is a strange and beautiful place.
~ Rachel Carson

Measuring coastline is not as simple as it appears. The aggregated length depends on the size of the ruler.

The smaller the measurement, the more coastline.

How can that be?

Measured length increases without limit as the measurement scale decreases towards zero.

Mathematician Benoît Mandelbrot discussed self-similar curves, laid groundwork for fractal dimension, and examined the Coastline Paradox in a 1967 paper published in Science.

The Coastline Paradox is that the measured length of a stretch of coastline depends on the scale of measurement. Empirical evidence suggests that the smaller the increment of measurement, the longer the measured length becomes. If one were to measure a stretch of coastline with a yardstick, one would get a shorter result than if the same stretch were measured with a 30cm (one-foot) ruler. This is because one would be laying the ruler along a more curvilinear route than that followed by the yardstick.

~ Wikipedia, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.

Resources

1. Benoît Mandelbrot, 1967, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science, New Series, Vol. 156, No. 3775. (May 5, 1967), pp. 636-638. doi:10.1126/science.156.3775.636, JSTOR.
2. Coastline Paradox, Wolfram Math World.
3. The Coastline Paradox, Wikipedia.

Tsunamis & Seafloor Topography

Seafloor features affect how tsunami waves build in intensity as they approach a coastline. Scientists have hypothesized that seafloor mountains and chasms also affects the strength and height of the moving ocean wave fronts.

NASA JPL scientists and researchers from the Ohio State University used satellite altimeter data that had recorded sea levels changes to within an accuracy of a few centimeters following the 2011 Tōhoku earthquake to confirm this hypothesis.

Jason-2 Satellite

Data from Jason-1, Jason-2, and Envisat satellites that passed over the wave fronts caused by the the 2011 Tōhoku earthquake confirmed that islands, ocean floor chasms, and submerged mountains play a role in deflecting or amplifying tsunami waves even at distances of thousands of miles from the source of the disturbance.

Sea level changes from the satellite data following the Tōhoku tsunami were verified with results from GPS sensors and buoy data recorded by NOAA's DART program.

(1)	Wave peaks are shown in reddish-brown.
(2)	Seafloor depressions are depicted in bluish-green.
(3)	Mid-ocean ridges, peaks, and islands are depicted as grayscale outlines.

The simulation above, created from the 2011 satellite data, shows refraction, bending, and merging of waves as they propagate thousands of miles.

Source: The Seafloor Focuses and Merges Tsunami Waves published March 12, 2012 by NASA Earth Observatory.