HOW LONG IS THE COAST OF BRITAIN?

HOW LONG IS THE COAST OF BRITAIN? I learned from an obituary for Benoit Mandelbrot that one of the important mathematical papers of the last century was entitled “How Long is the Coast of Britain?” Mandelbrot’s article was an early application of fractals (which had not yet been named by Mandelbrot). This wikipedia article describes the Coastline Paradox. The Coastline Paradox is that “the coastline of a landmass does not have a well-defined length.” The article has two pictures of measurements of the coastline of Great Britain which make the problem clear. The smaller the unit of measurement, the longer the measured length becomes. This wikipedia article discusses Mandelbrot’s paper. To take an example close to home, if the Eastern coast of New England is being measured in miles, a mile-long segment across the coast of Connecticut will ignore many of the indentations going into Long Island Sound. Foot-long segments would pick more of them up; the ruler would have to track the ins and outs of the crannies. The article says that: “The empirical evidence suggests a rule which, if extrapolated, shows that the measured length increases without limit as the measurement scale decreases towards zero.” Mandelbrot’s paper showed how mathematical equations of the form that would later be called fractals could be used to approximate the length of a coast line.