How Long is the Coast of Britain?

Benoit B. Mandelbrot (1924-2010) was a Jewish Polish-born French-American mathematician and polymath. “What is the essence of a coastline?” he once asked. Mandelbrot asked this question in a paper that became a turning point for his thinking: “How Long is the Coast of Britain?”

Mandelbrot had come across the coastline question in an obscure posthumous article by an English scientist, Lewis F. Richardson. Wondering about coastlines and wiggly national borders, Richardson checked encyclopedias in Spain and Portugal, Belgium and the Netherlands, and discovered discrepancies of twenty percent in the estimated lengths of their common frontiers. Mandelbrot argued that any coastline is, in a sense, infinitely long. In another sense, the answer depends on the length of your ruler.

“Consider one plausible method of measuring. A surveyor takes a set of dividers, opens them to a length of one yard, and walks them along the coastline. The resulting number of yards is just an approximation of the true length, because the dividers skip over twists and turns smaller than one yard — but the surveyor writes the number down anyway.

“Then he sets the dividers to a smaller length — say, one foot— and repeats the process. He arrives at a somewhat greater length, because the dividers will capture more of the detail and it will take more than three one-foot steps to cover the distance previously covered by a one-yard step. He writes this new number down, sets the dividers at four inches and starts again.

“This mental experiment, using imaginary dividers, is a way of quantifying the effect of observing an object from different distances, at different scales. An observer trying to estimate the length of England’s coastline from a satellite will make a smaller guess than an observer trying to walk its coves and beaches, who will make a smaller guess in turn than a snail negotiating every pebble.”

If we measure our ascent on the spiritual ladder of our life like a snail, we will become disillusioned very quickly, for life has many twists and turns and setbacks. But if we take the satellite view, each one of us can follow in the footsteps of our father Yaakov — the ladder that is set on the ground but whose head reaches the heavens.