When it comes to your country, it's vitally important to know what the borders are. Measuring the land's perimeter, and knowing how much of it is ocean coast and how much of it is land-locked, is an excellent way to understand the geography of your nation. Pretend that you're a geographer in Mexico; how would you measure the exact length of its coastlines? It seems like there's an easy (albeit labor-intensive) solution, but the reality is far more mathematically complex.
If you look at an aerial photograph of a landmass, you can see that the coastline provides a fairly definitive boundary; the water begins at some point and the land ends. But according to modern mathematics, the truth is quite counterintuitive; depending on the method used to measure a coastline, the final length can vary greatly. Incredibly, there is no single well-defined perimeter to any landmass; the true boundary approaches infinity. This is called the coastline paradox, and it was the start of an entirely new branch of mathematics.
In 1951, Lewis Fry Richardson (a mathematician) first became aware of the coastline problem. He decided to see if there was a relationship between the length of the common border between two countries and the probability that they would go to war. He realized however, that as data was gathered from various countries that there were many differences in the measurements of borders. For instance, when examining the Spain-Portugal border, Portugal's maps said it was 613 miles long, and Spain reported 754 miles. The two nations were measuring the exact same piece of land, and coming up with wildly different results.
Richardson realized that the problem lay in the basic idea of Euclidean geometry--the very notion of measuring something as complicated as a coastline. In standard geometry, you measure things as if they are a straight line, or approach one. If you decided to view an aerial photo of Mexico's coastline, then take a yardstick to it to find the perimeter with a pen, you would find that your shape would only be approximate, and you would not have included all the bays and promontories. But that's not very accurate, so perhaps you zoom in a little and re-measure with a twelve-inch ruler. Once you made a bit more precise shape, you could get a more accurate measurement. Your precision would be even more accurate if you used a one-inch long ruler. Zoom in again, and you could get even more meticulous as tiny jutting rocks and miniature tide pools come into view.
This complexity continues no matter how much you zoom in on a coastline; it will never approach a perfectly straight Euclidean shape. Self-similarity is what this property is called; the same general configuration (increasingly complex features of a coastline) will occur at any scale. The rough pattern of bays and outcroppings will appear all the way down to the individual grains of sand on the Mexican beach. The smaller line segments that you use for your measurements, the more you will find that measured coastline length will approach infinity.
More than a decade after Richardson's initial observations, Benoit Mandelbrot invented fractal geometry, which described incredibly complex and self-similar patterns in nature--like infinite coastlines. He wrote a paper about the coast of Britain to illustrate his ideas: coasts are self-similar fractals instead of simple polygons, and their length can be very closely approximated rather than measured precisely. Mandelbrot's work was a revolutionary turning point in how humans looked at mathematics--and all because of a deceptively simple question about border length. So if you find yourself on the beach in Cancun, Mexico, know that you're relaxing on the surface of an infinitely complex fractal, whose true length approaches infinity. That sort of epiphany calls for another drink!
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