12 July 2007

why Four Corners is in the middle of nowhere

Mark Chu at ScienceBlogs writes about fractals. The classic example of a fractal is the idea that it is difficult to measure the length of the border between two countries -- Chu's example is Portugal and Spain -- because the length of the border depends on the scale at which you measure it.

In technical terms, the border is nonrectifiable. For those of you who don't know that word, basically what you should know is that the only thing we can "really" find the length of is a straight line segment. To determine the length of any curve* which isn't a straight line segments, we approximate it by straight line segments and add up their length to get an approximation of the legnth of our original curve. For a circle, this means that we approximate the circle as a square, an octagon, a 16-gon, a 32-gon and so on. The square has length the ancient Greek mathematician Archimedes knew how to do this, and he famously showed that π, the circumference (i. e. length) of a circle of diameter 1 was between 3+10/71 and 3+1/7. To four decimal places, these are 3.1408 and 3.1429, respectively; the actual value is about 3.1416.

The total lengths of the square, octagon, 16-gon, and 32-gon are 2.8284, 3.0615, 3.1214, 3.1365 -- you can see how these approach 3.1416.

But for a lot of the objects that exist in "real life", that sequence doesn't approach anything. You might see that each time you halve the length of the line -- which corresponds to doubling the number of sides in the example above -- the length increases by 10%. As you measure on finer and finer scales, the length gets longer and longer.

Natural features tend to have this sort of behavior -- mountains are not cones, clouds are not spheres, and so on -- while artifical features tend to be straight lines or curves with simple mathematical definitions. Take a look at a map of the United States, for example. The eastern states for the most part have boundaries which are natural features -- rivers, the crests of mountain ranges, and so on -- because rivers and mountains were the natural barriers that separated population centers. The western states, which were divided up for the most part before they were settled and by people who didn't have much of an idea of the population patterns anyway, are often straight lines. There's a reason that "Four Corners", the point where Utah, Colorado, Arizona, and New Mexico, is remote. (Link goes to a Google map/satellite image.)

* The mathematical use of "curve" includes what normal people would call "curves", but also lots of other things. For example, a straight line is a "curve" to a mathematician. It is an exceedingly boring curve, but it is still a curve.

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