05 August 2007

A puzzle

It occurred to me that there are 64082 points on the surface of the earth whose longitude and latitude, calculated in degrees, are both integers: 360 degrees of longitude · 178 degrees of latitude (from 89ºN to 89ºS) + the two poles.

How many of these gridpoints are on land? It would be tedious to count, but probably about 18600 of them, since 71% of the earth's surface is covered in water (64082 · 0.29 = ~18583).

Now for the generalization (which can be developed in various ways): take a simple closed curve on the surface of a sphere, defined by an equation of a particular form but with parameters unspecified, subject to the constraint that the curve encloses some fraction j of the total surface area of the sphere; assuming randomly chosen (real-valued) parameters satisfying the constraint, calculate the probability distribution for the number of gridpoints lying within the curve. At least for garden-variety, well-behaved curves, one imagines that the distribution will peak at 64082j and taper off as one goes above or below that. If one allows in too many strange curves, there might be a problem even defining a probability measure on the set. I don't know exactly how one would go about beginning to solve this problem except for the simplest cases.

(Am I right, by the way, in thinking (1) that there are a total of beth-two simple closed curves on the surface of a sphere in R³ and (2) that at least some of these will enclose no well-defined area, for Vitali/Hausdorff reasons?)

(Disclaimer: I am not a mathematician. For all I know, there might be an extensive literature on this problem. )

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