schoolyard subversion

From: Bruce Litow
To: Aaron Swartz

Hi Aaron. I sympathize with the denseness of teachers. Being a teacher I know that I am sometimes dense. However, so far as the matter of surface area to volume goes, the matter is a bit trickier than your report of it. It is indeed true that one cannot compare quantities with differing units, but the objection that a ratio of area to volume cannot reveal anything sensible is wrong. A bit of history, Newton had just this difficulty in justifying to himself the sense of adding tp terms like x, x^2, x^3, etc. since as you point out were x to have units, the additions would be absurd.

Now imagine a sphere filled with small marbles. Let the sphere have radius R, but regard R as unit-less. It is the case that, subject to the fact that packings of marbles leave little unused volumes (by the way, determining packings that minimize the total gap volume is still open, I think) a good case can be made for saying that the number of marbles (a unit-less number) is directly proportional to the sphere's volume. We just put the number at R^3 because I don't want to worry about constant coefficients and the volume is 4*pi*R^3/3.

Imagine that the sphere's surface is now magically permeable to marbles, but the marbles still cannot pass through each other. The number of marbles just at the surface is again directly proportional to the area, which is 4*pi*R^2 will be taken to be R^2, again to simplify things.

The fact that marbles cannot move through each other makes it reasonable for us to imagine that the marbles exit the sphere in an orderly way; marbles underneath the current `skin' of marbles at the surface regrouping to form the next skin until the sphere is empty of marbles. We can get rid of R^2 marbles in one `wave', so it will take R such waves to do the job. This provides a physical justification for taking seriously the statement that the area/volume ratio of a sphere is inversely proportional to its radius. As we look at bigger and bigger spheres it will take longer and longer to empty them of marbles. Now one can complain and say that the marble metaphor might not fit all physical processes where something is radiating away from a sphere through its skin, but it works well in a lot of cases.