One of the Christmas presents from my sweetheart was John D. Barrow's "One Hundred Essential Things You Didn't Know You Didn't Know." Perhaps it is a little "Math-Lite", but it is still a nice read.
Anyway, reading a section on "A sense of Proportion" in which he points out that if you plot world-record weight lifting records against the weights of the lifters, they fall into a line along the 3/2 power rule, that is (weight of athlete)2=(weight lifted)3. OK, that isn't THAT surprising, but it got me thinking about other things that fall into a square to cube relation, that don't seem to be as easy to understand.
The obvious first choice, is Kepler's third law... The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit, or as I think of it to make it easy, year2=(distance from sun)3. That has always seemed unexpected to me. One is dependent on the circumference (sort of) and one on the radius, so they might be expected to be essentially equal.. but that inverse square effect of gravity, somehow makes it work.
From there I wondered about other examples I had heard of..For instance the Beaufort scale gives a relationship between wind speed and wave height on the oceans as v = 0.836 B3/2 m/s .. ;
There is another that shows up in forestry... I couldn't remember it offhand, so I found a note that says, "The 3/2 power law states that the relative rate of self-thinning with respect to biomass growth per unit area is a universal constant" so maybe there is more of a geometric relationship here that would make sense.. Maybe you can explain it to me.
ADDENDUM: I got a tweet from tom@liverbubble that told me about a similar result with the 3/4 power called Kleiber's Law. The principal is named after a agricultural chemist named Max Kleiber. In 1932 he came to the conclusion that the ¾ power of body weight was the most reliable basis for predicting the basal metabolic rate (BMR) of animals and for comparing nutrient requirements among animals of different size. (Send more examples please)