Sometimes, we look for relation between objects of different dimensionality, search for proportionality rules and try to factor away constants, or, at least, figure out what they are made of. A cute example of which came to me in the shower…
Knowing your weight, can you know how much soap you use?
Let us abstract the fact that weight is not mass; and consider mass only. So, let be the mass, the average density, the average height, the average width, the average thickness (and assuming a suitable face separation).
Mass, density, and volume are related by:
where , . Surface is also related; of course. We have:
Using with :
So, the amount of soap you use is directly proportional to your surface, which is proportional to .
This reasonning holds for “ordinary” objects that are close to most objects we see in the world around us, but eventually breaks when considering exotic objects like, say, a sheet of paper. In the case of a sheet of paper, surface greatly exceeds volume as the volume of an ideal sheet of paper goes to zero as it approximates a region of the plane. In this case, the approximation doesn’t hold at all—it does, but with a degenerate proportionality constant.