Tuesday, June 16, 2009
Nahin on Torricelli’s Funnel
On the beach I finally got a chance to start (if not finish) Nahin’s When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. So far it is an engrossing survey of work on maximum/minimum problems that leads one gently into the mathematical intricacies of the history without being overly technical.
One of the first examples that Nahin discusses is Torricelli’s Funnel (also known as Gabriel’s Horn). One can think of it as the result of rotating the "first quadrant branch" (positive x, positive y) of the hyperbola xy = 1 around the x-axis. If we consider its surface, then we can show that the total surface is greater than any finite number. But if we consider its volume, then we can show that it is finite. For this case, the volume is pi. While I had come across this example before in Mancosu’s book, Nahin raises a paradox which I think should be more widely known for those working on mereology and "intuitions" in philosophy. As Nahin puts it, if the surface is infinite, then we cannot paint the funnel with a bucket of paint, no matter how large the bucket is. But if the volume is finite, we can paint the inside of the funnel by filling the funnel with paint! Do we have a proof that this figure is impossible?
No! Nahin points out that we are tacitly working with two different notions of paint: "mathematical paint" and "real paint". If we consider real paint, it is composed of molecules of some finite size. So, if we pour the real paint into the funnel, then it will not coat the inside of the funnel because at some distance its molecules will no longer fit. But if we consider "mathematical paint", then a finite amount can coat the outside of the funnel, or the inside of the funnel, because a finite volume can be spread out to any degree over a surface if we place no limit on the thickness of the coating. This elegant example shows just how liable we are to make mistakes when we consider these sorts of mereological questions if we fail to pay attention to the mathematical subtleties.