Good Physical Demonstrations of Abstract Mathematics I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to use.
I'm looking for nontrivial ideas in abstract mathematics that can be demonstrated with some contraption, construction or physical intuition. 
For example, one can restate Euler's proof that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ in terms of the flow of an incompressible fluid with sources at the integer points in the plane.
Or, consider the problem of showing that, for a convex polyhedron whose $i^{th}$ face has area $A_i$ and outward facing normal vector $n_i$, $\sum A_i \cdot n_i = 0$. One can intuitively show this by pretending the polyhedron is filled with gas at uniform pressure. The force the gas exerts on the $i_th$ face is proportional to $A_i \cdot n_i$, with the same proportionality for every face. But the sum of all the forces must be zero; otherwise this polyhedron (considered as a solid) could achieve perpetual motion.
For an example showing less basic mathematics, consider "showing" the double cover of $SO(3)$ by $SU(2)$ by needing to rotate your hand 720 degrees to get it back to the same orientation.
Anyone have more demonstrations of this kind?

How about probability & statistics?  Not exactly physics, but lots of applications which can be demonstrated with empirical data.  Any example where "taking an average" seems reasonable is amenable to finding a distribution.  Many examples: frequencies of arrival (traffic, say) as Poisson or negative binomial; arrival times as geometric; insurance claims as lognormal or gamma (or other more complex skewed distributions, but no need to get that complicated); percentiles as beta; human physical characteristics as normal.  Depending upon your course, you could even take empirical data and try fitting distributions using various techniques, which employ calculus, numerical methods, power series (e.g. moments), etc.
