# 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?

I cannot resist mentioning the waiter’s trick as a physical demonstration of the fact that $SO(3)$ is not simply connected. For those who don’t know it, it is the following: you can hold a dish on your hand and perform two turns (one over the elbow, one below) in the same direction and come back in the original position. I guess one can find it on youtube if it is not clear.