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 1n2=π26 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 ith face has area Ai and outward facing normal vector ni, Aini=0. One can intuitively show this by pretending the polyhedron is filled with gas at uniform pressure. The force the gas exerts on the ith face is proportional to Aini, 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.

To see why the two things are related, I borrow the following explanation by Harald Hanche-Olsen on MathOverflow:

Draw a curve through your body from a stationary point, like your foot, up the leg and torso and out the arm, ending at the dish. Each point along the curve traces out a curve in SO(3), thus defining a homotopy. After you have completed the trick and ended back in the original position, you now have a homotopy from the double rotation of the dish with a constant curve at the identity of SO(3). You can’t stop at the halfway point, lock the dish and hand in place, now at the original position, and untwist your arm: This reflects the fact that the single loop in SO(3) is not null homotopic.

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Andrea Ferretti

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