# 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?

• No clue what to suggest, but it's an intriguing question! – Noldorin Jul 22 '10 at 10:48
• @Katie Banks wrote: 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. Could somebody please elaborate or provide a reference? Thanks. – Shibi Vasudevan Aug 13 '10 at 3:04
• Good question . – Mehper C. Palavuzlar Oct 27 '10 at 11:38
• Here's a blog post I wrote a while ago about proving Vieta's formula with basic physics: arcsecond.wordpress.com/2010/09/17/… – Mark Eichenlaub Mar 5 '11 at 18:24
• Could you provide a reference on how one solves the Basel problem with fluid flow? – Potato Nov 11 '11 at 21:22

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.

• I think the most interesting part about the waiters trick is that it demonstrates an element of finite order, not just a nontrivial element. – Sean Tilson Dec 2 '10 at 21:06

From wikipedia article about Gauss Theorema Egregium:

An application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction perpendicular to the fold, an attribute desirable when eating pizza (since it prevents the pizza toppings from falling off).

The book "The Mathematical Mechanic" by Mark Levi is a very good source of such examples, which Levi has been collecting for some time. The first two here are in the book, if I recall correctly.

In the article Ronald N. Bracewell. The Fourier Transform, Scientific American, 1989 there is a nice picture of a sunbeam resolved into a spectrum by a prism. So it can be regarded as a physical analogue device performing Fourier transform. White light as a complicated function to be decomposed into the sum of harmonics. There is the hairy ball theorem, which states that no even dimensional sphere admits a nowherevanishing continuous vector field. In the case of $S^2$, the physical demonstration of this is that one cannot comb the hair of a ball without getting a cowlick.

• A modest enhancement of which is the Poincare-Hopf index theorem. – Ryan Budney Nov 1 '10 at 18:19
• I wonder if there is a place where one can buy hairy balls, for classroom demonstration purposes. (Hmm, that sounds really bad out of context.) – Nate Eldredge Nov 8 '10 at 18:54
• Also, that an cyclonic system must always exist. On a planet. – Allen Nov 11 '10 at 6:29
• @Allen: except that the air mass velocity is a rank three bundle (not two) and that the theorem only implies there is no non-vanishing field, not that there must be a field with isolated zeros (in particular, vanishing vector field is certainly admissible...). – Marek Jul 1 '11 at 21:05
• Sine you can "comb the hair" on a coffee mug (i.e. torus), they should sell mugs with vector fields on them. – Akiva Weinberger Jan 20 '16 at 1:29

There's a whole room devoted to physical demonstrations of mathematical concepts in the Museum of Science in Boston. I'll list the ones I can remember, and others who have been there please feel free to add to the list.

• A little train that goes around on a track that's on a Möbius strip.
• A "Pachinko" type machine where the balls fall into different slots and form a bell curve as the slots fill up.
• A pendulum that is free to swing forward and backward as well as left and right. At the bottom is a container of sand with a small hole in it, so as the sand leaks out, it forms Lissajous curves. (Video of that exhibit)
• Something about the Riemann Zeta function that I didn't really understand.
• A straight, solid bar, rotated around an axis that is skew to the bar itself, that fits right through a hyperbola-shaped slot. I'm pretty sure the Exploratorium in San Francisco has this too, and here's a video of a similar exhibit thanks to commenter Rahul.
• +1 for the Pachinko-type machine, which is a nice demonstration of the central limit theorem. Here's a video of the hyperbolic slot: youtube.com/watch?v=OJGkdO9gcdY – user856 Oct 14 '10 at 16:51

Some principles of inversive geometry can be demonstrated using Peaucellier's Inversor, a linkage device designed to transform circular motion into motion in a straight line and vice versa. There are various online demonstrations however, you may wish to build a "real" one for yourself. The book "Mathematical Models" by Cundy and Rollett has an entire chapter devoted to making mechanical models, including the above and related likages.

• The go-to book for linkages is still Kempe's "How to Draw a Straight Line". – J. M. isn't a mathematician Nov 12 '10 at 12:00

The Fourier transform of the Dirac delta function is a constant function.

I'm not sure how good of an example this is since the original problem already has a natural physical interpretation. Nonetheless this is an example of something that can be proved using Euclidean geometry or mechanics.

Starting on one of the edges of a plane polygon $P$ one can set into motion a point-mass which moves along a straight trajectory except on collision with another edge, when it is subjected to an elastic collision response (i.e., its velocity vector is reflected in the edge it has collided with). The trajectory traced out by such a point-mass is its orbit, and an orbit is periodic if the point-mass eventually returns to its starting spot with its starting velocity.

One of the most basic questions one can ask is whether periodic orbits exist in acute triangles. The answer is yes, and a particularly nice periodic orbit in an acute triangle can be constructed by drawing the three altitudes and connecting their bases.

This can be proved using geometry, but it can also be seen in the following physical manner. Assume the triangle's edges are thin wires. Place around each wire a small ring which is free to move along its edge. Now thread a rubber band through the three rings. Tighten this rubber band. This system reaches equilibrium when the rings occupy the positions of the bases of the altitudes.

In particular, in the equilibrium state the triangle whose vertices are the rings will be the inscribed triangle of minimal perimeter. There's no such minimum in an obtuse triangle, so the construction doesn't work in that case. As of today it is unknown whether every obtuse triangle admits a periodic orbit, although much partial progress has been achieved.

I recall what I now remember as "the Metereological theorem" from college:

At any point in time there exist two antipodal points on the equator that have the same temperature.

Trivial to prove. Only requires the mean value theorem and continuity.

You could demonstrate it by stating that you can not draw a simple closed curve around a circle without having two points 180 degrees apart being exactly the same distance from the circle.

The same is true of any continuous physical property like pressure, humidity, etc. Also, it can be extended to any three metereological properties on a sphere.

It's rather amazing what you can do with just continuity and the mean value theorem.

• You don't even need the Mean Value Theorem; Intermediate Value Theorem will suffice. – Joe Johnson 126 Oct 26 '11 at 16:00

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.

Well, one Stokes theorem surely deserves to be viewed in physical terms. It's hard not to think about these matters in terms of physical flows (be it fluid, or electrical fields), sinks, sources, etc. and this can give intuition when dealing with arbitrary differential forms.

Another physical concept often used in abstract mathematics is conservation of energy, or more generally integrals of motion. Used all the time in analysis of PDE, as far as I know.

I think one that comes up often in discussions like this is Brouwers Fixed point theorem which I think can be demonstrated by putting a map of your country on a table and asserting that one of the points on the map is directly above the corresponding point in your country (ie where you are right now).

Another way i think is simpler is to take two sheets of paper, crumple one up and place it on top of the other.

I remember someone telling me that when you stir a cup of coffee, one point in the fluid will always end up at the same place as it started by this theorem, but im not convinced this is really a continuous map since the particles in the fluid are discrete objects.

• Perhaps it is continuous from the Navier-Stokes perspective? – akkkk May 24 '12 at 10:00

This video and those that it links to are impressive demonstrations of spectral properties of differential operators. (You could involve more or less abstract mathematics as per audience.)

Coupled harmonic oscillators. Stretch a cord between two chairs, attach three equal weights to the cord by cords of equal length, so you have three pendulums hanging from the cord. If you start one weight swinging back and forth, it will over time slow down, transferring energy to the other two, which will start swinging, and over time the three will take turns speeding up and slowing down. But there are three stable modes: if you start by drawing all three back by the same amount and releasing simultaneously, they will go back and forth in unison, with no transfer of energy; if you start by drawing the left one back and the right one an equal distance forward, that's also stable; if you start with the left and right ones forward, and the middle one back twice as far, that's the third stable configuration.

These can be explained as eigenvectors of a matrix describing the system.

Surely there's a video of this somewhere on the web. I found some with two oscillators, and one with 5, but I didn't find the one I'm describing here.

Physical interpretation of the Mean Value Theorem: If the average speed of a car between two locations was V km/h, then there was at least one instant where the speed indicator displayed V km/h.

In physics and certain branches of engineering, there exists a device called the Möbius resistor, which consists of two conductive materials separated by a dielectric and twisted to form a Möbius strip. It is very useful because is essentially a resistor with no residual self-inductance, that is, there is no magnetic interference while electricity passes through it. When I first heard about it, I was actually quite impressed that concepts from Topology had a direct physical meaning. More can be read up on the Möbius resistor here http://en.wikipedia.org/wiki/M%C3%B6bius_resistor

How about a Galton Board? Don't miss the video.

The linearity of the wave equation can be demonstrated by the fact that harmonics are played together with fundamental tones on stringed musical instruments. Probably on woodwinds too, but they are visible on strings.