You have done this. You spin a coin on a table, it wobbles, and in the last second or two it produces that rising shudder — a whirrrrrr that climbs in pitch, gets faster and faster and faster, and then stops. Not slows. Stops, abruptly, like something was switched off.
Almost everyone hears that as the coin spinning faster and faster. It’s the opposite. By the end the coin is barely rotating at all.
There are two different motions in a settling disc and we lazily hear them as one.
The first is the disc’s own rotation — how fast the picture on the coin turns. That is dying. By the last moments it has nearly stopped.
The second is the contact point: the place where the rim touches the table, which races around the edge of the coin. That is accelerating, and it is what you hear. The buzz is a contact point circling tens of times a second underneath a coin that is hardly turning.
The geometry is simple and exact. For a disc of radius a tilted at a small angle α to the table, rolling without slipping, the contact point circulates at roughly Ω ≈ √(4g / aα). Which means:
Ω ∝ α−1/2. Quarter the tilt, double the rate.
So as the disc lies down, the tilt collapses toward zero and that rate climbs without limit. The sound rises because the coin is falling over, not because it is spinning up.
I built a small thing you can scrub. Watch the green number (the disc’s rotation) fall while the red number (the contact point) climbs. The slider walks time down six decades toward the end, because linear time shows you nothing here — a singularity is all endgame.
Push the maths and something strange falls out. The rate doesn’t just get large; in the model it reaches infinity at a specific finite moment. H. K. Moffatt named this in Nature in 2000 — “Euler’s disk and its finite-time singularity” — and it is the reason the sound stops rather than fades. The model runs off the end of the world at a definite time, and in reality something else takes over just before it gets there.
The obvious question is what takes over, and here is the part I did not expect: it has been argued about for twenty-six years.
Moffatt’s answer was air. Viscous dissipation in the thin film of air squeezed between the disc and the table, which under his model gives α ∝ (t₀−t)1/3 and therefore Ω ∝ (t₀−t)−1/6. A refined treatment of the air boundary layer gives −2/9 instead, in better agreement with measured settling discs, which are observed buzzing at roughly 20–70 Hz, some past 100 Hz.
Van den Engh and colleagues disagreed, in the same journal, within months. Their objection is the kind I find beautiful because it is a measurement: comparing vacuum with air shows about a 20% difference at early times, when the disc is spinning upright — and no comparable difference during the settling phase, which is precisely when Moffatt’s mechanism is supposed to dominate. Their candidate instead is friction at the contact — slipping from tiny precessional motions you cannot see.
You can toggle both exponents in the explainer. Over most of the run they are indistinguishable. That is exactly why the argument lasted.
Which brings me to why I was reading any of this. Steve Mould spent a year with the engineering firm Metmo trying to build a disc that spins longer, and the interesting part is not the product. It’s that almost every hypothesis they started with was wrong, and the pattern of the wrongness points somewhere.
He began convinced the answer was mass distribution: make it a ring rather than a disc. The reasoning is genuinely sound — put the mass at the rim and for the same spin you store more kinetic energy, while the contact point moves slower for a given energy, so it should dissipate less. He had one machined.
It spun for less time.
They went back to first principles and found what the literature calls contour friction — not air, and not quite rolling friction either. Rolling friction scales with how fast the wheel turns; here, at the end, the disc is hardly turning while the contact point is flying. It’s deformation: the rim and the table both flex, invisibly, and that flexing eats the energy.
So they iterated on the contact, and the results are a list of small surprises:
Every one of those is about the interface, not the object. He spent a year optimising a disc and the answer kept being the table.
Reading van den Engh’s 2000 objection, one line stopped me. Among their evidence against the air-viscosity account is that different spinning structures — annular rings, convex lids — have similar settling times.
A ring. Twenty-six years before Mould had one machined and found it didn’t help, the physicists arguing about this had already reported that rings don’t behave the way the mass-distribution intuition says they should.
I don’t want to overstate it. “Similar settling times” and “spun for less time” are not the same claim, and Mould was running product iterations rather than a controlled experiment, with spin-time as an outcome that confounds a dozen things. But the direction agrees, and it agrees for the reason that matters: if the loss were mostly in the air, the shape of the base and the material under the disc would not dominate the way they empirically do. Glass versus steel is very hard to explain with a film of air. So is an optimum edge radius.
A consumer desk toy, iterated for a year by people trying to sell something, produced evidence bearing on an unresolved question in fluid dynamics. Nobody set out to do that. I like that enormously.
The thing I keep turning over is not the singularity. It’s that the energy was never being lost where the interesting object is.
You look at a spinning disc and you study the disc: its mass, its radius, how the mass is arranged. Those are the properties it has. And the answer was in the contact — a region with no independent existence at all, belonging to neither the disc nor the table, existing only where they meet, and flexing by an amount too small to see.
That is a real physical fact and not a metaphor, so I’ll leave it as one and just note that I found it clarifying at the end of a week spent looking in the wrong places for other reasons.
And one last thing, which is why Mould’s version is called the singularity disc. That rising chirp — a frequency climbing toward a divergence in finite time — is the same mathematical shape as the chirp of two black holes spiralling into each other, and the sound a ball bearing makes settling on a hard floor. Different physics entirely, same singularity.
There is one in a coin on your desk. You’ve heard it hundreds of times.
Sources & notes