Maths of a tossed coin can leave heads spinning

Coin Cartoon

Get a coin, set it spinning on a table and watch what happens. Within seconds you will be witnessing a classic demonstration of the maxim of the Nobel-prizewinning physicist Philip Anderson: that there is no baffling problem which, when looked at in the right way, does not become even more baffling.
As the coin spins. it rapidly settles down onto the table with a rattling sound. A trivial enough phenomenon, one might think - but one which lives up to Anderson's law, as the mathematician Keith Moffatt from Cambridge University found recently.
In a paper published in Nature, Prof Moffatt showed that if gravity is taken to be the only force at work, a spinning coin should carry on spinning forever - a conclusion somewhat at odds with reality.
Prof Moffatt, casting around for effects that could stop the coin, looked at air resistance - and this time got a much more sensible result. The equations predicted that the coin does eventually end up flat on the table. Better still, they also showed that the spin-rate increases as the coin settles - which accounts nicely for the rising pitch of the "rattle" as it comes to rest.
The trouble now was that Prof Moffatt's equations went too far, predicting that the rate of spin would increase forever. Indeed. they showed that a 2p coin would be spinning infinitely fast within a couple of seconds.
Prof Moffatt reckons he has this covered, pointing out that the coin cannot settle on to the table faster than gravity can pull it - a fact his equations failed to take into account. To prove his equations work, he used them to predict the behaviour of an "executive toy" called Euler's Disc, a chrome-plated disc spinning on a mirror- like surface. Moffat's equations predict that, once set spinning, the disc should continue to spin for well over a minute - which it does.
Now I suspect that readers can be divided between those who think this is a neat bit of science and those who think Prof Moffat needs to get out more. I would point out to those sniggering at the back of the class, however, that studies of spinning discs have a distinguished place in the history of science.
In 1859,James Clerk-Maxwell, another Cambridge mathematician, used the laws of mechanics to show that the rings of Saturn cannot be solid discs - a prediction confirmed more than a century later by planetary probes. The American physicist Richard Feynman also claimed that seeing a dinnerplate spinning through the air one day in 1946 led him to research into quantum electrodynamics that won him a Nobel Prize.
In all these cases, theory has later been confirmed by observation - which has yet to happen with my own favourite bit of spinning disc research, published in 1993 by yet another distinguished Cambridge mathematician, Sir Hermann Bondi.
In his paper in the European Journal of Physics, Sir Hermann attempted to calculate the probability that a coin dropped on to the floor will land on its edge, After five pages of tightly-argued mathematics, he arrived at a formula which predicts that the chances of a 2p coin landing on its edge must be less than 1 in 2,000.
And a lot less, by all accounts - for we don't hear of many accounts of referees having to re-toss coins that landed on their edge the first time. Sir Hermann wasn't able to put a much tighter limit on the probability because the maths involved is so tricky that he had to assume the coin just tumbled to the ground from a random starting position. Predicting the fate of a coin tossed high in the air is quite another proposition - and the way we toss coins may militate against on-edge landings.
It doesn't entirely rule them out, though - as I discovered recently after receiving a first.hand account of perhaps the only well- attested case of tossed coin landing on its edge.
On October 9, 1972, the mathematician Dr Jeffrey Hamilton from Warwick University wanted to show his students the effect of chance by tossing a coin. Taking a 2p coin out of his pocket, he tossed it, then watched as it hit the floor, spun round and came to rest on its edge.
Prof Hamilton tells me that dozens of students witnessed the amazing event, and after a stunned silence they all broke into wild applause. As well they might, for you don't need to be a distinguished Cambridge mathematician to postulate that none of them will see such an event again.

Robert Matthews





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