Until very recently, the densest known packing of pound coins in the plane was the hexagonal lattice packing of circles, which achieves a density of π/√12 ≈ 0.9069. This was proved by Axel Thue to be the densest arrangement of points subject to the constraint that the minimum distance between any pair of points is *d*.

However, recently a new method of packing pound coins has been realised, which attains a marginally increased density of 4√3 − 6 ≈ 0.9282. Specifically, we take the following packing of dodecagons induced by the *truncated hexagonal tiling*:

This increased density is a consequence of a new dodecagonal pound coin being introduced, and not that there is a denser packing of circles (which is, of course, impossible). The coin has been designed on the basis that it is considerably more difficult to forge, and rather satisfyingly still includes one of the earliest anti-counterfeiting precautions: Isaac Newton’s milled edge.

Apparently there is a competition to decide the obverse design of the coin; maybe we could deliberately design and submit something with an even more subtle fundamental flaw than the famous gear train parity error of the two pound coin? It would be absolutely hilarious if we manage this. I feel that an overbalanced wheel or other perpetual motion machine would be too obvious, but an impossible cube with a portcullis on one face might just get past the quality control process…

(This is also the first recent British coin to lack a constant diameter, and will therefore necessitate a major overhaul of all coin-recognising machines.)

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You could try to get a mobius strip of an even number of gears, claiming to have fixed the problem in the last coin, but they’d probably catch that one.

“This is also the first recent British coin to lack a constant diameter, and will therefore necessitate a major overhaul of all coin-recognising machines.”

Ouch. Why did they do that? Is this some weird form of economic promotion?

The coin is based on the (pre-decimal) threepenny bit, so it was probably an attempt to reinforce nostalgia and patriotism. But it would indeed generate revenue for manufacturers of coin-recognising machines…

Has anyone determined the densest packing of our other non-circular coins – the heptagonal 20p and 50p?

I very much doubt it. The 50p and 20p coins are constant-diameter shapes composed of piecewise circular arcs, so the problem would be strictly harder than for discs. And even the optimality of the hexagonal packing of circles wasn’t proved until the 19th century (by Axel Thue), and the three-dimensional version has only just been resolved.

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