The Problem with the New £1 Coin
Today marks the launch of the new £1. It actually looks quite pretty, like a smaller version of the £2 coin with its two different metals. The reason for the change is the same reason that forces them to upgrade banknotes: counterfeits. The Royal Mint estimates that around 1 in 30 of the classic £1 coins are fakes, which is a far higher rate than other coins, which usually aren't worth the effort (except the £2 which is too hard) and banknotes which receive more scrutiny.
The new coin has a list of security features to make it more secure, which on top of being bimetallic include a fancy “latent image” which switched from displaying 1 and £ when viewed from different angles, some tiny detail such as the milling on some of the edges and a mystery hidden security feature. Quite what this last one means is up to debate, but I enjoy the idea that it may be an elaborate bluff.
However the problem I have with the coin is the number of sides that it has. Curves of constant width are those that have the same diameter no matter which way you measure them. So a circle would meet this criteria, but so would a triangle with rounded edges such as the reuleaux triangle below. Notice that although the pointed corners jut out, they are opposite the flatter bits which keeps the distance across the shape the same.
This is a really useful property because it means that these shapes can roll. So if you make a curve of constant width into a prism then by putting a book on top of it you will be able to roll it without the book rising up and down. 20p and 50p coins are both curves of constant width (with 7 sides) which is not by accident. Imagine you are designing the coin intake for a vending machine. If the shape was a different width depending on how far through its spin it was then it would be a nightmare. You would have to check for a range of sizes.
It is exactly this problem that vending machine makers have been facing for the last few months getting ready for today. 12 sided shapes such as the new £1 coin cannot be curves of constant width because the 2 corners will always be opposite each other, which will always be greater than the distance between the two flatter bits. In practice the vending machines have relied on mostly lasers with software updates rather than mechanical trickery since the launch of the £2 coin, which has made it less of a headache than such launches have been in the past.