Curves of constant width – The 50p story...

When all the Pi is Wrong nonsense stormed the internet I had many responses to my video and it wasn’t all just hate mail and “how did this guy get a PhD when he finds dividing by 2 difficult?”. One assertion in particular gathered a lot of interest. I got comments ranging from bewilderment to abuse. My inflammatory statement: a 50 pence coin has constant width. It’s an amazing counterintutive idea that there exist shapes that have constant width but are not circular. The 50 pence coin – which to non-Britons probably looks more like a martial arts weapon than national currency – has such a shape. It’s a wonderful design and now it and the 20 pence coin are considered design classics. Nonetheless, Alex Bellos notes in his recent Royal Society Winton Prize nominated book that initially there was a campaign against it because the Queen would be “insulted by this heptagonal monstrosity”. (If you haven’t read Bellos’s book, then go buy one now!) There is more to it than just looks. It is practical. The shape allows visually impaired people to tell what the coin is without having to carefully judge the size. (Judging coin size can be difficult – many old people suffer from both sight problems and arthritis.) Of course, if you wanted a coin that is an easily detectable shape, then why not use a square? Well, then the problem is how do you get a vending machine to detect coin values? You will need an extra mechanism to detect shapes and one to orient the coin to check its size in case it is a fake. The underlying mathematical beauty of the 50 pence is that wherever you measure its width it is the same and hence the...