I found this board at the International Centre for Mathematical Sciences (ICMS) during a conference about shape optimisation and shape geometry. Talking to some of the delegates there I found out that the field is all about finding the best shapes to suit a purpose, and these purposes might come from physics, engineering, architecture, or simply pure mathematics. For example, the Reuleaux triangle is a shape of constant width (it has the same diameter wherever you measure it) but has an area 12% less than a circle of the same diameter. This makes it more efficient for making coins (our 50p piece is a similar shape but with 7 sides), manhole covers and even buildings.
I asked conference delegate Jimmy Lamboley what his favourite shape was, and he laughed and said “Anything but a sphere!”. He explained: ” The sphere is so often the answer to minimisation problems that I love to find the problems where it isn’t the case.”
The Reuleaux triangle is a great example of a shape more efficient than a circle. What’s amazing is that the corresponding question for 3D shapes (which shape of constant width has minimum volume?) is a problem still waiting for a solution.