What's that about square pegs and round holes?

By Burkard Polster and Marty Ross

The Age, 23 June 2008

Why are manhole covers round? This question, famously asked at job interviews, has a number of good responses. Our favourite? Covers are round in order to cover up the round holes!

The mathematical answer is less funny, but much more intriguing: manhole covers are round so that they cannot fall through the hole. No matter how you position a circular cover, one of its diameters will be horizontal, and so it will be wider than the (slightly smaller) hole. So, it will always get stuck before falling completely through. By comparison there is definitely a danger with, for example, a square cover. If you align the square with the diagonal of the square hole, it could fall straight through. And the same is clearly true for most shapes.

This leads us to ask: is it possible to make safe manhole covers of any other shape? The surprising answer is that we can. Here is one solution, a shape known as Reuleaux’s triangle. To form this shape, start with an equilateral triangle. Now, draw three circles, each centred at one vertex with radius the sidelength of the triangle. Then the region trapped in the middle is Reuleaux’s triangle.

The special property of Reuleaux’s triangle is that it is a shape of constant width (SCW): no matter how Reuleaux’s triangle is trapped between two parallel lines, the distance between these lines remains the same: see the square picture below. This is the critical property which makes for a safe manhole cover. And there are many other SCW’s we could use: we’ll leave that for you to ponder, or contact us if you want some hints.

Reuleaux’s triangle has other beautiful properties. If you place it inside a square of just the right size, it will roll around smoothly. It is the principle behind an ingenious machine which drills (almost) square holes. And, the famous Wankel engine is based upon a similar principle.

Another property of SCW’s is that they make very good coins. A coin machine identifies a circular coin by its width, and this means that the machine could recognize an SCW just as easily. And in fact the British have issued some beautiful SCW coins.

Which is the best coin? Well, out of all SCW’s of a given width, Reuleux’s triangle has the smallest area. So, we should make Reuleaux coins, and save all that precious metal!