Q) Why is it that manhole covers are round?

The question achieved notoriety when it started being used by microsoft as an interview question. The answer that I’ve always liked is that a circular manhole cover can never fall into a circular hole. If you were to use a square, for example, it could easily fall through the diagonal.

So here’s another puzzle: is there any other shape for which this property holds – that you can make a manhole cover out of it that will never fall into the manhole?

Well, I wouldn’t be writing this post if it wasn’t possible. It turns out that there is another simple shape that can do this. It’s called the Reuleaux triangle. It’s an example of a surface of constant width. If you play a guitar, chances are that you’ve already held on in your hand.

The Dunlop stubby triangle has the neat property of being a curve of constant width. If you contain it within two parallel lines and rotate it, the parallel lines won’t move apart. This shape can be used to drill approximately square holes. It also has an ingenious application in the Wankel engine, where it’s used to cycle between the different strokes of an Otto cycle!

The way you make one of these is you take a triangle and draw an arc around each side centered on the opposite vertex. While the circle is the simplest two dimensional curve of constant of width, it’s easy to see that by generalizing the above construction we can get infinitely many such shapes. You just start with a pentagon, or a hexagon, or any other regular polygon, and inscribe each of the sides in arcs. As you increase the number of sides, you get close and closer to a circle. The circle is, if you like, the infinitely stubby triangle. A neat result is that all such curves have the same perimeter.

Such heptagonal shapes are used in the British 20p and 50p coins. Since they have a constant diameter, vending machines can verify their width no matter how you insert them. Another example of why British currency is more awesome than that of their American cousins. As if we needed another reason.

Which brings me to the subject of my post. One of my friends was in a rather creative frame of mind when he had the idea of the chubby pyramid — a shape kind of like a regular pyramid, but more rounded at the triangles. It was a piece of concept art, complete with a charcoal drawing and a song to go with it! And so, the questions eventually arises, are there any surfaces in three dimensions other than the sphere which have a constant width?

It turns out that if you try to generalize the Reuleaux triangle (i.e., take a tetrahedron and inscribing spherical caps on each end), this doesn’t quite work. Such a shape is called the Reuleaux Tetrahedron, and it needs to be modified slightly to have a constant diameter. But there’s still a neat intuitive way of constructing our shape: take the surface of revolution of the stubby triangle around one of it’s axes of symmetry. You have to sort of squint at it and wave your hands to convince yourself that it works. And there you have it, the mathematically perfect chubby pyramid!

References:

Poster on curves of constant width by David Wills, Durham University.