My post on “Why are manhole covers round” was made in all innocence. I’m interested in sewers and long ago someone had mentioned this little factoid to me and I thought it was interesting. Little did I know.

Little did I know, what? First, that this is a notorious question. Allegedly it came to notoriety because this was a question asked by Microsoft on job interviews. In addition it is supposedly a Mensa question (couldn’t find the cite) and some companies claim to have used it before Microsoft (McKinsey & Co).

Wikipedia has a good entry on manholes with 11 good reasons they are round (the first of which is the one we gave, they can’t fall through the hole). Which is what interested me, and I posed the question what other shapes had this property. I mistakenly suggested an equilateral triangle was one, but it isn’t, as pointed out almost immediately by a commenter. In fact the hivemind performed admirably, coming up with the right answer within 30 minutes (an answer I should have remembered; of course there are quite a few things I don’t remember these days, like the names of people I’ve known for decades, but that’s another story, one I can’t remember).

The answer to the mathematical question is either a Reuleaux triangle, or more generally, a curve of constant width. Since a Reuleaux triangle is a particular curve of constant width (as is a circle), let’s deal with that.

It’s easy to see that if a curve had the same width no matter which way it was turned, it would be like a round manhole (which also has that property). It couldn’t fall through a hole that had the same shape and size (well, maybe a tad smaller with a lip for it to sit on). So how do you measure the “width of a closed curve”? The most intuitive way to do it would be to use calipers, a pair of parallel jaws:

Source: Wikipedia

If the closed curve has constant width, the caliper’s jaws would be the same distance apart no matter how you placed it placed it across the curve. The smallest such closed curve is the Reuleaux triangle, and it looks like this:

You can see an ordinary equilateral triangle inscribed in it. One way to construct the Reuleaux triangle is to put one tip of a pair of compasses on a vertex of the ordinary triangle and draw an arc from one of the other two to the third. One application is as the single rotor of a Wankel engine. It can also bore holes with straight sides (although the corners are slightly rounded). You can make a curve of constant width from any polygon with an odd number of sides, and indeed that is just what the British 20 and 50 pence coins are. This allows automatic coin machines to recognize them no matter what their orientation. The same is true, of course, of a round coin. Circles are not only curves of constant width but are the largest such curves for a given width. All curves of constant width are convex, so a five pointed star wouldn’t do for a manhole.

The biggest surprise from the post, however, was that there is a thriving, large and vibrant “manhole subculture” out there. There are books, websites, histories, collectors. People devoted to manholes. There is a thriving business in trading, buying and (therefore) stealing manholes. One commenter pointed us to a link of photos of Japanese manholes, although it turns out there are several sites with pictures like this from all over the world. Some of the them are really stunning. There’s also a book of quilting patterns taken from decorated manholes. I’m not kidding. You can buy it on Amazon. Here’s one example:

You can find other photos here.

So I’m not so bad, after all. I was just interested in sewers. Little did I know.