The Math of Manholes, Part Two

(Continued from part one)

Before we start, I want to clear up a small point: Reuleaux is pronounced RUH low.

Moving on. The answer to the question posed at the end of the post (“Why can we turn some shapes into Reuleaux polygons but can’t for others?”) is: if the polygon has an even number of sides, it will turn into a circle, and if it has an odd number, it will form a Reuleaux polygon. The reason for this is simple. Go back to the intersecting circles construction of the Reuleaux triangle. Here it is, so you don’t have to go back to the other post:

And I’ll remove some bits of the circles:

Now, each side of the triangle is a circular arc centered at the corner opposite from it. Right? Right.

Now, take a look at this square.

Focus on an edge. Find the corner opposite from it.

Do you see the problem? There’s no corner to draw an arc from. Persisting and choosing one corner we get this:

Or, picking the midpoint of the edge:

Remember that the diagonal is \mathit{\sqrt{2}}, so this isn’t constant-width.

So you can’t make a Reuleaux polygon with an even number of sides.

Anyway, we are getting a little far from the point, which is manholes.

When we have talked about the width being the same regardless of rotation we have nearly neglected the fact that some manhole covers aren’t round. So why?

What’s interesting is when you look up “why are some manhole covers not round” or “why are manhole covers square” on Google almost all search results say “Manhole covers are round because so and so”. The internet seems to be denying the existence of square manhole covers. Anyway, the main reason seems to be that the covers are easier to lift.

Well, that’s enough for square manholes. It’s time for manhole theft!

Did you know? People have stolen manhole covers before for the scrap metal!1

These thefts occur when scrap metal prices are high.2According to the two sources in this article, when thefts occurred in 2004 they were priced at either $6.32 (in China) or $183.45 each (in Britain, both converted to 2022 US dollars). My research led to the conclusion that covers cost roughly USD$300, but keep in mind that this would be considerably less if you consider that manhole thieves would be selling just the metal and not the labor associated with the production of the cover. They should not be taken lightly — people have died when they fell down manholes, although it should be noted that it wasn’t the fall, or the sudden stop at the end, that killed them; it was all the sewage at the bottom.3 (Note to potential manhole thieves: you would do well to remember this. Save lives. Don’t steal manhole covers.)

That seems to be about it for manholes, actually. Stay tuned for a follow-up post on Franz Reuleaux and the Reuleaux tetrahedron!


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