Effect Measure

The shape of man hole covers (reprise)

The Revere troop is still on the road (we arrive at our beach destination later today), and while WiFi in motels is convenient, it’s not so easy to blog without the usual creature comforts (a library, good coffee, my own workspace and lots of unread/half read papers with great sounding titles that might become blog posts). However I do have Mrs. R. for company and our old and hobbling dog is along to be a literal creature comfort for both of us. So I’m going to reprise an oldie but goodie from the archives (January 2007), this one the follow-up to an earlier post asking “why are man hole covers round?” I came to the question because I am interested in sewers. Some might consider this a character defect, but I came by it honestly, having once edited a volume of historical documents on the history of sewers for a large project on history of public health in America. Anyway, the man hole cover post generated a lot of traffic so I did a sequel, cleverly entitled:

Why are manhole covers round? The sequel (from Effect Measure, January 6, 2007)


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 the reason that 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:

i-c242a1c55b799f43bd88b0a840113fe6-250px-DigitalCaliperEuro.jpg

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:

i-2082f5201fbce104d333d38759ad6807-ReuleauxTriangle.png

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:

i-0cd081bd5460b451d6ed83f5dd54d4e0-manholes14.jpg

You can find other photos here.

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

Comments

  1. #1 george.w
    August 1, 2009

    OK, before looking at Wikipedia’s entry, I’d have to guess that manhole covers are round because they probably weigh 200 lbs, and would you want to have to think about getting them in the right orientation while wrestling with something that heavy? Lot of smashed fingers in that equation.

    (Goes off to Wikipedia to look…)

    Ah! that was one of the reasons, more or less. Also a couple I hadn’t even thought of, including one that made no sense to me at all. I guess Microsoft will never hire me now.

  2. #2 Jimmy
    August 1, 2009

    I still read you, Revere, after 5 years, because you can mix whimsy with science so eloquently.

  3. #3 neil
    August 1, 2009

    Thanks for the unusual (reprise) diversion from our Influenza Obsession this Saturday Revere!

    I aslo never saw this on any Mensa test. 🙂

  4. #4 Don Marti
    August 1, 2009

    Have you seen the big casting at the top of the manhole that the cover fits into? The shape that uses the least iron for a given manhole area is a circle. If you made a different shape you’d need more iron for the rim.

  5. #5 kagiso
    August 1, 2009

    It is often very difficult to seat manholes in their bases properly, due to slight failures of tolerance and presence of grit. As a result square manholes, with four contact points, one at each corner, are usually problematic as they rock when vehicles pass over them. This causes noise pollution and potentially damage to the manhole and road structure.

    So most manholes are made with only three points of contact (a three legged stool never needs paper under one leg); which can’t rock, which favours a triangular or circular shape. If you only use three points on a square or rectangle, you can still get rocking on the corner(s) without a designed contact point.

    Where a square or rectangular cover is required, this is normally done by making up the shape with two (non-rocking) triangles.

  6. #6 Hilarious...
    August 2, 2009

    “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.”

    If your mind’s in the gutter, this is hilarious.

  7. #7 CanadianChick
    August 3, 2009

    and here I was thinking that manhole covers were round because manholes are round…