Effect Measure

Why are manhole covers round? The sequel.

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:

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 John Wilkins
    January 6, 2007

    Don’t ask any questions about steam trains… you’ll regret it even more.

  2. #2 revere
    January 6, 2007

    John: LOL. I’ll remember that. I have one more manhole related one and then I’m done.

  3. #3 Dave S.
    January 6, 2007

    What I have discovered is that no matter how mundane the subject, there is still going to be someone, perhaps many someones, passionately interested in it. I wouldn’t be surprised to find people crazy about potato chip bags or who collect clothespins.

  4. #4 Ken Hirsch
    January 6, 2007

    Strictly speaking, any shape could be used for a manhole cover and you could still have the property that it couldn’t fall through the opening. All that is necessary is that the maximum width of the inner hole (the one that people have to fit through) is smaller that minimum width of the manhole.

    The inner hole and the outer hole (where the manhole cover fits) don’t even need to have the same shape. For strictly geometric considerations, with circular manholes and circular manhole covers, an infinitesimal difference in the inner and outer diameter would do, but that “lip” has to support the weight of not just the manhole but also the traffic that passes over it. I just searched and it appears that they are supposed to support 16,000 pounds and are tested to 40,000 pounds.

    And of course there have to be allowances for manufacturing tolerances, wear, possible different rates of thermal expansion, etc.

  5. #5 revere
    January 6, 2007

    Ken: Of course you are right, strictly speaking. But who would speak that way? Why would you have a manhole cover of much larger area than the hole? Your other points are well taken, from an engineering/manufacturing standpoint, but the question had more of a mathematical flavor. I gather most manholes these days are made in India. I wonder who does their customer service calls?

  6. #6 Mustafa Mond, FCD
    January 6, 2007

    I note that the Zen-like purity of my response didn’t make an impression on you.

  7. #7 revere
    January 6, 2007

    Mustafa: I’m not religious. :) Moreover, as Ken’s response shows, the hole and the cover needn’t be the same shape.

  8. #8 Dave S.
    January 6, 2007

    We could always get really persnickety and say the top of the cover needn’t be the same shape as the bottom, which needn’t be the same shape as the lip, which needn’t be the same shape as the hole. :)

    P.S. Apparently someone actually makes Reuleaux triangle drill-bits to produce square holes (Watts Brothers Tool Works, Wilmerding PA), but the holes only almost-square, even after modifying the Reuleaux shape a bit. The also make drills to make pentagonal, hexagonal and octagonal holes. To make a truly square hole, its easier to drill a round one and square it up with a chisel, or use a drill-chisel combo on a drill press.

  9. #9 Judy
    January 6, 2007

    Revere: I followed your link to the other pictures. They are really quite lovely and wonderfully detailed. Who would have thought? I did notice, however, that they were not all round…

  10. #10 Susan Och
    January 6, 2007

    When we travel, we’re always excited to see manhole covers from East Jordan Iron Works, about 40 miles from here, as the crow flies, and still in business.

  11. #11 revere
    January 6, 2007

    Judy, Susan: One of the things the post did for me was make me look down as I walked. I also discovered that there are some interesting a varied patterns on manhole covers and drains. As you note, Judy, they are not all round, either, although they are mostly round. So if Microsoft interviews you and asks you why manhole covers are round, you can answer that not all of them are. Then when you don’t get the job you can use your Mac with a clear conscience.

  12. #13 revere
    January 8, 2007

    sam: Yes, of course. I don’t think that was the point, though, was it? There is no shortage of manhole covers with other shapes, true. But the spirit of the question was, shall we say, a little more on the intellectually playful side? And ournd covers predominate. Why?

  13. #14 Easy Hiker
    January 8, 2007

    I got a great laugh reading this post and subsequent comments. I’m a Civil Engineer and actually knew from experience why manhole covers are (most often) round.

    This series exemplifies why it is so difficult to gain concensus on ANY subject. This is a SIMPLE one, and made in jest. Whew!! But now try to make posts and comments on some complex and ambiguous subjects. I’m typically amazed on the variety of responses. Human beings and their responses are so very interesting.

    I shouldn’t do this, but here are some other mental puzzles.
    Water from a faucet come out as the diameter of the faucet, and then get smaller and smaller as it falls to the sink. Eventually it breaks into droplets. Why does this occur?

    When you press symetrically on the ends of a non-symmetrical plastic ruler (any length), it always bends to the same side. Why does it do this?

    I’m sorry, perhaps you all should just ignore these puzzles.

  14. #15 revere
    January 8, 2007

    Easy: No, I love science questions. I have one more manhole cover puzzle, and this one is real. I have yet to find someone (an engineer, particularly) able to give a satisfactory answer. On the other hand, most people are very sure they know the answer. Watch this space.

  15. #16 Victoria
    January 8, 2007

    Off the wall here. Could a man hole cover be used as a grill on a bar-b-que?

  16. #17 revere
    January 8, 2007

    Victoria: Well, it’s pretty heavy and it’s pretty thick, and cast iron develops hot spots and it has few holes, but other than that, I guess so. You guys really like your barbies, don’t you.

  17. #18 Victoria
    January 9, 2007

    Oh! Revere you know. A shrimp here, a Kangaroo there.

  18. #19 Victoria
    January 9, 2007

    Oh! Revere you know how it is – a shrimp here – a kangaroo there.

Current ye@r *