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Circular manholes are great because the cover can not fall down the hole. If the hole were square, the heavy metal cover could fall down the hole and kill some man working down there.

Circular manhole: Circle

Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?

Semi rigid math formulation:

Let us say that we have an infinite matematical 2D plane in 3D space. In this plane is a hole of some shape. Furthermore we have a flat rigid 2D figure positioned on one side of the plane. This figure has the same shape as the hole in the plane, but infinitesimal larger. Is it possible to find a shape, where there is no path twisting and turning the figure that brings the figure through the hole?

Here is one such shape (only the black is the the shape):Five side

But if one put the restriction on the shape, that it needs to be without holes (topological equivalent to a circle in 2D), then I can not answer the question!?

Edit:

Because of the huge amount of comments and answers not about math, I fell the need to specify that:

I am not interested in designing manholes. I am interested in the math inspired by the manhole problem.

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I believe the notion that they were designed to avoid falling through is urban legend, popularized by interviewers. I believe the justification evolved, rather than the design. –  copper.hat Jul 30 '12 at 3:37
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A few plausible alternate answers I've heard are 1) The symmetric shape makes the road with the hole in it more structurally sound, minimizing stress concentrations. 2) It's easier to fabricate/cut the cover and drill the hole, 3) you can roll it into place instead of carrying it. –  Nick Alger Jul 30 '12 at 5:18
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its actually super easy to find square ones, of course the question still stands from a maths point google.co.uk/… –  jk. Jul 30 '12 at 7:28
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A related MO question. See also this and this and this. –  J. M. Jul 30 '12 at 13:33
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Look, people rarely put the cover on when other people are inside, and these things do not slip easily. Its not an issue of real concern. Dropping drills, hammers and kicking stuff in is a bigger issue. I have spent quite a bit of time in sewers funding my education (and entertainment):->. Its a cute problem, and entertaining in an interview situation, but that's about it. –  copper.hat Jul 30 '12 at 21:21

7 Answers 7

up vote 81 down vote accepted

Any manhole cover bounded by a curve of constant width will not fall through. The circle is the simplest such curve.

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"Curves of constant width are also the general answer to a brain teaser: "What shape can you make a manhole cover so that it cannot fall down through the hole?" In practice, there is no compelling reason to make manhole covers non-circular. Circles are easier to machine, and need not be rotated to a particular alignment in order to seal the hole." –  Pedro Tamaroff Jul 30 '12 at 0:14
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@Peter. A minor nit--I'm pretty sure manhole covers are cast, rather than machined from a large blank, so ease of manufacture wouldn't be a determining factor. –  Rick Decker Jul 30 '12 at 0:22
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I'd sure like to see somebody do manholes in the shape of a Reuleaux triangle... –  J. M. Jul 30 '12 at 3:22
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@J.M.: Look here. The one on the left is almost a Reuleaux triangle (its curvature is slightly higher, if you want to nitpick). –  Generic Human Jul 30 '12 at 7:30

A manhole cover can't fall into the hole if the minimum width of the cover is greater than the maximum width of the hole.

For example, consider a one-meter square cover over a square hole slightly smaller than $1\over\sqrt 2$ meter on a side. The diagonal of the hole is slightly less than 1 meter, so the cover won't fit into it.

The point is that manhole covers aren't the same size as the manholes they cover; they have flanged edges.

EDIT :

Oops, I missed this sentence in the question:

This figure has the same shape as the hole in the plane, but infinitesimal larger.

so my answer, though it does have real-world applications, doesn't really answer the question as stated.

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+1 This answer is much more correct and to the point that that upvoted one over there. –  Mr Lister Jul 30 '12 at 7:12
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@MrLister: OP specified that the cover is only a small amount larger than the opening. This one is almost half again in linear dimension. Real ones do have flanges, but they are small. –  Ross Millikan Jul 30 '12 at 13:27
    
This is indeed a practical answer applicable to the real world - more so than the upvoted one. On the other hand, this is the Math SE. –  DJClayworth Jul 30 '12 at 16:41
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That infinitesimal gap is going to get annoying when it gets a bit of dust in it... –  naught101 Jul 31 '12 at 2:19
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@Hans-PeterE.Kristiansen: The answer itself is my fault; I didn't read the question closely enough. Perhaps the upvoters had the same problem. (I'm not going to complain about it too much.) –  Keith Thompson Jul 31 '12 at 20:30

This question was frequently asked on technical interviews for software engineering positions, up until developers started using counterfactual reasoning.

There is an excellent article "If Richard Feynman applied for a job at Microsoft" showing that there is actually very little practical link between manhole shape and it's conventional representation as a circle.

If I may, I would like to throw a few quotes:

Interviewer: Why are manhole covers round?
Feynman: They're not. Some manhole covers are square. It's true that there are SOME round ones, but I've seen square ones, and rectangular ones.


Interviewer: I mean, why are there round ones at all? Is there some particular value to having round ones?
Feynman: Yes. Round covers are used when the hole they are covering up is also round. It's simplest to cover a round hole with a round cover.
Interviewer: Do you believe there is a safety issue? I mean, couldn't square covers fall into the hole and hurt someone?

Feynman: Not likely. Square covers are sometimes used on prefabricated vaults where the access passage is also square. The cover is larger than the passage, and sits on a ledge that supports it along the entire perimeter. The covers are usually made of solid metal and are very heavy. Let's assume a two-foot square opening and a ledge width of 1-1/2 inches. In order to get it to fall in, you would have to lift one side of the cover, then rotate it 30 degrees so that the cover would clear the ledge, and then tilt the cover up nearly 45 degrees from horizontal before the center of gravity would shift enough for it to fall in. Yes, it's possible, but very unlikely. The people authorized to open manhole covers could easily be trained to do it safely. Applying common engineering sense, the shape of a manhole cover is entirely determined by the shape of the opening it is intended to cover.

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Your answer surely is interesting reading, but it does not help solve the problem. –  Hans-Peter E. Kristiansen Jul 31 '12 at 20:29

A solution nobody has mentioned yet is to make the cover in the shape of a cone. The hole can be any shape at all as long as the cover is an appropriately-shaped cone; if the hole is square, for example, then the cone is actually a square pyramid. Such a cover can fall into the hole, but not all the way in, unless the hole is sufficiently large that the base of the cone fits through, in which case the results could be spectacular.

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"First, assume a spherical manhole cover." –  Keith Thompson Jul 30 '12 at 7:45
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Now explain to me what happens when somebody tries to drive over said manhole –  Snakes and Coffee Jul 30 '12 at 17:53
    
I imagine such a cover would be incredibly heavy, and even if it was made to be light, it would be quite unwieldy and relatively expensive to manufacture, while not offering much of an aesthetical or practical advantage, since you couldn't see the part below the ground most of the time, anyway. ;) –  tomasz Jul 30 '12 at 19:38
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People complaining about this solution are overlooking the advantage that one needn't manufacture many different sizes of cover; one size of cone fits a hole of any size less than some maximum. –  MJD Jul 30 '12 at 20:35
    
I specified the cover to be flat rigid and the same shape of the hole. Obviously you can cover a small hole with a big cover. –  Hans-Peter E. Kristiansen Jul 31 '12 at 20:32

Q Are non-circular manholes possible?

A Yes


Q A better question is why are non-circular manholes less practical?

A Corners are weakest part of a lid and consume more material cost. Round means no corners.

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"Are non-circular manholes possible?" is the title meant to intrigue you. I am interested in the math. If you wanted to you could build a triangular purple manhole cover in styrofoam - I do not care. -1 –  Hans-Peter E. Kristiansen Jul 31 '12 at 20:35
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kindky ammend your question to indicate what you expect. "A mathematical proof of a non-circular manhole that fits , easy to remove, and will not fall in the hole with complex shapes added,or proof that it is impossible." –  Tony Stewart Jul 31 '12 at 23:47
    
You need to define the boundary requirements. lip edge width etc. manhole specs. etc. –  Tony Stewart Aug 1 '12 at 0:04

Here are two examples of shapes, that does not fit through their own holes (I think). But it does not help to get any closer to the general answer.

many sticks euro sign

Edit: Ok - second shape is not good. But maybe the basic idea is still good - I will think some more.

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I could insert that second shape you have by starting to insert one of the curved tips into the straight part of the hole... –  J. M. Aug 1 '12 at 7:30
    
Yeah that 2nd one doesn't work, but this is still the only answer so far that address the full question. –  Nick Alger Aug 1 '12 at 8:26
    
@J.M. I isn't clear: doing it in the naive way would get the cover will stuck when you try to pass the straight part through the hole. –  Bruno Le Floch Aug 1 '12 at 13:11
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Hans-Peter, a spiral cover to a spiral hole would not pass through I think: doing what J.M. describes would get the spiral stuck after one turn, since the spiral would bang into the ground from below. Actually your way of stating the problem with an infinite plane is a bit odd, since that means that the ground is entirely hollow. A slightly different (but more realistic) situation is if the hole below ground level is $\mathbb{R}^+ \times \text{shape}$ where the shape is part of $\mathbb{R}^2$. –  Bruno Le Floch Aug 1 '12 at 13:16
    
@Bruno, even if I tilt the second shape slightly? Try cutting the shape out of cardboard... –  J. M. Aug 1 '12 at 13:20

I hope manhole of equilateral Triangle Shape will also work.

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It won't. Since the altitude of an equilateral triangle is shorter than the side, you can drop the triangular cover down the triangluar hole by putting it to one side of the hole, with one of the sides of the cover being vertical. –  Henning Makholm Jul 30 '12 at 11:50
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@HenningMakholm, It will so long as the hole is smaller than the altitude. The cover has to rest on something. –  zzzzBov Jul 30 '12 at 14:13
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@zzzzBov The altitude of an equilateral triangle is $\frac12\sqrt 3\approx 87\%$ times the length of the side. A triangular cover with a side length of 1m would require a flange nearly 7cm wide to prevent the cover from falling into the hole. –  MJD Jul 30 '12 at 14:33
    
I never said that it was a good idea, I'm just trying to be technically correct. –  zzzzBov Jul 30 '12 at 14:34
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Try cutting one out from cardboard and see what happens if you try dropping it into the hole that you cut it from. –  J. M. Jul 31 '12 at 13:34

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