This question was posed by my grandfather during dinner with him my first night in Key Largo.

The answer to this question can, naturally, be answered using just a little bit of math.

As a table, we collectively came up with two solutions.

1) By using a circular design, the manhole cover can be oriented in any direction, thereby making it easier to place because, being made of cast iron, they aren't particularly light.

2) They can't fall through. If one were to use a polygonal shape, such as a square, where the distance between two opposite points is greater than the length of any one side, then the cover would be able to fall through the hole.

Later, reflecting upon the question further, I came up with another answer using the semantics of geometry where I realized that the shortest path that can be taken where one starts at a single point and finishes at that same point is, in fact, a circle. This solution would allow for the least amount of iron possible to be used in the production of a manhole.

Math is cool :)

Dork!

ReplyDeleteThinking about your last point, a circle might not be the most effective shape to use the least amount of iron. You should choose the shape of the thing going through the opening. If using the least amount of steel was your concern, then you would make it it a rectangle a little larger than the top down profile of the average man. Also a rectangle is much easier to cut uniformly and repeatedly in accurate dimensions for both the hole and the cover (but the cover is cast, so that's irrelevant). But, considering the first two points, this is impractical so a circle makes more sense.

Just my thoughts :-)

I thought about this for a while, and circles do actually require the least amount of iron for someone to get through. If the manholes we being manufactured to fit an average person, the holes would in fact need to be round (because let's be honest, the average person is round).

DeleteTo address your second point, an equilateral triangle is a polygon that you could use and it not fall through. It would also use less iron than a circle if the diameter were equal to the side length.

ReplyDeleteFalse! An equilateral triangle actually wouldn't work. However, what's known as a Reuleaux triangle would because all points on one side are the same distance from the opposite vertex, therefore making it impossible to fall through, much like a circle.

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