3 ms·

I'm pretty sure a triangular one can fall down its hole if tilted properly.

by eliben 16y ago

suraj 16y ago
Oldie http://hebig.org/blog/003029.php http://hebig.org/blog/003029.php
StavrosK 16y ago
I think so too, but how? It doesn't seem like there's a distance in an equilateral triangle longer than its edge.
krschultz 16y ago
They can't. First off, there will be a small lip to keep it from falling in flat. This means that the hole is actually smaller than the manhole cover by an inch or more. Second, no matter how you orient an equilateral triangle with on point facing down the hole, there is an edge across the top that is too big to fall in. The way to visualize this, if you take one edge of an equilateral triangle and sweep it in a circle around one end, it will be outside of the triangle at all points except where the corners meet. So trying to drop it in any position other than with the edges lined up will make it impossible to fall through. And since there is a lip when the edges line up, it can't fall through. In contrast with a square or rectangle, if you sweep one of the edges it will be inside of the square or rectangle at some points (on a rectangle, you have to use the small sides). This means all you have to do is shove one edge of the cover down the hole and it will fall in.
- michaelcampbell 16y ago
  > First off, there will be a small lip to keep it from falling in flat. This means that the hole is actually smaller than the manhole cover by an inch or more. Exactly, and with a big enough lip/flange, ANY shape will work.
  - krschultz 16y ago
    Not a square. The diagonal dimension of a square is sqrt(2) * the edge length, so roughly 1.414*edge length. So if you rotate the square cover 45 degrees, you can kick it right down the hole no problem. Sure you could have a cover that was several inches bigger than the lip, but assuming the lip is only going to be on the order of .5-3" and the cover is going to be around 24-36" wide, you can always rotate the square one ot fall down there.
    - michaelcampbell 16y ago
      > with a big enough lip/flange Your flange is not "big enough". For an absurd example, consider a hole 3" to a side, and a flange 48' in width. That said, I honestly meant this as a mathematical/rhetorical statement, not a practical one. So yes, you're correct that for any reasonable flange width (given the assumption we're talking about steel/iron manhole covers, and accepted values of the material's strength, etc.), some shapes wouldn't qualify. My point was to show that the question, taken at its face value, isn't even a good question. It assumes a flange, or no shape would work as a cover, as the cover would simply fall into its own hole. Thus, if we're to assume a flange (or a taper; it could be argued wine bottle stoppers and the holes they "cover" are both "circles"), we should be able to assume one of arbitrarily large size, and then ANY shape will work.

I'm pretty sure a triangular one can fall down its hole if tilted properly.

by eliben 16y ago

I'm pretty sure a triangular one can fall down its hole if tilted properly.

suraj 16y ago

Oldie http://hebig.org/blog/003029.php http://hebig.org/blog/003029.php

StavrosK 16y ago

I think so too, but how? It doesn't seem like there's a distance in an equilateral triangle longer than its edge.

krschultz 16y ago

They can't. First off, there will be a small lip to keep it from falling in flat. This means that the hole is actually smaller than the manhole cover by an inch or more. Second, no matter how you orient an equilateral triangle with on point facing down the hole, there is an edge across the top that is too big to fall in. The way to visualize this, if you take one edge of an equilateral triangle and sweep it in a circle around one end, it will be outside of the triangle at all points except where the corners meet. So trying to drop it in any position other than with the edges lined up will make it impossible to fall through. And since there is a lip when the edges line up, it can't fall through. In contrast with a square or rectangle, if you sweep one of the edges it will be inside of the square or rectangle at some points (on a rectangle, you have to use the small sides). This means all you have to do is shove one edge of the cover down the hole and it will fall in.

michaelcampbell 16y ago
> First off, there will be a small lip to keep it from falling in flat. This means that the hole is actually smaller than the manhole cover by an inch or more. Exactly, and with a big enough lip/flange, ANY shape will work.
- krschultz 16y ago
  Not a square. The diagonal dimension of a square is sqrt(2) * the edge length, so roughly 1.414*edge length. So if you rotate the square cover 45 degrees, you can kick it right down the hole no problem. Sure you could have a cover that was several inches bigger than the lip, but assuming the lip is only going to be on the order of .5-3" and the cover is going to be around 24-36" wide, you can always rotate the square one ot fall down there.
  - michaelcampbell 16y ago
    > with a big enough lip/flange Your flange is not "big enough". For an absurd example, consider a hole 3" to a side, and a flange 48' in width. That said, I honestly meant this as a mathematical/rhetorical statement, not a practical one. So yes, you're correct that for any reasonable flange width (given the assumption we're talking about steel/iron manhole covers, and accepted values of the material's strength, etc.), some shapes wouldn't qualify. My point was to show that the question, taken at its face value, isn't even a good question. It assumes a flange, or no shape would work as a cover, as the cover would simply fall into its own hole. Thus, if we're to assume a flange (or a taper; it could be argued wine bottle stoppers and the holes they "cover" are both "circles"), we should be able to assume one of arbitrarily large size, and then ANY shape will work.