“A circular manhole cover isn’t the only shape that won’t fall into the hole. A cover
shaped like an equilateral triangle won’t fall in either.”
An equilateral triangle manhole cover will easily fall in. I can’t describe the process eloquently in words but if you hold it upright with one side perpendicular to the ground and move it to one of the sides of the hole, it’ll fall right in.
You’re referring to Why are manhole covers round?, I assume.
I think you’re correct. The “height” of an equilateral triangle is sqrt(3)/2 or approximately 87% of the length of the edges, so if you dropped the cover into the hole so that the “height” of the cover slid along one of the edges, it would fit in.
Perhaps Cecil made a boo-boo (perish the thought).
The article in question on here. Why are manhole covers round?
I just ran a little demo with the nifty solid modeler on my computer and it appears you are correct, minghai. With one minor caveat; the thickness of the cover plays a factor in this. At some point the cover will become thick enough to prevent in from fitting through the hole.
The sample I ran assumed an equilateral triangle circumscribed about a 2" diameter circle with a thickness of 1/4". The actual numbers aren’t important here. What matters is the ratio of the cover diameter to the cover thickness.
Hey Unc, I beat you in the simulpost by three minutes. Neah Neah Neah. 
An equilateral triangle will fall in the hole, but a Rouleaux triangle will not. This is the “puffy” triangle you get when you take the intersection of three circles that each pass through the centers of the other two. I have heard that there are a few cities that make their manhole covers in this shape, although I’ve never seen one.
Sorry, but that weirdo-shaped cover will also fall in. Put in one of the “puffs”, to the indention between it and the next one. Then the opposite “puff” will have plenty of room.
I think minghai should get a tee-shirt for pointing out Cecil’s error.
We can save ourselves some trouble trying to find other shapes that won’t fall in, because what it would require is that the shortest distance across the cover is smaller than the largest distance across the cover. There is only one shape that won’t fall in.
There is no opposite “puff”, opposite a “puff” is a point.
You forget about the lip around the inside of the hole. That will keep a Rouleaux triangle and perhaps even an ordinary triangle from falling in.
Dan Tilque
Plus, a round manhole cover is easy to put back. Just keep sliding it until it locks in. With a triangular manhole cover, you’d have to take a lot more care to make sure it lines up right.
Sorry, I read the description wrong. I pictured the union of the three circles, not the intersection. I agree, there’s no way for this shape to fall in.
It’s an equilateral triangle whose sides have been curved out, so that the altitude is equal to the length of a side. In other words, each side is a segment of a circle whose center is the opposite apex.
I guess you could do the same with any regular polygon with an odd number of sides. A Rouleau pentagon also wouldn’t fall in.
Could this be what Cece had heard of when he wrote about triangular covers?
And what’s this about popping up when a car drives over? I like Feldman’s books, but I wouldn’t say they’re a reliable source. An example of his needing to get out more comes to mind - he wrote about how traffic lights are always configured with the red above the yellow and green, never side-to-side, so color-blind people can tell the difference. I tried to count the number of side-to-side traffic lights I pass through on my way to work each morning, but there were too many.
I think it’s likely that the Rouleaux triangle was the shape Cecil was actually remembering, especially since I’m pretty sure that there are such manhole covers somewhere in the world. One property specifically of the triangular version is that it’s the shape with the minimum area that won’t fall through the hole.
Shapes having this property are generically called “shapes of constant diameter,” or other words to the same effect. Playing with a cart with wheels in one of these shapes is pretty fun - it rolls level but jerkily.
During WWII, when they were “rounding out” dents in submarines, they used to do it by rotating an arrangement that had two scaffolds fixed a constant width apart around the sub, and pounding out the dents so that the sub would have the same width when measured across any diameter. They reasoned that only a circle would have this characteristic, when in fact, many shapes can have it, and they ended up with some pretty weird looking cross-sections for the subs. 
Just to show that people don’t always learn, there’s also a reference to the same mistake in “What Do You Care What Other People Think?”, by Richard Feynman, when he’s talking to workers about getting solid rocket boosters that have gotten out of true to fit together.
On this evening’s news from Washington DC - something under the streets has been popping manhole covers - last week in Georgetown-up to thirty feet in the air - this week only 3’ in the more central areas on the city (just a few, no one hurt). One of the ones that weould have been a three foot pop was directly under a car. Mess.
I’m 99% sure they said the manhold covers weighed 80 lbs???
Are you driving with your eyes open or are you using The Force? - A. Foley
Manhole covers are round because manholes are round.
Jois, that’s probably true - 80 lbs. They aren’t so much lifted out of the hole, as they are pried out with a large pry tool and rotated to the side, then slid out of the way.
Manhole covers (the round ones) are iron 2 ft diameter and a half inch thick minimum (vague guestimates here so I may be wrong, going by my memory of seeing one as a kid). Big and heavy. But you’d want that, if it has to support the weight of a car driving over it without flexing.
Are you driving with your eyes open or are you using The Force? - A. Foley
they are round because this is the simplest way to make them. do you really think the guys who started making them would thing of “fatty triangles”?
In China manhole covers and gutter grids are chained to prevent them from being stolen
Quick question.
Concerning the Rouleaux shapes. Would this only work with shapes with an odd number of sides?
Or could I create a square that won’t fall in by drawing four circles centered half-way between the points?
Yes, a polygon-based Rouleaux shape should start with a polygon with an odd number of sides. The square you describe would be fatter from top to bottom than diagonally. Basically, a shape has constant width when every set of two parallel tangents on either side of the shape are exactly the same width apart. In this formulation, “tangent” is taken to mean a line that intersects the shape at exactly one point (so cusps have multiple “tangent” lines).
ENugent wrote:
Are you sure it would roll ok? The diameter is constant, but not the radius. The distance from center to a side is shorter than from center to a corner, and the wheel would tend to a position where one of the three cornes was pointing up, and you’d use some effort to roll it to raise the radius high enough to allow a corner to point down.
Jerky indeed! Where have you seen such carts?
I’m pretty sure it was at the Exploratorium in San Francisco. And yes, I misdescribed it - the cart goes up and down, but the wheels still run “smoothly” - you just need a little momentum to get it going. The one that stays level is when you have a Rouleaux triangle gear, rolling along a fixed level toothed track, with a movable level toothed track on top. The top track “wobbles” a little unless you have two wheels.
That exhibit at the Exploratorium is on a narrow table with a low partition at the rear. On the table there are rollers consisting of pairs of irregular equal-radius figures (not Rouleaux triangles) connected by axles. You place a flat plate on two or more of these rollers. You then move the plate so that it rolls on the rollers and observe the plate stays level with a line drawn on the partition at all times.
You wouldn’t want to build a cart with these shapes because the axle moves up and down irregularly. But you could with Rouleax polygons (that is, if I’m visualizing currectly).
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