Why are most manhole covers round? Sure, it makes them easy to roll and slide into place in any alignment but there’s another more compelling reason involving a peculiar geometric property of circles and other shapes. Imagine a square separating two parallel lines. As it rotates, the lines first push apart, then come back together. But try this with a circle and the lines stay exactly the same distance apart, the diameter of the circle. This makes the circle unlike the square, a mathematical shape called a curve of constant width.
Another shape with this property is the Reuleaux triangle. To create one, start with an equilateral triangle, then make one of the vertices the center of a circle that touches the other two. Draw two more circles in the same way, centered on the other two vertices, and there it is, in the space where they all overlap. Because Reuleaux triangles can rotate between parallel lines without changing their distance, they can work as wheels, provided a little creative engineering.
And if you rotate one while rolling its midpoint in a nearly circular path, its perimeter traces out a square with rounded corners, allowing triangular drill bits to carve out square holes. Any polygon with an odd number of sides can be used to generate a curve of constant width using the same method we applied earlier, though there are many others that aren’t made in this way. For example, if you roll any curve of constant width around another, you’ll make a third one. This collection of pointy curves fascinates mathematicians.
They’ve given us Barbier’s theorem, which says that the perimeter of any curve of constant width, not just a circle, equals pi times the diameter. Another theorem tells us that if you had a bunch of curves of constant width with the same width, they would all have the same perimeter, but the Reuleaux triangle would have the smallest area. The circle, which is effectively a Reuleaux polygon with an infinite number of sides, has the largest. In three dimensions, we can make surfaces of constant width, like the Reuleaux tetrahedron, formed by taking a tetrahedron, expanding a sphere from each vertex until it touches the opposite vertices, and throwing everything away except the region where they overlap.
Surfaces of constant width maintain a constant distance between two parallel planes. So you could throw a bunch of Reuleaux tetrahedra on the floor, and slide a board across them as smoothly as if they were marbles. Now back to manhole covers.
A square manhole cover’s short edge could line up with the wider part of the hole and fall right in. But a curve of constant width won’t fall in any orientation. Usually they’re circular, but keep your eyes open, and you just might come across a Reuleaux triangle manhole.