Why are manhole covers round? - Marc Chamberland

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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. Marc Chamberland explains curves of constant width and Barbier’s theorem.

Lesson by Marc Chamberland, animation by Pew36 Animation Studios.

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.