It makes intuitive sense that objects like circles and spheres are round

- but what is it about a sphere that makes it round?

If you're talking about how effective a shape is at enclosing a large volume, you mean "sphericity".

And a sphere is the shape that contains the most volume with the least surface area,

that is, a sphere is the most "spherical" shape.

But roundness has more to do with rolling than with volume, right?

A round object can roll smoothly - like a wheel or ball bearing.

And the main feature that ball bearings need is to be the same width from one side to the other.

It turns out there are plenty of non-circular 'shapes of constant width' that are excellent bearings as well.

The Reuleaux rotor, for example,

is made up of segments from three circles intersecting at their centers -

so every point on a side is the same distance from the opposite corner,

and the rotor rolls around wonderfully.

In fact, the Reuleaux rotor can even turn smoothly in a square hole!

But don't try to use a Reuleaux rotor for the wheel on your car

those points will make for a bumpy ride!

In fact, the points on a Reuleaux rotor are just that - pointy.

Doesn't that go against the idea of roundness?

Geology has the answer:

stones with sharp or rough edges that are worn away become "rounded."

So we might say a Reuleaux rotor is round but not rounded.

On the other hand, certain British coins are round AND rounded,

though they're still not circles:

the 20 and 50-p coins are heptagons of constant width,

which means they look un-circularly cool but don't get stuck in coin machines!

And isn't if funny that "rouleau" means "roll" in French

and Franz Reuleaux invented a rolling rotor?