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 in order to roll smoothly is to be the same

width from bottom to top/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 - since they're not

a constant distance from the axle, 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 shapes 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?