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?