Terence Tao

It's a problem I've worked on a lot in my early research.

Historically, it came from a little puzzle by the Japanese mathematician Soichi Kakeya in 1918 or so.

The puzzle is that you have a needle on the plane.

Think of it like driving on a road.

You want to execute a U-turn.

You want to turn the needle around.

but you want to do it in as little space as possible.

So you want to use this little area in order to turn it around.

But the needle is infinitely maneuverable.

So you can imagine just spinning it around its, as the unit needle, you can spin it around its center.

And I think that gives you a disc of area, I think pi over four.

Or you can do a three-point U-turn, which is what we teach people in the driving schools to do.

And that actually takes area pi over eight.

So it's a little bit more efficient than a rotation.

And so for a while, people thought that was the most efficient way to turn things around.

But Bezekovic showed that, in fact, you could actually turn the needle around using as little area as you wanted.

So 0.001, there was some really fancy multi back and forth U-turn thing that you could do.

that you could turn a needle around.

And in so doing, it would pass through every intermediate direction.

Is this in the two-dimensional plane?