Terence Tao

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?

This is in the two-dimensional plane.

So we understand everything in two dimensions.

So the next question is what happens in three dimensions.

So suppose the Hubble Space Telescope is a tube in space, and you want to observe every single star in the universe.

So you want to rotate the telescope to reach every single direction.

And his unrealistic part, suppose that space is at a premium, which it totally is not.

You want to occupy as little volume as possible in order to rotate your needle around in order to see every single star in the sky.

How small a volume do you need to do that?

And so you can modify it based on Copic's construction.

And so if your telescope has zero thickness, then you can use as little volume as you need.

That's a simple modification of the two-dimensional construction.

But the question is that if your telescope is not zero thickness, but just very, very thin, some thickness delta, what is the minimum volume needed to be able to see every single direction as a function of delta?

So as delta gets smaller, as your needle gets thinner, the volume should go down, but how fast does it go down?