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Comment by jan_Inkepa | Hacker News Reader

"Consider a pencil lying on your desk. Try to spin it around so that it points once in every direction, but make sure it sweeps over as little of the desk’s surface as possible."

I'm really stuck at the start here - moving a pen so that it pointing in all directions is basically impossible - the space of directions is two-dimensional and you can only trace out a one-dimensional curve (or pair of curves).

Ok, wikipedia makes it clearer:

"In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction."

Quanta writers are generally very good at explaining things, but wikipedia wins hands down in this case...

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It’s not pointing in all directions at once, it’s pointing once in each direction. So if you spin the pencil so it does exactly one complete rotation that works, doesn’t it?

Yes, and spinning the pencil on its centre like that shows that the circle (of pencil length diameter) is such a set. (I think you're thinking about it the wrong way around: it's which containing shapes allow this, not how can it be done at all.)

I think OP is thinking about covering the sphere of directions in 3D space, not just directions in a 2D plane. No matter how hard you spin the pencil, you're drawing a one-dimensional curve that has no area, so any finite amount you draw will cover zero percent of the area of the two-dimensional sphere surface.

> No matter how hard you spin the pencil, you're drawing a one-dimensional curve that has no area

The object doesn’t matter, using pencil as the example was what threw you off - it’s not about what the pencil “draws”. Consider a thin cylinder, or rectangular prism, or just a stick - if you spin it around, its endpoints trace out a circle whose diameter is the length of the stick. You can move and spin such an object in another way where the shape traced out by its endpoints has smaller area than that circle.

in 2d, but not in 3d though (like in the video on top of the article)!

The Wikipedia entry [1] has some good illustrations for the 2-dimensional case.

[1] https://en.wikipedia.org/wiki/Kakeya_set

> Ok, wikipedia makes it clearer:

> "In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction."

Is that definition correct/complete? It leaves open the option that such a set isn’t connected. I think there’s an additional requirement that, for any two directions D and E, you can move a line segment oriented in direction D so that it’s oriented in direction E without any point on it ever leaving the set.

The minimal set is trivially connected. Though showing any two unit line segments have a homology between them is trickier.

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