Kakeya Needle problem video

Question

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid:

Needle turning through hypocycloid

What I'm searching for is a visualisation of the same process but on a Besicovitch set like:

Besicovitch set

because I can't quite imagine it.

That doesn't seem likely to be visualizable—I think the needle would be jumping back and forth infinitely many times. — dfeuer, Jul 27 '13 at 16:44
@dfeuer: Not infinitely many times. For any $\epsilon > 0$, we can construct such a set with area less than $\epsilon$. Any such construction is finite and well-behaved, but the smaller the value of $\epsilon$, the more spiky it gets. — TonyK, Jul 29 '13 at 17:26
A bit late, but you can find some animation in this Numberphile video, slightly before 12:00, and in this Mathologer video — Del, Jun 27 '18 at 09:37

score 1 · Answer 1 · 2015-04-22T14:59:17.730

1

I understand the perron tree construction of a Kakeya set but I do not fully understand the visualisation of the second diagram (but I have seen it before).

Here is a link to a lecture by Timothy Gowers that may help from the talk he gave on the importance of mathematics:

https://www.youtube.com/watch?v=gjA8S82Q810

Something that myself & one of my professors are looking at is trying to get our hands on films of lectures that Abram Besicovitch gave on this problem. These may shed some light on this difficulty so I can post other stuff here if we get anywhere.

edited Apr 22 '15 at 14:59

answered Apr 22 '15 at 12:16

That'd be lovely! – Marcus Johnson Apr 23 '15 at 05:39
The second diagram may just be a fancy visualisation of the Perron tree construction. – Apr 25 '15 at 09:21

Kakeya Needle problem video

1 Answers1