# Can a fly visit every point of the unit square

Can a fly (a point) visit every point of the open unit square in finite time? Its motion traces out an continuous curve and it has a finite velocity at every point in time.

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No. The path must have have an infinite length and with finite velocity and finite time you can only travel a finite length. –  A.P. Jan 31 '13 at 11:27
The Peano Curve actually fills the square in a continuous fashion, but the finite issue sounds unlikely to me. –  busman Jan 31 '13 at 11:29
@A.P. Not true, the velocity may approach infinity as the time comes to end –  user59761 Jan 31 '13 at 11:32
May approach it but do not have the infinite velocity. So no. –  A.P. Jan 31 '13 at 11:32
Finite but unbounded velocity and finite time may result in infinite length, yes. –  user59761 Jan 31 '13 at 11:34

If the fly is a point and have visited every point of an open square, then for any $N > 0$, it have visited the $N^2$ points $(\frac{i}{N+1}, \frac{j}{N+1})$ for $i, j = 1..N$. To move between any two points, the fly need to travel at least a distance $\frac{1}{N}$. So the total length of the path $\ge \frac{N^2-1}{N}$. Since $\frac{N^2-1}{N} \to \infty$ as $N \to \infty$, no finite path can cover the whole square.

This argument is not mine. I just rephrase another post Why isn't R2 a countable union of ranges of curves? I stumbled across a few days ago.

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Right but it can have an infinite long path. –  user59761 Jan 31 '13 at 11:47
If you allow unbounded velocity, then you need more advanced ideas to show it is impossible. Look at the post I cited in the answer, in particular those answers that uses Baire category theorem. –  achille hui Jan 31 '13 at 12:06
So you know its impossible? Doesnt it follow directly from the second answer there? "The image of a C1 curve has measure zero by Sard's lemma " –  user59761 Jan 31 '13 at 12:11
I know its impossible. You can either use Baire category theorem or Morse Sard's theorem to show that. In fact, I have answered a similar question using Morse Sard's theorem just a few days ago. Now I understand why no one but me answer that post.... –  achille hui Jan 31 '13 at 12:23