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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
up vote 1 down vote accepted

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

Hint : That depends on ratio of fly size in points to the number same sized points square is made of.

Further more, with finite velocity it will take an unountably infinite amount of time. With infinite speed it will still take an infinite amount of time. Weather that is uncountable infinite amount of time is the question.

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The fly is just a point – user59761 Jan 31 '13 at 11:23
@T97778 : how many fly points is the square? – Arjang Jan 31 '13 at 11:26
A normal filled open square in R^2, uncountable – user59761 Jan 31 '13 at 11:26
Too subtle for me. An uncountable infinity of points in the square, but also in any curve traced out by the fly. This needs more than cardinality. – Gerry Myerson Jan 31 '13 at 11:27
@T97778 : in that case it will take not a countably infinite amount of time , but an uncountably infinite amount of time for the fly to cover the square. That is even if the fly was moving at an infinite speed still it will not do it in finite amount of time. – Arjang Jan 31 '13 at 11:31

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