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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.

  • 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 –  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
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    Finite but unbounded velocity and finite time may result in infinite length, yes. –  Jan 31 '13 at 11:34
  • No matter how big is the velocity at any moment of time and no matter does the velocity changes continuously or not, because of the finite time request the answer is no. – A.P. Jan 31 '13 at 11:36
  • @A.P. Then you are wrong –  Jan 31 '13 at 11:37
  • Hardly, you are practically claiming that infinity can be constructed by finite number of "finities". It is very hard to believe that. – A.P. Jan 31 '13 at 11:39
  • @A.P. No, infinitely many finites, for instance walk n meters forward at time 1-2^-n –  Jan 31 '13 at 11:49
  • @T97778 : you are correct about infinite length, but not about finite area. – jimjim Jan 31 '13 at 11:55
  • I understand what you mean, the unbounded velocity may result in infinite path in finite time, I agree because integral of the velocity dependent on time is the path traveled and there are such functions that give infinite value of the integral. So my previous comments are not in the spirit of this comment, but to prove the above problem there must be additional information on the velocity function, it is very important is the velocity function differentiable, so try to edit your question by adding additional information. – A.P. Jan 31 '13 at 11:56
  • So essentially, you're asking if there exists a space-filling curve with defined speed at every point in time? (I changed "velocity" to "speed".) – Tunococ Jan 31 '13 at 12:00
  • Yea, it seems like it was not my best idea to phrase it in such colloqial language. –  Jan 31 '13 at 12:04
  • It may have to go back to how you define speed. I'm under the impression that a function that has properties similar to Volterra's function may be the only way to achieve the desired trajectory if speed is defined conventionally. However, you might want to blow it up a little. (Volterra's function has a bounded derivative, but I think you don't want that.) – Tunococ Jan 31 '13 at 12:17

2 Answers2

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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.

Community
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achille hui
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  • Right but it can have an infinite long path. –  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 " –  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
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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.

jimjim
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  • The fly is just a point –  Jan 31 '13 at 11:23
  • @T97778 : how many fly points is the square? – jimjim Jan 31 '13 at 11:26
  • A normal filled open square in R^2, uncountable –  Jan 31 '13 at 11:26
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    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. – jimjim Jan 31 '13 at 11:31
  • This needs a proof that a curve that fills all of the square must have an infinite length and than the answer is no. – A.P. Jan 31 '13 at 11:31
  • Guesses and speculations. Where is the math? there is none... –  Jan 31 '13 at 11:39
  • @GerryMyerson : a unit circle is made of uncountably unit lengths. – jimjim Jan 31 '13 at 11:41
  • @T97778 : if are you saying cardinality of unit length respective to unit square is not a mathematics then according to you there is no math involved. – jimjim Jan 31 '13 at 11:45