Something fun for a change…Ant on a rubber band.

Does someone with an open mind want to try to tackle this? Its about a theoretical ant on a rubber string. If the ant is crawling on the string, but the string is expanding, can the ant ever reach the end. In the video they claim it can, because the ant’s progress gets stretched towards the end, at the same time the string is stretched. But this is a bizarre interpretation in my opinion. This is only true if the string is being stretched in only one direction, that is, the direction that it is approaching. So the string is secured at one end, and streched at the other end, so indeed it keeps getting closer to the end.

But if neither end is secured, and it is stretched, there is no reason for the ant to be stretched in the direction of its progress, thus it would never go anywhere. But neither the narrator, nor anyone in the comments seems to have a problem with this. Why?

And is this in any way analagous to spacetime stretching?

196 thoughts on “Something fun for a change…Ant on a rubber band.

  1. Just a general question, anyone can answer.

    Make the rope an inextensible wire. However long the wire and however slowly the ant moves, allow enough time and the ant can cover any finite distance. Now slow down the progress of the ant at a regular rate (start with 1 cm per second and, say, halving the velocity of the ant wrt the wire every second), given enough time, there’s again no limit to how far the ant can go.

    Is that at all analogous to “ant on a rubber rope”?

  2. Alan Fox: The key fact is that the ant moves together with the points of the rope when the rope is being stretched.

    When its stretched both directions?

    Jesus, I can see Omagain going along with this stupidity, but holy shit.

    What do you think happens when you stretch an equator Alan, where does the ant go? Thank you Alan, thank you so much for this entertainment-its been exhilarating. I don’t even have to insult you and Jock, you guys are doing it to yourselves.

  3. Or!

    Billy’s stride is proportional to the length of her legs. What happens if instead of stretching the rope we keep the rope fixed and we shrink Billy and her progress is constant in the sense of paces taken? Again, so long as Billy shrinks harmonically and thus never becomes zero length, she can cover any finite distance given enough time.

  4. phoodoo: When its stretched both directions?

    You have yet to explain your problem of a relative distance between two points changing being different when you change the reference frame. Unless…

    Is that the joke?

  5. Alan Fox: halving the velocity of the ant wrt the wire every second

    Different problem, different answer.
    You are now describing a geometric series, viz.
    a + ar + ar2… where -1 < r < 1
    they have a finite sum S = a/(1-r)
    so 1 + 0.5 + 0.25… etc sums to 2.000
    The whole thing that confuses the heck out of people is the fact that the sum of any harmonic series
    a/x + a/(x+1) + a/(x+2)… is infinite..

  6. I am curious as to why phoodoo included x0(0) in his equation 3:
    x(t)=x_0(t)\cdot \lambda(t)+x_0(0)=\frac{v_aL_0}{v}\ln{\left[1+\frac{vt}{L_0}\right]}\cdot \left(1+\frac{vt}{L_0}\right)+x_0(0)
    it’s almost like he is treating it like it is a co-ordinate in space. It isn’t.
    If only he had divided both sides of equation 1 by L0
    x_0(t)=v_a\int_0^t {\frac{dt'}{\lambda(t')}}=\frac{v_aL_0}{v}\ln{\left[1+\frac{vt}{L_0}\right]}
    oh well.

  7. DNA_Jock: The whole thing that confuses the heck out of people is the fact that the sum of any harmonic series
    a/x + a/(x+1) + a/(x+2)… is infinite..

    Yes, exactly.

    This thread was posted under false pretenses. The title suggested that it would be fun. But there’s no fun in watching someone trying to dominate the thread with nonsense.

  8. Looking at equation 1

    x_0(t)=v_a\int_0^t {\frac{dt'}{\lambda(t')}}=\frac{v_aL_0}{v}\ln{\left[1+\frac{vt}{L_0}\right]}

    So this equation has some variables that need defining. Is \lambda(t)=1+\frac{vt}{L_0}? Does it represent the stretch factor?

  9. What is s in the second equation? Distance between the ends of the rope at time (t) Indeed what is the second equation telling us other than velocity is the differential of distance?

    v(t)=\frac{dx_0}{dt}=\frac{dx_0}{dx}\cdot \frac{dx}{dt}=\frac{1}{1+\frac{1}{\text{s}}\cdot t}\cdot 1\,\frac{\text{cm}}{\text{s}}\enspace.

    ETA: nope, x is the distance metric. All it is saying is v is the differential of x which is 1\text{cm/sec}

  10. Alan Fox,

    Yes.
    So x0(t) is a measure (in units of length) of how far the ant has traveled along an unstretched rope — immediately a confusing metric — think of it as how far from the starting end the ant would be if the rope were instantaneously returned to its original length.
    A FAR more useful metric would be to divide this metric by L0 to yield the proportion of the rope that the ant has traversed. Which we covered a while back.
    The relationship between the sum of a harmonic series and the log function is well covered ground.

  11. Alan Fox,

    The second equation may not go anywhere. It is calculating the ant’s velocity along that “returned to its original length” rope. It’s always positive, so both equations 1 and 2 confirm the ‘Billy will get there’ result.
    ‘s’ appears to be the original length divided by the rate of stretch, but phoodoo introduced an ant speed of 1 cm/s, so he is mixing the straight algebra with some plugged in parameters; that’s a particularly good way to confuse yourself, especially if one of your parameters is set to 1.
    Yikes.

  12. phoodoo: Jesus, I can see Omagain going along with this stupidity, but holy shit.

    When people disagree they can either work towards a mutual understanding, or not.

    The topic of the disagreement are not important. What is important is working towards a shared understanding of who is right about what and who is not, and agreeing on that.

    DNA_Jock and Alan have just had such a conversation. And shared understanding and clarity have improved.

    You, on the other hand, phoodoo are simply unable to do that. You are simply unpleasant in demeanor, in all ways in fact.

    You are also, it seems, unaware of many Fairy Tales.

    Little boys who act like asses don’t get to play nice after years of poking the bear. You don’t get to have a “fun” thread, you are simply not capable of it.

    phoodoo: I was going to turn over a new leaf and not be insulting here anymore, but I think I might still make an exception in your case. Because the “math” guy still has no concept of reality.

    phoodoo: Go get your rubber band, try it, and you will maybe increase your brain by 50%. That still won’t be enough, but….

    From your forth comment on this thread.

    You can’t play nice. It’s not natural for you. Nobody said anything insulting to you to justify that.

    You are a lot like Trump in that regard I think. Somebody disagreeing with Trump not only makes them wrong it makes then an idiot. That seems to be how you think.

    Nobody can disagree with you about anything without making you angry it seems.

  13. phoodoo,

    But one good thing has come of it, now you too are skeptical of Wikipedia. Congratulations-you might become a real skeptic just yet!

    I haven’t said one word in this thread about Wikipedia, pro or con, so I don’t know what you are on about.

    There is a stretching threshold beyond which the ant cannot get to the end, if you are correct. Is it too much to ask that you calculate it? You don’t have to ‘educate’ me, just verify your contention with the aid of the maths you yourself posted. Or bluff, that’s an option too.

  14. Interesting illustration of the ‘intuitive’ approach here

    Of course he’s ‘stretching’ the rope from one end. But it makes absolutely zero difference if it loops back round off-screen, as it were, to conform to the Rubber Band/Balloon Variation.

  15. Allan Miller,

    Is it too much to ask for you to actually watch the video you are posting here, and watch what is said at exactly 2 minutes in? And then try to figure out what that means?

    So you don’t understand what he is saying at 2 minutes, Jock doesn’t understand, Alan doesn’t, Cornell doesn’t, Omagain (obviously ) doesn’t, now Neil doesn’t, Graham doesn’t (actually Graham probably does and that is why he stopped posting), Flint doesn’t….

    As if that concpet is so hard to understand, holf cow.

    Jock can’t even tell which that math is representing (and he has a PHD..hoho).

    So Allan you have now provided the THIRD source that shows I am right-the orginal video expalins why the end must remain stationary, the Wikipedia article explains the end must remain stationary, and now this video explains it and you all STILL don’t believe it.

    Oh, but, but, but…Jock knows math, so…..

  16. When everyone disagrees with me, especially if they are known to be smart, educated, capable people, I don’t usually conclude that everyone has been struck stupid except me. Chances are I’m either misinformed or not fully understanding what I’m saying.

    Conversely, if everyone except me IS wrong, I’d be powerfully motivated to find the clearest and most rigorous argument for my position possible, since I’d possibly be making some sort of breakthrough. I certainly wouldn’t support my unique position by telling everyone else to go watch a video!

  17. Do explain, phoodoo: why did you include x0(0) in equation 3. Does it, in your view, represent the co-ordinates of the starting point? If not, why is it included? What does it represent?
    You keep claiming that I do not understand the math, but you seem strangely reluctant to demonstrate that I am wrong.
    Have you so little confidence in your abilities? Coz that’s how you are coming across. Like you know you are wrong.

  18. phoodoo,

    yadda! YADDA!!!

    So at least you agree that if one end remains ‘stationary’ (in a particular frame of reference), the ant will get to the end regardless how fast the free end is stretched?

    This is progress. It actually seems intuitively even less likely (even though it is in fact the same) when all the extra distance is seemingly piling up in its direction of travel! And it would seem that we’d need different mathematics to describe the two situations. In one, there is a limiting case, in the other, there is not.

  19. phoodoo,

    Is it too much to ask for you to actually watch the video you are posting here, and watch what is said at exactly 2 minutes in? And then try to figure out what that means?

    At 2 min, he says “when the rope is being stretched”. I think we are all agreed that the rope is being stretched. I presume, as above, that your issue is that it is stretched ‘at one end’. But I already referenced that issue in the post you responded to. I was aware of the issue, and mentioned it myself. Is it too much to ask that you read comments before responding?

  20. Flint:
    When everyone disagrees with me, especially if they are known to be smart, educated, capable people, I don’t usually conclude that everyone has been struck stupid except me. Chances are I’m either misinformed or not fully understanding what I’m saying.

    Conversely, if everyone except me IS wrong, I’d be powerfully motivated to find the clearest and most rigorous argument for my position possible, since I’d possibly be making some sort of breakthrough. I certainly wouldn’t support my unique position by telling everyone else to go watch a video!

    This resonates with me. I was the phoodoo on a board where I offered some musings on the evolution of sex. The reaction was uniformly, over-the-top hostile. “If everybody thinks you’re wrong, maybe you’re wrong”. I quietly withdrew, and spent several years on it, because I remained unconvinced by the consensus view. The two ‘evolution of sex’ threads here are part of my effort to clarify my position – which remains that of a crank! They’ll see, though. There will be show trials where my opponents are forced to publicly recant. 😁

  21. DNA_Jock: You keep claiming that I do not understand the math, but you seem strangely reluctant to demonstrate that I am wrong.

    This is a pattern that phoodoo constantly holds to.

    He claims that people are in camps in china because they have conspired to commit terrorist harm, but seems unable to demonstrate the truth of that via e.g. court cases or other such evidence that might be available to any interested citizen.

    He claims that Uri Geller is a real PSI.
    He claims that the FBI uses PSI.
    He claims that he makes decisions via free will in a way that materialistic robots cannot.

    The difference between the above and what is under discussion here is that we can objectively demonstrate via mathematics who is and who is not correct. But, like all of his ilk, phoodoo refuses to make.a statement he can be held to, that can be shown to be wrong objectively.

    Baby does not take well, like his idol Trump, to being corrected.

  22. Allan Miller: At 2 min, he says “when the rope is being stretched”.

    Why didn’t you finish the sentence? The sentence starts at 1:55 and goes until after the 2 minute mark. Are you trying to stretch the truth? I mean you could have just claimed he said “when”. Why are you being disingenuous?

    The problem with the math-heads here is they don’t seem to understand physics. Do you understand what stretching entails? Do you know why rubber stretches? What is actually happening, are the molecules inside the material just becoming further apart? Is that what you think stretching is? Because that is not what stretching is. Stretching is actually uncoiling, the molecules are basically wound coils, that unfurl. So why does that matter? Good question Allan, I am glad you asked!

    Because what stretching does is NOT make points just become further apart, as if in between those points there is some void. No, no, that is not what stretching is. Stretching is ACTUALLY adding more points! Because we must fill in the space where those points existed. Otherwise we would just jump from point to point, and not have to count the void. But we must count the void! And the void is NOT unidirectional..Its is bi-directional if we stretch from both ends! The void is being filled on all sides. So this so called point, that you all claim the ant gets to stay on, that’s a myth. The point isn’t travelling east or west, and giving the ant a ride. The point is being filled in by new points, and more, and more points.

    So to calculate all these new points, you don’t get to let the ant “skip” all the new points, and just get a free ride. He doesn’t go anywhere during the stretch, some points are being filled in to the right of him, and to the left of him….etc…

    So here is where the really interesting philosophical point comes in in my opinion. Let’s say you are going to teach this little paradox to a class of elementary school kids. And one of the kids in the class understands a little bit about physics-or let’s just say they think a little. And when you tell that kid that

    “the ant is crawling on the rope, so it moves together WITH the points of the ROPE when the rope is being stretched. And what this means is that once the ant completes a segment, it will never have to complete it again, even if it stops moving for a while.”

    , then that kid might say, well, wait a minute, stretching isn’t points moving, stretching is new points being added. Why the fuck are you lying to me? When the universe is stretching, are points just moving further apart, and a void is being created, is that what is happening to the universe, or is it in fact creating new points everywhere, thus there is no ONE direction?

    And the kid might be super bright, and he also might play basketball. And he might say when I inflate my basketball the hole where I am inflating my basketball doesn’t start stretching sideways. The name in the middle of my basketball doesn’t suddenly become the bottom of my basketball if I blow it up more. The middle stays in the middle, the top stays on top, the bottom stays on bottom. And if I keep blowing it up, the ant is going to find it harder and harder to get to the bottom of the ball from the top. So teacher, you are full of shit. Mr. Jock, why are you lying to me, don’t you understand physics? If you didn’t understand physics, why did you make up a physics problem to solve a silly math one? Couldn’t you have just said none of this is true, but follow along anyway? Then we will make up math that is also not true, so you don’t have to ever believe math is real. Because what I know as a dumb elemenatry school kid is that is you keep blowing up that basketball, that ant is fucked, he is never, ever, ever going to get home! Please show me the math for that.

  23. phoodoo: The problem with the math-heads here is they don’t seem to understand physics.

    It’s a math problem; it is not a physics problem.

    If you want to look at it in terms of physics, then the elastic breaks and the whole thing becomes absurd.

  24. Neil Rickert: It’s a math problem; it is not a physics problem.

    And it precisely why the paradox requires stretching in only one direction, with one end fixed, and pretending the ant is “stretched” in a direction-exactly as I have said.

  25. phoodoo,

    Why didn’t you finish the sentence? The sentence starts at 1:55 and goes until after the 2 minute mark. Are you trying to stretch the truth?

    No, I’m just assuming that when you said ‘exactly 2 minutes’ you meant ‘exactly 2 minutes’. The subsequent sentences contradict you, and explain exactly why the ant is always reducing the percentage left to travel, so I didn’t think you meant them.

    It would have been a fucking sight more straightforward to have just said the words you think are relevant, rather than dressing it up in your characteristic pissiness.

    Do you accept that the ant will get to the end of the rope if only one end is stretched?

  26. phoodoo:And if I keep blowing it up, the ant is going to find it harder and harder to get to the bottom of the ball from the top.

    Until, presumably, he finds it impossible. What stretch rate ensures that? Given that he does find it possible at lower stretch rates?

    what I know as a dumb elemenatry school kid is that is you keep blowing up that basketball, that ant is fucked, he is never, ever, ever going to get home! Please show me the math for that.

    You’ve been shown the maths for that. Or, at least, the maths does not show that; it shows the ant will make progress at all (constant) rates of stretch. If you are right there must be a threshold rate of stretch at which the ant is ‘fucked’, because it certainly isn’t fucked when you blow slowly. And it’s more than the ant’s rate of progress, because expanding at 1cm/s won’t put a whole cm in front of an ant moving at 1cm/s. So what is it?

  27. Allan Miller: Do you accept that the ant will get to the end of the rope if only one end is stretched?

    AND if you pretend that stretching is moving points and not adding new ones you mean? Isn’t this what I have said from the very beginning?

  28. Allan Miller,

    Oh isn’t it just trivial Alan?

    The ant won’t get there at some rates, don’t you agree? Of course you do. But then it will require you to swallow your pride (and to stab Jock in the back) so you would NEVER do that.

  29. phoodoo: AND if you pretend that stretching is moving points and not adding new ones you mean? Isn’t this what I have said from the very beginning?

    Stretching is moving points. You’re happy with stretch rates of a km/s and ants that outlive the universe by billions of orders of magnitude but the stretching‘s a bit unrealistic? 🤣

    OK, let’s add this new condition:
    1) stretching is moving points, rope tethered one end.
    2) stretching is adding points, rope tethered one end.
    3) stretching is moving points, rope stretched both ends.
    4) stretching is adding points, rope stretched both ends.

    In which of these scenarios will the ant get to the end at all rates of stretch?

    Oh isn’t it just trivial Alan?

    It is rather. Certainly not worth getting yourself worked up over.

    The ant won’t get there at some rates, don’t you agree? Of course you do.

    No, I don’t. I’ll thank you not to misrepresent my position. It is you who thinks that; I’m asking you what threshold separates those rates from the rates at which we both agree it will succeed.

    But then it will require you to swallow your pride (and to stab Jock in the back) so you would NEVER do that.

    You sound like my wife.

  30. Imagine phoodoo like this at work. I’d have fired him on the first day.

  31. Allan Miller: In which of these scenarios will the ant get to the end at all rates of stretch?

    phoodoo does not do such questions. We know this.

  32. phoodoo: Because what I know as a dumb elemenatry school kid is that is you keep blowing up that basketball, that ant is fucked, he is never, ever, ever going to get home! Please show me the math for that.

    Well, you did ask nicely, so here goes:
    We are going to use the harmonic series this time, for fun.
    Let’s suppose that in the first minute, the ant manages to get one tenth of the way from the bottom to the top, covering 18 degrees. BUT, the basketball is now 10% larger, so in the next minute, he can only cover 1/11th of the distance from pole to pole (i.e. 180/11 degrees). In the third minute, he can only manage 1/12th of the distance (15 degrees). and so on. (Remember, linear expansion — the basketball’s circumference increases 1, 1.1, 1.2, 1.3 etc)
    He’s going to reach the top if the sum of
    1/10 + 1/11 + 1/12 + 1/13 etc… gets above one.
    Which of course it does after 17 minutes. It took 7 minutes longer than the non-expanding case, and the basket ball is 2.7 times larger than it started at.
    One might say, “Gee Jock, you made it too easy on the ant: instead of 10% larger, make it 50% larger each minute” (1, 1.5, 2, 2.5 etc)
    18 degrees in the first minute, then 12, then 9…
    He’s going to reach the top if the sum of
    1/10 + 1/15 + 1/20 + 1/25 etc… gets above one.
    He’s never getting there, right?
    Wrong.
    It’ll take 226 minutes, and the ball will be over a hundred times larger (in its linear dimensions… its volume is up 1.48 million-fold ) than when we started, but he’s gonna get there.
    So the question becomes, phoodoo, for what values does Billy never get there?
    Do tell.

  33. phoodoo: So teacher, you are full of shit. Mr. Jock, why are you lying to me, don’t you understand physics?

    that aged badly.

  34. DNA_Jock:
    One might say, “Gee Jock, you made it too easy on the ant: instead of 10% larger, make it 50% larger each minute” (1, 1.5, 2, 2.5 etc)

    Uh, maybe I’m slow, but if it becomes 50% larger each minute (than it was the previous minute) this is exponential expansion, not linear. A 50% per minute expansion would be 1, 1.5, 2.25, 3.375, 5.0625, etc. And I don’t think the ant would ever reach the top.

  35. Flint: Uh, maybe I’m slow, but if it becomes 50% larger each minute (than it was the previous minute) this is exponential expansion, not linear. A 50% per minute expansion would be 1, 1.5, 2.25, 3.375, 5.0625, etc. And I don’t think the ant would ever reach the top.

    Quite. That’s why, to avoid ambiguity, I gave the first four terms of the series. I suspect that imagining an exponential (or at least super-linear) expansion is one of phoodoo’s errors.

  36. “the ant is crawling on the rope, so it moves together WITH the points of the ROPE when the rope is being stretched. And what this means is that once the ant completes a segment, it will never have to complete it again, even if it stops moving for a while.”

    Jock doesn’t know what this means.

  37. graham2:
    Jeez guys … still at it ?I think phoodoo is just winding us up.

    Joining the mind-reading club for a moment, it seems a plausible explanation for phoodoo’s output.

    On the other hand, maybe phoodoo has yet to play his trump card and tell us why he is right and everyone else is wrong.

  38. phoodoo: This is for Corneel, who thinks the math will make it easier to understand-and was certain I was wrong because a “math” person told him so.

    Thanks, but I do not believe the math makes it easier to understand for everyone, nor did I conclude you were wrong because someone else told me. I tend to decide those things for myself.

    phoodoo: I have nothing against math, but claiming you are right because you think the math says so is often a dubious claim. Math is supposed to model reality, not reality is supposed to model the math.

    And therefore I have to unquestionably swallow what you are telling me? I have nothing against words, but claiming you are right because you say so is a dubious claim. Words are supposed to represent reality, not the other way around.

  39. graham2: I think phoodoo is just winding us up.

    I hope so. Otherwise I fear that at some point he will start replying with just “haha”.

  40. So I grabbed the rope and twanged it, sending the ant sailing off into space. Its movements will trouble us no more..

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