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

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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. phoodoo,

    If I’m wrong, I’m wrong.

    Imagine two nuclear powered microrockets operated by tiny, spherical, weightless but highly intelligent cows. Imagine the microrockets linked by an elastic rope. The cows attempt to stretch the rope by driving the microrockets away from each other. Does their location in space have any effect on the stretch of the rope, other than their distance apart?

  2. phoodoo: WTF???

    This is why it’s unrealistic to explain it in a way you can understand.

    Are you equipped to make such assessments prior to producing any explanation?

  4. OMagain: But phoodoo implys he’s more then a layman elsewhere. And here we have people offering to engage with phoodoo at a mathematical level but he prefers to brush them off with insults instead. Hence demonstrating he’d rather deal with it at the rhetorical level then engage with the modelling side.

    That’s typical phoodoo, but probably not typical ID supporter. The IDers that persevere here at TSZ may not be representative.

  5. Alan Fox,

    What in the heck are you talking about? And what does this have to do with anything I have written?

    I think you need to first provide proof that a single other person knows what you are talking about before it can be shown to have any meaning whatsoever.

  8. phoodoo: If you have any idea what he is saying, and how it applies to my comments, please do tell?

    I fear I will not be more succesful than the others, but here is my attempt: Did you watch “The Matrix”? Remember the part where that bald kid tells Neo “There is no spoon”? This is a bit similar. There is no ant, no rubber band, no dots, no one stretching stuff. That’s just window dressing.
    There is an expanding line segment and a point moving along it. That’s the puzzle. It’s inherently mathematical.

  10. Corneel,

    Because if you are suggesting that when Alan writes

    I doubt that and seems perverse of phoodoo to suggest it. Still, you could try explaining why an ideally elastic rope stretch is affected by anything other than the distance between the two ends.

    what he means is -There is no rope, no ants, no one stretching anything-just believe the math, I have to question your stability as well.

    But I will take up your issue.

    If one says 2 tires (t) 2 carburetors (c)= 4 cars(^), I can claim that the math is right all day long, but that doesn’t make it so.

  11. phoodoo: If one says 2 tires (t) 2 carburetors (c)= 4 cars(^), I can claim that the math is right all day long, but that doesn’t make it so.

    Tires, carburettors and cars are no part of mathematics. Again, you are taking issue with the application of mathematical theory to real life objects and events. This can be an important discussion but not in solving mathematical puzzles.

    A point that travels along a line segment that is expanding at a constant rate will reach the end. Period. Whether that can be usefully applied to ants traveling down rubber bands, or photons traversing spacetime or Evel Knievel driving down a rope stretched between two cow-steered microrockets is a different question.

  12. Corneel: A point that travels along a line segment that is expanding at a constant rate will reach the end. Period

    No, that’s not true. This is precisely why the paradox specifically adds in the caveat that one end is secured, and also that the point is “stretched” in the direction of the end it is moving towards (these are just made up qualifiers that has nothing to do with anything other than to make it a paradox). Why do you think the paradox qualifiers on wikipedia state that? For fun?

    You aren’t really paying attention.

    You can just say its trivial if you like. I will know what that means.

  13. Corneel,

    I think you let phoodoo off the hook. The mathematics provides a precise language to the analysis. If a person believes that the math is being mis-applied, they should be able to show how, with math.
    But when someone’s math disagrees with phoodoo’s intuition (and it often does…), all he can muster is hand-waving and insults, whether the topic is surveys of Veblen goods, M&M’s, GPS clocks, trains in tunnels or ants on rope.
    Here’s how it would look:
    Claim:

    2 tires (t) 2 carburetors (c)= 4 cars(^),

    Counter:

    Your dimensions are off.

    Also, it is hardly “sleight of hand” when I explained explicitly what I was doing here. Given that I was trying to explain the situation to someone who thinks the ant will never get there, I thought the ‘conservative’ approach was most generous.

  16. DNA_Jock: I think you let phoodoo off the hook. The mathematics provides a precise language to the analysis. If a person believes that the math is being mis-applied, they should be able to show how, with math.

    Mathematics is a highly formalised and quantitative language, so precise it is and I prefer it over verbal models (like you). But who can decide whether some model is appropriate for a given scenario using only math? Ants are not dimensionless points and rubber bands are not line segments. How did you decide that the model would suffice regardless? With math?

    DNA_Jock: But when someone’s math disagrees with phoodoo’s intuition (and it often does…), all he can muster is hand-waving and insults, whether the topic is surveys of Veblen goods, M&M’s, GPS clocks, trains in tunnels or ants on rope.

    Yeah, phoodoo is not the most polite debater here at TSZ. Yet he came up with the argument that it matters which end of the band was being stretched. It’s wrong for sure, but it demonstrates that phoodoo has been grappling with the facade of the puzzle. He missed what the puzzle was really about.

  17. Corneel: It’s wrong for sure,

    Oh..then what is with the wikipedia page and the narrator saying otherwise?

  18. Corneel,

    You know Allan likes fixing wikipedia pages. Maybe we should remind him to go correct that page.

    Wanna bet he won’t?

  19. phoodoo: Oh..then what is with the wikipedia page and the narrator saying otherwise?

    “You aren’t really paying attention.”

  20. phoodoo: What Jock is claiming is that no matter the expansion, as long as the ant keeps walking in one direction he can always make it back to the other end of the circle.

    Humm. I never made that claim, but phoodoo is (in this regard) correct: It is a logical entailment of the claim that the ant can get to the end for all values of L, S, and A.
    So, upsetting as this must be, it is in fact true that the ant can circumnavigate the rubber band and get back to his starting point. Whichever end is getting stretched.
    For the 20, 10, 5 youtube parameters:
    the outward journey takes 12.778 seconds, by which time the ends are 147.78 cm apart. Returning immediately takes Billy a further 94.418 seconds.
    Round trip in 107.2 seconds.
    But Billy’s 1cm/s cousin SlowPoke Rodriguez is in trouble: the outward journey takes 12.2 hours, and the return will take 30 years. He should bring snacks.

  23. Alan, how many times have I been insulted, in this very thread? To find out why Jock is wrong, I guess we have to go to the moderation thread.

  24. DNA_Jock: Humm. I never made that claim, but phoodoo is (in this regard) correct: It is a logical entailment of the claim that the ant can get to the end for all values of L, S, and A.
    So, upsetting as this must be, it is in fact true that the ant can circumnavigate the rubber band and get back to his starting point. Whichever end is getting stretched.
    For the 20, 10, 5 youtube parameters:
    the outward journey takes 12.778 seconds, by which time the ends are 147.78 cm apart. Returning immediately takes Billy a further 94.418 seconds.
    Round trip in 107.2 seconds.
    But Billy’s 1cm/s cousin SlowPoke Rodriguez is in trouble: the outward journey takes 12.2 hours, and the return will take 30 years. He should bring snacks.

    There is no outward journey. The rope has become an equater. Why would you call circling the globe an outward and inward journey? Its a circle that keeps expanding.

    Now you see why you are wrong? Of course you do.

  25. The reason Jock is wrong is being hidden by Alan. You have to go to moderation to see it.

  26. phoodoo: Its a circle that keeps expanding.

    What a great way of thinking about it! I wish I had thought of this before. Thank you phoodoo.
    Billy is marching around the equator of an ever-(linearly)-expanding globe. No matter how big the sphere, nor how fast it is expanding, Billy’s angular velocity remains positive, declining in a harmonic series, and he will therefore circumnavigate. [The more numerate readers will notice that this is identical, mathematically, with the “calculate the percentage” approach that we advocated at the very beginning, but phoodoo’s description here is a better way of visualizing this behavior.]
    Of course, this also entails that Billy will circumnavigate the globe as many times as you can imagine, which is gonna really upset phoodoo. Counter-intuitive results are not something he handles well.

    phoodoo: Now you see why you are wrong? Of course you do.

    But I really, honestly do not see why I am wrong. I am quite confident that I am correct. phoodoo is welcome to move past the random insults and try to demonstrate exactly how I am wrong. We all shiver with anticip…

  27. DNA_Jock,

    First off keep in mind there is a reason why the video started off using 20, 10 , and 5 units. Don’t be fooled by this. Because later in the video he goes on to say that the lengths don’t actually matter, he will still reach the end. And there is the rub. Only when you use their trickery.

    Now think about it some more and you will see what it won’t work for other measurements once we get rid of their trickery.

    You will never completethe circle if the expansion is faster than his walking. Because the expansion does nothing to shorten the distance left to travel. Without the trickery.

  28. Think of it this way. The time it takes for him to cover one percent will keep rising exponentially the larger the circle gets. Or to put it another way, with every second that passes, the more time he needs to cover one more percent. That time will keep increasing not decreasing.

  29. phoodoo:
    Think of it this way.The time it takes for him to cover one percent will keep rising exponentially the larger the circle gets. Or to put it another way,with every second that passes,the more time he needs to cover one more percent. That time will keep increasing not decreasing.

    By this logic, an ant would not get back to its starting point even if the expansion rate was a centimetre a year and the loop initially one metre long. Something’s not right here. You presumably argue that there is a (constant) rate of expansion that defeats the ant, then – a threshold below which it gets there and above which it doesn’t.

    So, for an ant moving at 1cm/second round a loop initially 1m long (or, likewise, the surface of a balloon 1m in circumference), what (constant) rate of expansion defeats it?

  30. phoodoo,

    I have some sympathy with phoodoo regarding how mathematical paradoxes are framed. Achilles and his race with a tortoise “proves” Achilles can’t win but it has no bearing on a real race.

  31. phoodoo,

    You will never complete the circle if the expansion is faster than his walking. Because the expansion does nothing to shorten the distance left to travel. Without the trickery.

    This can’t be right. Your answer to my ‘threshold’ question above would presumably be 1cm/second – an ant walking 1cm/second would never complete a circuit of a loop expanding at 1cm/second.

    OK, break it down. After 1 second, the ant has moved 1cm and the loop has expanded 1cm in total. But after walking 1cm, part of that 1cm expansion is ‘behind’ it. It doesn’t have an extra 1cm to go, but about 99% of that (on a 1m loop). Same for the next 1cm, and so on. It’ll take longer, but it must get there, because the amount of extra distance ahead of it is increasing by less per second than it can shift.

    So the answer must be that the threshold expansion rate, that which prevents it ever returning to Start, must be greater than its speed. So: how much greater?

  32. phoodoo: Think of it this way. The time it takes for him to cover one percent will keep rising exponentially linearly the larger the circle gets. Or to put it another way, with every second that passes, the more time he needs to cover one more percent. That time will keep increasing not decreasing.

    Otherwise, what you wrote is correct.
    Billy will get there eventually, whatever the values of L, S and A.
    You appear to believe that Billy will get there for some values of L, S, and A, but not for others, absent ‘trickery’. What is the threshold? S > 3A? S > 10A?
    Show your work.

  33. DNA_Jock,

    The problem is that you seem to think that the ant’s speed is also increasing, which it is not. This is the crux of the one end being stationary while the other end stretches matters. The point all along which you have been saying DOESN’T matter. Has the light gone off in your head yet?

    Values matter.

  34. Allan Miller,

    The question is not why I think he wouldn’t reach the end, the question is why Jock thinks he would. Because Jock doesn’t understand the basic difference between a fixed end and a non-fixed end. Just because someone thinks they know math, it doesn’t mean they understand the math. String theory isn’t correct just because someone says the math is correct.

  35. 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]}

    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.

    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)

    x'(t)=v_a+v_a\cdot\ln{\left[1+\frac{v}{L_0}\cdot t\right]}\enspace .

    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. 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.

    But I am just a dumb rice farmer.

  36. phoodoo: The problem is that you seem to think that the ant’s speed is also increasing, which it is not.

    The ant’s speed, relative to the section of rope he is currently on, does not change at all.
    But, the ant’s speed, as measured relative to the start of the rope, or the end of the rope, or the midway point of the rope, IS changing continuously.

    This is the crux of the one end being stationary while the other end stretches matters. The point all along which you have been saying DOESN’T matter. Has the light gone off in your head yet?

    Values matter.

    I agree that values matter. How about you walk us through the 20, 10, 5 case with values: describe, using numbers, the positions of each end of the rope and of the ant at t = 0, t= 1, etc., up to t=17
    If you do this for the ‘start stationary’ and the ‘target stationary’ versions of the puzzle, you will be able to show us how their ‘trickery’ affects the outcome.
    I am sure you must be champing at the bit to definitively prove me wrong.
    Just to be safe though, within any one scenario, use the same co-ordinate system to describe the start, target and ant locations. Otherwise you might confuse yourself.
    I think that describing all positions in terms of {distance from target} will highlight where you and I differ…

  37. phoodoo,

    The question is not why I think he wouldn’t reach the end, the question is why Jock thinks he would.

    Worth pursuing my train of thought though, which may help illustrate why Jock thinks he would. Jock (presumably) thinks there is no threshold of expansion relative to the ant’s speed that would see it fail, given enough time. You think there is, so as an exercise, try and find that threshold, for parameters of your choosing.

  38. Allan Miller,

    I showed you the math above. Have at it.

    You are a Jock apologist, so even though you know he is wrong, you are hesitant to call him out.

  39. phoodoo,

    But the math that you “showed above” has distance traveled increasing with ln(1+vt/L0), so Billy reaches his goal under all circumstances. You just proved yourself wrong…
    Awkward.

  40. DNA_Jock,

    That’s the math for it attached at one end. You understand nothing.

    DNA_Jock: The ant’s speed, relative to the section of rope he is currently on, does not change at all.

    This is where you still really don’t understand.

  42. phoodoo:
    Allan Miller,
    I showed you the math above.Have at it.

    I’m afraid I’m not very good at maths. Can you not do the sums for me?

    You are a Jock apologist, so even though you know he is wrong, you are hesitant to call him out.

    I think he’s right, though. I don’t think there is a ‘stretching threshold’ beyond which the ant cannot get there in any amount of time. You could prove me wrong by calculating what it is.

  43. Allan Miller,

    It appears I am now having to give you and Jock a free education. It feels I should start charging.

    Do you think the ant’s speed is measured by the stretching of the rubber band? Because that is what Jock is doing.

    I have pretty much gotten all I need out of this thread. At first I was unsure what I was missing. Now I can see from the responses that I wasn’t missing anything. I am completely correct. I didn’t really post here to teach you and Jock how to think. Alan was helpful in showing me a wikipedia post that showed me exactly why I was right. Even with the wikipedia post confirming it, you still don’t believe it, so I guess its that you don’t trust Wikipedia-which is understandable, since you apparently spend much time doctoring it.

    Beyond that your all’s posts have become more and more nonsensical. I have also seen other posts online elsewhere where there are other people who actually understand the math, instead of just scribbling numbers, or making up phony graphs, and they have confirmed exactly what I thought.

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

    Tell Jock I am sorry a poor rice farmer had to teach him.

  44. phoodoo: Alan was helpful in showing me a wikipedia post that showed me exactly why I was right.

    Well thank-you for the acknowledgement for my contribution that involved typing “ant on a rubber rope” into Google search, however…

    I seem not to be spotting how you get from posting the teaser to claiming “I was right”.

    I appear to be witnessing a variation on the South Park underpants gnomes syllogism:
    1. Phoodoo presents maths teaser
    2. ?
    3. Phoodoo claims “I was right”.

    You need more detail in step 2.

  45. Alan Fox: Well thank-you for the acknowledgement for my contribution that involved typing “ant on a rubber rope” into Google search, however…

    However, you could have then read the aforementioned page.

    If you were so inclined to want to learn.

  46. phoodoo,

    The reason I searched was to find a clear exposition of the maths puzzle as the video presenter had a manner and accent too annoying for me to persevere. The essential point of disagreement appears to be that phoodoo thinks it matters whether and which end of the rubber rope is fixed. Nobody else contributing in the thread agrees and that the answer is the same in all cases. Reduce the puzzle to its bare bones – three points on an expanding straight line (or indeed two points on an expanding circle) – as Jock has done. Show your working.

  49. phoodoo:

    Show your working.

    I can’t do better than Jock’s three graphs showing the same result from three different frames of reference.

    here

  50. phoodoo:
    Alan Fox,

    Show your working.

    And looking at the intuitive explanation on the Wikipedia page:

    Regardless of the speed of the endpoint of the rope, we can always make marks on the rope so that the relative speed of any two adjacent marks is arbitrarily slow. If the rope initially is 1 km long and is stretched by 1 km per second, we can make marks that are initially 5 mm apart along the whole rope. The relative speed of any two marks is then 5 mm per second. It is obvious that an ant crawling at 1 cm per second always can get from one mark to the next, and then to the next again and so on, until it eventually reaches the end of the rope. The same reasoning works for any constant stretching speeds, ant speeds and rope lengths.

    The key fact is that the ant moves together with the points of the rope when the rope is being stretched. At any given point of time we can find the proportion of the distance from the starting-point to the target-point which the ant has covered. Even if the ant stops and the rope continues to be stretched, this proportion will not decrease and will in fact remain constant as the ant travels together with the point on the rope where the ant stopped (because the rope is stretched uniformly). Therefore, if the ant moves forward this proportion is only going to increase.

    makes sense to me in that the uniform stretching of the “rope” between its two ends is the essential fact and the absolute locations of those points in math space is irrelevant; only the relative distance between them.

    What do you disagree with, phoodoo?

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