Talk:Ant on a rubber rope

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I got so carried away writing up a full differential equation solution (Ant on a rubber rope#An analytical solution) that I didn't notice that this is of course hugely overcomplicated -- a much simpler solution can be framed by considering the ant's progress proportionally. I shall replace the long-winded solution when I get time! Mooncow (talk) 02:27, 8 April 2008 (UTC)

that's now done, and a reference to the source for the simpler solution included. Mooncow (talk) 23:51, 19 April 2008 (UTC)

counter example?

I think it would help me understand this article better if an example was given of how the parameters could be changed such that the ant does not ever reach the end of the rope. If I understand it properly, if the length of the rope kept doubling (which would imply exponential expansion), then the ant would never reach the end. I think adding this might make the solution easier to understand for non-math people (like myself). 66.66.149.221 (talk) 04:48, 16 April 2012 (UTC)

Yes that's right, or at least it is if the expansion is fast enough and/or the rope is sufficiently long, but I can't think how to justify it without referring to infinite series. For the ant on the steadily growing rope, the ant moves (say), 1/10 of the rope in the first second, 1/20 in the next, 1/30 in the third etc etc, so the series you add up is (one tenth of) 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... which (perhaps surprisingly) grows without limit, so the ant can get to the end of the rope however long. For a rope which doubles every second the equivalent sum you have to do is 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... which converges (which also surprises some people, see Zeno's paradox), and so the ant can't get to the end of the rope if it is long enough. — Preceding unsigned comment added by 92.27.55.215 (talk) 12:39, 26 April 2012

Wow. The above (not my contribution) actually explains the paradox much better than the article itself. 195.241.164.104 (talk) 22:42, 19 October 2012 (UTC)

applying this to universe

the author itself tells this question is an application for the metric expansion.Let me ask,is the speed of expansion greater than the speed of light? if yes,then why cannot we think of ejecting an automated spacecraft to search for external universe? anyway it will reach the end,right? — Preceding unsigned comment added by 221b in (talk • contribs) 06:49, 14 November 2012 (UTC)

I removed this complete section as it falsely draws an analogy with the expansion of the universe.

Since objects recede in proportion to the distance between them, the target object is accelerating away rather than moving away at a constant speed, as is required for the solution.

For more details, see Hubble_volume#Hubble_limit_as_an_event_horizon. — Preceding unsigned comment added by GreyLabyrinthMember (talk • contribs) 11:41, 14 January 2014 (UTC)

Please don't remove entire sourced sections just because you disagree with certain claims in them. Paradoctor (talk) 12:55, 14 January 2014 (UTC)

Can someone justify this sentence?

The following sentence is at the end of the article.

"But the metric expansion of space is not analogous to the ant on the rubber rope problem, because the metric space expansion is occurring everywhere isotropically, not stretched from a fixed point like the rubber rope. So the light from some galaxies will never reach us."

This point is not intuitively obvious to me and, I believe, not true. I don't see how an isotropic expansion would be any different than the rubber band receding at a linear velocity, since the ant will locally always be at rest. Unless someone can show (or cite a source) to show that the rate of isotropic expansion would be faster than the rate of the rubber band in the problem, maybe we should mark this as citation needed or unverified?

Thanks. — Preceding unsigned comment added by 216.250.37.44 (talk) 23:04, 24 November 2012 (UTC)

See section below. Imagine the rubber rope being between two distant galaxies. The farther apart the galaxies are, the faster they recede from one another. Thus they do not move apart at a constant rate of speed. Vanyo (talk) 17:43, 15 January 2014 (UTC)

This is NOT analogous to the expansion of space.

The entire section on claiming the ant on a rubber rope is analogous to photons travelling in an expanding universe is wrong. The section was earlier removed, but has reappeared with the edit comment that it has been "toned down". Toned down or not, it is incorrect. The caveat in the section states that "This, however, ignores the fact that the metric expansion of space is accelerating", but this is not the issue. The analogy is false even if the metric expansion of space were constant. Even with constant expansion, any two points in space still recede from one another at a rate proportional to their distance from each other, and that distance is increasing, therefore the rate of recession of the two points is increasing. Thus, this is not analogous to the rubber rope being stretched by pulling its endpoints apart at a constant rate of speed.

The entire section need to be removed, as well as the reference allegedly supporting it.

Vanyo (talk) 17:39, 15 January 2014 (UTC)

The Koelman reference does not "allegedly" support the section, it supports it in fact. That the analogy fails is already stated by Koelman and, accordingly, in the section. You have given an additional reason to reject the analogy, which is fine. But without reliable sources, this is WP:OR, and can't be added to the article. And it does certainly not justify removing the section. I've amended the wording so as to state the failure of the analogy at the beginning. If you have any further problems with the article, please let me know. Paradoctor (talk) 18:11, 15 January 2014 (UTC)

Regardless of whether the Koelman reference supports the section, it is clearly false and misleading. This section, as it stands, states "This, however, ignores the fact that the metric expansion of space is accelerating. An ant on a rubber rope whose expansion increases with time is not guaranteed to reach the endpoint." This implies that the acceleration of the metric expansion of space is responsible for the failure of the analogy, so that if the metric expansion of the universe were not expanding we would expect this analogy to hold.

Even with a linearly increasing cosmic scale factor—a constant ${\dot {a}}(t)$—with which the universe is not accelerating, this analogy would not hold, as the distance between the two points would still grow exponentially, not linearly.

The above quoted statement is certain to confuse the vast majority of readers who are not well-versed in Cosmology—either about the nature of the universe, or the nature of the problem. It is only correct in the most technical and pedantic sense, and must either be removed or altered to reflect the true nature of the metric expansion of the universe with clarity. (that the acceleration of the metric expansion of the universe has absolutely no bearing on whether the problem is analogous; it is due to the isotropic nature of the expansion) bgg1996 (talk) —Preceding undated comment added 01:16, 16 January 2014 (UTC)

Welcome to Wikipedia. I'll post more later, right now just this: What would you tell someone who says "I don't see it"? If someone claims that in the case of constant metric expansion the ant still reaches the end, it just takes longer, say $T\propto e^{({\frac {v}{\alpha }})^{({\frac {v}{\alpha }})}}$, how do you prove them wrong? Paradoctor (talk) 06:04, 16 January 2014 (UTC)

Disclaimer: The following calculation, while mathematically correct, bases on a wrong assumption. See below. That is easy to see. Take the Hubble law $v=H_{0}D$ and take two points in space: $x_{0}=0$ where you start at t=0 and $x_{f}$ where you want to travel to. Further assume that when you start (t=0) these are 10 km away (i.e. D(t=0)=10 km) and $x_{f}$ moves away with ${\frac {10km}{s}}$. The Hubble's law tells us that $H_{0}={\frac {v}{D}}={\frac {10km/s}{10km}}=1/s$. Solving the differential equation $v(t)=H_{0}D(t)$ is easy and gives us for the distance D between $x_{0}$ and $x_{f}$ the relation $D(t)=D(0)\cdot e^{H_{0}t}=10km\cdot e^{\frac {t}{s}}$, i.e. the distance between both points grows exponentially (that is the main difference to the ant on a rubber rope as pointed out before; remember that we here assume a non-accelerating expansion of space!).

Let us now use a variation of the analytic solution in the article. The relative position we are at time t (between starting and end point) is $\psi (t)={\frac {x(t)}{D(t)}}$. Furthermore, let's say we move with a constant _local_ speed of $\alpha =1km/s$. Then the relative speed measured in our $\psi (t)$ coordinate is $\psi '(t)={\frac {\alpha }{D(t)}}={\frac {1km/s}{10km\cdot e^{\frac {t}{s}}}}={\frac {1}{10s}}e^{-{\frac {t}{s}}}$. It follows that $\psi (t)=\int _{0}^{t}dt'\psi '(t')={\frac {1}{10s}}\int _{0}^{t}dt'e^{-{\frac {t'}{s}}}={\frac {1-e^{-{\frac {t}{s}}}}{10}}$. Taking the limit $\lim _{t\rightarrow \infty }\psi (t)={\frac {1}{10}}<1$ we see that after an infinitely long time we can at most accomplish 10% of the journey. Doing the calculation with our variables we find that in general $\psi (t)={\frac {\alpha \left(1-e^{-H_{0}t}\right)}{D(0)H_{0}}}$, which in the limit gives $\lim _{t\rightarrow \infty }\psi (t)={\frac {\alpha }{D(0)H_{0}}}={\frac {\alpha }{v(t=0)}}$. That means that we can only reach the endpoint if the speed of our rocket is at least as large as the speed $v(0)$ with which the start and end point move away from each other at t=0. This is obviously in contrast to the result with the ant on the rubber rope which is independent from the relation of the speeds.

Even if we allow for a slight deceleration of the expansion of space by making the Hubble parameter time-dependent, say $H(t)={\frac {H_{0}}{(1+t)^{\gamma }}}$ for some $\gamma >0$, we find that for $\gamma <1$ the limit $\lim _{t\rightarrow \infty }\psi (t)={\frac {\alpha }{D(0)}}e^{\frac {H_{0}}{1-\gamma }}\int _{0}^{\infty }dt'e^{{\frac {H_{0}}{\gamma -1}}(1+t')^{1-\gamma }}$ still yields some finite value proportional to $\alpha$ which means that we can always choose $\alpha$ sufficiently small so that we do not reach the end point even for infinite times. Only if we choose $\gamma >=1$ the integral diverges, implying that we can always reach the end point, analogous to the ant. Although I haven't given an external source, this is relatively simple math and should be enough to reject the implicit assumption in the text that the analogy would work if the universe wasn't expanding. Jan Krieg (talk) 00:44, 21 March 2014 (UTC)

"this is relatively simple math" Everything is simple once you have mastered it. Please refer to WP:CALC and Wikipedia:These_are_not_original_research#Simple_calculations for what constitutes "simple math" in Wikipedia articles. To include this argument, we need a reliable source using exactly this loin of reasoning. Paradoctor (talk) 01:56, 21 March 2014 (UTC)

Haha, I should have refreshed my cosmology terminology before writing this. Thanks to Paradoctor for making me rethink this. Searching for external sources I realized that, while the math is correct, I wrongly assumed the Hubble parameter H to be constant in a non-accelerating universe, while in that case it indeed scales like $H(t)\propto {\frac {1}{1+t}}$. This corresponds to the "decelerating" (in my wrong terminology) case $\gamma =1$ I mentioned above, being just the limit in which the analogy with the ant works (because the system is exactly the same). However the assumption of a constant H is correct for a "de Sitter" universe, which is an extreme version of an accelerating universe. But even a slight acceleration (e.g. with some $\gamma <1$ in the formalism above) suffices for the analogy with the ant to break down. Perhaps this now helps somebody else falsely assuming H to be constant or something; at least writing this helped my understanding . Jan Krieg (talk) 04:51, 21 March 2014 (UTC)

Glad to be of service. Been there, done that, so I can feel your pain. Though in my case, one of my errors took me about 15 years to realize. Still puts me ahead of Dingle, I guess. Paradoctor (talk) 06:37, 21 March 2014 (UTC)

No pain involved, you learn best from mistakes if you make them yourself. Guess you won't make that particular one of yours again^^. Jan Krieg (talk) 12:27, 21 March 2014 (UTC)

I really, truly hope I've learned my lesson. Paradoctor (talk) 13:32, 21 March 2014 (UTC)

"if the metric expansion of the universe were not expanding we would expect this analogy to hold" Yes, that certainly is implicit in Koelman's writing: Now this would be a nice analogy for photons traveling in an expanding universe, were it not that the universe is not expanding linearly. Rather, the universe is undergoing an accelerated expansion. And that makes all the difference.

"certain to confuse the vast majority of readers who are not well-versed in Cosmology" I'd say it's just the other way around, only readers with some knowledge in cosmology might notice this and be confused. IME, people who don't understand even the basics of cosmology would be more likely to either accept anything they're told, or reject it for invalid reasons (which would not concern us). That said, I'll admit that at least some readers might get confused. But there is nothing we can do, because the confusion originates from the source.

You have provided a prima facie plausible argument that Koelman is wrong. But we can't use it! We need a reliable source that contradicts Koelman.

How about the following compromise: Without a reliable source, we cannot claim that the analogy fails even for constant expansion. But if we can find a reliable source for the statement that constant expansion leads to exponential distance increase, we could add a statement to the effect that this exponential increases violates the assumption made in the proof of the paradox, hence the paradox has not been proved to arise in the constant expansion scenario. Not a perfect solution, since it is somewhat WP:WEASELy, but there are no good compromises. If it puts this discussion to rest, I could live with that, though.

Side note: "the isotropic nature of the expansion" Isotropy has no bearing on the exponential distance growth. Rather, it is implied by homogeneity in space and constancy in time. Note that this is a one-dimensional problem. ;) Paradoctor (talk) 13:51, 16 January 2014 (UTC)

[User:GreyLabyrinthMember|GreyLabyrinthMember]] (talk) Johannes Koelman himself says that states that it is not analogous as "the universe is not expanding linearly" without any reference to metric expansion. Below is a quote from an article already referenced in this entry.

Now this would be a nice analogy for photons traveling[sic] in an expanding universe, were it not that the universe is not expanding linearly. Rather, the universe is undergoing an accelerated expansion. And that makes all the difference. — Preceding unsigned comment added by GreyLabyrinthMember (talk • contribs) 16:08, 16 January 2014 (UTC)

Figure

Details of the puzzle can vary. This depiction is discrete rather than continuous: An ant steps 0.25 cm, after which the rope, which begins with a length of 1 cm, is stretched 0.5 cm.

There could be a picture in this article. Hyacinth (talk) 07:27, 7 February 2020 (UTC)

Yes. I don't think the figure you were adding was a good choice, however. --JBL (talk) 16:42, 7 February 2020 (UTC)

I could improve upon the image or a create a new one. Hyacinth (talk) 00:50, 10 February 2020 (UTC)

Here are some more detailed comments on the image you've added in this section (thanks for doing that, for discussion purposes):

1. The size of the ant is large, not point-like, whereas in the problem the ant should be point-like. This is exacerbated by the fact that the location of the ant is given by its left edge, rather than its center. (The second point is not obvious, i.e., it took me a long time to figure out the nature of the difficulty I was having matching the image to its description.)
2. The font in the image is very small and difficult to read.
3. The image discretizes the process, but not far enough: it would be much clearer to actually split the two steps (stepping and stretching) out from each other.
4. At the location you placed the image, there is no discussion of the discrete version; so if an image like this is going to go somewhere, it should be in the correct section.

Of these four points, I feel 1 and 4 are more important than 2 and 3 -- if you created a version that addressed them, I would support including it. Maybe Deacon Vorbis has other comments. --JBL (talk) 19:17, 10 February 2020 (UTC)

That about covers it; I'm actually using this as an excuse to try to learn at least just enough Gnuplot and Inkscape to cobble a graph together, if not a full diagram at least. No ETA though. –Deacon Vorbis (carbon • videos) 20:36, 10 February 2020 (UTC)

How does the ant walk without legs? Hyacinth (talk) 02:14, 18 February 2020 (UTC)

????? --JBL (talk) 03:13, 18 February 2020 (UTC)

I put an animation together; it's at File:Ant on a rubber rope animation.gif. I had planned on slapping it at the end of the lead scaled to 90% of the width if it was otherwise too large to fit (it's 1200px wide nominally). But I can't seem to find an appropriate template to let me do this. While I try to figure something out, in the mean time, if anyone's got any requests for tweaks, please let me know. Even at 80% width or so, it feels a little tall to me. I might scale down the "Elapsed time" bit to help with that at least. –Deacon Vorbis (carbon • videos) 16:05, 18 February 2020 (UTC)

And added; I'm a bit unhappy with how large the text is, but I can tweak it later. Another option is to just have it as a .webm that sits off to the side as a normal, non-animated thumbnail until clicked. –Deacon Vorbis (carbon • videos) 16:02, 21 February 2020 (UTC)

Sorry for the belated response. It's really big -- on my default view, it takes up essentially as much visual real estate as the entire lead section. That's probably not ideal. I also must admit that I see a lot of relative appeal in the discrete version, because it allows one to visually separate out the movement contributions from the stretching of the band and the walking of the ant. By comparison, in the continuous version, there's no real way for a reader to "check" that it's doing the right thing -- all I can tell by eye is that the ant is speeding up. --JBL (talk) 02:02, 22 February 2020 (UTC)

Thanks for the feedback. For just a quick touch-up, I tweaked it some so that it should be significantly smaller in the vertical direction. As far as doing a discrete version, that should certainly be possible, but it will take a bit longer, so I just wanted to do this quick update first. I'm thinking along the lines of showing the ant crawling from t=0 to t=1 with the timer running, then a quick pause, then showing the rope expand with the timer stopped, then another quick pause, and then switching back to the ant crawling, etc. –Deacon Vorbis (carbon • videos) 04:19, 22 February 2020 (UTC)

Okay, there's a new version for the discrete variant of the problem at File:Ant on a rubber rope animation discrete.gif. I used the same setup for the graphics, but since the rope doesn't stretch as far by the time the ant reaches the end, that means there's some extra wasted space at the end. I don't think it's too terrible, and if want both versions at once, it would probably be good to keep the same scale for comparison. Currently, the article lead only mentions the continuous version, so this might be weird to include as is. However, Gardner's original posing of the problem give the discrete version, so it can probably be reworked to talk about that instead (or both really). At some point, I want to clean the article up a bit, and this would probably all work together for that. –Deacon Vorbis (carbon • videos) 23:43, 22 February 2020 (UTC)

ambiguity in conceptual explanations

The phrasing in the article at several points suggests that the logic is (to paraphrase) "ant making continual progress implies eventual success", a reason that would be independent of practically all the details of the formulation of the problem, when the actual reason is that the ant makes sufficiently fast progress relative to expansion of the rope. I'm not sure how to best clarify that without creating very longwinded explanations that are not much better than giving a formal solution. 73.149.246.232 (talk) 08:14, 16 February 2020 (UTC)

Well, I'm hopefully planning on revamping the article a bit at some point in the near future. I know I've seen this in an old puzzle book somewhere, which would have been a good source to add, but I checked all the ones I had lying around and I can't find it for the life of me, so that's a bit frustrating. I'm pretty sure it confirmed what's in the § Intuition section, which I think would be a better informal solution than what's presented first, because it doesn't rely on the divergence of the harmonic series.

As for your gripe, it seems like both of the informal solutions appeal to the specific setup of this problem and argue that the ant makes enough continual progress, not just any amount. Hopefully moving that Intuition section up closer to the beginning, along with the animation I've got (almost) ready will help. Another option is to also include an example where the rope stretches fast enough so that the ant never catches up. The original Gardner source suggests a rope that doubles in length each second will work. More modest growth rates would work too, but I'd rather not stray too far into WP:OR territory. –Deacon Vorbis (carbon • videos) 15:27, 16 February 2020 (UTC)

It looks like this has not been fixed yet. The lede and the second paragraph of the "Intuition" section are especially misleading.

In the lede, the sentence

"Once ... and enabling the ant to make continual progress"

should perhaps be extended with something like

"While continual progress by itself does not garantee that the ant can reach the end, a more detailed analysis (see below) shows that this is actually the case."

In the "Intuition" section the first paragraph is a valid argument but maybe it should be formulated a bit clearer (e.g. "Regardless of the (fixed) speed of the endpoint of the rope, we can always" and "The relative speed of any two neighbouring marks...") given its (apparent) intended audience.

Perhaps the second paragraph in "Intuition" should be dropped because while everything it says is true, it is also irrelevant: none of it implies that the ant will reach the end. 175.39.25.134 (talk) 09:51, 13 April 2021 (UTC)

I noticed the same conceptual issue and tried to clarify it. I also added a description of an example where the ant never catches up (the rope growing quadratically), though I didn't give the details. Kmill (talk) 20:11, 7 August 2021 (UTC)

Skeptic

"An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on). At the same time, the rope starts to stretch uniformly at a constant rate of 1 km per second, so that after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc. Will the ant ever reach the end of the rope?"

What happens in the last seconds, when the ant has almost reached the end but kilometers continue to add up?

"An ant (red dot) crawling on a stretchable rope at a constant speed of 1 cm/s. The rope is initially 4 cm long and stretches at a constant rate of 2 cm/s."

This is comprehensible, not counterintuitive at all, because the speed of the ant is the same as the speed of the stretching rate in front of her. But make the latter greater than one centimeter per second, and the ant, at the last second, will still have more to crawl than it is able to do. No?

--Robert Daoust (talk) 20:33, 25 April 2021 (UTC)

This page is for discussing improvements to the article, not for general questions about the content of the article. A better venue for your question would be Wikipedia:Reference desk/Mathematics. --JBL (talk) 21:01, 25 April 2021 (UTC)

Thank you very much, JBL. I'll do as you suggest and will come back here with reference if my skepticism is justified. --Robert Daoust (talk) 20:48, 26 April 2021 (UTC)

Well... I understand now that the ant is brought forward by the stretching rope at the speed of the stretching rate and she has also her own speed, so that she advances on the rope at a faster speed than the speed at which the rope is getting longer in front of her. In other words, kilometers are adding faster behind her than in front of her, and in the last seconds there is less than 1 cm of rope to get through while kilometers continue to add up. Should I suppress my comment on this page? --Robert Daoust (talk) 14:57, 27 April 2021 (UTC)

That's nicely put. No, there is no need to remove your comments. --JBL (talk) 15:32, 27 April 2021 (UTC)

Removed section on expansion of space

I've removed the section which attempts to draw a parallel between the ant on a rope and the expansion of space because it is incorrect.

The rope's length is analogous to the distance to an object in space. The rope's length expands - or in other words, the end of the rope recedes from the beginning of rhe rope - at a constant rate, in units of m/s, but distances to objects in space do not. They recede at a rate measured in units of m/s/m (technically a frequency, 1/s), because the speed of recession is dependent on distance. As expansion makes an object recede, the speed of the recession increases.

This can be seen in the first animation by looking at where the end of the rope starts. It starts at x=4cm, and after one second it reaches x=6cm. However, consider the rightmost end of the second grey block in the rope. When it eventually reaches x=4cm, it is moving much slower, and 1 second later it has not even reached x=5cm. This is not the case with the expansion of space, where any object will always recede at the same speed when it is at a certain distance.

While it is true that light from 'some' galaxies receding at a rate greater than the speed of light can reach Earth, the mathematics of the ant-on-a-rope is not sufficient to prove this.

David (talk) 23:15, 12 June 2021 (UTC)

I don't find the basis of this removal particularly compelling, given that the section in question explicitly discusses varying rates of expansion. (The issue of units is a red herring: the expansion in either case could be expressed in the language of the other.) I would appreciate input from other editors of this page about whether the removed source Koelman, Johannes (2012). "Beam Me The Farthest, Scotty!". Archived from the original on 6 April 2013. Retrieved 26 December 2012. meets WP:RS; if so, I would be inclined to restore the removed section, after checking that it accurately reflects the reference. --JBL (talk) 12:44, 7 September 2021 (UTC)

Incorrect explanation

In the Intuition section, this sentence is incorrect: "At any moment, we can imagine putting down two marks on the rope: one at the ant's current position, and another 1 mm closer to the target point. If the ant were to stop for a moment, from its point of view the first mark is stationary, and the second mark is moving away at a speed of 1 mm/s."

This is only true if the marks are set at T=0. If the marks are set at, say T=10s, the second mark would be moving away at a speed of 0.1 mm/s. This doesn't break the logic or the intuition of the argument as the ant can still reach the second mark because it is moving away at a constant speed that is equal or lower than 1 mm/s.