Talk:Coastline paradox

Last paragraph contradicts the first

The last paragraph could, I suppose, be made precise, but as it stands it contradicts the first. Coastlines have no defined length, but this one is longer than that one? Tom Permutt (talk) 21:39, 10 November 2008 (UTC)

Tides

"Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below"

Tides negate the need to measure a coastline down to millimeter and below precision. The variability is so high between tides, that we don't need to go below the meter resolution. —Preceding unsigned comment added by 65.44.114.33 (talk) 01:49, 30 January 2010 (UTC)

The paradox is not one of necessity (how closely do we need to measure), but one of possibility (how closely can we measure). catsmoke (talk) 02:55, 21 June 2019 (UTC)

The "paradox" relies on going way beyond what we can reliably and meaningfully measure. Real measurements of coastline lengths are by no means guaranteed to continue growing with further shrinking of rulers. Tides, waves, work effort requirements, measurement uncertainties, and eventually, perhaps, the quantization of space itself, will make further increases of resolution pointless. That is, while the problem does have real practical consequences, the paradox is more of a theoretical exercise. I wish the article shed some more light on how (and why) to solve the practical problem too. Elias (talk) 12:30, 23 March 2023 (UTC)

Removed note

Actually it does mean that the coastline is of infinite length... or at least that successive approximations diverge. As you measure on smaller and smaller scales, the perturbations increase in magnitude, instead of vanishing. In a moment I'll be removing the section at the end of the article that confuses the whole issue, and adding a citation from Mandelbrot's "The Fractal Geometry of Nature". Andrew Rodland 06:07, 16 June 2007 (UTC)

For clarity, this is my comment on a section of the article I removed and rewrite, and not a note which was removed. I apologize for the poor heading but I'm not going to change it now. Andrew Rodland 06:27, 16 June 2007 (UTC)

It seems to me that the notion of coastlines being fractals and of infinite length and so on confuses the actual nature of coastlines with representations on them on maps. Even then it is not obvious to me that mapped coastlines are necessarily fractals. Even in the most fjordy, rock-ridden places there are tides, waves, mudflats, estuaries, marshes, etc. The coast "line" changes with tides and waves, and is ill-defined in estuaries, marshes, mudflats, and so on. In short, coastlines are only two-dimensional lines in the abstract, and dependent upon human-defined boundaries (eg, just where is the exact line between land and sea in Louisiana?). The fractal nature is obvious when looking at maps -- especially at larger, zoomed-out scales -- but in the real world it is not so simple. I realize there are sources that claim coastlines are fractals and infinite in length, and I have yet to find a source that offers a more realistic perspective (of course I haven't really looked). Still I can't help but comment on the topic. It seems obvious -- just try using an inch-long ruler to measure the "coastline" on a broad stretch of beach where the waves wash 50 feet in and out, while the tide is turning. Wouldn't the main problem be not that coastlines are fractals, but that they are subject to waves, tides, and are not "lines" at all? Pfly (talk) 06:19, 17 April 2008 (UTC)

Regarding the sentence at the beginning of this section, i.e. "Actually it does mean that the coastline is of infinite length... or at least that successive approximations diverge". In terms of this paradox being being valid, stepping from one thing being 'infinite in length' and two being 'divergent' is stepping across the divide of this article being valid and it not being valid. Since distances on maps have been read using calipers people have known that narrowing the caliper width yields closer approximations to a length with the length, usually, tending to increase as the caliper width narrows. This effect though is entirely intuitive since you introduce an increasing amount of detail each time you narrow the caliper gauge. Much, in fact, like increasing the sampling resolution and discovering higher frequencies in the data set. Whether or not it truly tends to infinity or simply increases in apparent length is the subject of the debate. From physics we have the planck length, which is the smallest distance possible; in any two dimensional region then there must be a finite number of planck lengths. Also, at the smallest useful scale you have atoms, this suggests the problem eventually devolves to a microscopic dot-to-dot whose solution must be finite.

The trouble with the idea of the paradox is that it is only useful when referring to mathematical fractal shapes, some of which will tend to infinity and some of which won't. When referring to real coastline the analogy breaks down when you reach the molecular scale. Even there you must make 'political' decisions about which molecules are 'onshore' and which are 'offshore'. Given the arguments, if the paradox in true then the measurement of the perimeter of any real-world object would suffer the same tendency to infinity. The truth is though, nobody believes the perimeter of a bicycle tyre may be usefully said to be infinite, you simply place it in a bounding shape and measure that instead. Also, making tape measures would be impossible because any piece of tape would be infinite in length for the same reason as the coastline should be if this paradox were true. Personally I think this is a reassignment of Zeno's paradox into a realm into which it doesn't apply.

You're almost certainly correct that measurement breaks down or changes substantially at the molecular scale. The 'paradox' is still important because it illustrates that measurements of a continuous nonlinear path can differ wildly based on the 'resolution' (or 'magnification' or 'size of measuring stick', you get the idea...) chosen by the measurer. "Paradoxes" can't be universe-breakers or else we wouldn't be around to write Wikipedia articles about them. groupuscule (talk) 11:22, 6 July 2013 (UTC)

""Paradoxes" can't be universe-breakers or else we wouldn't be around to write Wikipedia articles about them". This is because paradoxes usually prove not to be paradoxes at all but faults of reasoning. The reason why I made the 'universe breaking' argument is that the reasoning clearly betrays the reductio ad absrdum of what would happen if the paradox were to be an existential fact. Clearly, the article must be a faulty piece of reasoning or the 'universe' would not operate the way it does, nobody would be able to measure the circumference of a real-life circle using real-life measuring tools with any level of consistent meaning and also tape measures would be impossible to reliably print because they'd all be infinite in length, owing to having a crinkly surface, but this is obviously not true. In what way is this supposed paradox any more valid than arguing you can't obtain a reliable measure for the distance between London and Manchester by road?

Because distance = line. You can only ever produce (approximately) one measurement of a straight line. But, as Andrew wrote in 2007, measurements of coastline (and any other non-linear continuous path, which is many of the paths you'll find out there!) diverge in successive approximations. groupuscule (talk) 12:42, 6 July 2013 (UTC)

But, for the final time, your statement is Not the subject of the paradox. This type of divergence is neither counter intuitive, not paradoxical, for the reason given above which refers to sample frequency. Obviously the data will naturally extrapolate with a higher linear density of measurements, but as in the Nyquist limit or the resolution of aliasing distortion from signal processing, this definitely does Not mean the divergence continues ad infinitum, it does not mean it tends to infinity and does Not mean a narrowing caliper width, over successive data sets, ultimately produces a paradox. For another example: It is Not true to say the more closely you follow the contour of a coast road the more your journey will tend toward infinite length. One you have reached the curvature of a contour the length increase ceases. Using the caliper measurement with decreasing gauge's for each journey only more closely approximates an ideal length and then stops.

Perhaps not counter-intuitive to an intellect such as yours! We generally agree with your points, though not certain about the nature of this ideal length. We would like to check out some secondary literature on the question. Do you have thoughts about how the article should change? It doesn't currently make the "infinite length" claim. And although the examples given seem a little beside the point, they do emphasize the practical over the mathematical side of the issue. groupuscule (talk) 15:04, 6 July 2013 (UTC)

Contradiction

First, the article states that the coastline of a landmass does not have a well-defined length (which seems correct to me). And then, at the end, there appear coastline lengths in miles and km again (which seems useless if the lenght of the approximation steps is not declared). Even a relationship (percentage) is useless, and even if the same (unknown) step length is used to measure both of them. Can't we delete all those coastline lengths all over Wikipedia...?--Panda17 (talk) 08:33, 30 August 2011 (UTC)

Images need units clarification/simplification

The map images on this article currently have a line on the bottom which is alternately colored black and grey. (What's the word for that line?) Presumably, each black and grey line represents a length, but the maps don't say what. They should either provide units for those lengths, or simply remove that particular line and label the remaining yellow line with its length. The second option seems preferable for its simplicity; the yellow line means that no other units of scale are really needed, and that line is more what this article is about. ± Lenoxus (" *** ") 23:10, 20 September 2011 (UTC)

Confused mess?

This whole article reads like a confused mess. I can recall reading Mandelbrot's writing in The Fractal Geometry of Nature about how to expect the measured length of a coastline to increase as the gate on the measuring device narrows but I don't recall any actual experimental data suggesting it can be expected to tend towards infinity and I don't recall it being referred to by the title in this article. The whole thing sounds bogus, and largely because this result is neither counter-intuitive nor a paradox. Anybody who had used callipers to take map measurements knows the length between points increases as you narrow the stepping gap between the calliper points. I don't think anybody believes the result tends towards infinity, the assertion below that it does and the reason given read like a personal claim, rather than fact.

It is hard to argue there is a real-world test scenario for the assertion since, once you reach molecular lengths deciding where the line of the coast actually is becomes absurd. The test case is limited by real world limitations of what can usefully be called the coastline. This is usually thought of as a region between the sea and habitable land and there is no literal line. The chosen line though is usually a conceptual line that may be measured by a vehicle journey since this is the meaning of the length of the coastline "How far must you travel to get around it". The thought experiment suggested is exactly that, a thought experiment for which, after a point of detail, there is no longer any literal analogue or genuine test case. Even as a thought case though, the line stays finite, however it is measured, after decisions about which line fairly represents the coast has been measured. Hence the difference between water bodies we think of as rivers, mouths or deltas and water bodies we think of as the seas.

What evidence, outside this article are there of any serious scientific research stating the 'coastline paradox' exists? This article reads like a hobby horse for the writer and not a realistically backed phenomenon.

Does the Coastline Paradox Apply to Coastlines?

I have nagging problem with this paradox. Coastlines are not really like this. The East Coast of the United States is dominated by barrier islands and peninsulas. These features are fairly consistent in width, so they are not self-similar. The beaches of southern Long Island are quite straight (except for the occasional inlet). In addition, coastline becomes a meaningless concept in the zone of wave and tidal action. — Preceding unsigned comment added by Roger J Cooper (talk • contribs) 00:30, 16 September 2013 (UTC)

they may appear straight at a certain scale, but as you zoom in that can change. That's the whole point about the fractal aspect of it all. The overall appearance might be straight, but in reality there are countless little serpentines and zigzags. And when you get down to grain-of-sand level, nothing is straight. PurpleChez (talk) 13:15, 10 June 2019 (UTC)

Not precisely worded and quite irrelevant.

> More concretely, the length of the coastline depends on the method used to measure it.

That is not true. Coastlines are easily and precisely MEASURED by having a guy roll an odometer equipped unicycle all along the shore, recording the revolutions. Regrettably, most countries are too mean or lazy to do that and they try to GUESS the lenght instead, based on mere teodolite points. Anyhow, the whole issue is rather moot in the era of 0.5 meter resolution public satellite imagery available on the Google Earth Maps site. 82.144.172.66 (talk) 23:05, 5 April 2014 (UTC)

An "odometer equipped unicycle" will not measure in the "fractions of a millimeter and below" mentioned in the lead. HiLo48 (talk) 23:13, 5 April 2014 (UTC)

Not only for coastlines!

"More concretely, the length of the coastline depends on the method used to measure it. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be measured around, and hence no single well-defined perimeter to the landmass."

These same troubles apply to every endeavor of measuring perimeter in the "real world"! Ariel C.M.K. (talk) 13:24, 3 June 2014 (UTC)

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Practicality section contradicts rest of article

The Practicality section feels a lot like someone who came up with their own "solution" to the Coastline Paradox; it has no sources at all.

In addition, it says:

"[...] the problem of measuring the boundary of an atom—which ultimately does not have a defined boundary—arises."

This problem would apply to any measurements, directly contradicting the statement in Mathematical Aspects section that's says:

"[...] unlike with the metal bar, there is no way to obtain a maximum value for the length [...]"

I think this entire section should be removed because it doesn't feel like it contributes to the article, it's more like someone's opinion.

JollyTurbo1 (talk) 05:00, 23 March 2019 (UTC)

I think you're absolutely right. I'm removing. -- St.nerol (talk) 15:50, 9 May 2019 (UTC)

unclear... to me at least

"When measuring a coastline, however, the issue is that the result does not increase in accuracy for an increase in measurement —it only increases;" It doesn't increase... it only increases. I suspect something has been lost in translation. PurpleChez (talk) 13:19, 10 June 2019 (UTC)

@PurpleChez: something seems amiss, what it means to say is: The result of measuring the metal bar converges toward a number with an increase in accuracy, a little more or less, but the results of measuring coastlines will always increase with any increase in accuracy [because fractals and disorder]. cygnis insignis 18:08, 10 June 2019 (UTC)

I've rewritten the sentence. catsmoke (talk) 02:56, 21 June 2019 (UTC)

Fallacious Article: Incomplete Data and Jumping to Conclusions

True coastlines are ever-changing due to tides, waves, erosion, and more. To measure it, there must be some rule defining what line is to be measured. The definition of what is being measured is never given in the article. It just remains as the ambiguous coastline. The only option then is to use another source for the definition. A coastline is defined by Merriam-Webster as "a line that forms the boundary between the land and the ocean or a lake" which will work as a start. This changes over time, so it must be further defined. A specific moment in time could be used, or the average over a specific span of time could be used. The last pieces of the puzzle are the method and scale of measurement. Once these definitions are in place, the specific length can be measured. The coastline paradox, the idea that the perimeter of a land mass is infinite, appears to arise from certain assumptions in the article and incomplete data.

The first external link on the page seems to be the source of the article's problems. I'll refer to it as the source for the rest of this section. Most of the assumptions are made there, and very little information is provided. This source seems to be based on a book from 1961, but no details on the extent of the information in said book are given. Only a few scant examples are provided in the source. The Richardson section only shows six data points, which is where the apparent rapidly expanding length comes from. Unfortunately for the source, six data points can't actually tell much of anything. While it may be true that, over those six data points, the length of a coastline appears to increase faster and faster as the scale decreases, it doesn't show what happens at the extremes of scale, either large or small. A conclusion of any merit cannot be drawn from that, aside from the need for more data. Other examples of scale are given in other sections, but all of these are still in the realm of approximations. None of them actually go down to the smallest scales to see if the supposed paradox holds up. The smallest example labeled in the Examples section is 250m. The data points shown in the Richardson section don't label any of the points. They are just shown on a graph.

There is a true maximum measurable length to any coastline. It is determined by the coastline's smallest feature. If the measurement scale is larger than this smallest feature, the measurement will be an approximation that will get larger if the scale is reduced and more features are revealed. This growth of length will continue until the scale of the smallest feature is reached. If the measurement scale is smaller than this, it will result in a measurement that will be very close to the true length of the coastline, and it will stop rapidly growing as the scale is reduced further. At this point it will behave like simple curve approximations. There are no more features to expand below this scale, and thus coastlines are not truly fractals. They resemble fractals at some scales, but true fractals have no limits on how small their scales can get. This is a problem in the source. It assumes coastlines to be infinite fractals, but they are not. There is a limit to the smallest size of features to expand. That can delve into quantum mechanics, though, which is way beyond the scope of the article. A brief trip down that rabbit hole reveals that subatomic particles don't actually touch, though, which means there actually is no coastline below a certain scale. There must be a lower limit for what defines a coastline which is larger than subatomic particles, and there must be a smallest scale for features on that coastline.

Approximation can lead to other interesting false conclusions as well, such as the idea that pi = 4. This YouTube video, Rhapsody on the Proof of Pi = 4 by Vi Hart, goes into detail about how this idea works and how it's wrong. It shows that, while there is a specific perimeter length to a given shape, approximations can be very misleading when trying to get the exact measurements. This is a problem with the source as well. It assumes that a few approximations, the six data points, somehow represent the full truth of the situation.

All in all, the coastline paradox does not exist. It only appears to exist in a small data set. Closer examination reveals the flaws in the original arguments as well as the need for more data. This article really needs to be flagged in some way to show that it's not accurate. I do not know how to do that.

This last part is related, but it's about preventing the coastline paradox or others like it from arising. It tackles the very first point. How the coastline is defined determines everything about the measurement results. The only reason there is even room to argue about the accuracy of coastline measurements is that the coastline itself is not defined accurately enough. Is it the border between molecules? Is it larger than that? Perhaps it's best defined as the path a string follows when laid down along the water's edge, which would be quite usable for most people. The supposed paradox isn't a product of measurement scale. It's a product of poor definitions. Science has gotten quite a bit better at rigorous data collection since 1961 though. Maybe there will be a new data set to extrapolate from in the future. Caelen Tigris (talk) 05:03, 1 July 2019 (UTC)

Article misses its own point

This article claims to be about a paradox, but the content of the article seems to be just a jumbled presentation of various factoids and various discussions, leading to nothing but "Measuring coastlines is actually pretty hard".

I suggest either deleting the article outright, or doing a complete rewrite that contains only the following: a clear and definitive statement of the paradox, followed by explanations of each proof for it, each method of resolving it (if there are any), and each common objection to it (if there are any). If the article is going to fail to define its own paradox, and fail to show any proofs for it or ways of resolving it, why even bother trying to write any other parts? TooManyFingers (talk) 23:36, 29 August 2021 (UTC)

== Error in history wrt Portugal/Spain

The article states that Portugal reported a length of 987 km and Spain reproted 1214 km, but this research article (Stoa, Ryan. 2019. The coastline paradox. Rutgers University Law Review 72 (2): 351– 400. doi:10.2139/ssrn.3445756. https://ssrn.com/abstract=3445756.) reports it the opposite way. See p. 356. With the correct numbers, this is an instance of the phenomenon observed by Richardson that it was generally the smaller country reporting a larger value for a common border, which makes sense, since their country maps would generally have a finer scale, to produce a map of a normal size. JoelDavid (talk) 17:38, 9 January 2023 (UTC)

this is not a paradox...

it assume that coastlines are fractal to get the paradox part... and they are not fractal they are baryonic matter, so can be measured in discreet intervals.

so the basis of this is flawed, ergo the supposition is incorrect.

coastlines are 100% definable in a number of measuring ways, none of which involve fractals.

this is all just nonsense and should be deleted. 82.9.90.69 (talk) 14:29, 31 August 2023 (UTC)

The coastline paradox applies to cartography and attempts at modeling the physical world. When doing this, the unit used to define the boundary of the coastline will impact its final length. While you can, in theory, get to the subatomic level to measure a coastline, erosion and tides are going to create enough variation that your efforts will be meaningless. When two countries survey their coastline or boarder, they will come up with different values if they use different spatial resolutions and chosen discreet interval. The paradox is for cartographers trying to give a "correct" measurement. The paradox exists in cartographic peer-reviewed literature, and if you think it is non-sense I suggest you publish a paper in a cartographic journal, not try and get it deleted from Wikipedia. GeogSage ^{(⚔Chat?⚔)} 18:17, 2 October 2023 (UTC)

Merge page How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension with this one.

The Wikipedia article How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension covers one peer reviewed journal on a topic. This paper is fantastic, and could easily serve as a section on the page Coastline paradox. However, I don't believe single peer reviewed journals generally need to have their own dedicated Wikipedia page, as they are primary sources that stand alone. Thus, I propose moving the contents of that page into the coastline paradox page as a subsection and made into a redirect. I'm placing relevant tags on each page. Please discuss here, and we can see if we can reach a consensus.

I'll wait at least two weeks, so until October 1st 2023, for discussion until I make the move permanent.

GeogSage ^{(⚔Chat?⚔)} 18:35, 17 September 2023 (UTC)

That seems fine to merge, yeah. Elli (talk | contribs) 03:30, 24 November 2023 (UTC)

Thanks for the feedback! This is one I'm trying to figure out how to approach. Not sure how much content to pull from the journal article. GeogSage ^{(⚔Chat?⚔)} 04:39, 25 November 2023 (UTC)

  Y Merger complete.  ; I've move most it over, into the Measuring a coastline section. Feel free to refine it in situ. Klbrain (talk) 18:33, 4 January 2024 (UTC)

Thank you! I have been meaning to get to this and have been putting it off. Awesome job! GeogSage ^{(⚔Chat?⚔)} 18:35, 4 January 2024 (UTC)