Talk:Pi

Semi-protected edit request on 14 November 2023

I would like to include the first 314 decimals of Pi, due to the first numbers in pi being 3, 1 and 4. I don't know, the idea just came to my head. Here's the source to obtain the numbers: https://3.141592653.com JonJohnJimmy (talk) 18:52, 14 November 2023 (UTC)

I don't think that's a particularly good reason for including this content. —David Eppstein (talk) 19:09, 14 November 2023 (UTC)

Therefore ...

Towards the end of the lead we say:

In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics.

That therefore makes no sense. Rather,

Because it appears in areas having little to do with geometry, such as number theory and statistics, it is possible to define it without any reference to geometry, as is usually done in modern mathematical analysis.

However, I will not make that change; I hope someone else can come up with a better wording. Nø (talk) 08:42, 8 December 2023 (UTC)

In that context, because makes as little sense as therefore. How about Because it appears in areas having little to do with geometry, such as number theory and statistics, it is convenient to define it without any reference to geometry, as is usually done in modern mathematical analysis.? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:49, 8 December 2023 (UTC)

I've edited it using "and" rather than asserting causation in either direction; the body of the article doesn't, as far as I can tell, assert one caused the other so our WP:LEAD summary shouldn't. I couldn't see that "often" was supported by the body of the article so I dropped that too, rather than replace it with "usually". NebY (talk) 14:49, 8 December 2023 (UTC)

Good solution. Tito Omburo (talk) 16:16, 8 December 2023 (UTC)

This 'often' or 'usually' is probably a supportable claim, but "can be defined" also seems fine. –jacobolus (t) 19:07, 8 December 2023 (UTC)

Yes indeed, quite possibly supportable but maybe not worth the trouble? Thanks, both, and to Nø for spotting the problem. NebY (talk) 17:56, 9 December 2023 (UTC)

Woo

I expect the "pop culture" bit to be full of trivia, but the claim in Carl Sagan's novel seems flaky. I am not an expert, but the nature of the digits of pi suggests that there is probably a proof that given any message (or its logical converse), it is indeed encoded somewhere in the digits of pi. I removed the following sentence:

In particular, when Moses asks God for His name, God replies, "I Am who I Am." in the 3rd chapter, 14 and 15 lines of the Book of Exodus ($\pi \approx 3.1415\dots$).

Meaning is not very clear, but for (almost?) every book of the bible there is a 3rd chapter, which includes verses 14 and 15, which remarkably enough are always consecutive. Imaginatorium (talk) 11:13, 9 December 2023 (UTC)

Thanks for removing that. I'd just as soon take out the whole paragraph about the novel, song, and TV episode. –jacobolus (t) 17:23, 9 December 2023 (UTC)

That is a plot point in the novel Contact. So, it is the kind of thing we can mention if we have secondary sourcing that indicates it is significant enough to merit inclusion. Whether it's mathematically legitimate is a separate question. XOR'easter (talk) 18:54, 9 December 2023 (UTC)

In Carl Sagan's 1985 novel Contact it is suggested that the creator of the universe buried a message deep within the digits of π. This suggestion is supported by the Bible. When Moses asks the creator for His name, the creator replies, "I Am who I Am." in the 3rd chapter, 14 and 15 lines of the Book of Exodus ($\pi \approx 3.1415\dots$). Why cannot we add this information to support this valuable Carl Sagan's suggestion to this article? Or perhaps we should remove Carl Sagan's suggestion from this article as it is unsupported? Guswen (talk) 22:21, 9 December 2023 (UTC)

This is discussed above.

• The existence of a third chapter with verses 14 and 15 consecutive to each other is trivial (true of anything divided into sufficiently many chapters and verses) and the contents of that chapter are not particularly relevant, so this does not actually support the suggestion.
• We cannot include this material without secondary published sources discussing it. Without these sources, the idea that it has some connection to hidden messages in pi is original research.

—David Eppstein (talk) 22:43, 9 December 2023 (UTC)

The Bible numerology is clearly un-encyclopedic nonsense. That Carl Sagan's novel included π as a plot point is not controversial, and there are secondary sources discussing it. The only relevant question about it is whether it's important enough to include in this article, or whether it seems too trivial. –jacobolus (t) 23:09, 9 December 2023 (UTC)

Also, "3.1415" is not "deep within the digits of pi", so even if we include the material about Sagan, its connection to the Bible verse that Guswen wants to add seems extremely tenuous. —David Eppstein (talk) 23:14, 9 December 2023 (UTC)

Thank you for this feedback, gentlemen. I will think about it. For now, let me just say that for the man in the street $\pi \approx 3.14\dots$ not even $\pi \approx 3.1415\dots$, so the 3rd chapter 15 line of the Book of Exodus suffice. The 15th line is redundant.

And I don't know much about those "deep within the digits of pi". But perhaps we should create a Wikipedia article within the digits of pi to cover this seemingly overlooked issue (?) Guswen (talk) 00:08, 10 December 2023 (UTC)

I don't understand what your complaint is. The removed material should not be included in Wikipedia. An article within the digits of pi does not seem worthwhile to create. Newly invented numerology about the Bible and π does not have any encyclopedic value (cf. Wikipedia:No original research). Old, well sourced numerology with some attested historical significance may be relevant at the page biblical numerology, but is not relevant at pi. –jacobolus (t) 02:26, 10 December 2023 (UTC)

But I did not "invent" any numerology. This numerology has been known at least since the 6th century BCE, when the Book of Exodus was written, according to modern scholars.

Carl Sagan's suggestion is valid and reveals historically the first correct approximation of $\pi \approx 3.1415$. Earlier approximations of π dated around 1650 BC found in Egypt gave the value of π as ${\textstyle \pi \approx {\bigl (}{\tfrac {16}{9}}{\bigr )}^{2}\approx 3.16}$, for example.

Carl Sagan's suggested that "the creator of the universe buried a message deep within the digits of π". But what could be the content of such a message (according to Carl Sagan) if not the name of the creator of the universe? And this name is revealed in the 3rd chapter, 14,15 lines of the book of Book of Exodus.

Many books contain 3rd chapters with lines 14 and 15, but those lines do not carry any message from the creator of the universe, as Sagan suggested.

Guswen (talk) 08:49, 10 December 2023 (UTC)

Please stop writing garbage. (After all, you claim to have a degree in mathematics.) FWIW, John 3: 14, 15 also contains a "message from the creator of the universe", viz. blah blah, sorry, look it up yourself. What could be the content of such a message (according to you) if not something about "eternal life"? Sagan is (obviously) referring to the idea that peering inside the digits of pi there would appear a "message" encoded somehow in the digits. A message that was not known beforehand. Not that an index to the bible (the chapter and verse numbers, which were in any case added much later) would somehow point to a particular message. I pointed out above that because of the infinite elasticity allowed to interpreters of such claims, it is always going to be possible to find any desired messages if determined enough: for example "Donald Trump savio(u)r" or "Donald Trump harbinger of doom"; WP does not require editors to have any mathematical sophistication whatsoever, but you cannot claim have any and still believe this nonsense. Anyway, tactically, the discussion here is over. Imaginatorium (talk) 09:56, 10 December 2023 (UTC)

I do not need my degree in mathematics to see that $\pi \approx 3.1415$. But thank you for the hint! Indeed, the Creator's message in John 3:14,15 (14 Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up, 15 that everyone who believes may have eternal life in him.) relates to His earlier message in Exodus 3:14,15. So Carl Sagan's intuition was outstanding!

"We can judge our progress by the courage of our questions and the depths of our answers, our willingness to embrace what is true rather than what feels good." Carl Sagan.

Guswen (talk) 10:31, 10 December 2023 (UTC)

As far as I can tell having not read it, Carl Sagan book has nothing to do with any line 3:14–15 of the Bible. "since the 6th century BCE, when the Book of Exodus was written, according to modern scholars" – If you ever want to add this to some other page where it's relevant (e.g. a page about examples of pseudoscientific nonsense), please carefully cite these modern scholars in the discussion there. In my opinion biblical numerology is not relevant or appropriate at the page about pi the mathematical constant, and this conversation has become largely off topic for this discussion page. –jacobolus (t) 16:49, 10 December 2023 (UTC)

I've read it. You are correct in saying that it has nothing to do with any line 3:14–15 of the Bible. XOR'easter (talk) 18:32, 10 December 2023 (UTC)

If Carl Sagan discovered that the creator of the universe buried a message about His name and eternal life within the first five digits of π ([Ex 3.14.15], [J 3.14.15]), Sagan would have certainly written it in his book.

Unfortunately, he had not (He was agnostic. Agnostics have no motivation to read the Bible). However, he still had an outstanding intuition. And intuition is the only real valuable thing, according to your Master (Albert Einstein).

Guswen (talk) 21:46, 11 December 2023 (UTC)

@Guswen: Wikipedia talk pages are not social media sites or fora for general conversation, and conversations about your religion are all off topic here. –jacobolus (t) 15:30, 12 December 2023 (UTC)

Of course the claim in the Sagan novel is flaky, but the statement that the novel makes that claim is objectively verifiable and is clearly relevant to the popular culture section. That's a claim about the book, not a claim about the actual provenance of $\pi$. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:05, 12 December 2023 (UTC)

It's not entirely off topic, but I still maintain it's not really helpful or meaningful to readers looking to learn about π, or even "π in popular culture". This kind of topic is better to leave at the page about the book; Wikipedia doesn't need to cross-reference every time any subject is included in any novel. I'm sure if we hunted we could find a longer list of books/poems/songs/TV dramas/documentaries/... involving π, but such a list is pretty much trivia. In any event, the book is currently mentioned, and nobody seems to be trying too hard to remove it. –jacobolus (t) 15:30, 12 December 2023 (UTC)

Well, I am trying hard to remove it as an unsupported suggestion. If "the creator of the universe buried a message deep within the digits of π", as Sagan suggests, then we should inform the reader that this suggestion is supported by the Bible, as the creator of the universe buried a message about His name and eternal life already within the first five digits of π.

Otherwise, Sagan's suggestion is, indeed, a big Woo.

This leaves us with three options: 1. Remove Sagan's suggestion (and information about his book and movie) from this page, 2. Hunt for all books/poems/songs/TV dramas/documentaries/... involving π, as jacobolus proposes, and list them on this page in the "Popular Culture" section and 3. Give a reader a hint that they can find support for Sagan's suggestion, at least in the Bible (I'm a Catholic, but perhaps a message from the creator of the universe can be found in Quran as well).

Guswen (talk) 20:15, 12 December 2023 (UTC)

The claim made in the article is only that Sagan used this as a plot point in his book, and it is discussed here an example of the appearance of π in popular culture. No claim is being made about the truth of Sagan's plot device, which is obviously pseudoscientific nonsense. Whether Sagan personally believed it to be true (unlikely) is irrelevant. The current text does not need to be validated by evidence external to the book, beyond secondary sources noting this as a culturally important example of π appearing in a novel. The stuff about particular lines 3:14 in the Bible has literally nothing to do with Sagan's book. –jacobolus (t) 20:20, 12 December 2023 (UTC)

Indeed, there is a message from the creator of the universe also in Quran. 3rd Surah Al Imran, verses 14 and 15 ("The faithful, their character and reward") state that (Dr. Mustafa Khattab translation):

(14) "The enjoyment of worldly desires—women, children, treasures of gold and silver, fine horses, cattle, and fertile land—has been made appealing to people. These are the pleasures of this worldly life, but with Allah is the finest destination."

(15) "Say, O Prophet, 'Shall I inform you of what is better than all of this? Those mindful of Allah will have Gardens with their Lord under which rivers flow, to stay there forever, and pure spouses, along with Allah’s pleasure.' And Allah is All-Seeing of His servants,"

Therefore, Sagan had an outstanding intuition.

Guswen (talk) 20:37, 12 December 2023 (UTC)

Please stop using this talk page as a forum for your irrelevant nonsense. It is off topic for this talk page, which can only be about properly sourced content about π. —David Eppstein (talk) 21:28, 12 December 2023 (UTC)

Do you criticize the noble Quran Eppstein?

Guswen (talk) 22:58, 12 December 2023 (UTC)

Please stay on topic. XOR'easter (talk) 23:53, 12 December 2023 (UTC)

Do you criticize the Bible XOREaster?

Guswen (talk) 01:23, 13 December 2023 (UTC)

@Guswen If you persist I am going to ask that you be temporarily blocked from editing. You are making a disruptive nuisance of yourself. –jacobolus (t) 01:40, 13 December 2023 (UTC)

I'm aware of that.

A Christian cannot be afraid of losing face. The Son of God did this in the most shameful way.

Will you ask to temporarily block me from editing, for this entry too?

Guswen (talk) 22:24, 13 December 2023 (UTC)

Ok, see WP:ANI#Guswen on Talk:Pi —David Eppstein (talk) 23:51, 13 December 2023 (UTC)

The Bible didn't get chapter and verse numbers until much later --the Medieval Age. See "Chapters and verses of the Bible". These additions are just arbitrary organizational aids for the reader and are not considered canonical by major Jewish and Christian authorities . This stuff is not Biblical numerology. If the Almighty left us a message in the digits of pi, it's not a list of Biblical verse numbers.--A. B. ^{(talk • contribs • global count)} 04:55, 15 December 2023 (UTC)

Another thing to keep in mind is that the statement that pi is "approximately equal to 3.14159" is accurate to five decimal places. To be the most accurate to four decimal places, it's not 3.1415, it's 3.1416. ←Baseball Bugs ^{What's up, Doc?} carrots→ 10:20, 15 December 2023 (UTC)

References

Polygons inscribed in a circle of a diameter of 1 unit:

Polygons inscribed in a circle of a diameter of 1 unit:

Let there be an equilateral triangle inscribed in a circle and the measure of the sides of the triangle is a measure of an angle of sin of 60°, as the measures of the angles of the triangles decrease by a multiple of 2 the sides of the triangles increase by a multiple of 2. From 3 sides to 6 sides to 12 sides to 24 etc... The sides of the polygons are a measure of angles with isosceles triangles in each polygons and in Isosceles triangle you only need one angle to find the lenght of the sides of the triangles since the sides correspond to the measure of an angle.

The following statement holds true for isosceles triangles:

$\cos C=1-(\ (\cos A)^{2}+\ (\cos B)^{2})$

199.7.157.97 (talk) 14:04, 14 February 2024 (UTC)

This post was correctly removed recently, then re-added. To the poster: The talk page is not for discussing the subject, or for publishing your own research on the subject -- it is for discussing changes to the article. If you suggest your text should be added, we'd need quotable independent quality sources for it. As it stands, your post should be removed again -- but I suggest, to other editors, that we leave it here for a few days, to see if something relevant comes of it. Otherwise, after that, feel free to delete my post along with the post above! Nø (talk) 14:38, 14 February 2024 (UTC)

I don't understand the direct relevance to π. Seems like a subject for regular polygon (and indeed is already more or less discussed at Regular polygon § Circumradius). –jacobolus (t) 18:47, 14 February 2024 (UTC)

If it seems less discussed, there is a reason behind it. But It should be mentioned that at least why the following statement ${\frac {\sin 60}{2x^{2}}}\times (3)(2^{x})=\pi$ is not mentioned with polygons inscribed within a circle and that is exactly what it represent. 199.7.157.97 (talk) 20:47, 14 February 2024 (UTC)

correction$\ sin{\frac {60}{2x^{2}}}\times (3)(2x^{2})$ 199.7.157.97 (talk) 21:02, 14 February 2024 (UTC)

$\sin {\frac {60}{2^{x}}}\times (3)(2^{x})$ sorry 199.7.157.97 (talk) 21:29, 14 February 2024 (UTC)

Can you be specific about what change to the article you are proposing? I don't really understand what you mean here.

Are you trying to say that you can approximate π by forming a $3\cdot 2^{n}$-gon? So far as we know this was first done by Archimedes in Measurement of a Circle. Expressed in modern notation the relevant identity is ${\textstyle \cot {\tfrac {1}{2}}\theta ={}}$$\cot \theta +\csc \theta ={}$$\cot \theta +{\sqrt {\cot ^{2}\theta +1}},$ starting from $\cot {\tfrac {1}{3}}\pi =1{\big /}{\sqrt {3}},$ which leads to $\cot {\tfrac {1}{6}}\pi ={\sqrt {3}},$ $\cot {\tfrac {1}{12}}\pi =2+{\sqrt {3}},$ $\cot {\tfrac {1}{24}}\pi =2+{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}},$ etc. Archimedes eventually came up with $3+{\tfrac {10}{71}}<\pi <3+{\tfrac {1}{7}}.$ Over the following centuries this was taken to closer approximations, e.g. by al-Kashi who computed 16 digits. This is mentioned in the section Pi § Polygon approximation era. There is a bit more detail at Approximations of π.

We could plausibly say a bit more about it, especially at our article Measurement of a Circle which is currently not very complete. A source is

Miel, George (1983). "Of calculations past and present: the Archimedean algorithm" (PDF). American Mathematical Monthly. 90 (1): 17–35. doi:10.1080/00029890.1983.11971147. JSTOR 2975687.

–jacobolus (t) 22:01, 14 February 2024 (UTC)

I don't know what measurement of unit Archie's used and with polygons inscribed in a circle there aren't any, even considering triangles angles, so pi is without an SI unit.The only measurement to pi are the measures of an angle this is where polygons kick in and Archie's discovery was a guess and he was right.But his work was not finish and many people don't agree or are not sure.For me I am sure that pi is infinite and true. 199.7.157.97 (talk) 23:26, 14 February 2024 (UTC)

Quick question. Who is "Archie"? Dedhert.Jr (talk) 05:16, 15 February 2024 (UTC)

Presumably Archimedes (mentioned in my previous comment). –jacobolus (t) 07:18, 15 February 2024 (UTC)

I don't understand "pi is infinite and true". On the other hand, Archimedes' method can be translated in modern mathematical language as the observation that the half perimeter of the regular $3\cdot 2^{n}$-gons inscribed in and circumscribed to the unit circle are respectively (angles in degrees) $3\cdot 2^{n}\sin {(60/2^{n})}$ and $3\cdot 2^{n}\tan {(60/2^{n})}.$ So

$3\cdot 2^{n}\sin {(60/2^{n})}<\pi <3\cdot 2^{n}\tan {(60/2^{n})}.$

This is a special case of $\sin x<x<\tan x$ for small angles in radians. However, for computing the above approximations of π, one needs a method for computing trigonometric functions. This is this method that is described above by Jacobulus. As this method has only a historical interest, my opinion is that there is no need to give these details in the article. D.Lazard (talk) 15:56, 15 February 2024 (UTC)

Thanks.pi can be expressed in degrees and in radian.What I meant by infinite is that there is no end to the numbers of the circumference of the circle,and by true I meant it's accurate. 199.7.157.97 (talk) 17:07, 15 February 2024 (UTC)

Please note: wikipedia is not a forum. Nø (talk) 11:20, 15 February 2024 (UTC)

Discovery and Invention of Pi

Though pi has been mentioned in many Egyptian and Babylonian civilizations' sources, pi was first said to be discovered by the Archimedes of Syracuse in Greece more than 2200 years ago around 250BC. The Chinese mathematicians approximated π to seven digits, while Indian mathematicians (especially Aryabhata during the Gupta dynasty achieved a five-digit approximation using geometrical techniques. William Jones then devised the Greek symbol π to represent pi in 1706. It was then popularized by Leonhard Euler in 1737. Georges Buffon devised a method to calculate the value of pi based on probability. The invention of calculus allowed the calculation of the digits of pi up to a hundred digits which was sufficient for scientific purposes. Aanchal.Mishra20 (talk) 04:37, 9 March 2024 (UTC)

I don't understand what you are getting at. The article already discusses all of this in greater detail. –jacobolus (t) 06:52, 9 March 2024 (UTC)

Edit request: Mistake in arctan equation

In the section History->Infinite Series there is a mistake in the equation $\pi/4=5 \arctan(1/7)+2\arctan(3/77)$. This should be replaced by the correct version used by Euler $\pi/4=5 \arctan(1/7)+2\arctan(3/79)$, i.e. 79 instead of 77 in the denominator of the second arctan. 134.2.251.3 (talk) 18:11, 14 March 2024 (UTC)

 Done Tito Omburo (talk) 18:20, 14 March 2024 (UTC)

Thank you! 2A02:3038:600:F6A4:1049:1828:E0EC:9F34 (talk) 18:25, 14 March 2024 (UTC)

Thanks. I believe I was responsible for that typo. –jacobolus (t) 18:25, 14 March 2024 (UTC)

Mistake in definition

In the definition using integral, the function of the upper half of the circle is in the denominator. It should just be int_{-1}{1}sqrt(1-x^2)dx if I’m not mistaken. Marsi Viktor (talk) 22:14, 4 April 2024 (UTC)

This is the integral for arc length of the circle, not area. Tito Omburo (talk) 22:25, 4 April 2024 (UTC)

You are right, sorry :) Marsi Viktor (talk) 08:06, 5 April 2024 (UTC)

Mistake in meandering river

I think there's an error in this paragraph:

Under ideal conditions (uniform gentle slope on a homogeneously erodible substrate), the sinuosity of a meandering river approaches π.

But if we look at Posamentier & Lehmann (2004, p. 141):

[…] We then have a sum of semicircular arcs that will be compared to a single semicircular arc with a diameter equal to the distance the full distance the river will have traveled […].

• I = length of the river from the source A to the month B
• AB = (straight) distance between the source A to the month B
• […]
• a = approximation of the river's length […]

$${\frac {\pi }{2}}={\frac {a}{AB}}={\frac {I}{AB}}$$

So, shouldn't it be "the sinuosity of a meandering river approaches π/2"? Vinickw‍✉ 12:18, 11 April 2024 (UTC)

A sinuosity of pi/2 corresponds to gluing together two semicircles into an S shape. I think the claim in the article is that an ideal meandering river is pi, which would be more sinuous than this (so the river tends to close up more, and so there can be oxbows, for example). That seems intuitively reasonable to me, and also not actually contradicted by the above cited paragraph. I think one should consult the first cited source for clarity:

Stølum, Hans-Henrik (1996), "River Meandering as a Self-Organization Process", Science, 271 (5256): 1710–1713, Bibcode:1996Sci...271.1710S, doi:10.1126/science.271.5256.1710, S2CID 19219185.

Unfortunately, I do not have access. Tito Omburo (talk) 21:10, 11 April 2024 (UTC)

Yeah, I'm was talking about it with my professor yesterday, I'm going to read this article once again in detail. By the way, I think you can access it via meta:The Wikipedia Library, you seem to meet the requirements. Vinickw‍✉ 11:50, 12 April 2024 (UTC)

That source has "In the simulations ... [t]hese opposing forces self-organize the sinuosity into a steady state around a mean value of s = 3.14, the sinuosity of a circle (π).... The mean value of π follows from the fractal geometry of the platform." I see no mention of π/2. NebY (talk) 12:14, 12 April 2024 (UTC)

Great, thanks. Related question: is this claimed to be proven, or just conjectured based on simulations? Tito Omburo (talk) 12:49, 12 April 2024 (UTC)

It's explicitly what simulations with a fluid mechanical model show. That's not a direct answer to your question, because I wouldn't talk about such a thing as river sinuosity under ideal conditions being proven or describe such modelling as conjecture. I do fear that the modelling demonstrates that Posamentier & Lehmann's theoretical approach, at least as summarised above, may not be realistic. NebY (talk) 13:10, 12 April 2024 (UTC)

Yesterday my professor (pinging him, maybe he help @Cesarb89) found some files, like this one, it says on page 10 that the value 1.5 (note that π/2 ≈ 1.57) "arbitrarily divides rivers with high sinuosity (greater than 1.5) of those with low sinuosity (less than 1.5)". A meandering river (in Portuguese: canais meandrantes) is a single channel river with high sinuosity (this is also the definition on Meander). This makes sense, after all, if we look at the image on Posamentier & Lehmann (2004), the river is still far from creating oxbow lakes. Vinickw‍✉ 16:11, 12 April 2024 (UTC)

A meandering tale: the truth about pi and rivers by James Grime[1] found an average much lower than π, and despite some outliers (5.88!) relatively close-packed data. I found some of the comments interesting: should immature rivers be excluded, should a meandering river's length be measured with respect to the downhill direction(s), and is it a version of the coastline problem?

Perhaps, rather than our current confident statement that sinuosity approaches π, we should say that various attempts have been made to relate sinuosity to π. NebY (talk) 17:49, 12 April 2024 (UTC)

This is a huge finding. It's important to note that Stølum (1996) uses a simulation of rivers, so it's reasonable to assume that real-world conditions may yield different results. Although pimeariver.com is no longer active, the latest archive, from 31 May 2019, shows that the average of 280 rivers (22 more than what's written on the Guardian) is 1.916, still far from π, and the value is moving away from π/φ. Vinickw‍✉ 19:41, 12 April 2024 (UTC)

I would imagine that the steepness of slope makes a huge difference, and probably also the local geology, type/quantity of plant cover, amount of rainfall, seasonal variation in water quantity, etc.

I bet if you look up sources about hydrology / hydrographic engineering there is probably more detailed/careful technical material than in sources about mathematics per se. –jacobolus (t) 00:46, 13 April 2024 (UTC)

yeah, I'm a little concerned about the way our article approaches this, making it seem much more definitive. This is why I wondered to what extent there is something like a "theorem" as opposed to "someone ran a simulation once". It seems like in-text attribution would be warranted. Tito Omburo (talk) 15:46, 13 April 2024 (UTC)

Perhaps something along these lines?

Analyses of river sinuosity (length relative to distance) have found it to approach π, π/2 and neither.

Given that there are so few studies of any relationship between sinuosity and pi (though I see Stølum has published a little more) and that the results are so varied, I'm not sure our article should give more space to the idea. NebY (talk) 16:08, 13 April 2024 (UTC)

I agree that it is unfortunately too tenuous, and should be removed. Tito Omburo (talk) 09:45, 15 April 2024 (UTC)

You're right, simple removal's better than dwelling on the claim and its contradictions.  Done NebY (talk) 10:30, 15 April 2024 (UTC)

References