Talk:Pi

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

Pi in the Bible

By some people, pi is believed to be encoded in 1 Kings 7:23. The plain text gives a diameter D (1.5 foot) and a circumference 3D (4.5 foot), which would seem to indicate a value of 3 for pi (supposing both measures measure the same circle). However, the word for line/circumference, סָבִיב, is misspelled as סְבִיבָה. These two words have a gematric value of 106 and 111, respectively, so the word for circumference is "inflated" by 111/106. If one inflates the given value (3D) for the actual circumference, inflated the same amount, yields 3D×111÷106 = 3.1415..D 2A02:A45C:FF55:1:2F71:58D8:FF2D:3D8 (talk) 11:09, 21 June 2024 (UTC)

The use of gematric values is to some extent arbitrary and has little, if any, scientific basis. You need a reliable source, not original research, if you want to include this information in the article. Murray Langton (talk) 12:33, 21 June 2024 (UTC)

The תַּנַ״ךְ‎ Tānāḵ (Hebrew Bible) is a work of ethics, history, morality, poetry and tradition; it is not, nor does it pretend to be, a Geometry text. Further, the value given are only to one figure, and they are correct to one figure. Were the text to accurately give the measurements to 20 figures, they would still be incorrect, since π is irrational (in fact, transcendental). This is a long standing rebuttal of a claim that the text never made.

However, there might be a case for including various spurious claims in the popular culture section, including the notorious Indiana pi bill #246 of 1897. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:50, 21 June 2024 (UTC)

Mispelled? I checked online and the text says סָבִיב, not סְבִיבָה. :: -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:50, 21 June 2024 (UTC)

3.1415926535… 192.150.155.229 (talk) 21:35, 3 July 2024 (UTC)

References

Mistake in infinite series

In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss.

Looking at Arndt & Haenel 2006, pp. 194–195:

The Austrian mathematician Lutz von Strassnitzky (1803-1852) exploited an unusual opportunity which came his way. In 1840, the "famous mental computer" Zacharias Dase (1820-1861) visited him in Vienna and attended his lectures on elementary mathematics. [...] Strassnitzky persuaded him to do perform some research which "at least he would be able to use", namely the calculation of π to 200 decimal places. [...] Dase chose the following arctan formula which does not converge as well as Machin's formula[...]: ${\frac {1}{4}}\pi =\arctan {\frac {1}{2}}+\arctan {\frac {1}{5}}+\arctan {\frac {1}{8}}$

[...] He offered to calculate some mathematical tables, so Gauss suggested he should expand the existing prime factorisation tables. Dase took up this suggestion and, with financial support from the Hamburg Academy of Sciences, he calculated the prime numbers in all the numbers between 7 and 9 million.

Vinickw‍✉ 17:40, 27 June 2024 (UTC)

Should it be categorized as a Welsh invention

Two very well-known mathematical symbols, "=" (equality) and "π" (pi) originate from Cymru in the 16th and 18th century respectively. Only the equality sign is classified as a Cymru invention. The concept, calculation, and approximation methods for π far predate the actual symbol for π which we all know today. I am thinking about categorizing "π" as a Cymru invention, but I am unsure because the number, not the symbol, was discovered in antiquity, and much of the discussion concerns about this transcendental number of its decimal expansion.

Additionally, lowercase π could mean something entirely different depending on context, most notably that of the prime-counting function, which I don't recall any of them being introduced by a Cymro. Moreover, some formulas, notably that of the Riemann zeta function, involve multiple occurrences of π with different meanings!

--MULLIGANACEOUS-- (talk) 00:47, 24 July 2024 (UTC)

If π was "invented" by anyone, it was God. If you think God is Welsh, it's fine for you to think so, but you need an RS to put it in the article.

As for the symbol, that comes from the ancient Greeks (though they didn't use it with this meaning).

What you seem to be talking about is that it was a Welsh mathematician who is first recorded to have used π by itself (as opposed to something like ${\frac {\pi }{\delta }}$ or ${\frac {\pi }{\rho }}$) to denote the number.

That's ludicrously far from making π a Welsh invention.

Leave the nationalism where it belongs, which I probably shouldn't say where that is lest I violate the current WP proprieties. --Trovatore (talk) 05:31, 24 July 2024 (UTC)

If Jones was the first user of "π" in this particular way, which he does not claim and is in doubt, the usage originated in London. NebY (talk) 07:19, 24 July 2024 (UTC)

That seems pretty ridiculous to me, to be honest. People had some concept of the ratio between circumference and diameter of a circle going back to ancient Mesopotamia, Egypt, China, etc., and since then there have been hundreds if not thousands of small developments in conceptual/practical understanding and use of this idea. Plucking out the first published appearance of the symbol π used in precisely this way is quite arbitrary. –jacobolus (t) 08:56, 24 July 2024 (UTC)

Wrong symbol used for π

At the end of the "In computer culture" section, the last sentence uses τ instead of π.

Here is an excerpt that begins with the exact issue:

τ has been added to several programming languages as a predefined constant.

I believe this should be π instead. 7agonczi (talk) 19:18, 17 August 2024 (UTC)

 Not done: τ (tau) is correct. See the ref[2] you did at first include with this request and the description of τ (tau) two paras up from the passage you quote.

small typo in first section

π is found in many -->formula(e)<-- in trigonometry and geometry, 160.179.102.212 (talk) 01:40, 20 August 2024 (UTC)

There's no typo there. "Formula" is singular, "formulae" plural, and "many formulae in trigonometry" is correct. NebY (talk) 01:54, 20 August 2024 (UTC)

Are we exaggerating the claim about Weierstrass?

This article says "An integral such as this was adopted as the definition of π by Karl Weierstrass", citing Remmert (2012), but what Remmert explicitly says is "This identity is pointed out by Weierstrass as a possible definition for π" which is a weaker claim. And I don't read German but glancing at Weierstrass (1841) even that seems like it might be a mild exaggeration. In the place where I see this integral what Weierstrass says is (via Google translate) "The integral ⁠$\textstyle {\color {Sepia}\int _{0}^{\infty }{\frac {d\lambda }{1+\lambda ^{2}}}}$⁠ is known to be equal to ⁠$\color {Sepia}{\tfrac {\pi }{2}}$⁠; but it is sufficient to know that it has a finite value, which can be shown as follows." And then later on the page says, "If we now denote the definite integral ⁠$\textstyle \color {Sepia}\int _{-\infty }^{+\infty }{\frac {d\lambda }{1+\lambda ^{2}}}=4\int _{0}^{1}{\frac {d\lambda }{1+\lambda ^{2}}}$⁠ by ⁠$\color {Sepia}\pi$⁠, the value of ⁠$\textstyle \color {Sepia}\int x^{n}dx$⁠ in the sense explained above is equal to ⁠$\color {Sepia}2\pi i$⁠ for ⁠$\color {Sepia}n=-1$⁠ and equal to zero for any other integer value of ⁠$\color {Sepia}n$⁠." I guess this is sort of a definition of π, but it seems a lot more off-hand than implied by our language. (The claim was added in July 2015 by Slawekb/Sławomir Biały.) –jacobolus (t) 19:26, 25 August 2024 (UTC)

This seems consistent with the language in the article, but perhaps the definite article should be replaced by the indefinite: "...adopted as a definition...". For what it's worth, Hardy (1908, Course in pure mathematics) explicitly says "If we define $\pi$ by the equation $\textstyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {dt}{1+t^{2}}}$", without sourcing this to Weierstrass. Tito Omburo (talk) 20:14, 25 August 2024 (UTC)

I guess the way I read Weierstrass's paper is more like "here's an integral which equals π, which I assume every reader already knows how to define, so where convenient we can substitute the symbol π for the integral". I don't get the implication of something like "We shall define the constant π to be the result of this integral ...". YMMV. –jacobolus (t) 20:20, 25 August 2024 (UTC)

New algorithm for calculating Pi

While I'm competent in math, I'll leave this here for others to digest and incorporate into this article:

https://www.scientificamerican.com/article/string-theorists-accidentally-find-a-new-formula-for-pi

--Hammersoft (talk) 13:05, 4 September 2024 (UTC)

What do you mean? 69.166.117.13 (talk) 13:47, 15 September 2024 (UTC)

Chief of sekamaneng

Bereng majara 197.189.137.144 (talk) 17:16, 18 September 2024 (UTC)

Who's the chief of sekamaneng now 197.189.137.144 (talk) 17:18, 18 September 2024 (UTC)