Talk:Pi

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Better sources for "Total curvature" and "Projective geometry"?

The section pi#Role and characterizations in mathematics is pretty good, but there are a couple of subsections within it that seem only tangentially related to pi. A subsection should only be created for a given subject if the subject has a special relationship to pi. Merely using pi or depending on pi is not sufficient. To be included, the subject should have a particularly noteworthy relationship with pi; or it the subject should define pi in a novel way; or the relationship with pi is documented in sources that are about pi; or pi appears in the subject in a surprising way (and is noted by sources).

A couple of subsections do not appear to meet the standard of a strong relationship with pi: pi#Total curvature and pi#Projective geometry. The former has no citations, and the latter has a single cite that is not particular to pi (source is: Ovsienko, V.; Tabachnikov, S. (2004). "Section 1.3". Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups. Cambridge Tracts in Mathematics. Cambridge University Press. ISBN 978-0-521-83186-4. ).

Can anyone provide sources that demonstrate a unique & strong relationship between those two subjects and pi? Noleander (talk) 19:24, 31 March 2025 (UTC)[reply]

For total curvature, Arroyo Garay & Mencia "When is a periodic function the curvature of a closed plane curve?" (Amer. Math. Monthly) looks very relevant. It sources the material quoted here and a converse fact that a periodic function is the curvature function of a closed curve (possibly with different period) iff its integral over the period is a rational multiple of 2π (and that the period is indeed different if it is not an integer multiple). Another textbook source is Prasalov's Differential Geometry where this is one of the earliest topics covered and a whole section is devoted to it (Section 1.3, "The Total Curvature of a Closed Plane Curve", pp. 7-10). Prasalov names the theorem that the total curvature of a simple closed curve is ±2π the "Umlaufsatz" and credits it to Heinz Hopf (1933). The article on total curvature is badly sourced, but the lack of sources here is an artifact of the way its lead is excerpted to form the content here, and does not indicate that there is any lack of sourcing for this material in the literature. —David Eppstein (talk) 20:24, 31 March 2025 (UTC)[reply]

The excerpt template is in my opinion a terrible misfeature, an inappropriate use of transclusion. (Transclusion is excellent for many other purposes on Wikipedia.) We'd be better off copy/pasting the content here and then modifying the content to be relevant for each context (lead of Total curvature vs. subsection here) separately, including adding sources as appropriate here. The content appropriate for a lead section is almost never an appropriate summary for a subsection, and typical Wikipedia sourcing conventions differ between lead vs. body of articles. –jacobolus (t) 22:06, 31 March 2025 (UTC)[reply]

I agree. In fact probably we should just go ahead and do this here. —David Eppstein (talk) 22:16, 31 March 2025 (UTC)[reply]

After sources are identified, the next question (also for Projective Geometry) is: do the sources suggest that the relationship to π is sufficiently unique/important to include in the π article? Phrased another way: there are thousands of formulas that include π, what is the threshold rule to include a formula in the "Role and characterizations in mathematics" section? Do sources state that the relationship to π is especially important?

For example: regarding total curvature: the fact that the integral of the curvature is a multiple of π seems kind of intuitive. Do any sources say that the occurrence of π is especially significant? Do any sources state that π's presence is surprising? Or is the presence of π simply an artifact of the fact that the paths are looping around a point? Noleander (talk) 22:35, 31 March 2025 (UTC)[reply]

It looks like the Total curvature section was added by user:fgnievinski in 2023, so I posted a note on their talk page asking for their input. Noleander (talk) 23:16, 31 March 2025 (UTC)[reply]

Template:Excerpt served its purpose, which was to draw attention to a remarkable relationship between the two topics. Now that someone cares more about it, a custom text will get written. However, it'd be a pity if the main article is not improved with the new sources. I might be inclined to merge the improvements back into the main article. And in case the two texts are similar enough, template:excerpt can always be restored, to avoid future duplicate work. We're here to improve the encyclopedia, not just this specific article. fgnievinski (talk) 23:35, 31 March 2025 (UTC)[reply]

The {{excerpt}} template should, quite frankly, never be used for any purpose on Wikipedia. It is strictly inferior to simple copy/paste in every case. Among other problems, it breaks the link between article history and the article's actual content, making it impossible to read specific timestamped static versions of Wikipedia articles. It should certainly not be used to make texts that were intentionally changed to more appropriately match their separate contexts the same, forcing changes from one place into another where they are inappropriate. –jacobolus (t) 23:52, 31 March 2025 (UTC)[reply]

WP:TFDHOWTO. fgnievinski (talk) 00:09, 1 April 2025 (UTC)[reply]

I agree it would be a good idea to just eliminate the template and avoid the associated problems, but it surely has some dedicated promoters who are active in community discussions, and Wikipedia processes are notoriously incapable of changing anything with nontrivial adoption and motivated defenders, so this sounds like a lot of effort on a big political fight for a low chance of success. All I can do is remove it in articles I am paying attention to, where we can have a concrete discussion in each case, and hope that other editors in other areas of Wikipedia do the same so that the template doesn't infect the whole project too broadly. –jacobolus (t) 19:47, 1 April 2025 (UTC)[reply]

@Fgnievinski - You wrote "...a remarkable relationship between the two topics.." Can you provide a source that discusses the relationship, and specifies how it is remarkable? Otherwise, it is just another equation (out of tens of thousands) that has pi in it. Noleander (talk) 00:06, 1 April 2025 (UTC)[reply]

I think the "remarkable relationship" is that if you travel around a closed curve, the direction you are pointing ends up doing a whole number of turns. This seems relatively inevitable and unremarkable to me, but may still be worth mentioning in this article. –jacobolus (t) 19:49, 1 April 2025 (UTC)[reply]

I guess it's a stronger statement about radians being the natural unit of angles, as the result is not just N. fgnievinski (talk) 20:41, 1 April 2025 (UTC)[reply]

Thanks for the reply. A couple of questions: (1) do you have any sources that remark on the relationshiop (vs just your own thoughts)? (2) What is so special about pi in relation to total curvature, it seems like a trivial and unremarkable relationship.

My feeling is that the appearance of pi here is nothing special ... just one of thousands of formulas related to circles & loops around the origin. Curvature = 1/r. Total integral around a circle = 1/r * 2*pi*r = 2 * pi. If we include total curvature equation in this article, why not include thousands of others? Noleander (talk) 15:54, 2 April 2025 (UTC)[reply]

@Fgnievinski - Pinging on this ... did you find any sources that talk about the relationship between pi and total curvature? Any given editor might find dozens of equations (that include pi) fascinating, but it doesn't seem wise to include them in this article unless some source says that the relationship is somehow notable or speical. Noleander (talk) 00:35, 6 April 2025 (UTC)[reply]

I took a quick crack at clarifying the description of total curvature, but feel free to copyedit or rewrite if you have a better one. –jacobolus (t) 04:32, 1 April 2025 (UTC)[reply]

Trying to get closure on whether or not these two subsections are appropriate for this article: Personally, I don't think either have a very special relationship to pi. But it looks like editor D. Eppstein found a source (based on his comments above). What about the Projective Geometry subsection? I don't see a source identified that states that Projective Geometry has a special/unique relationship to pi. I don't think we should rely on the say-so of the editor that added the section - it is best to find a source that says "Wow! look at this amazing, unexpected occurrence of pi" or "Here is very unique appearance of pi that surprised the great mathematician J. Smith". Best would be a book/article about pi that identifies Projective Geometry (or at least that piece described in the subsection) as having a special relationship to pi. Noleander (talk) 03:33, 19 April 2025 (UTC)[reply]

I am worried that the "projective geometry" section might secretly just be an in-joke, a mathematics made difficult way of writing that the sin and cos functions have half-period π. —David Eppstein (talk) 07:27, 19 April 2025 (UTC)[reply]

I think the point here is that π rather than 2π appears in the spectrum of the Laplace operator on ${\displaystyle RP^{1}}$, which is why I think that this "mathematics made difficult" section serves a useful role. In a way, it's not that sine and cosine have some half-period, but that the most primitive kind of one-dimension geometry implicates π rather than 2π. I.e., π is a natural and necessary quantity in one-dimensional projective geometry. I could see this being dealt with in a different form in the article (e.g., alongside the discussion of Landau), but the inescapability of π even in *projective* geometry, seems worth noting. (Not sure about sourcing, to which I defer to others.) Tito Omburo (talk) 20:22, 19 April 2025 (UTC)[reply]

If it were trying to be expository rather than obfuscatory, it would describe the space of functions ${\displaystyle p\sin(x+q)}$ directly rather than only indirectly describing it as the solution to a certain second-order differential equation. And it would observe that these functions have roots that repeat with period π rather than indirectly defining the subfamily of these functions that have roots at a particular value ${\displaystyle t}$ without ever saying that these are the functions ${\displaystyle p\sin(x-t)}$, and then indirectly stating that the mapping from ${\displaystyle t}$ to this family is periodic with period π (because adding π to the offset negates the parameter ${\displaystyle p}$). —David Eppstein (talk) 20:54, 19 April 2025 (UTC)[reply]

That seems reasonable, although perhaps bringing tangent into the mix. Tito Omburo (talk) 20:59, 19 April 2025 (UTC)[reply]

I know less than zero about Projective Geometry, so have no opinion about how special the relationship is to pi. I'm interested in this mostly from the WP:RS, WP:V, and WP:OR angle. It's kinda cringe-inducing when an editor adds a subsection into this article simply because they think that the relationship is special. So, we need to find a quote from a source where the source says "This is a remarkable situation where pi unexpectedly appears" or "Here is an unusual formula that can be used to define pi in a unique way" or something like that. If no source can be found, maybe the Projective Geometry section should be pruned (until a source is located). Otherwise, the article could bloat over time. Noleander (talk) 21:35, 19 April 2025 (UTC)[reply]

I think the projective geometry section is trivial and should be removed. But I think your argument is bad. What you are looking for is not the sort of thing people write even when it is true. Good expository writing, the sort we should use as a reliable source, would show us that pi appears in some situation, and leave it to the reader to infer that it is significant that it appears because otherwise why would it have been described in that way, rather than telling the reader what to believe. Show, don't tell. —David Eppstein (talk) 21:47, 19 April 2025 (UTC)[reply]

Isn’t this just another way of saying that if we associate pairs of antipodal points on the circle we only end up with a half-circle’s worth of unique pairs? –jacobolus (t) 04:29, 20 April 2025 (UTC)[reply]

I think so, yes. —David Eppstein (talk) 05:46, 20 April 2025 (UTC)[reply]

I think not. A circle requires a metric, but there is no metric here. Tito Omburo (talk) 12:21, 20 April 2025 (UTC)[reply]

Yes there is, the metric on the one-dimensional space of parameters t, in which this paragraph follows trivially from the fact that the absolute value of the sine repeats with with half the period of the sine itself. —David Eppstein (talk) 16:22, 20 April 2025 (UTC)[reply]

There is a natural one-dimensional metric, but not a two dimensional metric. There is no circle: a plane figure in a two-dimensional Euclidean space. Tito Omburo (talk) 19:13, 20 April 2025 (UTC)[reply]

"There is no circle" is false. This passage explicitly considers (in obfuscatory language) the one-dimensional real projective line as being the circle modulo its central symmetry. —David Eppstein (talk) 21:28, 20 April 2025 (UTC)[reply]

Does it? The projective line is the quotient of the vector space V, not a circle inside V. There is a natural symplectic form on V, but I'm not sure about a metric. Maybe ${\displaystyle v^{2}+(v')^{2}}$? I guess the point is that the "circle" isn't obfuscated so much as it is emergent or unnecessary. Tito Omburo (talk) 12:11, 21 April 2025 (UTC)[reply]

I think the section about total curvature can be removed, as it speaks more of N than pi.

By the way: have you tried searching for "popularization of mathematics + pi"? It may turn up the kind of tertiary sources you're after. fgnievinski (talk) 05:08, 20 April 2025 (UTC)[reply]

Footnote with five redundant sources?

The body has the sentence: Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including with which he computed 20 digits of π in one hour. That sentence has a footnote which begins Sandifer ... . That footnote is currently number [82]. That footnote is bulleted and has five distinct sources. Bulleted cites are discouraged in FA articles, but they are acceptable when used in moderation. Footnote [82] is not moderate. I haven't started scrutinizing the five individual sources yet, but it seems like that could be pared down to one or two. If anyone wants to keep all five, speak up. Noleander (talk) 22:49, 6 April 2025 (UTC)[reply]

Surely you should check that the sources actually source what they are claimed to source rather than bean-counting.

We need sources at this point in the article for four separate claims:

• "popularized this series in his 1755 differential calculus textbook"
• "later used it with Machin-like formulae"
• the specific formula pi/4=5 arctan 1/7 + ...
• "computed 20 digits of π in one hour".

If it takes six sources for all of this (one secondary source for each of these four claims and two original publications by Euler retained for historical reference) then it takes six sources. If these claims are not in any of the five sources given, on the other hand, then we have a problem. Whether they are formatted as a single bulleted footnote or as six separate footnotes should be irrelevant.

The Sandifer source does not contain the 1755 date, and does not mention 20 digits in one hour. Its quote "Euler has a whole repertoire of such formula" will suffice for "later used it with Machin-like formulae" (p2), because the formulae described at this point of Sandifer are indeed Machin-like. It does not give the specific formula in exactly the form we use, but does give an equivalent formula with the denominator cleared (p6).

The first Hwang source, "Some Observations on the Method of Arctangents for the Calculation of π" (name mis-ordered; Hwang is the surname), now removed, does source the specific formula in the form we use, credited to Euler, and does also source the "20 digits in one hour" claim for Euler's use of this formula. Its only mention of 1755 is for a related arctangent formula, not this material.

The second Hwang source, "An elementary derivation of Euler’s series for the arctangent function", again mentions the 1755 arctangent formula but is otherwise irrelevant.

I conclude that the removal of the first Hwang source was a mistake; we need both that source and Sandifer. The removal of the second Hwang source was correct. And we still need a secondary source for "popularized this series in his 1755 differential calculus textbook". —David Eppstein (talk) 00:04, 7 April 2025 (UTC)[reply]

Of course the sources should be checked before deletion, as I said "I haven't started scrutinizing the five individual sources yet, ..." My post above was asking for input on the general principle of paring down sources when they are redundant or not needed. Did someone delete some of the sources? They appear to be all intact. Noleander (talk) 02:25, 7 April 2025 (UTC)[reply]

@Noleander someone else removed the two Hwang sources but they're back now —David Eppstein (talk) 04:23, 7 April 2025 (UTC)[reply]

Ah, thanks. Based on your research I suppose all those [82] sources should remain. I may tidy their appearance a bit cuz I recently got dinged in a feature article nomination because my I used bullets in a couple of my citations. Noleander (talk) 05:47, 7 April 2025 (UTC)[reply]

We just use separate footnotes elsewhere like [22][23], so maybe we should do the same here. But I think "An elementary derivation of Euler’s series for the arctangent function" can be removed even if the others can't. —David Eppstein (talk) 06:44, 7 April 2025 (UTC)[reply]

Hwang (2005) isn't necessary for verifying the claim in the article, but is a nice accessible derivation of this formula for any reader curious about mathematical, rather than historical, details. Maybe it would be helpful to add a line of prose to the footnote saying that directly. –jacobolus (t) 17:29, 7 April 2025 (UTC)[reply]

My recommendation is to just put {{pb}} between multiple sources in one footnote. It would probably be a good idea to consolidate some of the other consecutive footnotes in this article, as there's really not much benefit to putting 3 footnotes in a row when none of them is used elsewhere. –jacobolus (t) 17:18, 7 April 2025 (UTC)[reply]

It may be hard to find a source directly saying "popularized", but if you look around most people credit this to Euler, sometimes citing his book specifically (e.g. Bennett 1925, "Another familiar series for arctangent x is the following: ... given by Euler in 1755."; Dodge 1996 "Finally in 1755, using arctan formula, Euler found one that converged much faster than any other: ..."; Arndt & Haenel 2001, "But Euler discovered an elegant way to reduce the the amount of work. He derived the identity ... (5.26) [Euler, 1755], and inserted the arguments of (5.25) into it."; Hwang 2004, "If this identity is used together with Euler's series for the arctangent, discovered in 1755..."). You have to go digging to find out that Newton came up with the same formula 90 years prior. Roy (2021) says, p. 214:

Newton developed his profound ideas on interpolation and finite differences starting in the mid-1670s. In the early 1680s, he applied the method of differences to infinite series and in June/July of 1684, he wrote two short treatises on the topic.... The first chapter of the second treatise dealt with infinite series in a manner similar to that of his early works of 1669 and 1671. However, the second chapter employed the entirely new idea of applying finite differences to derive an important transformation of infinite series, often called Euler’s transformation.... Newton noted one remarkable special case of his transformation: ⁠${\displaystyle \textstyle \tan ^{-1}t=\ldots }$⁠ (10.4)

p. 215:

Newton's transformation (10.4) for the arctangent series is obviously important, so it is not surprising that others rediscovered it, since Newton's paper did not appear in print until 1970. In August 1704, Jakob Bernoulli communicated the ⁠${\displaystyle t=1}$⁠ case of (10.4) to Leibniz as a recent discovery of Jean Christophe Fatio de Duillier. Jakob Hermann, a student of Bernoulli, found a proof for this and sent it to Leibniz in January 1705. This proof is identical with that of Newton's when specialized to ⁠${\displaystyle t=1}$⁠. Johann Bernoulli, and probably others, succeeded in deriving the general form (10.4). Bernoulli, in fact, applied the general form in his 1742 notes on series and thereby derived a remarkable series for ⁠${\displaystyle \pi ^{2}}$⁠ found earlier by Takebe Katahiro by a different technique. In 1717, the French mathematician Pierre Rémond de Montmort (1678–1719) rediscovered Newton's more general transformation (10.3) with a different motivation and method of proof.

later p. 408

Bernoulli gave no further details in his letter, but in 1742 he offered an explanation in the fourth volume of his Opera Omnia. His method was to divide Newton's transformed series for ⁠${\displaystyle \arctan t}$⁠, (10.4), written up in the De Computo, by ⁠${\displaystyle \textstyle 1+t^{2}}$⁠ and then integrate. Recall that Newton had not published his work, so the alternative series for ⁠${\displaystyle \arctan t}$⁠ in powers of ⁠${\displaystyle {\tfrac {t}{1+t^{2}}}}$⁠ was Bernoulli’s rediscovery....

–jacobolus (t) 17:54, 7 April 2025 (UTC)[reply]

Instead of "Euler popularized", it might be more accurate to say something like "This series is now often credited to Leonhard Euler, who included it in his 1755 differential calculus textbook ...". –jacobolus (t) 18:19, 7 April 2025 (UTC)[reply]

I'm not sure where the 20 digits in under an hour claim originates. It can also be found in Borwein & Borwein 1987, p. 340. Edit: an earlier source is Beutel 1913, pp. 46–47. While we're here, I wonder if it's worth explicitly pointing out, in the article text, that Euler was able to do this because ⁠${\displaystyle \textstyle 7^{2}+1=50={\tfrac {1}{2}}(100)}$⁠ which facilitated doing computations with decimal arithmetic. –jacobolus (t) 22:03, 7 April 2025 (UTC)[reply]

I think it is usually worth linking both directly to original sources and to a secondary source about it, whenever possible. There's really no harm in adding links to original sources. Some readers may be curious to see what the historical sources look like and find them to be helpful – at least, I know I find such sources to be helpful, and I wish Wikipedia articles included them much more often. –jacobolus (t) 17:17, 7 April 2025 (UTC)[reply]

"Draft:Three-point-one-four-one-five-nine" listed at Redirects for discussion

The redirect Draft:Three-point-one-four-one-five-nine has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 April 28 § Various draftspace redirects until a consensus is reached. Xoontor (talk) 15:09, 28 April 2025 (UTC)[reply]

Pi article on WP main page: request submitted for pi day 2026

FYI: I nominated this pi article to appear on the main page of WP, on pi day, March 14, 2026. The request is at WP:Today's_featured_article/requests/pending. There is no guarantee that it will end up appearing, because (a) the TFA coordinators may receive other nominations that are more significant, or recently achieved FA status; (b) Another FA article with special attachment to March 14 may get nominated for that day; or (c) the TFA coordinators may think that pi day is too USA-centric, since most of the world uses "14-3" rather than "3-14" to designate March 14th. Noleander (talk) 18:08, 6 May 2025 (UTC)[reply]

You can just propose 31 April as a fallback day for Europeans. ;) –jacobolus (t) 19:18, 6 May 2025 (UTC)[reply]

😂 Noleander (talk) 19:38, 6 May 2025 (UTC)[reply]

Proposal: Expand π’s Role in Higher-Dimensional and Quantum Physics

Hello editors,

I would like to open a discussion on expanding the article's treatment of π to include its deeper role in higher-dimensional geometry and modern quantum theory.

At present, the article presents π primarily through classical lenses — as the ratio of circumference to diameter, and its appearances in trigonometry and basic physics. However, π features far more broadly in advanced mathematical and physical frameworks, particularly in:

- The generalized formulas for the volume and surface area of hyperspheres in n-dimensional Euclidean space, where π appears via gamma functions and dimensional recursion.

- Multivariate Gaussian integrals and normalization constants in higher dimensions, especially in statistical mechanics and field theory.

- Path integrals in quantum mechanics, where π enters through the measure of functional integration.

- Wavefunction normalization and vacuum energy expressions in quantum field theory, where π governs zero-point fluctuation integrals.

- Quantum tunneling and decoherence dynamics, particularly where analytic continuation and imaginary-time formalisms rely on π-dependent exponentials.

Recent theoretical work (Zaino, M., 2025. *The Role of π in Higher Dimensions and Quantum Physics*, DOI: [10.5281/zenodo.15635159](https://doi.org/10.5281/zenodo.15635159)) explores these appearances in the context of a higher-dimensional quantum projection framework. In this model, quantum phenomena like wavefunction collapse and uncertainty arise as consequences of 4D spatial projection, where π plays a geometric and statistical role in describing the curvature and distribution of projected quantum objects. These ideas connect π to entropy gradients, decoherence transitions, and tunneling amplitudes, framed within mathematically formal n-dimensional integrals and projection operators.

Would the community be open to including a brief, sourced subsection under "π in Physics" (or a new section such as "π in Higher-Dimensional and Quantum Theories") that touches on these deeper applications?

I'm happy to prepare a draft that strictly follows Wikipedia's sourcing and neutrality guidelines.

Looking forward to hearing your thoughts.

~~~~ Mazen-Zaino (talk) 21:51, 20 June 2025 (UTC)[reply]

Is that a real reference? Can you provide more bibliographic details? Are you the same Zaino in Zaino (2025)? Tito Omburo (talk) 22:35, 20 June 2025 (UTC)[reply]

The text of that reference appears to be AI-generated, as does the text of the posting here, and nothing by that author is listed in Google Scholar. —David Eppstein (talk) 22:39, 20 June 2025 (UTC)[reply]

Yeah, this should be instablockable honestly. (Not that you would be that brave administrator ;-) Tito Omburo (talk) 22:57, 20 June 2025 (UTC)[reply]

I have blocked people for AI use, but I don't think there's consensus for an insta-block at this level. —David Eppstein (talk) 00:55, 21 June 2025 (UTC)[reply]

AI wrote that text, and AI modifies the work in the paper, and there is a clear mention of AI contraption in the acknowledgment section.

That doesn't revoke the idea, but you do the paper is still a preprint, not yet peer-reviewed or published in a known journal.

I value your feedback on it, and I hope I can contribute effectively Mazen-Zaino (talk) 00:45, 21 June 2025 (UTC)[reply]

Why would you want to insert LLM-generated noise into Wikipedia? –jacobolus (t) 00:51, 21 June 2025 (UTC)[reply]

Thanks for asking. There is a peer-reviewed paper by Mazharimousavi (2025) titled “Schrödinger equation in higher-dimensional curved space” published in European Physical Journal C. It shows how π naturally appears in quantum mechanics when extended to higher-dimensional spaces through normalization integrals and volume calculations. This supports the broader idea that π plays an important role in higher-dimensional quantum physics, which is the foundation for my work.

Another relevant peer-reviewed source is Putz (2009), “Path Integrals for Electronic Densities, Reactivity Indices, and Localization Functions in Quantum Systems,” published in the International Journal of Molecular Sciences. This paper discusses how π plays a fundamental role in the path integral formalism used to describe quantum electronic densities and related properties. It shows that π is not just a classical geometric constant but is deeply embedded in the mathematical machinery of quantum mechanics, especially when dealing with many-body and higher-dimensional integrals.

My paper (Zaino, 2025) builds on these established concepts by proposing a 4D quantum projection framework where π has a direct geometric and statistical role in explaining phenomena like wavefunction collapse and tunneling. However, mine is still a preprint and not peer-reviewed yet. Mazen-Zaino (talk) 01:10, 21 June 2025 (UTC)[reply]

You know, continuing to use AI to formulate responses is not likely to be seen as constructive. —David Eppstein (talk) 02:30, 21 June 2025 (UTC)[reply]

The Putz paper seems to have been chosen because it contains "Path Integral (PI)" rather than any particular discussion of ⁠${\displaystyle \pi }$⁠. –jacobolus (t) 03:02, 21 June 2025 (UTC)[reply]

A Commons file used on this page or its Wikidata item has been nominated for deletion

The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:

• Pi-unrolled-720.gif

Participate in the deletion discussion at the nomination page. —Community Tech bot (talk) 04:52, 12 August 2025 (UTC)[reply]

• This bogus deletion request has already been ended. File remains. Imaginatorium (talk) 05:16, 12 August 2025 (UTC)[reply]

Calculating π value using sine function.

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

---

me@amadeus:~$ python3 -q
>>> 
>>> from math import sin
>>> 
>>> a = 3
>>> for i in range(3):
...     a += sin(a)
... 
>>> print(a)
3.141592653589793
>>> 
>>> # &•
>>> 
me@amadeus:~$

109.185.67.40 (talk) 04:20, 16 September 2025 (UTC)[reply]

Sorry. But we have a policy that it is preferable to provide a reliable source rather than relying on your own research. Also, I don't think it is necessarily important to include your work in the article as well. Dedhert.Jr (talk) 04:23, 16 September 2025 (UTC)[reply]

I do not know if this kind of source could be of help. I am not a Wikipedian. Please forgive me if I am doing something against Wikipedia rules.

Part 1: Zero is odd or even number? | by Aa | Medium: https://medium.com/@gvitalie/part-1-zero-is-odd-or-even-number-6e316b04bcbd 109.185.67.40 (talk) 04:26, 16 September 2025 (UTC)[reply]

Hi. I don't think you have violated the Wikipedia rules. You're basically templating your work in a talk page instead of the real one. Also, what I meant about reliable sources is that you need to provide books or journal articles that are peer-reviewed by reliable publishers. I recommended you to read WP:RS, or in any case, ask WT:WPM. Cheers. Dedhert.Jr (talk) 04:29, 16 September 2025 (UTC)[reply]

I think I found source from The American Mathematical Monthly.

This iterative approach for was considered by Daniel Shanks in a 1-page note: "Improving an approximation for pi." Amer. Math. Monthly 99 (1992), no. 3, 263.

Vol. 99, No. 3, Mar., 1992 of The American Mathematical Monthly on JSTOR: https://www.jstor.org/stable/i315219

A Sufficient Condition for All the Roots of a Polynomial To Be Real: The American Mathematical Monthly: Vol 99 , No 3 - Get Access: https://www.tandfonline.com/doi/epdf/10.1080/00029890.1992.11995845?needAccess=true 109.185.67.40 (talk) 05:49, 16 September 2025 (UTC)[reply]

This is true, and cute, relying on ⁠${\displaystyle \pi -x\approx \sin x}$⁠ for ⁠${\displaystyle x\approx \pi }$⁠ (or even closer, ⁠${\displaystyle \pi -x\approx 2\cos {\tfrac {1}{2}}x}$⁠ or ⁠${\displaystyle \pi -x\approx {\tfrac {2}{3}}\sin x-{\tfrac {1}{3}}\tan x}$⁠). I'm not sure where this would fit in the article though. Did you have a specific place you thought it should be thrown in? –jacobolus (t) 08:05, 16 September 2025 (UTC)[reply]

Let the editors of this article on Wikipedia decide where is the best place to put it, if useful.

Here is an image that shows why that happen: https://miro.medium.com/v2/resize:fit:4800/format:webp/1*TZ51Sv7iFJMzJd-k7if-gQ.png

This image is from article:

Part 1: Zero is odd or even number? | by Aa | Medium: https://medium.com/@gvitalie/part-1-zero-is-odd-or-even-number-6e316b04bcbd 109.185.67.40 (talk) 08:14, 16 September 2025 (UTC)[reply]

Put here in Approximations of π, and I suppose it has been discussed a long time ago. Dedhert.Jr (talk) 09:29, 16 September 2025 (UTC)[reply]

I have no necessary ability to disturb any article. I just come with an idea. Let the professional editors on Wikipedia to decide, where to put that info, if they consider that useful. I am not even trying to. :) 109.185.67.40 (talk) 09:49, 16 September 2025 (UTC)[reply]

P.S.: You could use that sequence of code in Python and also image that describe how the convergence are taking place.

Using sine function:

https://miro.medium.com/v2/resize:fit:4800/format:webp/1*TZ51Sv7iFJMzJd-k7if-gQ.png

Using cosine function:

https://miro.medium.com/v2/resize:fit:4800/format:webp/1*VHOxxmKC4cC2ANCPUp6-Rg.png

I will be glad to see that info made public to everyone, because it is really easy to calculate this very tricky number π. 109.185.67.40 (talk) 09:53, 16 September 2025 (UTC)[reply]

Forget to mention, images and code are made by me as open source. 109.185.67.40 (talk) 09:55, 16 September 2025 (UTC)[reply]

1. Calculating π value using cosine function.

:::::::me@amadeus:~$ python3 -q
:::::::>>> 
:::::::>>> from math import cos
:::::::>>> 
:::::::>>> a = 1
:::::::>>> for i in range(3):
:::::::...     a += cos(a)
:::::::... 
:::::::>>> print(a)
:::::::1.5707963267948966
:::::::>>> print(2 * a)
:::::::3.141592653589793
:::::::>>> 
:::::::>>> # &•
:::::::>>> 
:::::::me@amadeus:~$
:::::::

109.185.67.40 (talk) 10:12, 16 September 2025 (UTC)[reply]

As I have stated recently, please provide reliable sources instead of your own work, an original research. Dedhert.Jr (talk) 10:24, 16 September 2025 (UTC)[reply]

@Dedhert.Jr a reliable source for this was provided above:

Shanks, Daniel (1992). "Improving an Approximation for Pi". The American Mathematical Monthly. 99 (3): 263. doi:10.1080/00029890.1992.11995845. JSTOR 2325063. –jacobolus (t) 17:22, 16 September 2025 (UTC)[reply]

It's not a reliable source, but this blog post also discusses this topic, which it calls the "nerdiest bar trick ever", https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/ –jacobolus (t) 18:46, 16 September 2025 (UTC)[reply]

Another more recent (not RS) source is https://www.hvks.com/Numerical/Downloads/HVE%20Practical%20implementation%20of%20PI%20Algorithms.pdf – see the section about Newton's method. –jacobolus (t) 20:24, 16 September 2025 (UTC)[reply]

If you want a picture to show why this works, I think the clearest is to just show the graphs of ⁠${\displaystyle \pi -x}$⁠ and ⁠${\displaystyle \sin x}$⁠, and point out that they are tangent to each-other at ⁠${\displaystyle x=\pi }$⁠, for example. Something that could be mentioned is that this is the way to "compute" ⁠${\displaystyle \pi }$⁠ as a root of ⁠${\displaystyle x\mapsto \cot {\tfrac {1}{2}}x}$⁠ using Newton's method.

I don't think python code is particularly necessary though. If someone wanted to add this to the article I would recommend making it no more than 2–3 sentences of prose, optionally with a picture. I'm not sure if there's a particularly relevant spot in the article to put such a note though. It's not a practical method for computing digits of ⁠${\displaystyle \pi }$⁠, but more like a curiosity. Aside: to compute ⁠${\displaystyle \pi }$⁠ as a root of ⁠${\displaystyle x\mapsto \sin x}$⁠ by Newton's method, the iteration is ⁠${\displaystyle x_{n+1}=x_{n}-\tan x_{n}}$⁠. –jacobolus (t) 17:15, 16 September 2025 (UTC)[reply]

My first attempt to calculate π needed 10^6 iterations to get just 7 correct digits of π.

So, that sequence of code of just 3 lines of Python code with just 3 iterations for my was a miraculous discovery!

::::import math
::::p = 6
::::n = 10 ** p
::::sum_h = 0
::::for i in range(n + 1):
::::    h = (i * (n - i)) ** (1/2)
::::    sum_h += h
::::pi0 = 8 * sum_h / n / n
::::print(" ", math.pi)
::::print("~", pi0)
::::#   3.141592653589793
::::# ~ 3.141592650263593
::::

109.185.67.40 (talk) 18:16, 16 September 2025 (UTC)[reply]

Yes, forget to mention, that 10^6 iterations needed 2~3 minutes to finish! :D 109.185.67.40 (talk) 18:18, 16 September 2025 (UTC)[reply]

You are leaning on repeated invocations of the sine function, which is more expensive than alternative methods of computing ⁠${\displaystyle \pi }$⁠. If your goal is practical python code, and you want your approximation to be a double-precision float, try import math; math.pi. One place that your method is of practical value is if you want an double-double representation of ⁠${\displaystyle \pi }$⁠, then it is convenient to compute it as [math.pi, sin(math.pi)]. I'm pretty sure if you search you could find a mention of this trick in a reliable source. –jacobolus (t) 18:18, 16 September 2025 (UTC)[reply]

Newton method using tan x looks like that, also very clean and laconic.

::::me@amadeus:~$ python3 -q
::::>>> 
::::>>> from math import tan
::::>>> 
::::>>> a = 3
::::>>> for i in range(3):
::::...     a -= tan(a)
::::... 
::::>>> print(a)
::::3.141592653589793
::::>>> 
::::>>> # &•
::::>>> 
::::me@amadeus:~$
::::

109.185.67.40 (talk) 18:40, 16 September 2025 (UTC)[reply]

If we want to implement our sin x function, we could use e^(jx) = cos x + j sin x series expansion.

::::import math
::::def sin(x):
::::    a = 0
::::    for i in range(30):
::::        a += ((-1) ** i) * (x ** (2*i+1)) / math.factorial(2*i+1)
::::    return a
::::a = 3
::::print("{:<20} {:<20}".format(a, sin(a)))
::::for i in range(3):
::::    a += sin(a)
::::    print("{:<20} {:<20}".format(a, sin(a)))
::::#
::::#
::::/home/me/Projects/AI/.venv/bin/python /home/me/Projects/AI/Bb.py 
::::3                    0.1411200080598671  
::::3.141120008059867    0.0004726455123287083
::::3.1415926535721956   1.759734884346061e-11
::::3.141592653589793    3.3280566969799443e-16
::::Process finished with exit code 0
::::

109.185.67.40 (talk) 18:44, 16 September 2025 (UTC)[reply]

For what it's worth, this method does converge quite rapidly to ${\displaystyle \pi }$, in terms of black-box function evaluations (repeatedly computing the sine). For example, going up to seven iterations is enough for at least 1000 digits of pi. This depends on properly accounting for multi-precision floats, i.e., not using python's built-in "math" library (which introduces roundoff errors), and using enough terms in the power series of sine so that no precision is lost. But it's not really practical. Cute though. Tito Omburo (talk) 18:47, 16 September 2025 (UTC) - add: it looks to me like the absolute error in the approximation drops off like ${\displaystyle \exp(-e^{kn})}$ where ${\displaystyle n}$ is the number of iterations, and ${\displaystyle k}$ is slightly bigger than one. Tito Omburo (talk) 19:37, 16 September 2025 (UTC)[reply]

Yes. Cute though. I am agree. That made my day. I was extremely happy when I found that, because I come to this idea myself. :) 109.185.67.40 (talk) 18:56, 16 September 2025 (UTC)[reply]

Why? I tried to figure up the most impossible (oh, my God, π related!) of solving The Squaring the Circle and The Circling the Square opposite problems! :D Yes. Because of that. 109.185.67.40 (talk) 18:58, 16 September 2025 (UTC)[reply]

Thanks for sharing! Tito Omburo (talk) 19:07, 16 September 2025 (UTC)[reply]

You are always welcome.

I have e-mail correspondence with somebody from India, that spend all his entire life (70 years) to figure up The Squaring the Circle and The Circling the Square opposite problems. I wrote about Him an article. Yes, somehow π related.

Reddivari Sarva Jagannadha Reddy successfully succeeded to solve absolutely The Squaring the Circle problem and The Circling the Square opposite problem. | by Aa | Sep, 2025 | Medium: https://medium.com/@gvitalie/reddivari-sarva-jagannadha-reddy-successfully-succeed-to-solve-absolutely-the-squaring-the-circle-2acbc43f9a99 109.185.67.40 (talk) 19:28, 16 September 2025 (UTC)[reply]

I don't see how 'Squaring the Circle' can possibly be solved absolutely due to the fact that pi is transcendental and irrational and can not be calculated exactly. The reference you supply appears to calculate an approximation to pi, and then delves into philosophy. Can you explain a little more. Murray Langton (talk) 21:28, 16 September 2025 (UTC)[reply]

Most of the rest of the content on that blog is incoherent pseudoscience. The author has been repeatedly reverted for trying to start discussions about whether zero should be considered even or not. You shouldn't think too hard about it. –jacobolus (t) 01:45, 17 September 2025 (UTC)[reply]

Yes. We can't yet. But what we can using π value at least? What if trying to use π will reveal us a perspective to figure that problem? I do not know, that is everything I know, but I can try to figure it up. Socrate? 109.185.67.40 (talk) 05:17, 17 September 2025 (UTC)[reply]

This method has cubic convergence. –jacobolus (t) 20:02, 16 September 2025 (UTC)[reply]

An earlier source is HAKMEM (1972):

ITEM 142 (Salamin):

Some super-fast convergents to pi if one already has a super-fast computation of trig functions.

X approx pi:

• X <- X + sin X, epsilon <- epsilon^3/6
• X <- X - tan X, epsilon <- -epsilon^3/3

X approx pi/2:

• X <- X + cos X, epsilon <- epsilon^3/6
• X <- X + cot X, epsilon <- -epsilon^3/3

–jacobolus (t) 20:09, 16 September 2025 (UTC)[reply]

Oh mama mia! Cubic convergence! Extra super class! Thank You for information. So cool! 109.185.67.40 (talk) 20:15, 16 September 2025 (UTC)[reply]

I am not smart, therefore I prefer simplicity, things that I can do myself, understand easily.

I can't understand over-complicated thoughts.

${\displaystyle e^{x}=\lim \limits _{n\to \infty }\sum \limits _{k=0}^{n}\int ^{(k)}1\cdot dx}$

109.185.67.40 (talk) 06:38, 17 September 2025 (UTC)[reply]

Calculating π using imaginary part of exponentiation e^x.

:me@amadeus:~$ python3 -q
:>>> 
:>>> import cmath
:>>> 
:>>> a = 3
:>>> for i in range(3):
:...     a += cmath.exp(a * 1j).imag
:... 
:>>> print(a)
:3.141592653589793
:>>> 
:>>> # &•
:>>> 
:me@amadeus:~$
:

109.185.67.40 (talk) 15:22, 17 September 2025 (UTC)[reply]

This just looks like yet another way to calculate an approximate value of pi, possibly based on original research, unless you can come up with a reliable reference.. There are several such methods already in the article, with references. Murray Langton (talk) 16:51, 17 September 2025 (UTC)[reply]

# Euler and Newton relations make a tandem to calculate π value, using successive exponential tangent.

:me@amadeus:~$ python3 -q
:>>> 
:>>> import cmath
:>>> 
:>>> a = 3
:>>> for i in range(3):
:...     a -= cmath.exp(a * 1j).imag/cmath.exp(a * 1j).real
:... 
:>>> 
:>>> print(a)
:3.141592653589793
:>>> 
:>>> # &•
:>>> 
:me@amadeus:~$
:

109.185.67.40 (talk) 01:33, 19 September 2025 (UTC)[reply]

If calculations are made easily, only then we can have access to technological miracles!(talk) 04:49, 19 September 2025 (UTC)[reply]

Hi 109.185.67.40, You seem to have come up with numerous code fragments which calculate an approximate value of pi. What is the point of these? What changes would you like to see in the pi article? If you can't answer these questions, then I think that this discussion should terminate, since the idea of this talk page is to discuss improvements to pi. Murray Langton (talk) 06:18, 19 September 2025 (UTC)[reply]

Dear and lovely esteemed friend @Murray Langton, I am not a Wikipedian, so I put these sparks of knowledge here as a possible suggestion, that if useful, professional editors of Wikipedia, if they would consider that info somehow useful, will add them somewhere as a note. I have no ability to disturb the integrity of any article written by Wikipedia Professional Editors. I just bring what I saw, that it is easy for me, because I am not smart, I am not intelligent, I am not a Wikipedian. Please forgive my behavior, if what I did is against Wikipedia rules.

109.185.67.40 (talk) 06:51, 19 September 2025 (UTC)[reply]

Note: Ip has a two years block for talk page abuse at Talk:Parity of zero - DVdm (talk) 13:05, 20 September 2025 (UTC)}}[reply]

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

The Egyptians…

There is hardly an article about inventions that does not cite the Egyptians. It is so common and boring, as the claims are never made by specialists. So, there is this pi thing. The Rhind Mathematical Papyrus does not show that the Egyptians knew of π as a constant. What it contains is a practical rule for computing the area of a circle: reduce the diameter by one ninth, then square the result. In modern notation this gives A = (8/9 d)^2, which happens to correspond to π ≈ 3.16, but the Egyptians themselves did not isolate or conceptualize π. They were not seeking “accurate values of π,” only reliable procedures for land measurement and construction. Presenting this as knowledge of π projects a later mathematical concept onto a culture that did not think in those terms. Riyadi (talk) 21:39, 19 September 2025 (UTC)[reply]

It's a common problem: given that ancient civilizations did not really have a modern concept of number, how could they have a concept of a number such as π. The concept of number was shaped by historical need, like the horse collar or printing press, and came later than the invention of agriculture or writing. Tito Omburo (talk) 22:39, 19 September 2025 (UTC)[reply]

Feel free to rephrase here. It seems likely to me that Egyptian scribes knew that the circumference of a circle was proportional to its diameter and that the area was proportional to that of a circumscribed square. But this isn't written down anywhere, so we can only speculate. As you say, there's no indication that the method implicit in the solution of this problem is supposed to accurately represent a proportionality constant, but only that it's a good practical method for computing an (approximate) area. –jacobolus (t) 00:07, 20 September 2025 (UTC)[reply]

This is discussed in doi:10.1016/j.hm.2011.06.001. It and other sources give the Egyptian area formula as ${\displaystyle \left({\tfrac {8}{9}}d\right)^{2}}$. So for the Egyptians the multiplication by a factor (our ${\displaystyle {\sqrt {\pi }}}$) comes before the squaring, not after. Its author gives weaker evidence that the Egyptians used essentially the same approximation for circumference, but writes "However, there is no written evidence that the scribes either thought in terms of, or worked things out in terms of, the specific concept we call ${\displaystyle \pi }$." This is why our article writes that the Rhind Papyrus "has a formula for the area of a circle that treats ${\displaystyle \pi }$ as" rather than more directly that they approximated ${\displaystyle \pi }$ as something. But perhaps there is some other wording that makes this point more clearly. —David Eppstein (talk) 01:31, 20 September 2025 (UTC)[reply]

255-character redirect listed at RfD

A redirect with a name consisting of 255 characters (254 digits of pi) has been nominated for discussion aka deletion. Please see the discussion of the redirect. This was originally posted by A1Cafel but the standard message contained the full title in a couple of places which made it hard to follow. Johnuniq (talk) 12:12, 22 September 2025 (UTC)[reply]

π understood at its most fundamental level

Pi is the COUPLING CONSTANT between the 1st-to-2nd Dimension. Diameter is a 1-dimensional property. Circumference is a 2-dimensional property. The circle is the most basic 2-dimensional object, which can be defined by two parameters: a point, and a radius.

Square Root of 2 is the ORTHOGONALITY CONSTANT. It is the ratio which specifies the Right Angle relationship between two dimensions within the 2nd dimension. Two dimensional space, in general, need not be orthogonal. But in the special case where it is, then the square root of 2 gives this relationship.

i is not imaginary. There is nothing imaginary about i. It is the orthogonality between the 1st and 2nd dimension. So the much more accurate terminology would be to call it 'o', for orthogonal. And the constant of this orthogonality is, again, the Square Root of 2.

More most fundamental explanations:

e is the eigenvalue of the Derivative /Integral processes.

Similarly, 0 is the eigenvalue of the Addition /Subtraction processes. 1 is the eigenvalue of the Multiplication /Division processes.

The fundamental concept of the eigenvector is that the output of a process comes through essentially unchanged from the input.

This article can be improved by explaining that this is the most fundamental explanation of what Pi is. There are many times when pi appears, and people have scratched their heads as to what it has to do with circles. The answer is that it has to do with the 1-to-2 dimension coupling. Far more basic than the specific case of a circle. --Tdadamemd20 (talk) 12:13, 27 September 2025 (UTC)[reply]

Per WP:OR adding this to the article requires, at least, a WP:reliable source where this appears explicitly. Can you provide such a source? D.Lazard (talk) 13:22, 27 September 2025 (UTC)[reply]

The concept of a circle is directly related to the concept of distance, and vice versa (because the circle is the set of points at equal distance from a center point). The concepts of circles / distances are equivalently "basic", in the sense that either can be defined in terms of the other.

I'm glad your explanations are helpful for your own understanding, but adding this kind of thing would be very confusing for readers. Making up non-standard terminology and symbols is a mistake for resources intended for a general audience, because people who know the standard symbols will be confused by their replacement, and people who don't know the standard symbols but learn a different one will be confused when they encounter a conventional treatment of the same topic. Some of your phrases here, such as "eigenvalue of the Derivative /Integral processes", are not well defined and are incoherent on their face to anyone else, though I'm sure if you had an extended conversation with someone they could eventually figure out what you are getting at. –jacobolus (t) 16:48, 27 September 2025 (UTC)[reply]

I would argue that π doesn't really involve geometry, but is instead the constant relating the frequency spectrum of the generator of the Lie group R/Z to its dual group. (This would actually be 2π. The constant π is obtained projectively, although this content was recently excised from the article.) That is certainly more primitive than anything to do with the two-dimensional Euclidean geometry of circles. Tito Omburo (talk) 19:48, 27 September 2025 (UTC)[reply]

jacobolus: "...though I'm sure if you had an extended conversation with someone they could eventually figure out what you are getting at."

Eigenvalue /eigenvector was succinctly explained: "The fundamental concept of the eigenvector is that the output of a process comes through essentially unchanged from the input." My expection is that this would be crystal clear to anyone familiar with eigenvalues /eigenvectors.

You also: "...but adding this kind of thing would be very confusing for readers."

The intent was to help clarify. Not to confuse. So if this would be a step backward, I will not be making any edits to this article for the foreseeable future. Perhaps I will check back here after some time, and if a consensus appears to have gelled that this would actually be a benefit, then I will look at updating. --Tdadamemd20 (talk) 20:32, 27 September 2025 (UTC)[reply]

We (TINW) understand full well what an eigenvalue is. We also understand that the eigenvalue of the derivative on the function ⁠${\displaystyle x\to e^{x}}$⁠ is ⁠${\displaystyle 1}$⁠ and that the eigenvalue for ${\displaystyle x\to e^{2\pi {i}x}}$ is ⁠${\displaystyle 2\pi {i}x}$⁠; what we do not understand is what eigenfunction with eigenvalue ⁠${\displaystyle e}$⁠ you were referring to and how it relates to π. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:57, 29 September 2025 (UTC)[reply]

I agree that section § Definition is rather confusing: the habits of teachers has nothing to do there. It must made clear that π can be defined geometrically as the ratio as in the beginning of the section. It must also provide a definition in terms of analysis that is clearly independent from geometry, and explains the importance of π in analysis. IMO, such a definition is: π is the half period of every solution of the differential equation ⁠${\displaystyle y''+y=0}$⁠. These solutions include the sine, the cosine and ⁠${\displaystyle x\mapsto e^{ix}}$⁠. As the latter is a group homomorphism ⁠${\displaystyle \mathbb {R} ^{*}\to \mathbb {R} /\mathbb {Z} }$⁠, this makes easier to explain the definitions implying groups. D.Lazard (talk) 16:34, 28 September 2025 (UTC)[reply]

Shirley ⁠${\displaystyle \mathbb {R} \to \mathbb {T} }$⁠, although there is a natural isomorphism ⁠${\displaystyle \mathbb {R} /\mathbb {Z} \cong \mathbb {T} }$⁠. I'm not sure what ⁠${\displaystyle R^{*}}$⁠ is in this context. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:45, 28 September 2025 (UTC)[reply]

D.Lazard's italicized definition gets to the point of the connection to analysis much more directly than the version that was removed (see discussion in § Better sources for "Total curvature" and "Projective geometry"? and I would have no objection to including this wording. —David Eppstein (talk) 18:47, 28 September 2025 (UTC)[reply]

I think Lazard's account of the half-period is not quite right, and perhaps understated. The function ${\displaystyle x\mapsto e^{2\pi ix}}$ is a generator of the character group of ${\displaystyle \mathbb {R} /\mathbb {Z} }$ (the other being ${\displaystyle e^{-2\pi ix}}$). Thus it arises naturally in the Pontrjagin dual of ${\displaystyle \mathbb {R} /\mathbb {Z} }$, a group we all "know" (abstractly) as the group of integers under addition, but actually involves the constant 2π as the basic frequency unit. Now, one can ask the dual question, namely what is the "correct" one-dimensional torus, the one for which ${\displaystyle x\mapsto e^{ix}}$ is a character? (${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$.) The latter approach is that favored in basic analysis analysis textbooks (essentially going back to Landau), whereas Bourbaki emphasizes the former. While both views define the fundamental unit of frequency, I tend to prefer Bourbaki's view to Landau's, because although they are equivalent, the appearance of frequency seems a bit more natural (as the old saying of Kronecker goes about the natural numbers).

Now ${\displaystyle x\mapsto e^{i\pi x}}$ is not a character of ${\displaystyle \mathbb {R} /\mathbb {Z} }$, but is twisted by ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$. (This latter fact is especially important for constructing projective representations of certain groups such as ${\displaystyle SL(2,\mathbb {R} )}$.) The article doesn't really make this case, but the role of π (as opposed to 2π) is often enormously significant in representation theory, and I think is a view that deserves more treatment here. The advantage of the old projective section was that it constructed this fundamental projective representation naturally, with π, rather than 2π appearing, but not really leaving the old-school Landau perspective. This viewpoint seems almost too trivial in the Landau perspective to deserve mention, but is an important and deep-ish thing about π. (The projective representation is just sitting there, but it can be difficult to understand in other contexts like number theory.) Tito Omburo (talk) 21:37, 28 September 2025 (UTC)[reply]

The old version said the same thing you said was trivial, dressed up in obscurantist terminology and notation to make it both difficult to understand (WP:TECHNICAL) and have the appearance of nontriviality. —David Eppstein (talk) 22:00, 28 September 2025 (UTC)[reply]

Ok, so the content is good but the terminology bad? Also the connection with projective representations is important and deserves mention. Tito Omburo (talk) 22:46, 28 September 2025 (UTC)[reply]

I agree that the text of the section you added seems obscurantist and inaccessible. This is a fairly straight-forward idea – namely, if the full circle has length 2π, then if we consider antipodal points to be the same, our resulting semicircle of unique points now has only half the length – wrapped up in a bunch of unexplained jargon and symbols to make it sound like something more profound. I'd recommend this section be removed unless its main idea can be explained in a way accessible to a first year undergraduate student. –jacobolus (t) 01:41, 29 September 2025 (UTC)[reply]

Also needs to be backed by sources: the listed source (§ 1.3 of Ovsienko & Tabachnikov 2004) doesn't mention π at all. –jacobolus (t) 01:48, 29 September 2025 (UTC)[reply]

But the difference between π and 2π is actually extremely important, and subtle in Fourier analysis! (This article is about the former and not the latter, fwiw.) Tito Omburo (talk)•

Do you have a reliable source explicitly describing the subtle and important difference between π and 2π, ideally in a way that doesn't require two PhDs to understand? –jacobolus (t) 01:51, 29 September 2025 (UTC)[reply]

Kinda hoping I can get better engagement on this. It seems like a real deficiency of the article, that π appears in the Weil representation (as opposed to 2π). (I can, of course, cite sources that this is the case, although it can be found in any modern textbook om theta functions. This is how π appears in one of the sole-and certainly most famous-modern treatises on the subject of this article.) There is a comparatively trivial explanation of this, that doesn't require advanced mathematics. We could actually make something good? Tito Omburo (talk) 02:12, 29 September 2025 (UTC)[reply]

Sorry, I shouldn't be flippant. But seriously: do you have a source? –jacobolus (t) 03:12, 29 September 2025 (UTC)[reply]

I do, but the extent to which this is synthesis is indeed a question for the ages. It's something that "everyone knows" (because it's at some level "trivial"), which can be one of the most things to pin down. There is no shortage of references using both π and 2π in different places, and I suggest that we take that seriously here. Tito Omburo (talk) 03:16, 29 September 2025 (UTC)[reply]

That sounds entirely like synthesis to me. Wikipedia articles about basic topics don't seem like the appropriate venue for first publication of things that "everyone" with the relevant PhD knows. –jacobolus (t) 03:38, 29 September 2025 (UTC)[reply]

Again, let's try to be real here. Double covers are pretty standard fare in metaplectic/spinor worlds. π is important because it realizes this double cover. There are sources. (E.g., particle physics and spin one-half.) It is not synthesis to say anything that we currently do. The only question is how to glue things together without drifting into synth. Tito Omburo (talk) 03:42, 29 September 2025 (UTC)[reply]

Why can't that be said in ~2–3 sentences that point out that a semicircle only has half the circumference of a circle. –jacobolus (t) 04:25, 29 September 2025 (UTC)[reply]

What? Go and read what I wrote, from the beginning. (Hint: I said nothing about two-dimensional figures like circles.) Thanks. Tito Omburo (talk) 04:27, 29 September 2025 (UTC)[reply]

My take is that what you wrote is effectively equivalent, except you wrapped it up in like 5 layers of jargon, notation, and esoteric concepts, requiring years of dedicated study to even make basic sense of, the effect of which is to hide the basic 2:1 relationship under discussion rather than illuminating it.

Also: the projective line is entirely about circles. Or if you like, entirely about relative orientations in the plane. If you don't like a statement about circles with associated opposite points, we can instead say: oriented lines through the origin in the Euclidean plane can be represented by an angle relative to a reference oriented line, and this angle can in turn be represented by an angle measure (a signed circular arclength) between (−π, π]. Un-oriented lines through the origin in the Euclidean plane can likewise be represented by an angle, but a line is fixed by a half-turn rotation about one of its points, so all possible unoriented lines can be represented by an angle measure between (−π/2, π/2]. –jacobolus (t) 04:52, 29 September 2025 (UTC)[reply]

We may be talking past each other. I'm discussing the role of π and 2π in Fourier analysis and projective representations (e.g., spin-1/2 systems), which is not the same as talking about the arc length of circles. The 'half circumference' analogy isn't what I'm describing. 2π isn't really about circles: it's the fundamental frequency of Fourier analysis, whereas π is more subtle. (Yes, it happens to be true that π is half the arclength of a certain specific curve embedded in a certain specific metric space, but that's hardly the point here.) For example, the dual group of ${\displaystyle \mathbb {R} /\mathbb {Z} }$ requires the use of 2π. There are no circles anywhere: this is a one-dimensional object. There is no two-dimensional circle, one isn't dealing with any rectifiable curve in any mertic space. No integrals appear anywhere. The quantity π (as opposed to 2π) arises in the representations of the double cover ${\displaystyle \mathbb {R} /(\mathbb {Z} /2)}$. Why this representation is interesting is borne out by sources (Weil/Schrodinger/spinor). E.g., why have an article about ${\displaystyle 2\pi /2}$ and not ${\displaystyle 2\pi /7}$? It's because the former is genuinely more important. (Incidentally, this is already connected with the Heisenberg group, which involves area, rather than arclength, in a nontrivial way.) Tito Omburo (talk) 05:08, 29 September 2025 (UTC)[reply]

I disagree with you that "there are no circles" in a periodic interval. –jacobolus (t) 06:11, 29 September 2025 (UTC)[reply]

As an aside, Periodic interval should probably be an article. –jacobolus (t) 06:17, 29 September 2025 (UTC)[reply]

A circle, as an object of geometry, is a figure in two-dimensional Euclidean space. The group ${\displaystyle \mathbb {R} /\mathbb {Z} }$ is not this. For one thing, it's a group. That's something different than what most people mean when they say "circle". Tito Omburo (talk) 11:55, 29 September 2025 (UTC)[reply]

It's isomorphic to the group of rotations of a circle. It is commonly thought of as representing angles numerically, defined and understood as corresponding to circular arc lengths. So it's not exactly as if circles are deeply buried here. –jacobolus (t) 13:12, 29 September 2025 (UTC)[reply]

As an example turned up in a quick search, here's Pontrjagin 1934, JSTOR 1968438:

Let K be the continuous group of rotations of a circle, which we will consider as an additive group of real numbers defined up to an additive integer, ...

–jacobolus (t) 13:30, 29 September 2025 (UTC)[reply]

By this logic, the dihedral group is the same thing as a regular polygon. Tito Omburo (talk) 21:26, 29 September 2025 (UTC)[reply]

If someone said "the dihedral group isn't really about regular dihedrons" I would also disagree, and would point out that a dihedral group is (or is isomorphic to, if you prefer) the group of geometric transformations which fix a dihedron. –jacobolus (t) 22:36, 29 September 2025 (UTC)[reply]

I mean, the dihedral group is surely a group, and surely is defined independently of geometry. The fact that it acts on a certain set is a question of representation theory. Just as the group ${\displaystyle R/Z}$ happens to act on any periodic dynamical system. Tito Omburo (talk) 22:44, 29 September 2025 (UTC)[reply]

You could define a dihedral group in a wide variety of ways, but whatever definition you choose it is still fair to say that the group is closely related to the symmetries of a dihedron; indeed, if you pick some other definition, then congratulations, you have just found a 2-way relationship between some different mathematical system and the geometry of a dihedron, which allows you to answer questions about one system in terms of another (e.g. answer questions about the arithmetic of a certain set of complex numbers geometrically or answer certain geometry questions arithmetically). –jacobolus (t) 22:56, 29 September 2025 (UTC)[reply]

So we agree? All of the basic definitions of π should be included, including those that do not involve geometry? Tito Omburo (talk) 22:58, 29 September 2025 (UTC)[reply]

That really depends what you mean. I think we should try to make the top ~3 sections of this article accessible to middle or perhaps high school students, and defer anything which is not so broadly accessible to somewhere later. The current state of the "Definitions" section doesn't really seem acceptable to me. It is too long, too awkward, and too complicated. –jacobolus (t) 23:17, 29 September 2025 (UTC)[reply]

I am not sure that we talk of the same thing. I am talking of section § Definition, and Tito talks of things that have their place as subsections of § Role and characterizations in mathematics. Also, Tito's posts suggest that I should not have used "the half period" and I should have written: π is half the fundamental period of every solution of the differential equation. This should avoid any way of misunderstanding the sentence. D.Lazard (talk) 09:22, 29 September 2025 (UTC)[reply]

My first post above was regarding ${\displaystyle \mathbb {R} /\mathbb {Z} }$. The function ${\displaystyle e^{ix}}$ is not a group homomorphism of ${\displaystyle \mathbb {R} /\mathbb {Z} }$. 2π only appears as the fundamental spectral unit of this group, not its period (Bourbaki). Or, 2π is the period which determines the "correct" one-dimensional torus, for which ${\displaystyle e^{ix}}$ is a group homomorphism (Landau): the group is ${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$. My second paragraph in reply to you are musings regarding how to get π as a spectral value or period (not just 2π). The rest of this discussion is me attempting, rather poorly, to explain that the difference between π and 2π, while seemingly trivial, does have rather profound consequences in Fourier analysis. Tito Omburo (talk) 12:04, 29 September 2025 (UTC)[reply]

⁠${\displaystyle x\to e^{ix}}$⁠ is indeed a group homomorphism fron the additive group of ⁠${\displaystyle \mathbb {R} }$⁠ to the multiplicative group of ⁠${\displaystyle \mathbb {C} }$⁠. Its image is the unit circle in ⁠${\displaystyle \mathbb {C} }$⁠ and its kernel is ⁠${\displaystyle 2\pi \mathbb {Z} }$⁠. By the isomorphism theorem, this identifies the complex unit circle with ⁠${\displaystyle \mathbb {R} /2\pi \mathbb {Z} }$⁠. This suffices to explain why ⁠${\displaystyle 2\pi }$⁠ is more important than ⁠${\displaystyle \pi }$⁠ in many areas, including Fourier analysis. No need to introduce "spectral units", abstract tori, etc. for saying that ⁠${\displaystyle \pi }$⁠ and ⁠${\displaystyle 2\pi }$⁠ are not the same. D.Lazard (talk) 13:20, 29 September 2025 (UTC)[reply]

Again, you had incorrectly said that the kernel of e^ix was Z. This was obviously wrong, but it does point to a subtle difference in the two definitions in analysis: identifying the period of the torus (Landau) vs identifying the spectral units (Bourbaki). In the first, the group is R/2πZ, and in the second it is R/Z and pi appears via the spectrum. These are all mentioned already in the definition section. It would be nice if you would at least acknowledge that these are not the same thing, and are slightly different philosophically: in one π appears in the time domain, and the other, the frequncy domain. Tito Omburo (talk) 18:22, 29 September 2025 (UTC)[reply]

If you have uniform circular motion along a unit-radius circle, or some abstracted equivalent, possible choices include (1) set up the units so that the motion has constant speed of 1 radian per time unit (taking ⁠${\displaystyle 2\pi }$⁠ time units per revolution), or (2) set up the units so that there is 1 revolution per time unit (which makes the speed ⁠${\displaystyle 1/2\pi }$⁠ radians per time unit), or (3) pick some arbitrary other units. Being clear about which units are in use is important so that the arithmetic works out properly, but the difference between these choices seems pretty trivial, insofar as the change of coordinates is just a linear scaling. I don't think it's really worth belaboring this point in this article, since it's likely to be confusing to some readers unless explained in significant detail. There's some relevant discussion at Fourier transform § Angular frequency (ω). –jacobolus (t) 19:25, 29 September 2025 (UTC)[reply]

The article already includes the required detail, and references (to Landau and Bourbaki). Tito Omburo (talk) 19:54, 29 September 2025 (UTC)[reply]

Our section § Definition is getting increasingly awkward, wordy, and inaccessible, full of unexplained jargon, name dropping, etc. I think we should try harder to keep the top few sections simple and streamlined. (In my opinion, the version of this section in 2012 when this article was given its gold star, 495660335, was significantly better for most anticipated readers.) –jacobolus (t) 20:04, 29 September 2025 (UTC)[reply]

I think the section should summarize and cite the best sources, not withstanding you personal point of view that there is no difference between the definitions offered by Landau, Bourbaki, or Archimedes. Tito Omburo (talk) 21:24, 29 September 2025 (UTC)[reply]

Please do not attribute to people views that they never expressed and claims that they never made. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:15, 29 September 2025 (UTC)[reply]

My understanding of jacobolus's basic thesis is that all definitions of π ultimately boil down to the arclength of a certain curve in Euclidean space. Their every post appears to attempt to collapse all nuance in the three main definitions to this basic thesis, and it is not supported by the literature. Tito Omburo (talk) 22:19, 29 September 2025 (UTC)[reply]

I have three basic priorities:

(1) We should make the first 2–3 sections of this article (which appeals to a mass audience including, say, middle school students) as accessible and straightforward as possible, which might require splitting apart material that might otherwise seem to belong under the same heading, such as "definitions" of π.

(2) Following Wikipedia policies about verifiability, sources, and original research, we should pass along claims directly made in reliable sources instead of giving Wikipedians' personal interpretations about what they might have really meant but didn't say or making a combined interpretation of several different sources' claims which were not directly made together.

(3) We should try to reduce the amount of unexplained jargon and conceptual demand, even deep in this article, to the extent we can while still conveying the claims found in reliable sources. –jacobolus (t) 22:49, 29 September 2025 (UTC)[reply]

We do that, to some extent, although the lede currently leans heavily into the idea that "pi is all about geometry", which is not supported by modern sources. Although, it is true, if you squint hard enough at any source, you can find a circle there somewhere. Because pi is fundamental to real analysis, and therefore so are circles. Tito Omburo (talk) 22:55, 29 September 2025 (UTC)[reply]

You wote "Again, you had incorrectly said that the kernel of e^ix was Z. This was obviously wrong" as an answer to a post where I wrote "its kernel is ⁠⁠${\displaystyle 2\pi }$⁠. No further comment is needed, except that you should strike your comment yourself. D.Lazard (talk) 20:35, 29 September 2025 (UTC)[reply]

This diff shows that it's just possible you are missing something: [1]. There are three standard definitions of π which are popular: one is the circumference over diameter, which involves integral calculus; one is half of a period of a solution to a differential equation, which involves differential equations (group R/2piZ, dual group Z); the third is half the infinitesimal generator of the character group of the one dimensional torus, which involves representation theory (group R/Z, dual group 2piZ). These are all very different, mathematically and philosophically. Tito Omburo (talk) 21:24, 29 September 2025 (UTC)[reply]

I would not qualify as "popular" the third definition. It seems to me as a very complicated way to say that ⁠${\displaystyle x\mapsto e^{ix}}$⁠ is a group homomorphism with kernel ⁠${\displaystyle 2\pi \mathbb {Z} }$⁠. D.Lazard (talk) 16:02, 2 October 2025 (UTC)[reply]

Section "Definition"

Most of the preceding thread is about section § Definition. This resulted in dramatically shortening this section, and removing all other definitions than ⁠${\displaystyle C/d}$⁠. I expanded the section by explaining why this definition is not fully satifactory in modern mathematics and providing elementary non-geometric definitions.

As further improvements are certainly possible, I open this thread for discussing them. D.Lazard (talk) 16:18, 2 October 2025 (UTC)[reply]

In case it wasn't clear above, when I said that the version from 2012 was probably overall better for the article than the version from recently, I didn't mean that we had to revert to that, but only that I think we should prioritize keeping things somewhat streamlined and as accessible as possible in the top few sections, instead of trying to immediately trying to impress readers by how sophisticated we can be. I don't have a problem with mentioning alternate definitions early on or with unpacking trickier topics further into the article. I think we can still make this section more concise though. Perhaps along the lines of:

π is commonly defined as the ratio of any circle's circumference C to its diameter d: $${\displaystyle \pi ={\frac {C}{d}}}$$ The ratio of circumference to diameter is the same number for any circle in Euclidean space, whatever its size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference.

In modern mathematics, this definition is not fully satisfactory, because it relies on measuring the length of a curved line, which is difficult to define rigorously, requiring at least the concept of a limit, and typically based on the calculus concepts of derivatives and integrals. π also routinely appears in non-geometrical branches of mathematics, and the modern trend in mathematics is to establish geometry in terms of numerical coordinates rather than treating it independently.

Analytically, π can be defined as the smallest positive zero of the sine function; that is, ⁠${\displaystyle \sin \pi =0}$⁠ and π is the smallest positive number with this property. The sine function can, itself, be defined by an integral or by its Taylor series, or as a solution of the differential equation ⁠${\displaystyle f''+f=0}$⁠ which characterizes simple harmonic motion. π is half the fundamental period of any solution of this differential equation.

–jacobolus (t) 17:09, 2 October 2025 (UTC)[reply]

This is fine to me: the important ideas are kept, and the phrasing is much better than mine. D.Lazard (talk) 09:20, 3 October 2025 (UTC)[reply]