Talk:Pi/Archive 10

tau material?
I just reverted an edit that added a paragraph about tau. I don't have a big objection to some very minor mention of tau in the article, but there should be a discussion about the material first here on the Talk page (because it is relatively controversial and has been discussed before on the Talk page). I suggest that editors that want to add material supply a proposal here, including sources which focus on pi. --Noleander (talk) 03:23, 22 April 2012 (UTC)


 * That's odd: it looks like the Tau article has been changed to a redirect to this article.  Was there a merge notification here on the $\pi$ talk page?   I'm planning on submitting this pi article to FAC in 2 weeks, and I'd rather not have this disruption at the last minute.   The bottom line is that the sources on π do not discuss tau.  --Noleander (talk) 03:27, 22 April 2012 (UTC)


 * Regarding the merger: My concern there is that if the tau article is eliminated, the only article in WP to hold tau material is this pi article;  and that means it will balloon to  2 or more paragraphs ... but that is a clear violation of the WP:UNDUE and WP:FRINGE policies, since the math sources give scant mention of pi.  Again, I have no objection to a single sentence mentioning the  proposal for tau, but anything more than that is not consistent with WP policies.  I think the merge needs to be temporarily undone, and more notice given at the Math project and to this Talk page.    --Noleander (talk) 03:34, 22 April 2012 (UTC)


 * I agree. I'd also like to ask whether RunningOnBrains is considered sufficiently uninvolved to be the closer for the RfC discussion given that he advocated doing exactly this at last year's AfD discussion.  Thirdly, I'll have to double-check my numbers, but when I do what RunningOnBrains says he did (disregard votes for delete) I count 7 votes for keep where it is, 4 votes for merge with Pi. --Joseph Lindenberg (talk) 03:45, 22 April 2012 (UTC)


 * OK, I missed a few votes on both sides. I now count 10 for keep (David Eppstein, Greglocock, Rothschilde, Joseph Lindenberg, Kleuske, 71.206.24.106, Nageh, Waldir, Thepoodlechef, Thparkth).  I now count 6 for merge to Pi (Carl, Ruud, Waldir, Nyttend, V=IR, Reyk).  I don't see how RunningOnBrains found "a good majority supporting a merge to Pi" if he left out the delete votes as he said he did. --Joseph Lindenberg (talk) 04:51, 22 April 2012 (UTC)


 * That seems to be the crazy way that WP works. Knowledgeable editors discus a subject at length and give their opinions on it then a random (or worse still an involved) admin comes along and makes a decision based on their personal opinion. I think there should be a stronger preference for a straight vote in highly contested RfCs.Martin Hogbin (talk) 19:46, 22 April 2012 (UTC)


 * oooooh, I see what the problem is now. That stupid FAC bullshit rears it's ugly head again. Fine, screw the RFC then. Let Noleander feed his ego, and maybe after a few months have gone by this can all be rediscussed again. — V = IR (Talk&thinsp;&bull;&thinsp;Contribs) 05:18, 22 April 2012 (UTC)


 * Please refrain from personal attacks. See WP:NPA.  The purpose of the Talk page is to discuss improvements to the article. --Noleander (talk) 13:08, 22 April 2012 (UTC)


 * +1 Sławomir Biały  (talk) 13:09, 22 April 2012 (UTC)


 * This is exactly what I told you would happen. They merged the article, then deleted all the content. --Joseph Lindenberg (talk) 09:56, 22 April 2012 (UTC)


 * I second User:Noleander's proposal. One could add a brief sentence here about 2&pi; mentioning that B. Palais proposed adopting a single symbol for it (perhaps mentioning that others have proposed &tau; as the symbol).  This could be accompanied by one or two references (say, palais's paper as well as the physicist's manifesto).  That would be sufficient coverage for &tau;.  Extreme care should be taken to avoid citing fantastic reports in the tabloid media.  Tkuvho (talk) 08:50, 22 April 2012 (UTC)


 * That wasn't Noleander's proposal. He asked that we go away and leave him alone until this FAC thing is finished.  I've been doing exactly that even though I haven't been happy that all mention of tau (including the link to the tau page) was removed from the Pi page.  I knew they've been busy getting the article ready and they obviously wouldn't want to deal with it now.  I'd think you guys would be happy to keep tau out of sight for that. --Joseph Lindenberg (talk) 09:13, 22 April 2012 (UTC)


 * Please do not misrepresent what I said. I do not own this article, and anyone is free to edit it.  I have no objection to including  1 or 2 sentences on tau in this article if the sources support it. I've read a lot on pi, and do not recall seeing tau mentioned in any of the sources.  If reliable sources can be provided that relate tau to pi, I have no objection at all to 1 or 2 sentences in the Popular Culture section.   I also support the existence of a article dedicated to tau. --Noleander (talk) 13:08, 22 April 2012 (UTC)


 * Tau is utterly significant in the world of mathematics and I would be happy to see no mention of it here; one sentence maximum. Martin Hogbin (talk) 09:29, 22 April 2012 (UTC)


 * That's OK if you leave the tau article in place. But "merge with pi" does not mean one sentence.  Nobody could reasonably construe that people who voted "merge with pi" had that in mind. --Joseph Lindenberg (talk) 09:47, 22 April 2012 (UTC)


 * Merge with Pi does not mean anything for the amount of content merged. There are zero content merges. IRWolfie- (talk) 09:54, 22 April 2012 (UTC)


 * I think a short paragraph in the popular culture section would not be undue. IRWolfie- (talk) 09:54, 22 April 2012 (UTC)
 * Suggested paragraph:
 * Some individuals have proposed giving this number 2π its own symbol, τ, and using that instead of π in mathematics notation. This proposition has not been echoed in scientific publications nor by any scientific authority, though some have advocated the celebration of "Tau Day" on June 28, in a similar manner to the aforementioned "Pi Day".   The Royal Institution of Australia's Tau Day celebration in 2011 featured the performance of a musical work based on tau.


 * I'd leave out the last sentence. Otherwise it's good.  Sławomir Biały  (talk) 11:14, 22 April 2012 (UTC)


 * Seconded. In fact, leave out all references to newspaper reports and stick to scientific sources (here The Mathematical Intelligencer qualifies as a scientific sourse).  Tkuvho (talk) 11:31, 22 April 2012 (UTC)


 * But the The Mathematical Intelligencer  source is a primary source: the person that made the tau proposal in the first place, true?   We really need secondary sources that analyze and comment on the primary sources.  See WP:SECONDARY SOURCE.  Who as adopted Palais' suggestion?  What books have used tau instead of pi?   Has any math journal or conference had a session discussing tau?  etc. --Noleander (talk) 13:15, 22 April 2012 (UTC)


 * Answers: (1) yes, (2) no mathematical sources, (3) none, (4) no. These points were also raised first at the the AfD and then at the RfC.   Sławomir Biały  (talk) 13:25, 22 April 2012 (UTC)

Biased cabalistic POV-pushing admin poking his head in here. I had not realized that people would figure out that I was actually using my admin powers to advance my hidden agenda to....what exactly would I be advancing with this move? Truth is I haven't been following this article, and I did not recall !voting in the previous AFD...someone posted at WP:ANI that the RFC needed to be closed so I closed it. And if you really need to rehash the RFC votes here, I count 3 delete, 2 delete (sentiment was merge to pi), 1 redirect to pi, 1 merge to turn (geometry), 4 merge here, and 7 keep. I discounted 2 of the keep opinions per WP:ILIKEIT. The delete !votes were not completely discounted, as I noted in my closing statement, some of the delete !votes had the sentiment of merging. So with the 2 delete and merge, 1 redirect, and 4 merge to pi, that's seven in favor of this action, five against, and four "other". Not that any of this should matter, since the process is not a vote, but a determination of consensus, and it seemed clear to me that the consensus to merge had formed towards the end.

All that said, nothing has been done here that cannot be undone. If there is a clear opinion to revert my actions, DO IT. I'm an admin, not God, I was just performing my duties, not issuing a binding decree. If you think I did something wrong, by all means revert me, my feelings won't be hurt. - Running On Brains (talk) 20:52, 22 April 2012 (UTC)


 * Sorry if my comments came across too strongly on the matter. I didn't mean to accuse you of any intentional wrongdoing.  Just that by having previously voted for this outcome, you couldn't really be considered an outside party.  That doesn't mean an outside party wouldn't have come to the same conclusion as you did.  Thank you for clarifying your vote count. --Joseph Lindenberg (talk) 22:29, 22 April 2012 (UTC)


 * No offense taken, I'm glad to see there are still some people who realize us admins are human too :) - Running On Brains (talk) 01:49, 23 April 2012 (UTC)


 * For what it's worth, I think you made the right close, although it isn't my favored outcome.  Sławomir Biały  (talk) 21:19, 22 April 2012 (UTC)


 * TL;DR on the above, life's too short. Tau was a flash in the pan, it's a non-story that caught the attention of mathematical illiterates because to support tau-ism gave them the mistaken impression that they were for a moment more clever than mathematicians. File it under looney cults and religions, give it perhaps a line, then move onto interesting stuff. --Matt Westwood 21:50, 22 April 2012 (UTC)


 * For the record, I disagree with your closure. Mind you, I believe you made a reasonable and pragmatic interpretation of the !votes to boil the discussion down to one of a few possible outcomes. However, I would prefer a comparison of opposing arguments and an individual decision on each of them, rather than reducing them to their target !vote. Namely, I can't help but feel the points I presented in the discussion were somewhat ignored, as some of them were challenged, but the main idea I meant to put forward (perhaps I did so poorly) remained unchallenged: the position that there is sufficient notability to the historical usage and recent proposals of 2&pi; as a constant in itself (rather than a product) to have an article about it, and that the accessory topics of (1) naming the constant tau (a very recent proposal and thus unsusprisingly not (yet?) addressed by established mathematical publications, though unchallenged among 2&pi; supporters) and (2) celebrating "tau day", shouldn't be dissociated from it. But sure, it's just my opinion. I just wished that the flood of opinions didn't drown some good arguments and force a pragmatic closer to resort to reducing/coarsening methods to reach a meaningful outcome with a reasonable amout of effort. --Waldir talk 11:17, 24 April 2012 (UTC)


 * "the position that there is sufficient notability to the historical usage and recent proposals of 2&pi; as a constant in itself (rather than a product)." Possibly this is true, but one reason it was unchallenged is that none of us really knows.  The point about "historical usage" seems to be hypothetical rather than to have any particular bearing on the tau article as it stood.  I did suggest finding reliable sources (in the history of mathematics) that address this point, but none were located since the inception of the article, through the AfD, the intervening year between the AfD and the RfC, nor the RfC itself.  It can be difficult to prove a negative, and it certainly seems plausible that there have been attempts in the past to refocus the attention on 2&pi; as a mathematical constant rather than &pi;, but without reliable sources about this we simply cannot have an article about it (WP:V).  So, although this point wasn't exactly rebutted, it's not a point that was satisfactorily argued with the backing of sources.   Sławomir Biały  (talk) 11:44, 24 April 2012 (UTC)


 * I'm not sure I understand what you mean by "none of us really knows". The article did mention at least two past mathematicians who used 2&pi; as a single constant rather than as a multiple of &pi;: Al-Kashi and Laurent. Are you saying these don't count? Are you saying that since nobody linked them together in a secondary source before regarding the usage of 2&pi; means that it is WP:SYN to point this out? Or am I missing something?
 * Also: "to have any particular bearing on the tau article as it stood" -- as I mention above and had before, I don't believe the article should be kept as it stood, but instead focused on 2&pi;, the real core concept, without failing to mention tau, of course. --Waldir talk 10:31, 26 April 2012 (UTC)


 * It's not even an issue of SYN. It was a bald assertion, with nothing whatsoever to back it up.   Sławomir Biały  (talk) 12:02, 26 April 2012 (UTC)


 * So what do you say to this? --Waldir talk 00:26, 27 April 2012 (UTC)


 * Some random guy's website?  Sławomir Biały  (talk) 00:32, 27 April 2012 (UTC)


 * I meant the contents of that section (so I wouldn't need to repeat it all here). Specifically, the use of 2&pi; as a constant, rather than &pi;, by Al-Kashi, Laurent, Gregory, Bugge, etc. as listed there. Do you contest that?
 * Also, it's not "some random guy" — Harremoes is a respected academic with an [www.harremoes.dk/Peter/bibtex/listpub.pdf extensive publication record] and a background as editor of and reviewer for several journals, including Mathematical Reviews. --Waldir talk 15:04, 27 April 2012 (UTC)


 * Ok, some random academic's personal website. Also, I don't have to contest anything, just ask for reliable sources in textbooks or journals focused on the history of mathematics and the sciences. That's the basic non-negotiable minimal standard for this sort of thing.  Sławomir Biały  (talk) 16:30, 27 April 2012 (UTC)


 * Just slightly off-topic, but one other famous case I noticed that seems to possibly suggest an interest in C/r is Aryabhata. His famous quotation on the matter said that 62,832 is (approximately) the circumference of a circle with diameter 20,000.  Now, I fully realize that I'm reading it with biased eyes, but his choice to make the radius an even 10,000 instead of making the diameter 10,000 (and thus the circumference 31,416) strikes me as interesting.  It's not grounds for anything more than investigating it further, but if someone were interested in these historical cases (I'm not), that's one potential place to look. --Joseph Lindenberg (talk) 17:44, 27 April 2012 (UTC)
 * Also Waldir, if you actually want to pursue the historical angle, you should get in touch with Peter Harremoes. He has pursued this beyond what he has posted on his website.  He sent around an email on March 10 about some further findings involving Abul-Wafa' al-Buzagani (940-998) and Abur-Raihan al-Biruni (973-1048).  Supposedly using C/r was quite common in Islamic mathematics.  Although they often worked in base 60 instead of base 10. --Joseph Lindenberg (talk) 18:23, 27 April 2012 (UTC)


 * Thanks for the info, Joseph. I don't intend to pursue this subject, though: I was just trying to understand Sławomir's objections. I am not entirely sure yet, but it seems he is saying the primary publications by those mathematicians aren't acceptable, and secondary sources about those are necessary instead; well, I see plenty of publications about Al-Kashi's work in his article, and I'm sure similar secondary publications can be found for the other mathematicians on that list (and those outside it, as you mention). If I'm reading this correctly, the case for a 2&pi; article is merely lacking such sources to be considered acceptable for editors whose view of the policies are similar to Sławomir's. If that's the case, it might be a good/productive way to approach this topic, since the tau material will necessarily have to be mentioned in a section of that article (and, as I argued before, in the title as well, per WP:COMMONNAME). --Waldir talk 18:06, 28 April 2012 (UTC)
 * I'll definitely keep it in mind. But now that we've achieved consensus, I'm inclined to just get out of these guys' hair for a while. --Joseph Lindenberg (talk) 18:23, 28 April 2012 (UTC)
 * The next stage in deciding what our coverage should be will come from the analysis of reliable sources. What we say needs to be dictated entirely based on what these sources have to say, with as little analysis of our own as possible.  If there is no source linking Al-Kashi (and similar) to modern attempts to focus attention on 2&pi; rather than &pi;, then WP:SYN indeed becomes an issue.  Also, while there seems to be little doubt of the historical fact that Al-Kashi calculated 2&pi; rather than &pi;, this fact alone does not seem to be sufficient to base an article on unless some scholars have written on the significance of this vis-a-vis the prevalence of &pi; in the ancient world and Europe.  This is one reason reliable sources are important: there needs to be a suitable pool of encyclopedic things to say about (what should be) a serious topic before an article can be written about it.  (I'm paraphrasing here a line from WP:V which says: "If no reliable third-party sources can be found on a topic, Wikipedia should not have an article on it.")   Sławomir Biały  (talk) 14:30, 29 April 2012 (UTC)

Let us define upsilon as 4&pi; because there are 4&pi; steradians in a whole sphere. ;-) JRSpriggs (talk) 23:33, 22 April 2012 (UTC)


 * Go to sites.google.com/site/taubeforeitwascool and read the section titled "A different pair of formulas for every dimension". Tau isn't only fundamental in two dimensions.  All the n-sphere formulas come from multiplying/dividing by tau.  Seriously, go and look. --Joseph Lindenberg (talk) 00:10, 23 April 2012 (UTC)


 * But the formulas for the volumes (in terms of the gamma function) are more complicated with tau.  Sławomir Biały  (talk) 00:15, 23 April 2012 (UTC)


 * Read the section. You'll see the explanation. --Joseph Lindenberg (talk) 00:20, 23 April 2012 (UTC)


 * Yes, I saw that. It's an interesting observation.  I was just making a counterpoint.   Sławomir Biały  (talk) 00:28, 23 April 2012 (UTC)


 * (Responding to your counterpoint) True, but no choice of circle constant can simultaneously maximize the simplicity of ALL formulas. But tau simplifies more formulas than it complicates, and in the ones that it does complicate, there's a natural reason for that complication to be there.  Another section at sites.google.com/site/taubeforeitwascool shows this for the 2 in $$e^{i\tfrac{\tau}2} + 1 = 0$$.  Euler's Identity is just the n=2 case of the sum of all n nth roots of unity being zero.  When you look at it beside the n=3 and n=4 cases, it's clear the 2 belongs there. --Joseph Lindenberg (talk) 06:51, 23 April 2012 (UTC)


 * Joseph, I think you may be missing the point. It may very well be true that rewriting all formulas in terms of tau (or upsilon, etc) is a fantastic idea.  We just don't have any respectable scientific sources for that.  Tkuvho (talk) 08:31, 23 April 2012 (UTC)


 * No, I get your point. But we were just having a little to-the-side math discussion after JRSpriggs posted his clever remark.  From there down to here, we weren't discussing tau on Wikipedia.  We were actually discussing whether tau was a fantastic idea. --Joseph Lindenberg (talk) 09:22, 23 April 2012 (UTC)


 * OK, well, the probability density function for Cauchy distribution has a pi in it. Tkuvho (talk) 11:23, 23 April 2012 (UTC)


 * (@Joseph) I think your observation (and the graphic from your website) could be added to the N sphere article.  That would be something worth having in there.   Sławomir Biały  (talk) 11:57, 23 April 2012 (UTC)


 * Wikipedia is welcome to have the graphic — I'll upload it a little later. But I'm not sure I understand what specific observation you mean.  The very simple "recursive" formulas for n-sphere volumes (which use 2π instead of π) are already in Wikipedia's n-sphere article. --Joseph Lindenberg (talk) 22:37, 23 April 2012 (UTC)


 * Yes, I just saw it buried there at the bottom of that section. The section needs to be better organized.  It's a bit aimless at the moment.   Sławomir Biały  (talk) 01:08, 24 April 2012 (UTC)


 * Shall I regenerate the graphic with "volume" replacing "size of the interior" and "surface area" replacing "size of the boundary"? I was trying to make that section of my website easy to read for people unfamiliar with n-spheres.  Using the standard terms can confuse people who think of volume as a purely 3-dimensional quantity. --Joseph Lindenberg (talk) 01:41, 24 April 2012 (UTC)


 * Probably leave it as "size". I don't really think "surface area" is more proper, and calling it "volume" of the sphere (versus "volume" of the ball) is just going to upset most readers.  Sławomir Biały  (talk) 11:51, 24 April 2012 (UTC)
 * I figured that you'd probably want the "volume" and "surface area" labels, so I went ahead and made the changes before I saw your answer here. But I uploaded both versions to Wikimedia and put the name of the unused file in the article's revision history note.  So if anybody (including yourself) wants to switch to the other version, they can just copy the filename from there and use it to replace the filename that's listed in the article now.  I did put the new version in the article.  I actually think it's better because I used the labels "Volume (Vn)" and "Surface Area (Sn-1)".  That helps clarify which standard the article uses for the subscript on surface area.  Also, the article text does use those terms.  But let me know if something different is needed on the labels. --Joseph Lindenberg (talk) 14:24, 24 April 2012 (UTC)

Quotes from sources
Could an editor who is supporting inclusion of the material please post some quotes from secondary sources here? It would be best if the sources were mainstream mathematicians; or historians of mathematics. For contrast: pi day is described by historians of math in their books on π, so it would be best if similar sources described tau. The sources mentioned above show that a Utah professor made the suggestion in 2001 (not a secondary source), and a local Utah newspaper reported it. Are there more notable sources like NY Times, or something outside Utah? Or some math journal? I guess the absence of mainstream sources is why the proposers are suggesting that it go in the "Popular culture" section, true? Is there any evidence that people have actually celebrated tau in popular culture, or have there simply been suggestions that tau be used and celebrated? Exactly how many people out there in the world are advocated the use of tau instead of pi? One? Ten? A thousand? Who are they? (NOTE: I ask these questions not in the sense of skepticism, but rather a genuine desire to gather information to form an opinion of how much material should go into this article, in accordance with WP:UNDUE and WP:FRINGE). --Noleander (talk) 13:00, 22 April 2012 (UTC)


 * I raised the very same point at length at Talk:Tau (2&pi;), and there was some discussion about it there. My feeling overall is that the "tau day" thing is marginally notable, whereas the "tau" business is not (mathematically speaking).  There have been some other sources reporting on tau day.  I've just copied from an earlier revision of Tau (2&pi;), so I'm not endorsing any of these as "good" references.  In fact, treat these with a healthy dose of suspicion (most of these yellow media reports happened on tau day).   Sławomir Biały  (talk) 13:18, 22 April 2012 (UTC)





Here are some additional sources that were turned up at the AfD. Sławomir Biały (talk) 13:21, 22 April 2012 (UTC)

Okay, that is good information! The next step is to look at some quotes from those sources. The BBC source is a decent secondary source. Here is some text from that:

So, the 1 or 2 sentences in Popular Culture section should reflect what this BBC (or other secondary sources of comparable quality) say. --Noleander (talk) 13:29, 22 April 2012 (UTC)

Here's the list of news sources I've got: --Joseph Lindenberg (talk) 21:22, 22 April 2012 (UTC)

Noleander, I am providing this in response to your earlier question about people celebrating tau in popular culture. Though others here have been skeptical of its significance, MIT this year posted their admissions decisions on Pi Day at what they called "Tau Time", 6:28pm, to correspond to tau being 6.28. The announcement indicates they did this to honor both Pi and Tau equally, though some here have complained because they did the announcement in the form of a cartoon. You should really decide for yourself: mitadmissions.org/blogs/entry/i-have-smashing-news. To show they weren't just making fun of tau, I also have MIT Associate Director of Admissions Matt McGann opening a blog discussion for MIT applicants to discuss pi versus tau at mitadmissions.org/blogs/entry/pi-day-tau-time. The Boston Globe did an article about MIT's admissions decisions, with a quote by Dean of Admissions Stuart Schmill explaining, "Over past year or so, there has been a bit of debate in the math universe over which is a better number to use, whether it is Pi or Tau". You'll find that article at articles.boston.com/2012-03-14/yourtown/31167690_1_pi-day-decimal-places-mit-dean. Also MIT school newspaper The Tech had an article, though it just said, "On Pi Day at Tau Time (March 14 at 6:28p.m.), MIT granted admission to 1,620 eager applicants." The rest of the article is about the applicants. You'll find the article at tech.mit.edu/V132/N12/admissions.html. --Joseph Lindenberg (talk) 09:06, 23 April 2012 (UTC)


 * A quick attempt to explain why things are not going your way, Joseph: so long as your arguments are dominated by "newspapers", "blogs", "youtubes", etc., you are not going to get anywhere with impressing WPM. On the contrary.  Tkuvho (talk) 09:16, 23 April 2012 (UTC)


 * Tkuvho, I specifically prefixed the above post by saying this was about people celebrating tau in popular culture. Noleander asked above for "evidence that people have actually celebrated tau in popular culture".  Do I have to find evidence of that in math journals now too?  Please have your morning cup of coffee before posting a third time, because your last two criticisms of me missed what was going on entirely. --Joseph Lindenberg (talk) 09:42, 23 April 2012 (UTC)

proposed sentence on tau
How about this addition to the "popular culture" section:


 * Some individuals have proposed giving this number 2π its own symbol, τ, and using that instead of π in mathematics notation. This proposition has not been echoed in scientific publications nor by any scientific authority, though some have advocated the celebration of "Tau Day" on June 28, in a similar manner to the aforementioned "Pi Day". The BBC reported tau advocate Hartl as claiming that "people find themselves almost violently angry at pi. They feel like they've been lied to their whole lives,..."

That just about tells it all. Tkuvho (talk) 14:50, 22 April 2012 (UTC)


 * The latter sentence, plus the quote, is a bit much. Mentioning Hartl on the same level as Gauss, Newton, and Archimedes is not quite right.  Likewise for the quote.  Maybe just:


 * --Noleander (talk) 14:58, 22 April 2012 (UTC)


 * That seems about right. Should there also be mention of tau day? (Here my thinking is that this is the pop culture section, after all.)  Sławomir Biały  (talk) 15:08, 22 April 2012 (UTC)


 * "Some individuals have proposed replacing π by a new mathematical constant which equals two times π. According to proponents this new constant would be a more natural choice than π and simplify formulas. Robert Hartl proposed the Greek letter tau ($π$) for this new constant in his Tau manifesto, and declared 28 June as "Tau day" to celebrate the constant in resemblance to the Pi day. While these propositions have been widely relayed in the media, they have not been echoed in scientific publications or authorities."


 * I would prefer this version. It tells a bit more what it is all about without going into unnecessary details. Nageh (talk) 15:17, 22 April 2012 (UTC)


 * Alternatively, without mentioning Hartl:
 * "Some individuals have proposed replacing π by a new mathematical constant which equals two times π. According to proponents this new constant would be a more natural choice than π and simplify formulas. The Tau manifesto proposes the Greek letter tau ($τ$) for this new constant, and declares 28 June as "Tau day" to celebrate the constant in resemblance to the Pi day. While these propositions have been widely relayed in the media, they have not been echoed in scientific publications or authorities."
 * Nageh (talk) 15:31, 22 April 2012 (UTC)


 * The Tau Manifesto doesnt seem very important. How about


 * --Noleander (talk) 15:53, 22 April 2012 (UTC)


 * Excellent. Of course, references would have to be added to this.  I agree that Hartle needn't appear in the main text.  Tkuvho (talk) 16:21, 22 April 2012 (UTC)


 * I agree that the Tau manifesto itself is not really notable. Regarding the first two sentences, I was trying to avoid tau in the first place to indicate that the symbol tau has not been the only suggestion for the constant. So, my suggestion would be:
 * Nageh (talk) 16:29, 22 April 2012 (UTC)
 * Nageh (talk) 16:29, 22 April 2012 (UTC)


 * What "scientific authorities" are we talking about, and how would the "endorse" something? I think that the more accurate statement is: some have proposed rewriting formulas to mention &pi; to instead mention a new constant &tau; which is equal to 2&pi;. It is not &pi; beign replaced, it is the re-expression of formulas that involve &pi; as different formulas that involve &tau;. The constant &pi; itself is unchanged by all of this; it's not as if &pi; is being redefined. &mdash; Carl (CBM · talk) 16:49, 22 April 2012 (UTC)


 * The intention of that "scientific authorities" phrase is to indicate the WP:FRINGE nature of the tau proposal, i.e. that mainstream mathematicians are not (yet) adopting it. Can you think of a better way to word it, so readers don't get the impression that mathematicians are adopting tau to replace pi? --Noleander (talk) 17:00, 22 April 2012 (UTC)


 * It's been a while since I read the Tau manifesto, so I don't recall whether the suggestion is to get rid of pi in its entirety or have pi and tau alongside each other. In any case, you may simply remove "replacing pi by" as suggested by Noleander. Nageh (talk) 17:05, 22 April 2012 (UTC)


 * But you cannot replace &pi; by &tau; either. The diameter of a circle of radius r is 2&pi;r but not 2&tau;r. So &tau; does not just replace &pi;, the formula has to be rewritten as &tau;r. &mdash; Carl (CBM · talk) 17:21, 22 April 2012 (UTC)


 * Oh my, I didn't imagine that my phrasing was that ambiguous. I have just removed it. Nageh (talk) 17:24, 22 April 2012 (UTC)


 * It's no problme, just this topic may be of interest to people without much math background, so there's a reason to be as accurate and plain as we can be. "Some people wish that formulas that were written using &pi; were rewritten as different formulas involving &tau;, because they think the formulas using &tau; are more elegant." That sentence should be clear to someone who doesn't even know what &pi; and &tau; are. &mdash; Carl (CBM · talk) 17:38, 22 April 2012 (UTC)


 * Sorry, I disagree. That sentence is pretty dumbing down the issue, with "elegant" saying even less to someone as clueless as not knowing what pi is in an article about pi. In fact, the whole sentence is overly vague without saying anything. Just saying. :) Nageh (talk) 17:42, 22 April 2012 (UTC)


 * How about this: "A new mathematical constant tau (τ) has been proposed which equals two times π. Its supporters have argued that this new constant is a more natural choice than π and would simplify many formulae. While these proposals, including a suggestion of 28 June as "Tau day", have been reported in the media, they have not been reflected in the scientific literature."  This avoids the ambiguous claim about tau "replacing" pi, and also avoids the terminology of "endorsing".  Tkuvho (talk) 18:20, 22 April 2012 (UTC)


 * I like it, personally. &mdash; Carl (CBM · talk) 18:21, 22 April 2012 (UTC)


 * You are [//en.wikipedia.org/w/index.php?title=Pi&diff=prev&oldid=488688690 pretty quick] to determine consensus, Tkuvho. Anyway, I don't object. Nageh (talk) 18:51, 22 April 2012 (UTC)


 * This seems to be pretty good, modulo wording tweaks and the addition of references. Sławomir Biały  (talk) 19:03, 22 April 2012 (UTC)


 * It really should define tau as the ratio of circumference to radius, instead of circumference to diameter like pi. That's what everyone knows as the definition of pi (and is a central criticism of tau supporters for using diameter).  You can then go on to say that therefore it equals 2π. --Joseph Lindenberg (talk) 04:39, 23 April 2012 (UTC)

I thought Tkuvho's proposal was very well-written. It's just that it leaves out too much. So I expanded on what he wrote. Just a few more lines really isn't unreasonable, and it allows us to give a much fuller description. See what you think:

--Joseph Lindenberg (talk) 08:34, 24 April 2012 (UTC)

WTF? Someone added a paragraph on tau, without any sourcing at all. Ignoring that fact that the discussion is on-going (and only a few hours old), the absence of sources is not compliant with WP:V policy. Remember that many editors only login to WP every 2 or 3 days; one of those occasional editors may have a brillant suggestion on this issue. We can wait a bit. The article is not going anywhere. --Noleander (talk) 20:17, 22 April 2012 (UTC)

As a potential source, here is the chairman of the math department at the University of Massachusetts Lowell mentioning tau and Tau Day in their math department's alumni newsletter. It's at the bottom of the left-hand column on page 2. --Joseph Lindenberg (talk) 20:57, 22 April 2012 (UTC)


 * Of all the possible sources, an alumni newsletter is about the least reliable we could hope for. Some newspapers were listed above. They are still not ideal sources for a scientific topic, but since &tau; is mostly a media issue rather than a scientific one they should be OK. But things like emails, blog posts, etc. are really not what we want to use. &mdash; Carl (CBM · talk) 21:44, 22 April 2012 (UTC)


 * You do realize this was an actual printed and snail-mailed newsletter, don't you Carl? --Joseph Lindenberg (talk) 02:14, 24 April 2012 (UTC)


 * Fair enough criticism about it being an alumni newsletter, but it is the statement of the chairman of a university mathematics department. I'm just letting them know it's there if they want to use it. --Joseph Lindenberg (talk) 22:39, 22 April 2012 (UTC)


 * On second thought, why wouldn't the chairman's message in a university mathematics department's alumni newsletter (dead-tree, snail-mail) be a more reliable source than a newspaper? --Joseph Lindenberg (talk) 02:31, 24 April 2012 (UTC)


 * We need to keep things in perspective. This is an article on pi, not tau.  The article should not have more text on tau than on, say, the randomness of pi's digits.  We can stick a sentence or two into the Popular Culture section, but there is no need to belabor the topic.  All we need is to finalize the wording. How about:
 * --Noleander (talk) 03:21, 24 April 2012 (UTC)
 * --Noleander (talk) 03:21, 24 April 2012 (UTC)


 * I think the not ... mainstream wording is a little too aggressive; it seems to stop just short of calling tau advocates "crackpots". (I say this as someone who thinks the whole tau thing is a little bit silly.)  Maybe it would suffice to note that the notation has not caught on.  (I don't have a proposed wording at this time.) --Trovatore (talk) 03:28, 24 April 2012 (UTC)


 * Noleander, what you proposed is less than a third the length of the text on the randomness of pi's digits. Look, I know I'm not going to get much here, but you've got to define tau, and you can't say tau is defined as twice pi.  That's simply wrong.  It's defined as the ratio of circumference to radius.  I'm not even saying you have to point out the comparison to pi being defined as circumference divided by diameter, although it really would make sense to do so.  But obviously the text should also say that tau equals 2π. --Joseph Lindenberg (talk) 03:59, 24 April 2012 (UTC)

Arbitrary break
This (minus the last sentence) is the text that was here about tau for four months. I added the last sentence on March 13. The whole thing is a good amount shorter than the text on the randomness of pi's digits. Nobody voiced a problem with it before Slawomir did, and the Pi page was getting roughly 8,000 visits a day during that time. So there was plenty of opportunity for objection and editing. I would be satisfied with this text, but others here may have a different view. I actually do consider some of it a bit awkwardly expressed. (I didn't have final edit when we worked this wording out back in November.) Obviously we can reword things to address objections people may have, but this contains the basic key facts about tau. (By the way, the "usually represented by the Greek letter tau" part meant as opposed to the other symbols that had previously been proposed like Palais' three-legged pi. I think we can forget about those other symbols this time.)

--Joseph Lindenberg (talk) 05:25, 24 April 2012 (UTC)


 * Tau is the least interesting mathematical concept that I have ever heard of. The less said about it the better as far as I am concerned. Martin Hogbin (talk) 08:11, 24 April 2012 (UTC)


 * I vote for Tkuvho's proposal. Nageh (talk) 10:09, 24 April 2012 (UTC)
 * Could you take a look at the modified version of Tkuvho's proposal I suggested up above? --Joseph Lindenberg (talk) 10:15, 24 April 2012 (UTC)
 * It's too wordy for me. What is important is that it replaces 2pi, and that this is supposed to be more natural and simplify formulae. All the details can be looked up in any of the links we are going to add as references. Nageh (talk) 10:24, 24 April 2012 (UTC)


 * Either Tkuvho's proposal, or Noleander's second (?) proposal are acceptable: "Some individuals have proposed a new mathematical constant  tau ($τ$) which equals two times π, arguing that this new constant is a more natural choice than π and would simplify many formulae.  While these propositions, including a suggestion of 28 June as "Tau day", have been reported in the media, they have not been endorsed by scientific  authorities."  Sławomir Biały  (talk) 11:33, 24 April 2012 (UTC)
 * I'm still unhappy about the wording of the last phrase: who are these authorities, and what kind of endorsement do we expect them to make? It's not like the French language where there is an official mathematics academy that tells us what constant to use. I would prefer to replace "have not been endorsed by scientific authorities" by "have not been adopted by most mathematicians". —David Eppstein (talk) 17:53, 24 April 2012 (UTC)
 * Why not just drop the last phrase then and have, '"Some individuals have proposed a new mathematical constant tau ($τ$) which equals two times π, arguing that this new constant is a more natural choice than π and would simplify many formulae and have suggested 28 June as "Tau day" '.  Martin Hogbin (talk) 17:59, 24 April 2012 (UTC)
 * That would be ok with me too. That way we avoid having to find find sources to prove a negative. —David Eppstein (talk) 18:03, 24 April 2012 (UTC)
 * That is okay with me, although the grammar is a bit off. How about
 * --Noleander (talk) 22:14, 24 April 2012 (UTC)
 * Can't we say "Proponents have (called/named/referred to) 28 June "Tau day" since tau is approximately 6.28"? Do we really have to say proponents have only proposed it? --Joseph Lindenberg (talk) 09:34, 25 April 2012 (UTC)
 * Can't we say "Proponents have (called/named/referred to) 28 June "Tau day" since tau is approximately 6.28"? Do we really have to say proponents have only proposed it? --Joseph Lindenberg (talk) 09:34, 25 April 2012 (UTC)


 * Tkuvho (talk) 13:22, 25 April 2012 (UTC)
 * Tkuvho (talk) 13:22, 25 April 2012 (UTC)


 * (unindent) I prefer an even shorter version:
 * Let's keep the article focussed. Jakob.scholbach (talk) 14:46, 25 April 2012 (UTC)
 * Let's keep the article focussed. Jakob.scholbach (talk) 14:46, 25 April 2012 (UTC)


 * Remember we are in the "popular culture" subsection. "tau day" is tau's best claim to fame and should probably be mentioned.  Certainly speculations to the effect that "the radius is more fundamental than the diameter" should be kept out of polite discourse.  Tkuvho (talk) 14:54, 25 April 2012 (UTC)
 * "polite discourse" -- Huh? (Although, as a young teenager, I would've gotten my mouth washed out with soap for saying "that girl's got a great radius on her".) (I keep changing this in search of a funnier line.) --Joseph Lindenberg (talk) 15:31, 25 April 2012 (UTC)


 * I still vote for Tkhuvo's proposal: "A new mathematical constant tau (τ) has been proposed which equals two times π. Its supporters have argued that this new constant is a more natural choice than π and would simplify many formulae. While these proposals, including a suggestion of 28 June as "Tau day", have been reported in the media, they have not been reflected in the scientific literature.". Nageh (talk) 15:32, 25 April 2012 (UTC)


 * How about this one tweak to explain the natural choice bit: "A new mathematical constant tau ($τ$) has been proposed which equals two times π. Its supporters have argued that by being the ratio of circle circumference to radius, this new constant is a more natural choice than π and would simplify many formulae. While their proposals, which include celebrating 28 June as "Tau day", have been reported in the media, they have not been reflected in the scientific literature." --Joseph Lindenberg (talk) 16:29, 25 April 2012 (UTC)


 * One last tweak:
 * --Joseph Lindenberg (talk) 19:10, 25 April 2012 (UTC)
 * --Joseph Lindenberg (talk) 19:10, 25 April 2012 (UTC)


 * Regarding the straw poll on radius below. My proposed wording does not say that radius is more natural than diameter.  It says tau's supporters say that, which is undeniably true.  A basic fact of this article is that pi is circumference over diameter, so it makes sense to inform the reader that tau is circumference over radius.  I'm certainly not suggesting that Wikipedia endorse the claim that radius is more natural/fundamental than diameter.  But it should report that tau supporters make that claim.  At a bare minimum, it should tell that tau is circumference over radius.  That's tau's definition!--Joseph Lindenberg (talk) 19:47, 25 April 2012 (UTC)


 * Tkuvho, your proposed statement said that tau's supporters "have argued that this new constant is a more natural choice than π and would simplify many formulae". How is this issue any different?  That's not a rhetorical question.  Please explain how telling the reader about this claim is any different. --Joseph Lindenberg (talk) 20:09, 25 April 2012 (UTC)


 * That was my specific formulation. The essence of the proposition for tau is that 2pi frequently occurs in formulae and replacing it would simplify formulae, and not that using diameters instead of radii would simplify formulae. Adding this information would thus be unnecessary verbiage. Nageh (talk) 20:16, 25 April 2012 (UTC)
 * I see it showing above that Tkuvho wrote it. Is that an error?  Anyway, I disagree with what you've concluded the "essence" is.  But so you're saying your whole objection is all about eliminating a few extra words, not that we mustn't tell the reader about this claim regarding radius?  Is that right? --Joseph Lindenberg (talk) 20:30, 25 April 2012 (UTC)
 * The phrase is right from near the top of the previous subsection. I am saying that including this information is redundant: defining tau as C/r is the same as defining tau as 2π. But obviously, tau = 2π is the more general definition. So, the extra information is unnecessary. Nageh (talk) 20:38, 25 April 2012 (UTC)
 * Nageh and others, please read the 3rd post by Aficionada at hubpages.com/forum/topic/78332. ("they would still have to calculate pi first before they can find tau")  Yes, I facepalm when I see posts like that, but I've seen it enough times to know it can confuse some people.  Not everyone reading the Pi article will be as mathematically astute as the people writing it. --Joseph Lindenberg (talk) 12:48, 27 April 2012 (UTC)
 * No one was saying anything remotely like that in the discussions here. Nageh (talk) 13:59, 27 April 2012 (UTC)
 * Remotely like what? --Joseph Lindenberg (talk) 14:25, 27 April 2012 (UTC)
 * "they would still have to calculate pi first before they can find tau". Your post seemed to imply that one of us had said something like that. Maybe you posted that link just for... I don't know. Nageh (talk) 14:30, 27 April 2012 (UTC)
 * Oh no. My point was the last sentence.  Not everyone reading the article will be as knowledgeable as the people writing the article.  I know that none of the people here on this talk page fail to understand tau can be calculated and used just on its own, without calculating pi first.  But the article is meant for a much wider audience.  Granted, some sections might not be understandable to them because the topic is just inherently too complex.  But certainly for the "In Popular Culture" section, we shouldn't assume the reader is as sharp at mathematics as the people writing the article, or anywhere near it. --Joseph Lindenberg (talk) 14:55, 27 April 2012 (UTC)
 * {OK, I see now that a lot of the proposals were very similar in wording. I was working from Tkuvho's version.)
 * No, because defining tau = 2π makes tau dependent on pi, and it's not. Seriously, I've seen multiple people get confused by this and say that we'll still need pi because tau is determined from pi.  Or that pi is more fundamental because tau is determined from it.  This isn't a hypothetical issue.  And regarding the "essence" issue, tau supporters claim tau is a more natural choice because radius is a more natural measure for circles.  It's a central issue.  This isn't simply about which way will save more pencil lead.  Besides, my proposal is short enough as it is.  Let's not get into a big disagreement over saving just a few extra words.  I thought Tkuvho was objecting that we were somehow endorsing the claim. --Joseph Lindenberg (talk) 21:00, 25 April 2012 (UTC)
 * Nobody defines tau as 2pi. The only reason people usually say "tau is 2pi" is because it's shorter than saying "tau is the circumference of a circle divided by its radius".  It's 4 syllables versus 19.  Lots of people say "pi is 3.14" but that doesn't make it its definition. --Joseph Lindenberg (talk) 21:21, 25 April 2012 (UTC)
 * So what? It still equals 2π, and it is still the replacement of 2π by tau that is supposed to simplify formulae. This is the theme of the whole Tau manifesto with the exception of section 1.1. Ultimately, it is an editorial decision whether to include the extra text, derived by consensus. I don't see a need for it; others may disagree. Nageh (talk) 21:50, 25 April 2012 (UTC)
 * The core idea isn't that we've done a statistical analysis and after crunching all the numbers, it turns out that 2π just edged out 5π for maximizing the simplicity of all formulas. (And it's actually less about simplicity and more about clarity and understandability of formulas.  There are a few formulas that become less simple but more understandable.)  The core idea is that there's an underlying reason why that particular number works best in so many places.  Otherwise, a few decades from now, the formulas people most commonly use might change, and a different number (like 5π) might win in the statistical analysis.  But if you don't actually object to any of the content, please don't be overly restrictive about the number of words.  The article gives this same amount of space to the Palais de la Decouverte for having the digits of pi on its walls. --Joseph Lindenberg (talk) 22:37, 25 April 2012 (UTC)
 * I understand what you are saying, but really, it is all included in "this new constant is a more natural choice than π". Not in the detail, of course, but it can be looked up in the references. I don't see why stating "Its supporters have argued that a constant that equals the ratio of a circle's circumference to its radius would be a more natural than π" would add anything of substance to this article. Nageh (talk) 23:08, 25 April 2012 (UTC)

Hey guys, is this worth anything? The editor of Math Horizons magazine, which is published by the Mathematical Association of America, wrote a column in its April issue endorsing tau. Look here. --Joseph Lindenberg (talk) 23:13, 25 April 2012 (UTC)
 * Stephen Abbott's piece is a better source that Hartl for at least two reasons: (1) it avoids unnecessary "manifesto" verbiage; (2) Abbott happens to be a published mathematician (7 articles). If we ever reach agreement on this, we should probably provide two sources: B. Palais and S. Abbott.  I notice that Abbott has a real analysis textbook.  If Joseph can convince him to publish one based on tau, that may be a compelling reason to create a separate page on tau.  Tkuvho (talk) 09:10, 26 April 2012 (UTC)
 * I'm glad you're pleased. I do, however, feel it would be very strange for us not to use the Tau Manifesto as a reference.  Even if only for something non-technical, like Tau Day.  It is just such a central part of the tau movement.  Anywhere you read about tau, that's the one thing they always link to. --Joseph Lindenberg (talk) 11:05, 26 April 2012 (UTC)
 * Personally, I do support adding a link to the Tau manifesto as a primary source, in addition to about two reliable secondary sources. Let's treat this like any other primary sources, as a courtesy to the reader; after all, this is what the paragraph we are going to add is about. Remember, this is in the Popular culture section. Nageh (talk) 11:23, 26 April 2012 (UTC)
 * That's fine. Tkuvho (talk) 19:22, 26 April 2012 (UTC)

Tkuvho, for the sources, do you prefer we just not use any news-type sources? Assuming, as you suggested, we use the pieces by B. Palais and S. Abbott, and then the Tau Manifesto as I suggested, should that be it? I really have no preferences beyond those 3. Should we use news sources to support the "have been reported in the media" phrase? If so, which ones would you prefer to use? --Joseph Lindenberg (talk) 22:23, 27 April 2012 (UTC)

Is there any problem with italicizing radius? It might make the meaning clearer (i.e. radius, not diameter, the way π is defined) without making the text any longer:


 * --Joseph Lindenberg (talk) 16:18, 27 April 2012 (UTC)

straw poll on radius
Question: is radius more "fundamental" or "natural" than diameter?


 * No. Tkuvho (talk) 19:15, 25 April 2012 (UTC)
 * Absolutely. But it doesn't matter.  See my statement above.
 * A circle is defined as all points in a plane a certain distance — the radius — away from a center point.
 * Standard circle formulas use radius: $τ$  or  $τ$,  $τ$
 * The unit circle — note the word unit — has a radius of 1, not a diameter of 1.
 * Angles are measured in radians.
 * --Joseph Lindenberg (talk) 19:30, 25 April 2012 (UTC)


 * POV, but ultimately non-notable. Nageh (talk) 19:32, 25 April 2012 (UTC)
 * Impossible to answer without more context. For instance, in the context of curves of constant width, diameter is more natural, and I think that's true also in the context of quadrature of the circle, but there are other contexts in which radius is more natural. Anyway, it's irrelevant to the question of what exactly to say about &tau;, because the point there is not whether it actually is more natural, but merely that its proponents argue that it is. —David Eppstein (talk) 05:53, 26 April 2012 (UTC)
 * True, but what the proponents are arguing does not meet the blush test. This is the test whether a research mathematician blushes when he reads such stuff.  Anyway, any bicycle owner will tell you that diameter is more fundamental than radius.  Tkuvho (talk) 08:57, 26 April 2012 (UTC)
 * Actually, inch-for-inch, ride height is determined by a bicycle wheel's radius, not its diameter. Length of the spokes?  Also the radius. --Joseph Lindenberg (talk) 17:10, 26 April 2012 (UTC)
 * I really find this bizarre. Granted, I'm not a research mathematician, but I find the argument that a circle's radius isn't more fundamental than its diameter hard to fathom.  Yes, there's a physical ease factor that in practical terms, it's easier to slap a ruler up against something and measure its diameter than its radius.  And measuring diameter with a caliper is usually a more practical solution than sawing something in half so you can locate its center.  I don't dispute any of that.  In fact, it's the whole reason pi became the standard.  Not because ancient peoples had some ultra-advanced mathematical insight.  I'll give you more evidence.  Go look in the Name section of the Pi article.  Both William Jones and Euler didn't define π using diameter.  In fact, they seemed to go out of their way to use the radius.  They defined π as the half-circumference of a circle of radius 1.  It would be pure speculation to guess why, but I would guess that they, just like the mathematicians here, said, "well, it's no big deal, the 3.14 number is what everybody already uses, so why fight it to switch to a new standard." --Joseph Lindenberg (talk) 10:24, 26 April 2012 (UTC)


 * The defining feature of a circle is its radius, and so $τ$ = C/r (circumference divided by the radius) is the constant... --Stephen Abbott (in Math Horizons), by proxy via --Joseph Lindenberg (talk) 16:29, 26 April 2012 (UTC)

Straw poll on saying that tau is circumference/radius
Question: Should the entry about tau say that tau is circumference/radius? (Note this does not mean it should say anything beyond that, e.g. that supporters consider tau more natural because of it. We can vote on that separately.  This is just about whether it should say that tau is circumference/radius.)


 * YES. It only takes a few words, and it is tau's basic definition.  Also, it parallels/contrasts with pi's basic definition as circumference/diameter.  --Joseph Lindenberg (talk) 20:21, 27 April 2012 (UTC)

Based on the dwindling discussion, as well as the total lack of response to my attempt to take a straw poll on the circumference/radius issue, it's clear people want to move on to other things and have this over with. So I'm going to try invoking WP:BOLD to accomplish that. I took the lack of NO votes in my attempted poll to mean I had convinced people it would be OK to include the circumference/radius bit. If I'm wrong on that, let's all reconvene and hash it out. I used the 3 agreed upon sources, as well as the (seemingly most acceptable) BBC source to support the "have been reported in the media" phrase. So can we finally consider this tau issue resolved? If not, I'd suggest starting an "Arbitrary Break 2" section for the resumed discussion. But please consider whether it's really worth reopening the discussion. Thanks. --Joseph Lindenberg (talk) 11:25, 28 April 2012 (UTC)

Trying to develop consensus
I've picked a few of the choices in the above sections, trying to select the ones that editors gave as the result of comments made on earlier proposals. (Apologies if I've left out anyone's favorite. If so, please add it and WP:TROUT me.)  I'd like to try to get consensus on which version should be added. Sławomir Biały (talk) 12:21, 28 April 2012 (UTC)


 * Comment. I think that mentioning tau day is important.  I also feel that mentioning its media attention (without presence in the scientific literature) is important to provide balance.  I don't really have an opinion on the radius issue, but it is important to at least one other editor and no one else appears to feel that strongly about it.  So, with these considerations in mind, I would go with option 4, but I would prefer the first sentence to read as in the first two proposals: "Some individuals have proposed a new mathematical constant  tau ($τ$) which equals two times π."  Sławomir Biały  (talk) 12:23, 28 April 2012 (UTC)
 * That would be fine with me. --Joseph Lindenberg (talk) 12:29, 28 April 2012 (UTC)


 * I am not strongly opposed to the radius issue, and I will certainly go with the consensus. If option 4 is adopted the second sentence should be clarified. Taking into account Slawomir's suggestion above the new option would be:


 * Nageh (talk) 12:49, 28 April 2012 (UTC)
 * That seems like a good idea.  Sławomir Biały  (talk) 12:51, 28 April 2012 (UTC)
 * That would be fine with me too. I actually prefer it, but was trying to keep length to an absolute minimum.  One suggestion though.  Would it be better to say "defined as" rather than "based on"? --Joseph Lindenberg (talk) 12:57, 28 April 2012 (UTC)
 * I have deliberately avoided saying "defined as", to avoid arguments like we had above. The wording was also chosen in consideration of a reader recalling from the start of the article that such "definition" would only be valid in Euclidean geometry. Nageh (talk) 13:20, 28 April 2012 (UTC)
 * I believe that that is too much text, in light of the importance of this article (#488th read of all articles in WP). But in the spirit of consensus-building, I could live with it provided that (1) it is in the Popular Culture section; and (2) that it does not grow over time.  Any additional material on tau needs to go into an article dedicated to tau. --Noleander (talk) 13:12, 28 April 2012 (UTC)
 * .... and (3) the footnotes are limited to secondary sources, not primary sources (which would publicize the individuals who are promoting tau). --Noleander (talk) 13:17, 28 April 2012 (UTC)
 * I am not Michael Hartl. Above, Nageh and Tkuvho said they'd be fine with the Tau Manifesto.  Tkuvho specifically wanted Bob Palais' article. --Joseph Lindenberg (talk) 13:21, 28 April 2012 (UTC)
 * WP citation policies state that secondary sources are preferred to primary. In this situation, there are active promoters alive today who  are attempting to get tau, which is currently a fringe concept, adopted by the mainstream.  Publicizing the primary sources (the promoters) in this article is way too promotional.  We need to stick with disinterested, unbiased, secondary sources. WP is not a soapbox or an advertising forum. --Noleander (talk) 13:26, 28 April 2012 (UTC)
 * As I said above, it just seems strange, given how everyone who discusses tau links to this one document, the Tau Manifesto, it would seem strange not to reference it. Nageh was saying it should be done as a courtesy to the reader.  If you're absolutely opposed to it, I'll accept it, but I don't see the harm in listing it too. --Joseph Lindenberg (talk) 13:34, 28 April 2012 (UTC)
 * I already explained my reasoning above. If anyone wants to include primary sources as a citation, we'll need a Request for Comment (WP:RFC) to get consensus on that.  I would suggest getting consensus on the text first (in progress); then insert the text with secondary sources; then starting an RfC on whether the primary sources can also be included.  --Noleander (talk) 13:41, 28 April 2012 (UTC)
 * Agreed. So we just need to decide on the two stylistic questions I listed below, and the "based on" versus "defined as" thing. --Joseph Lindenberg (talk) 13:46, 28 April 2012 (UTC)
 * Whoops. Sorry, I just spotted Nageh's comment above about "defined as".  OK, "based on" it is. --Joseph Lindenberg (talk) 13:54, 28 April 2012 (UTC)
 * I certainly don't expect my website to be used as a source, if that's what you mean. --Joseph Lindenberg (talk) 13:23, 28 April 2012 (UTC)
 * I'll know better than to mess with it if we've achieved consensus. (What did people think about the "based on" versus "defined as" thing?) --Joseph Lindenberg (talk) 13:15, 28 April 2012 (UTC)


 * Two stylistic questions. Should "28 June" be reversed to "June 28" to match the next paragraph where it says Pi Day is March 14?  Likewise, should the d in "Tau day" be capitalized to "Tau Day" to match the next paragraph where the D in "Pi Day" is capitalized? --Joseph Lindenberg (talk) 13:40, 28 April 2012 (UTC)


 * Final proposed wording:


 * --Joseph Lindenberg (talk) 13:56, 28 April 2012 (UTC)


 * Agreed. --Joseph Lindenberg (talk) 15:08, 28 April 2012 (UTC)


 * As far as sourcing this, Stephen Abbott's piece seems optimal. He is a secondary source since he is responding to the primary proposals by B. Palais and Hartl.  Also, the publication venue is a step above "tabloid", and therefore preferable to numerous newspaper sources reproduced above.  Tkuvho (talk) 12:09, 30 April 2012 (UTC)


 * Good. Except I have another slight niggle: the word "supporters" in the second sentence seems a bit odd.  Would "proponents" be acceptable?   Sławomir Biały  (talk) 12:16, 30 April 2012 (UTC)
 * "Proponents" is better. Or perhaps "tauists" ?-) Tkuvho (talk) 12:19, 30 April 2012 (UTC)
 * The current version does not respond to Noleander's concerns about primary sources, namely B. Palais and Hartl. I think theses two references can be deleted without any loss of significant information. Tkuvho (talk) 14:52, 30 April 2012 (UTC)
 * When do we add a primary source as a reference and when not (in a non-mathematical/popular science context)? Are we saying that we are never accepting any primary source as a supporting reference in a non-mathematical/popular science context? Hartl's tau manifesto is widely mentioned/referred to in media reports on tau, so why shouldn't we include it as a courtesy to the reader (in addition to reliable secondary sources)? In contrast, I wouldn't add a ref to Palais's site for that matter. Well, just a few questions for you to answer. Nageh (talk) 17:23, 30 April 2012 (UTC)
 * You're just talking about Palais' website, right? Not his Mathematical Intelligencer paper?  (I'm requesting clarification because the pdf of the paper happens to be hosted on his website.) --Joseph Lindenberg (talk) 19:37, 30 April 2012 (UTC)
 * Personally, I am ok with the paper (also, to support the "some individuals" phrase). Nageh (talk) 08:42, 1 May 2012 (UTC)


 * Noleander's objection about "active promoters alive today who are attempting to get tau...adopted by the mainstream" doesn't apply to Palais. Yes, he's still alive, though I hope we don't start putting that sort of restriction on sources.  But he's not an "active promoter".  In fact, if you go to his website, he talks about having "passed the torch" to Hartl and Harremoes.  Furthermore, the source we're citing was indeed published in a legitimate math journal.  I wish Noleander would reappear so he can say if that satisfies him, but I really think the only question is whether to include Hartl's Tau Manifesto.  Hartl obviously would be considered an "active promoter".  Whether that means we should leave out the central document of the whole tau movement is another question. --Joseph Lindenberg (talk) 18:47, 30 April 2012 (UTC)

I think the tau paragraph should be the last in the section. Nageh (talk) 20:33, 30 April 2012 (UTC)
 * Noleander really wanted the transition between Tau Day and Pi Day. (Which is NOT happening with those two paragraphs reversed, Nageh.  Pi Day is at the beginning of the now-first paragraph, and Tau Day is at the end of the now-second paragraph.)  Pi Day is mentioned up in the introduction of the Pi article, so it's not like people won't have already read about it.  What's the big deal?  Other than Tkuvho's stated desire to juxtapose tau and the discredited Indiana Pi Bill? --Joseph Lindenberg (talk) 20:42, 30 April 2012 (UTC)
 * The objection is that it does not make sense to discuss pi, switch to tau, and then back to pi. And no, Pi Day has not been mentioned before (the lead section stands on its own, only summarizing the main article). I think most would prefer to have tau mentioned at the end of the section. This is what I am asking people to comment on. Nageh (talk) 22:35, 30 April 2012 (UTC)
 * Not a big deal to me either way. I was just trying to help Noleander get the smooth transition he wanted.  But I have no objection to putting tau last.  I think you should self-revert and get Noleander's input first though.  He was apparently fine with where it was, and like I said, there's no longer the nice segue from Tau Day to Pi Day.  I know he doesn't own the article, but considering how soon he intends to submit it for Featured Article status, we should probably try to let him know before moving things around.  (BTW Nageh, please answer my above request for clarification on Palais.  Thanks.) --Joseph Lindenberg (talk) 23:35, 30 April 2012 (UTC)
 * Noleander's objection to B. Palais still stands, not because he is an active promoter but because he is a primary source. Tkuvho (talk) 13:48, 1 May 2012 (UTC)
 * Why don't we let Noleander speak for himself? And speaking of Noleander, he hasn't been on Wikipedia since last Saturday afternoon.  That long an absence seems strange for him, especially now.  Anybody know what's going on, and whether he's alright? --Joseph Lindenberg (talk) 16:06, 1 May 2012 (UTC)

run-on sentence?
I shortened some of the verbiage in the current version by changing it to the following sentence: Some individuals have proposed a new mathematical constant tau (τ), which equals two times π, arguing that a constant based on the ratio of a circle's circumference to its radius rather than its diameter would be a more natural choice than π and would simplify many formulae. The change was reverted on pretense that this is a "run-on sentence". The fact is that this is not a run-on sentence. The question still remains whether it is preferable to the current version. Tkuvho (talk) 16:04, 1 May 2012 (UTC)
 * OK, you got me on not correctly remembering the definition of a run-on sentence from English class. Apologies to Tkuvho and to my high school English teacher.  Since I don't remember the proper term, I will just say the sentence is too long and complex.  Moreover, considering what we had to go through to achieve consensus on the wording, I would think that we'd all want to avoid reopening that discussion without a pretty good reason.  Saving 19 characters is not a pretty good reason. --Joseph Lindenberg (talk) 16:37, 1 May 2012 (UTC)


 * Shorter sentences are generally to be preferred. (That's all I'll say on this matter.) Nageh (talk) 16:48, 1 May 2012 (UTC)

Starting sentences with pi?
The manual of style (MOS) for math articles, MOS:MATH, says that sentences should not start with math symbols. This article has several sentences that start with π, such as "π is a transcendental number ...". I'm not sure if the MOS should be slavishly followed here, since the subject of this article is π itself, so it is natural that it should be the subject of many sentences in this article. Any time it starts a sentence, it could be changed to "The number π ..." or "The constant π ...", but that seems a bit unnatural. Can other editors say whether they think it is okay to deviate from the MOS since π is the subject of this article? Or should the MOS be scrupulously followed here? --Noleander (talk) 14:20, 3 May 2012 (UTC)


 * I agree with the sentiment. I follow the rule that if the sentence can be rewritten without π as the first word without sounding awkward, then we should do so, otherwise leave it be. It's hard to see how the sentence "π is a transcendental number ..." could be rewritten without sounding contrived, so I'd say this is an example of a case where it's okay as it is. And yes, the fact that π is the subject of this article does help define when it's appropriate to use the symbol in the article. — Loadmaster (talk) 15:36, 3 May 2012 (UTC)

Review
As requested by Noleander, I'm doing an FAC-like review here in order to spot problems of the article before the nomination process. I'd like to salute the authors of the article, most importantly Noleander, for their efforts. By and large I think the article is well done. However, it does have a number of weaknesses, most of all the global article structure, and secondly it should, IMO, improve on its depht of coverage of advanced mathematical material. That said, here are some more detailed comments: Jakob.scholbach (talk) 20:09, 22 April 2012 (UTC)

Introduction
In general I like it, it covers everything you need in the article. Try to find a copy-editor who goes over this. For example, I could imagine starting out like this "pi is the ratio of the ... The approximate value of this math. constant is 3.14159." (Many FAC reviewers only look at this part of the article, so be sure to be water-proof here. )
 * "Sometimes written pi" ?? Really, I have never seen a book not using the greek letter. [I agree.  I guess the editor that added that phrase was trying to explain to the reader why the   name of the article is "pi".   I have to admit that that question should be addressed in the first sentence or two.  I'll see if I can think of a better way to word it.  --Noleander (talk) 20:23, 22 April 2012 (UTC) ]
 * I looked again, and I think that it may be best to leave that phrase in the 1st sentence, for a couple of reasons: (1) the title of the article is "pi", and that really must be addressed in the first or 2nd sentence; and (2) the word "pi" is used commonly in non-math contexts, and even some math contexts, e.g. major book titles "Pi, a Source Book" and "A History of pi".   --Noleander (talk) 15:13, 24 April 2012 (UTC)
 * Hm, actually I think the article title should also be the Greek letter. This is technically possible, but I don't know how. Really, no-one, afaik, uses "pi". Jakob.scholbach (talk) 14:52, 25 April 2012 (UTC)
 * I believe Template:DISPLAYTITLE should do the job, but I cannot get it right at the moment. I really vote for removing the mention of "pi" in the first sentence. A welcome addition would probably be "pronounced pie [maybe a sound file]" in the lead section. Jakob.scholbach (talk) 10:26, 2 May 2012 (UTC)
 * DISPLAYTITLE does only allow changing letter cases and formatting, unfortunately; it is indeed a technical limitation. However, it should be possible to move the article to π. FWIW, pi is used when only the ASCII character set is available, e.g., in programming languages. Yep, could be helpful to stress that English doesn't pronounce pi like everyone else does. :P Nageh (talk) 12:25, 2 May 2012 (UTC)
 * I don't have a strong opinion about whether the article should be named pi or π.  However, 12 months ago there was an official Request for Move here, and it had about 30 participants, and the consensus was to keep the article title "pi".  If someone wants to initiate another RfM, I have no objection, but that prior RfM should be reviewed first. --Noleander (talk) 12:47, 2 May 2012 (UTC)
 * As a participant in that discussion, the main issue seemed to be that screen readers read &pi; as "pee", and apparently we can't put alt text into page titles.  Sławomir Biały  (talk) 13:05, 2 May 2012 (UTC)
 * Jakob: You wrote " I really vote for removing the mention of "pi" in the first sentence":  I moved that parenthetical mention of "pi" from the first sentence to the second sentence, to address your concern.  As long as the article is named "pi", that spelling needs to be included in boldface in the first couple of sentences.  Even if the article were renamed to π (which I would not object to, if that is the outcome of an WP:RFM) the optional (but rare) spelling "pi" would have to appear somewhere in the lead because it is used in so many book titles.  The only question is: where in the lead? --Noleander (talk) 12:53, 2 May 2012 (UTC)


 * "one of the most widely known constants" : I figure it is even the most widely known one. [Not Done - I agree, but an absolute statement like that needs a source, and I could not find one that said that.  Maybe they were thinking that 0 or 1 were more famous - Noleander]
 * This book, on page 160 says that the equation e^(i pi) + 1 = 0 refers to e, i, pi, 0, and 1 as the five most important mathematical constants. Maybe you can find a wording that reflects this status of pi? (Not necessarily in the intro, but down at the discussion of this formula). Jakob.scholbach (talk) 14:52, 25 April 2012 (UTC)
 * Done - Added " .... in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants" which exactly reflects the context in which that source mentions the importance (i.e. in relation to Euler's identity). --Noleander (talk) 13:01, 2 May 2012 (UTC)


 * "Several books" should be "several books". [Done - Noleander]

Fundamentals

 * The first paragraph with the digits is, to me, out of place. It starts out the section in an abrupt way. Also, the material here fits better elsewhere (see below).
 * Hm, I couldn't find where you are referring to. Nageh (talk) 19:48, 23 April 2012 (UTC)
 * "The first 100 decimal digits ..." I think section 1 should start with the definition of pi (say as pi = C / d). Then give an approximate value of pi. The fractions approximating pi should be moved to the discussion of the irrationality of pi. Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * Done - I moved that paragraph which contained the 100 digits into a sidebar box in the Properties section. I think the readers of the article will need a place to quickly grab some digits/fractions: since many may be visiting the article only to get a representation.  So "burying" the 100 digits (or fractions, etc) in prose is not helpful for that class of reader.  I think the sidebar box help solve lots of problems. --Noleander (talk) 13:43, 24 April 2012 (UTC)
 * Update - The Sidebar looked ugly, so I moved it into a dedicated subsection named "Approximate value". Let me know if it can be improved further. --Noleander (talk) 14:05, 2 May 2012 (UTC)


 * "The hexadecimal approximation" : insert "(base 16)" [ Done - Noleander]
 * I would always display important formulas on a separate line (e.g. pi = C / d, pi = Area / radius^2 etc.) [Done - I've put those two on dedicated lines; but the article used to have dozens of formulae that way, and there was a discussion above in the Talk page about making them more inline to help readability; so I suggest that only super important formulae get a dedicated line. - Noleander]
 * "This definition is only valid in Euclidean plane geometry". This material is too difficult to comprehend at this stage for most readers. I think you can remove it or move it to a footnote, at best. - [Hmmm. I hear what you are saying, but I'm not sure I agree.  This is not in the Lead, for one thing.  Also, it does explain somewhat why there are alternative trigonometric definitions. - Noleander]
 * The truth, as far as I know is, that no (professional) mathematician would define pi like C / d, exactly because of the problems you mention. In this sense, the Euclidean geometry definition of pi is usually (?) stated as a consequence of the calculus definitions. Of course, pedagogically it makes sense to start out with the circle, but non-Euclidean geometry will definitely be over 99% people's heads. Also, the terminology "circle" is not commonly used in non-Euclidean geometry, so it is questionable whether you need this explanation at all. Did you find this explanation like this in some book? Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * Yes, this section is based on page 8 of the Arndt source. The essence is (my paraphrase):  "For the layman, pi is C/d; but that has some issues for serious mathematicians, so there are these other, more obscure, definitions".     I've reworded the section to "This definition of π is not universal, because it is only valid in flat geometry and is not valid in curved geometries. For this reason, most mathematicians prefer a definition of π that does not rely on the circle, such as: π is twice the smallest positive x for which cosine(x) equals 0."  Does that seem better? --Noleander (talk) 15:31, 24 April 2012 (UTC)
 * Not really, IMO. The point is this: "circle" is terminology which is used only in Euclidean geometry. E.g. in Riemannian geometry, including, say, hyperbolic etc., it is uncustomary to speak of lines, instead one speaks about geodesics etc. Really, somebody not knowing non-Euclidean geometry will be unable to understand what you have written. Somebody who does know about it, will be confused by it. Jakob.scholbach (talk) 21:15, 27 April 2012 (UTC)
 * Jakob: Could you clarify the concern?   Are you saying that the new wording " it is only valid in flat geometry and is not valid in curved geometries" is too confusing and needs to be reworded?  Or that the circle-based definition needs to be deemphasize (moved below?) the non-circle-based definition(s)?  --Noleander (talk) 13:04, 2 May 2012 (UTC)

to "The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known, but it has been estimated as 7.6063." - Noleander]
 * "regardless of a circle's size" should read "regardless of the circle's size" [Done- Noleander]
 * Maybe an illustration for the claim with the circle of double size would be nice? (Just two circles rolling ...) [Not done - That picture is probably within my skill level :-)  But the illustrations are getting rather packed.  I think we should focus on the prose for now, get the prose  to FA standards, then see what kind of room is left for additional pictures.  I can draw some at that point. --Noleander (talk) 15:38, 24 April 2012 (UTC) ]
 * The wording is sometimes very repetitive, for example in the three trigonometric formulas (e.g., replace the 2nd one by "the smallest positive x satisfying sin(x) = 1, so that pi = 2 arcsin (1).") [Done - Reduced number of trig definitions from 3 to 1: so there is no more repetition.  Only 1 trig definition is needed.  --Noleander (talk) 15:39, 24 April 2012 (UTC) ]
 * "Mathematicians use the Greek letter": Math. is overly specific. Everybody uses it in this way. [Done - Changed to "The Greek letter π represents the ratio of a circle's ..." - Noleander]
 * Wording problems: the word "use" is used all over the place in the Name section ... [Done - "use" -> adopted or employed or utilized - Noleander]
 * OK. Keep watching for similar issues throughout. A copy-editor might be able to help out. Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * Italicize "Introductio ..." [Done- Noleander]
 * Wording in Properties section: "meaning" 3x [Done - That is a hard one: I went with "which means" and "in other words" ... But maybe someone else has some better ideas. - Noleander]
 * "The exact irrationality" This is not understandable without explanation. Give an explanation (this is related to structural problems, see below). [Done - Reworded
 * OK, better. However, an information like 7.6063 is not so useful unless one knows what this irrationality is. I'd suggest either adding a slightly more detailed explanation of the irr. measure or (rather) removing the 7.6063. If you prefer expanding it, it might be possible to "explain" the irrationality measure by mentioning ratios that approximate pi. Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * Done - Changed to "The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known, but it is estimated to be larger than the irrationality measure of other transcendental numbers such as e or ln(2), but smaller than the measure of Liouville numbers." --Noleander (talk) 15:47, 24 April 2012 (UTC)


 * "there is no polynomial" : insert "non-zero". Also, explain what this means (many people will not know polynomials...) [Done- Noleander]
 * This is a general problem of the article: it fails to explain mathematical terminology. For example, I think the average reader needs more detail to understand what a transcendental number is. True, we do have a separate article on this, but try and spend some more ink here. [Done- Added an example polynomial, so readers can see what pi is not the root of. - Noleander]
 * Consider explaining the irrationality by picking a linear polynomial, then take a quadratic one whose root is a bit closer (?) Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * Partially done - I changed the polynomial to illustrate transcendental to be a truncated Taylor series for sine, so that has more meaning now. I have not added a linear polynomial for irrationality:  irrationality is already well defined with fraction ...  adding a liner equation would be just embellishing unnecessarily, I think. --Noleander (talk) 16:05, 24 April 2012 (UTC)


 * The Nesterenko statement at the end of the section should be closer to the statement that pi is transcendental (the former is a sharpening of the latter). Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)  [Done - Noleander]
 * "area of a given circle" should read "area of a circle with radius 1" - [I believe the current wording is accurate:  "A consequence of the transcendence of π is the fact that it is not possible to "square the circle", in other words, it is not possible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle."  The principle is not limited to the unit circle.  - Noleander]
 * You are right. Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * "This follows from ..." : wording and sentence structure could be streamlined. [I'm open to suggestions on this, but it seems optimal to me: " ... it is not possible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This follows from the transcendental property, because no transcendental numbers can be constructed with the compass and straightedge technique.".  What is the specific issue?   - Noleander]
 * I'll try myself. If you don't like it, just rework it. Jakob.scholbach (talk) 21:15, 27 April 2012 (UTC)
 * In German "Squaring the circle" is a saying (meaning that it is utterly impossible). Is there a similar thing in English? If so, it might be worth mentioning in the discussion of this problem. [No. In English it is "putting toothpaste back into the tube" :-)  - Noleander].
 * What a shame :) Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * "putting toothpaste back into the tube" Love that one. Although, with a little bit of practice and patience :) Nageh (talk) 22:08, 23 April 2012 (UTC)
 * The section "Motivations for computing pi" is out of place here. I think smoothly merging it into the history section with mostly deals with this topic is more appropriate. [Sure, that can be done.  Did you have a specific recommendation as to where in the History section?   Right at the start?  Or at the end as a conclusion? or ?? - Noleander]
 * I would suggest putting it at / near the beginning of "Computer era and iterative algorithms". At this point, most readers will wonder, "why did they want to do that?". The earlier sections are somewhat more loosely correlated to pushing the number of digits to the extreme. Jakob.scholbach (talk) 22:00, 23 April 2012 (UTC)
 * Done - Noleander.

History

 * "1760 / 280 is approximately equal to 2pi". I wondered, how close? (I.e., is it plausible that the coincidence is a mere accident?) [Done - The sources indicate that it is mostly crackpots that believe the ratio is related to pi. I believe that it is very difficult to measure the size of the pyramids with certainty due to degradation & the fact that the baseline can be arbitrarily set at several levels; so by tweaking the measurements, you can get scores of possible sizes; hence you can get to any ratio you desire.   So, adding detail about the ratio would lend credence to the crackpot's claims.   However, to address your concern, I changed the wording to "The Great Pyramid at Giza, constructed c.2589–2566 BC, was built with a perimeter of approximately 1760 cubits and a height of 280 cubits; the ratio 1760/280 is about equal to 2π.". - Noleander]
 * OK, then just leave it as it is. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * The paragraph on the bible is, to me, too detailed. Mathematically it is not notable (3 is a crude approximation). I suggest removing this entire paragraph or at least trimming it down severely. [The Pyramid and Bible paragraphs are both reflecting material included in the major sources.  Just like the Indiana pi bill ... for some reason the sources talk about this stuff.  The article needs to reflect the sources.   The Bible is such a widely known book, that it seems important to keep in the article. - Noleander]
 * I'm almost 100% sure that the detailed discussion of the pool etc. is off-topic here. Maybe including the pi = 3 in the bible is appropriate, given the (non-mathematical!) importance of the bible, but such an inexact approximation is not so notable. I reiterate my suggestion to trim this down and move the rest to the subarticle. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * Done - moved the "discrepancy" sentence into a footnote, so now there are just 2 short sentences on the Bible topic. --Noleander (talk) 13:13, 24 April 2012 (UTC)


 * The whole section could be trimmed down without affecting the article's quality by introducing a table (year, mathematician, approximation, relative error, notes or something like that). Even consider creating a (logarithmic) chart showing the time at the x-axis and the logarithm (?) of the relative error at the y-axis. [That has been done, in a sense:    pi is a top level article, and there is already a subarticle Chronology of computation of π which has the table you are talking about.   That subarticle also has a graph showing accuracy over time .  But I was not able to include the graph in the pi article because its annotations are not in English.  I'm not a graph expert, or else I'd try to recreate it in English, but I dont think its absence is a bar to FA status.  As for the   prose in the History section:  I think prose is superior to a table for FA articles; and the level of detail in the History section is a good, encyclopedic summary of the sources: they too focus on the history, perhaps because it is a narrative, a story. The size of the pi article is still modest. - Noleander]
 * OK, this is certainly not a must. I would like it, but it should be OK without. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * "Infinite series": again, the article fails to explain this important terminology. This is something a reader can understand with a bit of help. Hence we should try to do so... [Done - Put a clear example right in the first sentence of the section; plus a few words of definition. - Noleander]
 * Better, but not yet FA-level. (I'm saying this because I think this is sure to come up in FA discussions. Have a look at other math FA candidacies to get a feeling...). I suggest this: try to avoid the sigma notation in the beginning, especially with superscript infinity. Rather say something like "the sequence 4/1 = 4, 4/1-4/3 = 2.66, 4/1-4/3+4/5 = ... gets arbitrarily close to pi. In the parlance of calculus, pi is therefore said to be the infinite sum ...". This is stuff that a non-mathematician can understand! Just serving the \sum_{i=0}^infty is not enough. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * Done - I added a table below that first infinite sum, which shows how each additional term gets it closer to π. What do you think?

--Noleander (talk) 18:03, 24 April 2012 (UTC)
 * Very good. Simple, clear and to the point. Jakob.scholbach (talk) 21:22, 27 April 2012 (UTC)


 * "Infinite series based on the arctan function generally converge much faster ..." You give a citation, so I hope that this claim is indeed found there (I did not check.) However, the sentence is mathematically vague. Can you sharpen its statement? [Done - Reworded to "Infinite series based on the arctan function converge faster than the series of Madhava or Leibniz, and were used to set records ..." - Note that the Leibniz/Mad formulae were described in the paragraph above. - Noleander]
 * "Lambert in 1761 proved" : how did he prove that? [Done - Added sentence " Lambert's proof utilized a continued fraction representation of the tangent function" including a link to existing article Proof that π is irrational.  The proof is non-trivial, unlike the sqrt(2) irrat proof.  Let me know if you think more should be done.  - Noleander]
 * OK, this is better. Maybe you could consider adding more mathematical material to the math section where you talk about that? I'm not sure what is the best modern proof, but mentioning an overview of proof ideas here seems a reasonable piece of the article. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * Partially done - I've double-checked the proofs described in Proof that π is irrational, and I just don't see how even the simplest proof could be summarized in the Properties section.  We could add a few buzzwords like reductio ad absurdum, but it would end up being superficial.  What I can do is add a statement to the effect that "the proof that pi is irrational is non-trivial" or "... requires calculus" or something like that, to let the reader know that the proof is not something that laymen could grasp?  I'll go ahead and do that.   Okay:  I added "There are several  proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique."  --Noleander (talk) 18:58, 24 April 2012 (UTC)
 * That looks fine to me. Nageh (talk) 22:57, 24 April 2012 (UTC)
 * To me too. Jakob.scholbach (talk) 21:22, 27 April 2012 (UTC)


 * "and it cannot be written as a fraction" just repeats what you said before. [Done - Noleander]
 * The "properties" subsection is very close to section 1.3. (with the same title). Merge the two. No reason to keep this separate, I think. [Let me think about that one.  I understand your point; but this is the classic conflict between organizing an article topically vs chronologically.  This article has both, as do many articles, and there is some inevitable overlap.  A little redundancy is a small price to pay for an outline that is inviting and understandable.  - Noleander]
 * It could help renaming the subsection to "Irrationality and transcendence [of pi]". Nageh (talk) 19:40, 23 April 2012 (UTC)
 * Done' - Implemented Nageh's suggestion: renamed the subsection (within the History section) to "Irrationality and transcendence" --Noleander (talk) 17:30, 24 April 2012 (UTC)


 * "the discovery of new iterative algorithms for computing pi" : use the greek letter consistently. [Done - Good catch. - Noleander]
 * "iterative approach": what does that mean? Explain! [Done' - Added "An iterative algorithm repeats a specific calculation, each iteration using the output from the prior step as its input, and produces a result in each step that converges to the desired value. " - That is a tough one to define concisely.  - Noleander]
 * That seems pretty inaccurate. An iterative algorithm does not have to converge (though it should ideally). Also, it may use the outputs from prior steps (plural) as its input. Maybe a source would help? Nageh (talk) 19:45, 23 April 2012 (UTC)
 * Done - I've improved it to " An iterative algorithm repeats a specific calculation, each iteration using the outputs from the prior step as its inputs, and produces a result in each step that – if properly designed – converges to the desired value." -  Let me know if that is not satisfactory. --Noleander (talk) 18:00, 24 April 2012 (UTC)


 * "multiply the number of digits at each step": insert "correct" [Done - Noleander]
 * Reword to "Canadian brothers Jonathan and Peter Borwein" [Done - Noleander]
 * "dozens of innovative new formulas" sounds a bit POV-like. Maybe just remove "innovative" [I concur that POV or WP:PEACOCK words should be avoided; but in this case an exception is okay:  every math historian comments on the extreme ingenuity of R's formulae; and it is appropriate to convey that sentiment in this article. - Noleander]
 * OK. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * "based on modular equations" : should be "modular forms", I think, but I'm not 100% sure now. [I have no personal expertise in that area; but it appears that both terms may be valid. The Arndt source uses "modular functions" or "modular equations" so I'd rather stick with that just to be safe. - Noleander]
 * "This equation converges much more rapidly". It is the series that converges, not the equation. Also, do explain the notion of rapidity of convergence (probably best done with two examples). This is needed to give the reader a sense of what's going on with all these series. [Done - I changed "equation" to "series".  Not too sure about adding convergence explanation here (didn't you above say the History section was already too long? :-)  What I did was add this at the first mention of "converge" in the article: "The Leibniz formula for π is simple, but converges very slowly (that is, approaches the answer gradually) "; and included a linke on "converge" to the WP article. - Noleander]
 * Hm. I currently don't have a strong feeling about length of the sections, but explaining carefully why people tried to improve the series is necessary. Maybe a thorough discussion of the math. background in "Rapidly convergent series" would be good? Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * Done - I added two inf series at the start of the "Infinite Series" section, and showed how one converged faster. --Noleander (talk) 01:06, 27 April 2012 (UTC)


 * "The Chudnowsky algorithm": is this really an algorithm? Or rather a formula? - [Done - I think either is appropriate.  I went with algorithm because there is a WP article on in "Chudnovsky algorithm", but Google hits suggests "formula" may be more common; so I 'll change to formula. - Noleander]
 * I think it is more appropriate. Be careful not to rely too much on other WP articles! Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * "The Chudnowsky algorithm developed by the Ch. brothers" reword to "The Chudnowsky algorithm developed". [Done - Noleander]
 * the PSLQ formula should end with a comma and then "where" with lower-case w. [Done - Noleander]
 * Are a,b,c arbitrary rational numbers? The text sounds like that. [No, that formula is representing a generic expression. Plouffe found several formulae that all fit into that template.  So a,b,c are not arbitrary; they are in fact very specific.  But they are rational, not irrational. - Noleander]
 * Rereading the paragraph I see what you mean. I think rewording this somehow would be good to make clear what you just told me. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * Done - reworded to "In 2006, Canadian mathematician Simon Plouffe, using the integer relation algorithm PSLQ, found a several formulae for π which conformed to the following template: ... where is eπ (Gelfond's constant), and  is an odd number, and a, b, c  are certain rational numbers that Plouffe computed" --Noleander (talk) 21:55, 24 April 2012 (UTC)


 * "because they can produce a sequence" : remove "can", I think. [Done - Noleander]
 * "providing instant feedback": series also do that, in a way. [Hmmm.  Maybe that can be worded better.  The point is that a normal series (all the ones discussed before in the article) do not produce single digits instantly:   they either produce clumps of digits or, so the sources imply, actually have to run to completion before they generate any digits (i.e. the high-accuracy arithmetic software they use is caching all the data somehow, and only produced output when it is fully complete). - Noleander. ]
 * All series produce "immediately" the first few (and then more...) digits. Typically the speed of convergence of a series is known, so the operator specify, e.g. I want to know 100 digits. However, during the course of the algorithm, the first 10 digits, say, are already known. The point is that a few particular series produce the n-th digit without computing (and therefore, without taking the time to compute) the n-1st one etc. This is the distinctive feature. Jakob.scholbach (talk) 11:52, 24 April 2012 (UTC)
 * Yes, I understand: I had the exact same thoughts.  However, the Arndt source defines two kinds of "spigot algorithms": (1) a digit extraction algorithm, which will produce any desired digit, without the preceding digits; and (2) an algorithm that produces one digit after another in sequence (vs. producing in clusters; or in one entire solution).  Wagon's algorithm is type 2;  BPP is type 1.  The "spigot algorithm" terminology is not widely used, and is a bit ill-defined.   I was tempted to use the power of WP to establish a new definition (i.e. define "spigot algorithms" as (1) only) but instead I was true and simply followed Arndt exactly.  But still, I can try to improve the wording; also I'll look for more sources to see if there  are other definitions. -Noleander (talk) 22:00, 24 April 2012 (UTC)
 * Ah, I see what is happening:  In 1995, Wagon published his algorithm in the article "A Spigot Algorithm for the Digits of Pi", American Mathematical Monthly, March 1995.  So, that established that definition.  Then came the any-single-digit algorithm, and the phrase "spigot" was already taken, so it is called a "digit extraction algorithm".  Arndt then lumped both under the rubric "spigot algorithm".   I don't see a huge problem here, since algorithm (1) has an official name  "digit extraction algorithm" and the article does use it.  I guess I could try to suggest to readers that the D.E.A. is not a spigot algorithm, but then that would be contrary to the Arndt source.   --Noleander (talk) 22:05, 24 April 2012 (UTC)
 * The original text I had for spigot alg was wrong, so I've improved it to: "Two algorithms were discovered in 1995 which opened up new avenues of research into π. The algorithms are characterized as spigot algorithms because, like water dripping from a spigot, they produce single digits of π and the digits are not reused after they are calculated.[  This is in contrast to infinite series or iterative algorithms, which retain and use all intermediate digits until the final result is produced.[  American mathematicians Stan Wagon and Stanley Rabinowitz produced a simple spigot algorithm in 1995 which generated individual digits of π in order: 3,1,4,1,5, ...., and prior digits are not used to compute later digits. .... The algorithm's speed is comparable to arctan algorithms but not as fast as iterative algorithms.  Another spigot algorithm from 1995 is the the BBP digit extraction algorithm discovered by Simon Plouffe." --Noleander (talk) 22:23, 24 April 2012 (UTC)

Usage

 * The first para. is an exact copy of the intro. A bit disappointing ... :) [Done - Noleander]
 * "is found in ... formulae" two times in a row. [Done  - Phrase still appears a few times in the article, but widely spaced. - Noleander]
 * "For any circle" : rather "For a" [Done  - Noleander]
 * The definite integrals rather belong to calculus, I think. [That integral could go either in Geometry or Integrals ... the former section is smaller and needs more prose so the pictures dont get too cluttered.  Unless you feel strongly about it, it may be better to leave it in Geometry. - Noleander]
 * Sure, somehow they belong to both sections. However, how about putting it to calculus, briefly explain that the integral is the area under the arc, explain that the sqrt(1-x^2) is the circle (link to Pythagoras), then explain how you calculate the integral using sin etc. This would make a more smooth transition. Again, try to explain as much as you reasonably can in this article. Every time a reader has to leave this page to, say, integral, there is a chance he will never come back. Given that integral is long and complicated, the chance is pretty high, I guess. Jakob.scholbach (talk) 11:56, 24 April 2012 (UTC)
 * Done - I added the following:
 * "In the above integral, the function $$\scriptstyle \sqrt{1-x^2}$$ represents a circle (the square root is a consequence of the Pythagorean theorem), and the integral $$\scriptstyle \int_{-1}^1 $$ is an operation which computes the area bounded by the circle's perimeter and the x axis. "
 * --Noleander (talk) 20:19, 26 April 2012 (UTC)


 * "Unit disk" : replace with "disk of radius one" [Done  - Noleander]
 * " A complex number z" : replace by "Any complex number"  [Done  - Noleander]
 * I would start out section 3.2. with "e^(i pi) = -1". The rest could serve as an explanation for this. [Most readers will probably benefit from a gradual progression of " here is a complex number; leads to Eulers formula; leads to Euler's identity.  Starting with The Identity then having a flashback is a bit dramatic :-)  - Noleander]
 * OK, if you want. Jakob.scholbach (talk) 11:56, 24 April 2012 (UTC)
 * That section should contain a mention of the Cauchy integral formula, I think.
 * Not done - I'd recommend against that. There must be  a zillion integrals and formulae that involve pi, and there is an entire subarticle List of formulae involving π on precisely that topic.  As the main, overview article, this article needs to stay above the fray so readers don't get bleary-eyed from excruciating detail.  In fact, it would probably be wise to go in the other direction and remove a couple of "usage" formulae, such as 1 or 2 from the Physics section.  --Noleander (talk) 20:22, 26 April 2012 (UTC)
 * Hm, I disagree. You know, the Cauchy formula is not just a random formula involving pi. Actually, all of the formulas here List_of_formulae_involving_%CF%80 are a consequence of the Cauchy formula. Other formulas, like List_of_formulae_involving_π these ones fall into a different regime (called special L-values). Machin-like formulae are often consequences of the addition theorem of sin or cosine, which ultimately boils down to e^(i x) = cos x + i sin (x). Another group of formulas, like Ramanujan's, is related to modular functions. On the other hand, I agree removing a physics formula might not hurt. Biot-Savart might be a candidate for some trimming. Jakob.scholbach (talk) 21:35, 27 April 2012 (UTC)
 * Done - Added new mention of Cauchy's integral formula including link & the formula. New wording is: "Cauchy's integral formula governs complex analytic functions and establishes an important relationship between integration and differentiation, including the remarkable fact that the values of within a boundary are entirely determined by the values on the boundary".    If anyone has any suggestion on how to improve that, let me know & I can help wordsmith. --Noleander (talk) 13:24, 2 May 2012 (UTC)
 * Done - Also, improved the physics section by reducing the number of examples from 6 to 4 (removed Biot-Savart) and converted bullets into prose. --Noleander (talk) 15:14, 4 May 2012 (UTC)


 * "and e is the base" : rather "and the contstant e = 2.7... is the base"  [Done  - Noleander]
 * roots of unity: insert "(complex numbers z satisfying z^n = 1)" [Done  - Noleander]
 * Move the Gaussian integral to proba. and statistics. [Done  - Noleander]
 * gamma function should read "Gamma function" throughout. (It is a capital Gamma). - [That may be a good thing to do, but the Gamma function article uses lower case. Maybe a query could be posted on the Math project to get a uniform convention established for WP articles. - Noleander]
 * Don't rely on WP articles :) Instead, a books.google query reveals that it is uniformely spelled with a capital G, very naturally.Jakob.scholbach (talk) 11:56, 24 April 2012 (UTC)
 * When researching capitalization in English, it is not terribly convincing to give results from German Google Books. It looks like a mix to me. --Trovatore (talk) 21:42, 27 April 2012 (UTC)
 * Done - Noleander


 * What is a "pinch point"? Spend some more time explaining this and trim down the part "based on this finding..." in order to keep it balanced. [Done -  Reworded to:  "He examined the behavior of the Mandelbrot set at the point (−.75, ε) located at the "neck" between the two largest regions of the set and found that the number of iterations until divergence, multiplied by ε, was equal to π. The point (.25,ε) at the cusp of the large "valley" on the right side of the Mandelbrot set " - Noleander]
 * The image on Riemann zeta can be removed. An almost 100% copy of the formula at the left.  [Done - Noleander]
 * "Finding a closed-form ..." is wordy. Also the paragraph structure can be improved.  [Done - Noleander]
 * Explain "relatively prime".   [Done - Noleander]
 * Don't use easter eggs :)
 * Done - Changed to "...the probability of two random numbers being relatively prime (that is, having no shared factors) ..." --Noleander (talk) 14:03, 2 May 2012 (UTC)


 * The formulas in probability and statistics tell relatively little. Why do they matter? [Done - Noleander]
 * The Cauchy distribution formula does not really strike a chord with me. Why is it important? (relative to pi?) Also, link and explain jargon such as density function. Jakob.scholbach (talk) 21:35, 27 April 2012 (UTC)
 * Done - Removed the Cauchy distribution formula. The sources do mention it a lot, but it is generally used as a pathological example, not a useful example.  --Noleander (talk) 13:38, 2 May 2012 (UTC)


 * Formulas on F_K and F_R should be on a separate line.  [Done - Noleander]
 * The length of a river: very nice! An animation would be even more awesome! [I agree. I'm not much of an illustrator, or I'd give it a go.   I think the article, fortunately, has plenty of good illustrations already. - Noleander]
 * The Fourier transform is related to the Gaussian integral. Jakob.scholbach (talk) 21:36, 27 April 2012 (UTC)
 * Done - The fourier xform was already mentioned in the "engineering" section. I expanded  the wording there.  I did not explicitly relate the FT to the Gaussian integral because that does not appear to be too  significant, particularly in the context of this pi article. --Noleander (talk) 13:51, 2 May 2012 (UTC)

Approximations

 * I would definitely prefer a graph showing the numbers of known digits of pi over the one of the recited digits!! Of course, memorizing so much is awesome. However, this is essentially unrelated to pi, people could (maybe they do?) also memorize digits of e, sqrt 2 etc. However, shedding as much light as possible on how to compute these numbers is a must for any article on a number such as this one. This is specific for the number you want to calculate. (Also, unrelated to this request, I'm amazed how much ink is spent on this tau question above...) Jakob.scholbach (talk) 21:41, 27 April 2012 (UTC)
 * I agree a graph of the digits of pi would be great. There is a picture File:Rekordaj_alproksimighoj_al_pi.png but the captions are not in English.  Plus, it is not sourced.  I'm trying to see if someone will volunteer to translate it to English. --Noleander (talk) 13:54, 2 May 2012 (UTC)

Jakob.scholbach (talk) 20:09, 22 April 2012 (UTC)
 * This section is entirely out of place, in my opinion. I would merge the Monte Carlo methods in the geometry section. (Especially the circle vs. square method is an immediate corollary of the geometry). Also, the continued fraction section should be close to the irrationality and transcendence section.  [Done  - I made the Monte Carlo material a 4-th level subsection under the Geometry section.  I moved the "Continued Fractions" into the Fundamentals section, next to Properties/Irrational material. - Noleander]
 * Explain "continued fraction" (first say, pi is not expressible as a fraction, then introduce continued fractions, then give examples.)   [Done  - Reworded to "Like all irrational numbers, π cannot be represented as a simple fraction. But irrational numbers, including π, can be represented by an infinite series of nested fractions, called a continued fraction: ... "  - Noleander]
 * Thanks for the feedback! I'll start implementing the suggestions soon. --Noleander (talk) 20:12, 22 April 2012 (UTC)

Nageh comments
I will just chain a few more comments of mine to this section:
 * Section Definition: "...because it is only valid in flat geometry and is not valid in curved geometries." This is no more clear than mentioning Euclidean geometry. Maybe it helps to simply say "...because it is only valid in standard geometry."?
 * I'm sure the term "standard geometry" would draw protests from many because it is not a generally used term. I don't mind "Euclidean", but JS above seemed to think it would be too confusing to lay readers.  I was hoping that "flat" and "curved" were a good compromise:  they are commonly used terms, especially in cosmology, and give a hint as to one being normal and the other abnormal.  But I'm open to suggestion.  The bigger question is: is it possible to craft a few sentences that explain the flat/curved issue, and why pi=C/d only works in one? --Noleander (talk) 01:11, 27 April 2012 (UTC)
 * I understand. Nageh (talk) 06:06, 27 April 2012 (UTC)


 * Can the cosine-based definition be given more prominence? E.g., ?
 * Can you clarify the concern? Do mean more prose to go with that sentence? Or are you suggesting displaying a math equation to illustrate the idea? --Noleander (talk) 01:13, 27 April 2012 (UTC)
 * Well, my thought was, if mathematicians prefer a definition of pi that does not rely on geometry then maybe such alternative definition should be given more prominence? But I think that display maths probably doesn't look too well. So I actually don't know how it can be given more prominence. Nageh (talk) 06:06, 27 April 2012 (UTC)
 * Okay, maybe it is best to leave it alone. Most of the sources on pi which are aimed at lay audiences start with the C/d definition, and the alternative "dont rely on Euclidian geometry" definitions are mentioned only in passing, or not mentioned at all. --Noleander (talk) 13:59, 2 May 2012 (UTC)


 * It seems that the sexagesimal approximation is a rather good one. Maybe you want to say "A good base 60 approximation is 3:8:30."?
 * Done --Noleander (talk) 01:14, 27 April 2012 (UTC)


 * Section Geometry and trigonometry: Please do not mix TeX with math for inline formulae in the same article.
 * Done --Noleander (talk) 01:26, 27 April 2012 (UTC)

Nageh (talk) 20:02, 25 April 2012 (UTC)

Some more quick comments:
 * Archimedes' constant redirects to this article. The name should be mentioned somewhere.
 * Done - Put it in the body of the article, in the History section. Perhaps it could go in the lead in bold ... but it is a very rarely used term (never used in modern era?) so it may be misleading to put it prominently in lead. --Noleander (talk) 21:00, 4 May 2012 (UTC)
 * Yeah, just leave it in the body, I think that's fine. Nageh (talk) 21:05, 4 May 2012 (UTC)


 * Milü and Zulü are the names given to the fractions discovered by Chongzhi. Since we have articles on them they should be linked in the text.
 * Done - Linked Milu. --Noleander (talk) 21:00, 4 May 2012 (UTC)


 * Maybe the table in the Rate of convergence section should state "π = 3.1415..." in the last row to ease comparison. How did you chose which number of digits should be displayed in the cells?
 * Done - Put 4 digits of pi in final column; also, made all other table entries 4 digits. --Noleander (talk) 21:00, 4 May 2012 (UTC)

Nageh (talk) 20:29, 4 May 2012 (UTC)