Wikipedia:Peer review/Pi/archive2

Pi
This peer review discussion has been closed.. I've listed this article for peer review because I"m planning on nominating it for Featured Article status. The reviewer should be someone familiar with the Featured Article criteria, and not afraid to nit-pick the article.  The reviewer need not have special mathematical skills (many other editors have taken care of that).  Thanks!  Noleander (talk) 12:44, 11 April 2012 (UTC)
 * Previous peer review

Comments from RJH – Overall it looks decent, although I continue to have concerns about the multitude of formulae being presented to what is likely a non-mathematical audience. I've attached a list of comments below; please pardon my somewhat terse style.
 * The lead contains the following long sentence:
 * $\pi$ is found in formulae related to the area of a circle, volume of a sphere, trigonometric functions, Stirling's approximation, Einstein's field equation, the Gaussian integral, the Cauchy distribution, Buffon's needle problem, Kepler's third law, the approximate period of a simple pendulum, the probability of two random numbers being relatively prime, the Mandelbrot set, and Euler's identity.
 * However, for most readers this is probably just going to be a list of things they have never heard of. Is there some way to put these in a context that is more meaningful for Joe reader? For example, many of these, but not all, have ties to the geometry of space. The odd ones out are perhaps Stirling's approximation and the Riemann zeta function.  [Done - Noleander]

Regards, RJH (talk) 21:10, 13 April 2012 (UTC)
 * Sure, that is a good idea. I'll take a stab at that. --Noleander (talk) 21:12, 13 April 2012 (UTC)
 * PS: I might collapse this discussion, so other potential reviewers don't see this and think you've volunteered to do a full Peer Review.  Is that okay? --Noleander (talk) 21:13, 13 April 2012 (UTC)
 * I was going to add more, if that's okay. Regards, RJH (talk) 21:14, 13 April 2012 (UTC)
 * Sounds good. The Featured Article criteria are at Featured article criteria.   I'd appreciate it if you could do a pretty detailed review and identify any shortcomings relative to those criteria.  Thanks!  --Noleander (talk) 21:16, 13 April 2012 (UTC)
 * I'll wait until you finish the review before I begin implementing your suggestions, so we don't get confused. Thanks again! --Noleander (talk) 21:19, 13 April 2012 (UTC)
 * Thanks. Yep I'm fairly familiar with the criteria. Regards, RJH (talk) 21:25, 13 April 2012 (UTC)


 * There are redundant terms: 'also', 'now', 'some', 'a variety of', 'several', 'many', &c. These need to be culled judiciously. See User:Tony1/How to satisfy Criterion 1a. [Done - Noleander]
 * "Another definition of π, which also relies on Euclidean geometry is $$\scriptstyle \pi = Area / radius^2.$$": this assumes the reader understands the parameters relate to the circle. It'd be better to make it clear. [Done - Noleander]
 * "Mathematicians use the Greek letter π to represent the ratio of circumference to diameter." Of any closed/convex shape? [Done - Noleander]
 * It looks unchanged. My question is whether π is specifically used for the circle, or for any closed/convex shape. The passage doesn't communicate which perception is correct.
 * Thanks for finding that. I did make the change here.  It looks like another editor may have undone it (or I somehow accidentally reverted??).  I'll just restore my wording. --Noleander (talk) 17:35, 15 April 2012 (UTC)
 * Ah, I see what happened. Two consecutive paragraphs in the Name section contain the phrase "ratio of circumference to diameter".  I fixed one, but not the other.  You were looking at the unfixed one; I was looking at the fixed one.   I've fixed them both now.  --Noleander (talk) 17:38, 15 April 2012 (UTC)
 * Thank you. Regards, RJH (talk) 23:14, 15 April 2012 (UTC)

Hopefully this is helpful. Good luck with your FAC. Regards, RJH (talk) 02:31, 14 April 2012 (UTC)
 * "Salikhov has given the approximation 7.6063" Who is Salikhov?  [Done - Noleander]
 * "An important consequence of the transcendence" is making a value judgment. Why is it important? See WP:YESPOV.  [Done - Noleander]
 * "...their efforts are sometimes ingenious, but doomed to inevitable failure..." This statement seems full of opinion and emotive language. From an impartiality perspective, it may be better if it were in the form of a quote. [Done - Noleander]
 * "The digits of π appear to be very irregular or random": 'very irregular' is vague. How does it differ from just 'irregular', or from 'random' for that matter?  [Done - Noleander]
 * "meaning that all possible sequences of digits (of any length) are equally likely": I think the likelihood is at a given length, rather than any length. Thus '111' does not have the same likelihood as '11'.  [Done - Noleander]
 * "They hypothesis that π is normal has not been proven or disproven." Who are 'They'?   [Done - Noleander]
 * "digits of π all consistent with normality": 'are consistent'?  [Done - Noleander]
 * "very evenly distributed" is an opinion, but I'm not quite clear what it is expressing. Is it just a judgment call on the part of the author, or did he perform a statistical significance test?  [Done - Noleander]
 * In paragraph the "Motivations for computing π" section, the point about breaking records is made twice. Once should be sufficient.  [Done - Noleander]
 * "Accounting for additional digits needed to compensate for computational round-off errors, a few hundred digits would suffice for any scientific application." This statement seems absurd for a number of reasons. [waiting for more detail from reviewer]
 * I think here it may just need a statement about practicality. For example: "In reality, the number of digits needed for any practical computation involving &pi; is less than 20." Not sure how you'd cite it though.
 * The article contains the text "For most calculations involving π, a handful of digits provide sufficient precision. Thirty-nine digits are sufficient to support most cosmological calculations, because that is the accuracy which is necessary to calculate the diameter of the universe with a precision of one atom...." There are several sources that mention the 39 digits; but I dont recall any saying a specific smaller number like 20.   I can look again. --Noleander (talk) 17:42, 15 April 2012 (UTC)
 * I'm probably making too much of a fuss about it. It's not a significant problem. Thanks. Regards, RJH (talk) 23:15, 15 April 2012 (UTC)
 * "Some Egyptologists conclude" is bordering on WP:WEASEL, but it does mention some in the reference. Still, it might be good to note that the idea originated with John Taylor (English publisher). [Done - Noleander]
 * "Some authors explain": 'some' is an indeterminate number. The reference only lists two. Does the reference say "some authors"? [Done - Noleander]
 * There are a number of uses of 'which' that seem to be missing a comma, but that's just an opinion. [Fixed one. The others, I read aloud, and the commas captured where my pauses were.  If any particular instances are brought to my attention, I'll double check them]
 * "but it is not certain how he arrived at the value, which he may have obtained from Archimedes or from Apollonius of Perga": the statement about uncertainty seems redundant here, since it is clearly implied by the "may". [Done - Noleander]
 * In some locations, names of persons are given with their nationality and profession; in others it does not. It would be good to be consistent and list the nationality and profession throughout. Some FAC reviewers seem to like that.   [Done - Noleander]
 * "Two new algorithms were discovered in 1995...": new is redundant.  [Done - Noleander]
 * "may be easily extracted": easy for whom?  [Done - Noleander]
 * In "Geometry and trigonometry" could discuss the formula for an n-dimensional sphere and show how it collapses to the expected formula in three dimensions . [I don't recall any sources mentioning that in relation to pi ... I'm trying to limit the article the the material that secondary sources about pi discuss. - Noleander]
 * No problem. It's already covered at Sphere.
 * "A complex number z can be expressed in polar coordinates as follows": you might put (r, φ) immediately after "polar coordinates" so that a non-mathematician isn't confused with the i (which is not explained until later).    [Done - Noleander]
 * "results in the remarkable Euler": for the non-mathematician, why is this remarkable?  [Done - Rewrote as "results in the Euler's identity, celebrated by mathematicians because it contains several important mathematical constants:" -  Noleander]
 * "the number of iterations needed for the point (-.75,ε) before escaping, multiplied by ε, was equal to π": escaping what?  [Done - chagned to "diverged".  The explanation is rather complex; Im not sure how much detail to put here.   The link to Mandelbrot set may be sufficient .. let me know if you think more is needed. Noleander]
 * In the "Physics" section, the parameters in the formula for the period of a pendulum need to be explained.  [Done - Noleander]
 * The "Probability and statistics" says "there are many distributions whose formulas contain π". This is both vague and unsourced.  [Done - Noleander]
 * "The approximation 355/113 was known in the fifth century in China" belongs in the history section. Does the citation for that statement apply to the sentence before?  [Done - Noleander]
 * The caption for the image in the "Continued fractions" section repeats a statement made earlier in the article.  [Done - Noleander]
 * "Poems for memorizing π is have been composed": 'is' or 'have been'?  [Done - Noleander]
 * What, no mention of the Indiana Pi Bill?   [Done - Noleander]
 * The references need some work for consistency.
 * Holton, David; Mackridge, – missing a first name? [Done - Noleander]
 * OED online", "pi" n.1; "pie", n.2.  [Done - Noleander]
 * V Kh Salikhov 2008 Russ. Math. Surv. 63 570 doi:10.1070/RM2008v063n03ABEH004543  [Done - Noleander]
 * C. Boyer, A History of Mathematics – more than one instance  [Done - Noleander]
 * Ogilvy, C. S., and Anderson, J. T., – 'and' instead of semi-colon  [Done - Noleander]
 * Halliday, David; Robert Resnick, Jearl Walker  [Done - Noleander]
 * Raz A, Packard MG, Alexander GM, Buhle JT, Zhu H, Yu S, Peterson BS  [Done - Noleander]
 * Several sources need expansion, such as a listing of the publisher, the work, &c. [TBD]
 * "5 Trillion Digits of Pi – New World Record", Oct 17, 2011.  [Done - Noleander]
 * Plouffe, Simon. "Indentities inspired by Ramanujan's Notebooks (part 2)". Retrieved April 10, 2009.
 * Can this be addressed? Bellard, Fabrice. "A new formula to compute the nth binary digit of pi". Archived from the original on September 12, 2007. Retrieved October 27, 2007.
 * Thanks so much for the scrupulous review. I'll start implementing your recommendations tomorrow.  My plan is to get it to FA status, then nominate it to appear on  the WP main page on pi day, March 14, 2013 :-)  --Noleander (talk) 02:44, 14 April 2012 (UTC)


 * Is this of interest per the history section? In 1775 the French Academy banned the submission of papers regarding the squaring of the circle in order to "keep the lunatic fringe in check" (per the author). If not, it might be useful on the Squaring the circle article instead. RJH (talk) 23:39, 15 April 2012 (UTC)

Comments by Nageh
I understand that I am probably not as competent as your fellow WikiProject Mathematics colleagues. Nonetheless, I'll give it a try:


 * Definition: It is being pointed out that pi is sometimes defined using the trigonometric functions to avoid the particularities of geometry. But it is not being mentioned that in order to avoid these particularities the trigonometric functions must be defined over their infinite series representations. In this regard it should be noted that such definition is more common in analysis. Also, the German pi article knows that Edmund Landau has defined pi as twice the smallest positive root of of the cosine(x).
 * I believe that you are correct ... but the sources I read did not explicitly say that, and for Featured Article status, everything has to be given a footnote which names a source. Ditto for the Landau fact.  --Noleander (talk) 21:37, 17 April 2012 (UTC)
 * A quick Google books search brought up this. This site includes the series and has a reference to a book this is supposed to be in, which I unfortunately don't have access to. You referenced Beckman, does it include a reference? Nageh (talk) 22:08, 17 April 2012 (UTC)


 * Polygon approximation era: "Archimedes' upper bound of 22/7 may have led to widespread belief that π was equal to 22/7", ...a value which was frequently used as a convenient approximation before the advent of the computer and electronic calculator.
 * True. But I think lots of values that were "frequently used".  Later in the article, it says "Archimedes' upper bound of 22/7 may have led to widespread belief that π was equal to 22/7" which is from a source. --Noleander (talk) 21:37, 17 April 2012 (UTC)


 * Maybe figure File:Archimedes_pi.svg could be used centered at the end of the Polygon approximation era so it doesn't get lost within the flow of images at the right border of the page. This would also move some of the other images and boxes further up near the text they belong to.
 * Done: I centered that picture; fairly near the bottom of the section. --Noleander (talk) 21:37, 17 April 2012 (UTC)


 * Properties of pi: When was it first speculated that pi was irrational?
 * I don't recall any of the reliable sources mentioning that. I Googled it, but found nothing except a few unreliable web sites that discuss it. --Noleander (talk) 21:37, 17 April 2012 (UTC)


 * Usage section:
 * Subsection Geometry: hint to Euclidean geometry?
 * Of course, the Fourier transform method has applications in several science fields, including telecommunications, multimedia encoding, signal analysis, in algebra for improving the speed of algorithms, etc. This gets a bit lost in this Usage section, e.g., it isn't mentioned in the Engineering section below.
 * I added the Fourier transform to the engineering section. --Noleander (talk) 21:58, 17 April 2012 (UTC)
 * Now it appears twice. :( I meant that its application should be better highlighted resp. the article better organized. Nageh (talk) 23:41, 17 April 2012 (UTC)
 * Ah, I did not see the other mention ... I remove the other one. Again, I'm not sure I see that kind of detailed information about the FT in the sources that are about pi. --Noleander (talk) 23:58, 17 April 2012 (UTC)


 * Related to Stirling's formula, pi is a re-appearing constant in formulas expressing cycle length values in functional graphs. Indeed, Graph theory wrt. cycle detection should probably have its own subsection.
 * I don't recall seeing any sources that specifically mention graph theory related to pi. Can you give some sources on that? I can take it from there if you start me off. --Noleander (talk) 21:58, 17 April 2012 (UTC)
 * Turns out this is rarely discussed under aspects of functional graphs but commonly under random mappings. A Google search for "random mapping cycle length" reveals tons of sources. Do you think you can identify some good ones? Nageh (talk) 23:54, 17 April 2012 (UTC)
 * Hmmm.  The sources I see are not about pi.  If you look in the References, those are sources about pi.  I think the article should be limited to that kind of material.  Other topics that use pi are in the thousands, and this article cannot hold them all, of course. --Noleander (talk) 23:58, 17 April 2012 (UTC)
 * Surveying has nothing to do with restricting itself to sources that explicitly are only discussing the use of pi. That is why it is called surveying. A more valid argument of course is missing notability of a particular use compared to other uses. And in this regard I probably agree with you – I thought it was being discussed within graph theory but it is not, so no need to cover. Nageh (talk) 00:12, 18 April 2012 (UTC)


 * Approximations: I'm a bit confused about this section. Approximations are discussed at length in the History section. Now it gets its section on its own but it is rather short. Can this be improved organizationally?
 * Yes, that needs to be fixed. I'll work on that. --Noleander (talk) 21:58, 17 April 2012 (UTC)
 * Okay: I've moved the summary of hex/dec/60 approximation up into the Fundamentals section ... which is good, since a reader quickly looking at the article to grab some digits will see them near the top. As for the MonteCarlo & ContinuedFractions:  I dont think they fit comfortably in the history section. It is probably best to leave them outside that, as stand-alone auxiliary sections, not tied to any particular historical era. --Noleander (talk) 12:49, 18 April 2012 (UTC)


 * Continued fractions: It should be noted in a sentence that continued fractions provide the best rational approximations of a number given a maximum denominator, and further that the values 22/7 and 355/113 determined in the antique correspond to two of these approximations.
 * Done. --Noleander (talk) 12:37, 18 April 2012 (UTC)


 * In popular culture: I think it would be fair to note that the legislature was accepted by the House of Representatives before it was rejected by the parliament.
 * Done. --Noleander (talk) 12:08, 18 April 2012 (UTC)

I hope you did find some of these comments useful. Nageh (talk) 19:07, 15 April 2012 (UTC)

One more: Why are the formulas in the Rapidly convergent series section more important than the other ones? I would suggest that you pick some particularly important formulas for centered display presentation, and box the other ones (without overwhelming the right border). Nageh (talk) 19:15, 15 April 2012 (UTC)
 * That is a good question. I think the article should be mostly words, a narrative.  Formulae should be de-emphasized, and place in boxes, generally.  However, in that section, I cannot find a way to move those to the right, without eliminating the photo of Ramanujan, which I consider very informative and pleasing aesthetically.   I also tried making the formulae smaller (still centered in the text), but that looked bad.  Another option is to eliminate the boxes altogether and make all formulae large and centered; but then the article is harder to read for the layman; and may discourage readers from reading the text.  I'm open to new ideas. --Noleander (talk) 12:19, 18 April 2012 (UTC)

Another one: Section Complex numbers and calculus could also mention pi's appearance in formulas for contour integration in complex analysis. Nageh (talk) 19:28, 15 April 2012 (UTC)
 * That is a tough call.  I think the "Usage" section could get very dry and ugly if it turns into a large list of things that employ pi.  The pi article has  a subarticle  List of formulae involving π ... I'm wondering if additional examples would be better off in there?  --Noleander (talk) 12:41, 18 April 2012 (UTC)

And another one: Should the Feynman point be mentioned? Nageh (talk) 19:30, 15 April 2012 (UTC) ← Possibly it could be mentioned in relation to the question whether pi is normal. Nageh (talk) 19:32, 15 April 2012 (UTC)
 * Done. --Noleander (talk) 12:08, 18 April 2012 (UTC)
 * Thanks for the feedback. Your suggestions are good. I'll start implementing them in a day or two.  --Noleander (talk) 20:08, 15 April 2012 (UTC)

I know you haven't added this text, but can the first paragraph of the Name section be phrased in a better way? It reads a bit awkward currently, and "curves" for serifs is probably not the best choice of word, either. Nageh (talk) 13:05, 18 April 2012 (UTC)
 * Sure, I'll see what I can do. --Noleander (talk) 13:07, 18 April 2012 (UTC)