Talk:Golden ratio/Archive 6

Aughost's christmas present
Today we got to Golden ratio looking quite garish, and a bunch of large-scale rearrangement that's hard to review. I think we've rejected that big complicated image before (or something much like it) as being not very informative and way too busy and complicated looking. Aughost should say here what he's up to, and why such big changes need to be made so fast. I think we should revert and consider more incremental changes. Dicklyon (talk) 17:20, 26 December 2012 (UTC)

Binksternet has reverted only the last step in Aughost's changes, leaving most of what I'm referring to. Dicklyon (talk) 17:54, 26 December 2012 (UTC)


 * Aha, because of holiday festivities I did not see the big changes made by Aughost. I don't think the reader is served by complicated images or convoluted text. We are here to break it down and make it comprehensible as much as possible. Binksternet (talk) 18:17, 26 December 2012 (UTC)
 * See also Pythagorean tiling and its talk page for past problems of a similar nature with the same user. —David Eppstein (talk) 19:24, 26 December 2012 (UTC)

Structural dynamics
I took out the section "Structural Dynamics", which seemed too odd. Nobody had bothered even to convert it to WP style (heading case, refs, etc.), and it seemed to give too much prominence to one minor occurrence of phi in obscure mechanicals systems, not well described or illustrated or relevant to any reason to think it interesting. Does anyone else think it should go back in? Dicklyon (talk) 22:42, 28 December 2012 (UTC)
 * I don't care for the section. It seems to be written in a singularly unilluminating manner.  If no one can be persuaded to rewrite it, then it should be removed altogether.  Sławomir Biały  (talk)|
 * Well, Tibbits has put it back with summary "This is not obscure to anyone with a smattering of engineering or scientific education. Suggestions for improving clarity are welcome." He ignored my suggestions implicit in my statement above, but I agree it probably can't be rescued by anything simple.  And yes I do have "a smattering of engineering or scientific education." Dicklyon (talk) 18:02, 3 January 2013 (UTC)
 * If it's not obscure then it would be covered in a text book somewhere, which should be given as a ref. instead of the journal article that is cited. There are certainly lots of quadratic equations that appear in science and certainly some have solution φ just by coincidence, so it seems to be that including every occurrence is not encyclopedic. There was a similar paragraph I tried to delete recently from the Right triangle article; when you worked out the solution to the quadratic equation you got φ3, but otherwise the connection to the golden ratio was tenuous at best. The golden ratio appears in many popular math books and most of the material is total crap, long since debunked but reappearing over and over to appeal to the kind of folks who want to believe aliens built the pyramids and Mayans knew when the end of the world will happen. The golden ratio is a very interesting number, but material consisting of coincidental connections to other scientific phenomena just make the article appear less reliable. It's like adding to the article on the Philippines that the population is the same as the number of miles from the Earth to the Sun; true enough but ultimately nonsensical trivia.RDBury (talk) 20:07, 3 January 2013 (UTC)

The "obscure physical system" is the harmonic oscillator. It's significance is phi's appearance in time as well as in space. No more coincidental than it's appearance in the line segment division problem. — Preceding unsigned comment added by Tibbits (talk • contribs)
 * A system of two masses and two springs and a support is a two-mode system, not a harmonic oscillator. The description, without specifying the one-dimension constraint that you have in mind, is obscure.  The particular system is itself "obscure" if you don't say why or where it comes up in an important way.  An illustration would help, if it's an important system, but probably it's not worth bothering.  Dicklyon (talk) 22:27, 3 January 2013 (UTC)
 * The two masses, two springs and one support are not important here unless they are shown to model an important practical system. We don't need trivial solutions in an encyclopedia. Binksternet (talk) 22:34, 3 January 2013 (UTC)
 * Well it would certainly help if the section would elaborate exactly how &phi; appears in the problem, rather than fetishically dwelling on things like page and figure numbers in books that a reader is not very likely to have access to. It isn't much good to tell the reader that the golden ratio appears on such and so page of such and so book.  This conveys no useful information.   Sławomir Biały  (talk) 23:25, 3 January 2013 (UTC)
 * It's the 31st of 35 problems at the end of a chapter in this book. The Moorman and Goff paper mentions a previous solution in which the golden ratio went unmentioned.  So why are they mentioning it now?  Hard to see.  And it has nothing to do with structural, just simple dynamics.  It's not obvious that it can be made into something useful for the article; perhaps a brief observation when a small illustration.  Dicklyon (talk) 00:18, 4 January 2013 (UTC)
 * I think an illustration and brief comment is worthwhile, at least if it's done properly.  Sławomir Biały  (talk) 01:21, 4 January 2013 (UTC)

I will be happy to rework the description, provide an illustration, and an equation or two. The references can be footnoted. Couple two identical harmonic oscillators. The two degree of freedom system then divides time with two frequencies whose ratio is the golden ratio. — Preceding unsigned comment added by Tibbits (talk • contribs) 03:09, 4 January 2013 (UTC)
 * That's not even correct. What does it mean to divide time here?  And the two frequencies are in the ratio of the square of the golden ratio.  Dicklyon (talk) 04:18, 4 January 2013 (UTC)
 * Yes, the ratio is the square of phi. To understand the division of time, consider the pendulum of a clock. The pendulum is a harmonic oscillator, which divides intervals of time into equal segments. If one pendulum were hung from the end of another, identical pendulum, would they divide time with frequencies in the ratio of phi squared? An example of an engineering problem in which this relationship of two modal frequencies appears is a three-story building, where the second and third floors have the same mass, and where they are supported on columns of equal stiffness. — Preceding unsigned comment added by Tibbits (talk • contribs) 14:36, 4 January 2013 (UTC)
 * But the two-mode system is not generally going to oscillate periodically like the pendulum. Only very special cases excite one periodic mode and not the other, and you don't get both time divisions at once in any case.  Do to the irrational frequency ratio, the motion is in general not periodic.  Dicklyon (talk) 18:31, 5 January 2013 (UTC)

The book by Morin cited above by dicklyon contains ten references to the golden ratio. The author lectures (lectured?) on physics at Harvard. Perhaps that addresses the concern about the system not appearing in a text. It also opens the possibility of a more general description of the occurrence of phi in the time domain of dynamic systems. Suggestions are not being ignored. It will take time to learn the syntax of including equations, references, and illustrations, as well as signatures.Tibbits (talk) 15:32, 4 January 2013 (UTC)

Solid matter can be approximated as an assembly (lattice) of point masses (atoms) coupled by springs (bonds), i.e., a multi-degree of freedom system of coupled harmonic oscillators. Texts such as Kittell or Ashcroft and Mermin use the phrase "harmonic potential" for the interatomic energy vs distance relation. It happens that quasi-particles in electronic enviroment of the solid state lattice can exhibit modal phenomena in which the golden ratio appears as the ratio of the first and second energy peaks. The ratio is not simply a numerical coincidence, but is predicted by theory. SCIENCE VOL 327 8 JANUARY 2010.Tibbits (talk) 17:55, 4 January 2013 (UTC)

Trimmed earlier speculation. Included text suggested by dicklyon. Included examples of significant systems. Relegated references to footnotes. Ilustration and equations still to come. Tibbits (talk) 18:16, 5 January 2013 (UTC)


 * Please fix heading case, put references after end punctuation, including URL in refs via templates such as cite book, and tell us what "mode shape" means (I'm pretty sure it's inappropriate, since the "mode shape" is not different from the modes of any other LTI system). The comments on "division of time" still seem appropriate, as well as separately commenting on period as if it was not obvious from frequency.  Dicklyon (talk) 18:22, 5 January 2013 (UTC)
 * Reviewing the book and paper, I'd say the book is worthless; it's just a bunch of problems cooked up that happen to involve golden ratio in the solution. The paper is more interesting. It would make some sense to show the equation being solved, maybe plot the x1 and x2 for the two modes and for a mixed solution.  But not go through the derivation, which anyone interested could easily do.  As for the building as an example of this equation, it's not a good one.  Dicklyon (talk) 18:43, 5 January 2013 (UTC)


 * I've taken it out until such time as someone is willing to make a decent section of it. Dicklyon (talk) 22:10, 10 January 2013 (UTC)

A one-word value judgement does not constitute a rational argument. — Preceding unsigned comment added by Tibbits (talk • contribs) 15:03, 14 January 2013 (UTC) The author of one of the references has committed to contributing his version of the discussion. Tibbits (talk) 17:58, 14 January 2013 (UTC)

This has been submitted to dispute resolution. Tibbits (talk) 20:05, 19 January 2013 (UTC)
 * Dispute resolution should not be used for a matter like this (where there is a clear consensus). The proposed text is in . One problem is that an article like this could be indefinitely extended with material where the golden ratio appears. The material would be of great interest to people with some knowledge of the topics mentioned, but as written the text is very unclear to the general reader. There are many technical articles where a general reader could barely understand any of the text, so difficulty is not an insurmountable problem, however to add an unclear and specialized observation would need acknowledgment in secondary sources that the observation is of particular interest. As proposed, it is unclear whether the result is a mere coincidence, or, as described above, there is some more intimate relation with the golden ratio definition. Johnuniq (talk) 02:32, 20 January 2013 (UTC)

Additional discussion appears at http://en.wikipedia.org/wiki/Wikipedia:Dispute_resolution_noticeboard/Archive_61#Golden_Ratio. None of the other editors is willing to restore any portion of the section to the page. I do not intend to make further contributions. Tibbits (talk) 22:19, 7 February 2013 (UTC)

Could Some One Sort Out the Reference to Plato's Timeaus
As it stands it is vague, and probably irreverent as Plato's Timaeus (as the article states) predates the invention of the concept. The description of the dodecahedron and icosahedron in terms of the golden number is probably fairly recent. What might be better would be a reference to the golden section as the ratio of a side and diagonal of a regular pentagon if any one knows when that was discovered. — Preceding unsigned comment added by 86.27.193.180 (talk) 22:10, 6 January 2013 (UTC)
 * Not irreverent, but possibly irrelevant. The timeline item is probably intentional vague, to make it correct.  Probably you're right that it doesn't belong, since there's no mention of extreme and mean ratio in it (as far as I can tell).  Dicklyon (talk) 23:29, 6 January 2013 (UTC)

Proposable Links from the German Version of the Article?

 * Georg Markowsky: Misconceptions about the Golden Mean. The College Mathematics Journal, Volume 23, Edition 1, January 1992
 * Clement Falbo: The Golden Ratio: A Contrary Viewpoint. The College Mathematics Journal, Volume 36, Edition 2, March 2005 — Preceding unsigned comment added by 91.21.43.149 (talk) 08:25, 1 February 2013 (UTC)


 * Proposable for what exactly? Adding them to external links or further reading? Or using them as sources to augment the article? Btw. Markowsky is already used as a reference for quite a while.--Kmhkmh (talk) 12:54, 1 February 2013 (UTC)
 * I think perhaps the anonymous IP has gone ahead with what he or she intended. I would like to assert that, while I certainly share an eagerness to discredit claims of the "divine" ratio in nature and art, I don't really think that this edit conforms to basic standards of academic scholarship.  For one thing, the reference to Sadowski is now being used to say basically the opposite of what it says in that text, a clear violation of WP:V.  Secondly, I was not previously aware that there was much controversy in the role of the golden ratio in the works (for instance) of da Vinci and Agrippa.  To suggest otherwise sets off something of a red flag that, I think, should require more thorough sourcing than that currently provided&mdash;particularly to say that these are "spurious claims" as the proposed revision asserts.  I think this is a violation of WP:NPOV, and in particular WP:WEIGHT.  Why is the entire weight of factuality being placed on a single source in the College Mathematics Journal (not exactly the gold standard in academic publishing&mdash;not necessarily to diminish the point that's being made there)?  It's not clear.  I have reverted the edit, pending discussion and further edits.   Sławomir Biały  (talk) 21:16, 1 April 2013 (UTC)

Mean of Phidias?
I just removed "mean of Phidias" from List of scientific constants named after people, after someone just added it, since I couldn't find much basis for it; just one 20th century book saying it has "also been called" that. I came here and found 3 sources cited for it, but except for the one book they looked unlikely, so I investigated the history. Back in 2006, some bozo added it with one ref, to the same one book I just found. But the refs were not arranged sensibly for what they support, and migrated to looking like they support mean of Phidias. This seems way to little to qualify "mean of Phidias" as another name for the golden ration, though it's everywhere now after 6+ years in WP. Can I just take it out? Dicklyon (talk) 21:01, 9 April 2013 (UTC)
 * I trust your scholarly sleuthing skills. __ Just plain Bill (talk) 22:40, 9 April 2013 (UTC)
 * Phi (the symbol for the golden ratio) is derived from Phidias, that info you can find in various sources (see for instance, , , ), meaning the you can certainly argue that golden ratio (more precisely its symbol) was named after a person (Phidias) and hence that entry in the list was correct. Another question is how common the use of the exact phrase "mean of phidias" is, that is much less common than the usage of the letter phi.


 * That is aside I somehow wonder, how adding some sourced information gets a fellow editor the title "bozo".--Kmhkmh (talk) 00:39, 10 April 2013 (UTC)
 * P.S. The article had that info (phi after phidias) sourced at thend of the history section.--Kmhkmh (talk) 00:47, 10 April 2013 (UTC)


 * No problem on the phi from Phidias. And the bozo was me so I figured I could get away with that one.  Dicklyon (talk) 14:31, 10 April 2013 (UTC)

sourcing
The current sourcing for the exact phrase "mean of phidias" is however unsatisfying/inappropriate in its current form. It cites 3 books/works and all without a page number.--Kmhkmh (talk) 00:51, 10 April 2013 (UTC)
 * It cites them, but that's because of the poor editting history. None of those three books contains the phrase, though ref 4 does (see my bozo edit linked above).  Dicklyon (talk) 14:31, 10 April 2013 (UTC)

Cairoun mosque and golden section.

 * A 2004 geometrical analysis of earlier research into the Great Mosque of Kairouan reveals a consistent application of the golden ratio throughout the design, according to Boussora and Mazouz.[25] They found ratios close to the golden ratio in the overall proportion of the plan and in the dimensioning of the prayer space, the court, and the minaret. The authors note, however, that the areas where ratios close to the golden ratio were found are not part of the original construction, and theorize that these elements were added in a reconstruction.

Earlier the our article correctly points out that (almost) all use in architecture and art before the moder era is "false or highly speculative" but then it continues to simply cite Boussora and Mazouz. Their article however false exactly in the "false or highly speculative" category, their analysis is essentially only numerical (and hence questionable) and they do not cite any historical evidence for their claims and the math sources they do cite (Heath, van der Waerden) however are somewhat outdated and essentially considered as "false"/"questionable" from current perspective.

In addition it is at first unclear to me whether the Nexus Journal in which the paer was published can be considered as sufficiently reputable for any claims on the golden ratio.--Kmhkmh (talk) 15:12, 11 April 2013 (UTC)


 * That's why it says "according to Boussora and Mazouz." We merely report their analysis; we do not suggest that it is right or important or sensible.  Dicklyon (talk) 16:39, 11 April 2013 (UTC)
 * Sure, but it begs the question why mentioning it at all and why it would relevant to the article, if it is just another questionable claim by 2 not particularly reputable authors.
 * Another problem is that content organization unfortunately suggests somewhat differently. If the article immediately after debunking such approaches and theories describes one of them without further context other than an author attribution, it may suggest to readers that this in an exception to the rule.--Kmhkmh (talk) 18:13, 11 April 2013 (UTC)
 * I won't miss it if you take it out or pose it better. Dicklyon (talk) 20:59, 11 April 2013 (UTC)

The whatwhat?
The first line... The what-whatofthewhatofthewhat? too many nested of the's. an this to this equals this to that is fine. a this of this of this of this t othis of this, equalls... wtf? 5.64.57.217 (talk) 21:39, 27 June 2013 (UTC)
 * I just tried rewriting it with fewer noun phrases. Better? —David Eppstein (talk) 22:50, 27 June 2013 (UTC)

I agree, the first line is incomprehensible even if it is mathematically correct: 'In mathematics and the arts, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to their maximum.' I'd be pretty sure that is NOT how the Greeks defined it, and its hard to see that the Arts would use such an awkward formalism. Much the better explanation is on the page for 'Golden Rectangle' and goes along the lines that a golden rectangle is one such that after a square is  removed from it, the remaining portion is also a golden rectangle, and hence has the same aspect ratio (proportions) as the first rectangle. It follows, referring to the figure and considering the whole rectangle (blue and red parts) that the aspect ratio of the sides of the rectangle is (a+b)/a. Similarly the aspect ratio of the remaining portion (red) is a/b. The ability to repeatedly subdivide a golden rectangle means that these two ratios must be equal,thus (a+b)/a = a/b, and this is solved only by a particular value of a/b, which is called phi. John Pons (talk) 09:09, 7 August 2013 (UTC)

Noted that Gandalf61 removed the above insertion on the basis that it was 'too specific' to be included here. I find that surprising given the highly specific nature of the other mathematical content on the page. I think there is a place for descriptive explanations alongside mathematical ones, and feel that Wikipedia is all the poorer when the mathematical perspective dominates. John Pons (talk) 07:56, 8 August 2013 (UTC)


 * I said it was too specific to be included in the lead . The lead section of an article is meant to be a brief summary of its contents. If you want to add a description of the properties of the golden rectangle, this should be in the body of the article, not in the lead. Gandalf61 (talk) 08:07, 8 August 2013 (UTC)

Can editors find citations that show how this recurrence relation approximates The Golden Ratio?
It can be shown that xn + 1 = 1 + (1 / xn) converges to The Golden Ratio with x1 = 1 as n → ∞. This is strikingly similar to the expression used to generate the number e. The differences in the two are that the expression used to approximate e is raised to the power of x and that it is not a recurrence relation.108.84.132.212 (talk) 22:05, 19 October 2013 (UTC) — Preceding unsigned comment added by 108.84.132.212 (talk) 22:03, 19 October 2013 (UTC)
 * This is just the standard continued fraction for the golden ratio. See the "continued fraction" line in the infobox in the Calculation section. —David Eppstein (talk) 22:39, 19 October 2013 (UTC)

Add
The Golden Ratio can also be expressed as the solution to the quadratic equation x^2 + x - 1 = 0 by completing the square to obtain x = sqrt(1.25) + 1/2 50.93.106.3 (talk) 04:46, 14 November 2013 (UTC)


 * that's just another way of solving the quadratic though. As one method is already mentioned we don't need another: the curious or those who prefer completing the square can use that instead.-- JohnBlackburne wordsdeeds 04:53, 14 November 2013 (UTC)

You're right, the square root of five is equal to twice the square root of one and a quarter. 50.93.106.3 (talk) 14:44, 14 November 2013 (UTC)

Golden Ratio on Architecture
For some reason, a good image showing the use of the golden ratio as a proportion rule in modern architecture has been removed twice by Binksternet. Though at first he removed it under the pretence that the architect himself had nothing much to do with the page, it was once again removed after his suggestions were met.

The topic is very well discussed on the page, and after a talk with David Eppstein, it was provided strong reference backing up the image. I think it is DISGRACEFUL that Binksternet is so arrogant as to prevent people to CONTRIBUTE to the page. His reasoning being 'original research' is an excuse for his large ego, since the reference I provided, being a doctoral dissertation, is the most reliable, though anyone with basic knowledge on Mid-Century modernism would know about the use of proportion rules.

Stop being a child. I am a new editor, and your actions are discouraging me from contributing or even learning further on how to. — Preceding unsigned comment added by RPFigueiredo (talk • contribs) 23:01, 20 June 2014 (UTC)


 * RP, you are free to question and discuss the decisions of other editors, but to call their actions "disgraceful", call them "arrogant", say they have a "large ego", and tell them to "stop being a child" are actions that are unlikely to advance your position, and probably transgress WP:NPA, and may lead to sanctions against you. And there's no evidence of your claimed discussion with David Eppstein, the other editor who reverted you.  So calm down and explain what you've got.  It looks to me like you added material from a 2012 primary source, to a mature article based mostly on secondary sources, which are the preferred sources for wikipedia.  See WP:WPNOTRS.   That's probably why the material was removed.  Dicklyon (talk) 02:10, 21 June 2014 (UTC)
 * The only interaction with me was what you can see on the edit summaries in the article edit history. —David Eppstein (talk) 03:11, 21 June 2014 (UTC)

I would be happy to see an analysis of Niemeyer's use of the golden ratio brought into this article. Such an analysis should be a representative summary of multiple writings about Niemeyer, all of which agree that he used the golden ratio for one his buildings. In that case, we could also show the reader an image of Niemeyer's golden ratio work. It would also be best for the encyclopedia to have even more detailed information at the Niemeyer biography. However, none of this should be brought into the article if there is only one little-known paper describing Niemeyer as employing the golden ratio. This topic is very broad, and we should tell the reader the major themes, not a minor sidebar. — Preceding unsigned comment added by Binksternet (talk • contribs)

The image is an example of the use of the GR in modern architecture. But I see your thought - examples must be cited on the text. Then I believe adding a image of Corbusier's work would be fine then, since it is mentioned?

I don't intend to 'advance my position' here, as opposed to what other users seem to be trying to do. I want to share knowledge. It's not helpful to remove representative information. But it is helpful to get in touch and try to help people in validating it. Just because someone is high rank on wikipedia doesn't mean they know all subjects, and that's were humbleness should kick in - Believe me, though I cited a doctoral dissertation, any first year architecture student could provide you secondary sources. But someone just don't seem to care about that...

I'm sorry for the term I've used about the interaction with David Eppstein - but his reversion was helpfully explained, I have no objections to it.

I would suggest someone adding an image of the regulating lines of the Parthenon - that would be much more explanatory to readers. But I won't dare to do any alterations after all that went through here... Just hope someone would.RPFigueiredo (talk) 05:24, 21 June 2014 (UTC)


 * Per WP:BRD, you need not hesitate to make a bold edit. Just don't get upset when/if it gets reverted; it can be a good discussion starter.  Now, if you could find sources that actually support the idea the the Parthenon was designed with the golden ratio, as opposed to the myriad of sources that just blindly repeat it even though there's no evidence for it, we'd have something interesting to talk about.   A Corbusier image is a good idea, if you can find one that's freely available and illustrates a use of GR.  Dicklyon (talk) 05:43, 21 June 2014 (UTC)

The issue I have is that people revert it before they even try to include it in the article. The approach is the opposite of what en encyclopedia should be. But that's fine.

Sourcing the use of GR in the Parthenon would be like sourcing F=ma. It's there for everyone to measure. Same with Niemeyer's congress building. But apparently I need secondary source for common knowledge.

Anyway, I could look for it, but I just don't care anymore. I hoped Binksternet would have encouraged my to do that on the first place, rather than instantaneously discarding contribution. — Preceding unsigned comment added by RPFigueiredo (talk • contribs) 06:20, 21 June 2014 (UTC)
 * Niemeyer at least probably left some documentation of his design choices, beyond the buildings themselves, unlike the designers of the Parthenon. And (without having researched the matter myself) it doesn't seem unlikely that he would have deliberately used the golden ratio. But there's a reason we're quick on the trigger with this article, more than a lot of others on WP: there's so much credulous repetition of bad information and not-even-close matches even in published sources (say, for example, the supposed occurrence of the golden spiral in the nautilis shell — it's a log spiral but with completely different parameters) that we need to sift through them to find the examples that bear up under scrutiny. —David Eppstein (talk) 07:23, 21 June 2014 (UTC)

Semi-protected edit request on 29 June 2014
Near the end of the section "Relationship to Fibonacci sequence", immediately before the sentence and paragraph beginning "However, this is no special property...", please insert the following (I.e., right after the 3-line equation sequence):


 * The reduction to a linear expression can be accomplished in one step by using the relationship
 * $$\varphi^k=F_k \varphi +F_{k-1},$$
 * where $$F_k$$ is the kth Fibonacci number.

208.50.124.65 (talk) 23:14, 29 June 2014 (UTC)
 * Red information icon with gradient background.svg Not done: please provide reliable sources that support the change you want to be made. — &#123;&#123;U&#124;Technical 13&#125;&#125; (e • t • c) 15:04, 1 July 2014 (UTC)


 * This standard equation appears as the second equation in Fibonacci number. That article gives no source for it, probably because it is so standard and easy to verify. However, one source that states it without proof is Mitchell, Douglas. "Powers of $$\varphi$$ as roots of cubics", Mathematical Gazette 93, November 2009, 481-482. 208.50.124.65 (talk) 15:33, 1 July 2014 (UTC)

Egyptian Pyramids
"Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ."

"the 3:4:5 triangle was the only right triangle known to the [ancient] Egyptians" is unverifiable and illogical. If an individual can create two measured lines (sharing an endpoint) and make them perpendicular to one another, then that person can construct any right triangle.

"they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ." is unverifiable.

AnonymousAuthority (talk) 05:51, 15 November 2013 (UTC)
 * Take it up with Eric Temple Bell's ghost. We are merely reporting what he said. —David Eppstein (talk) 06:05, 15 November 2013 (UTC)


 * The word "since" divides what he claimed from what is implied as some known facts. The sentence either needs to be rephrased to clarify that it is not a known fact, or a direct quote needs to be provided. AnonymousAuthority (talk) 20:07, 16 November 2013 (UTC)

That the Ancient Egyptians did know about irrationals, and that they did base their entire number system and way of doing mathematics on them, as grounding principles is outlined in detail in R A Schwaller de Lubicz's The Temple of Man - 2 Vols - Trans. Lawlor & Lawlor - ISBN 978-0892815708. Your article should take the timeline back far enough to include this information.86.160.8.214 (talk) 21:18, 17 January 2014 (UTC)
 * This article is neither about irrational numbers in general nor about fringe theories of Egyptian mathematics published by woo astrology/tarot publishers. That material does not belong here. —David Eppstein (talk) 21:56, 17 January 2014 (UTC)

Do you need pi in order to know about the Golden Section?? I don't think so. The Timeline section of this article arguably must start with the Egyptians, in any case, as it has been fairly well established that the proportional ratio is present in the pyramids, regardless of any controversy about their mathematical prowess. see for instance math.iit.edu/~mccomic/420/presentations/goldenRatio.ppt; see http://milan.milanovic.org/math/english/golden/golden3.html notwithstanding the poor English, it is written by a Serbian engineer with decent credentials; see also http://goldenratio.wikidot.com/egyptian-art, and http://jwilson.coe.uga.edu/EMAT6680/Parveen/ancient_egypt.htm, excerpts here: "The Egyptians thought that the golden ratio was sacred. Therefore, it was very important in their religion.  They used the golden ratio when building temples and places for the dead.  If the proportions of their buildings weren't according to the golden ratio, the deceased might not make it to the afterlife or the temple would not be pleasing to the gods.  As well, the Egyptians found the golden ratio to be pleasing to the eye.  They used it in their system of writing and in the arrangement of their temples.  The Egyptians were aware that they were using the golden ratio, but they called it the "sacred ratio."" Cesca1910 (talk) 09:13, 3 July 2014 (UTC)

The Theorem of the Golden Mean
Add a reference to Thean'so "The Theorem of the Golden Mean" in the Pythagorean school? it would be useful to show works on the golden mean at that time. — Preceding unsigned comment added by 78.137.177.20 (talk) 10:06, 20 August 2014 (UTC)

not well fleshed-out
I understand the rejection of my edit, especially given the general style of mathematical articles. First though - are there any other such numbers, whose inverted value plus one equals the value before inversion ? I'm not a mathematician, but I have though read (and used) more maths than most other people, including at university level. Although it was some years ago now. I just find the math-related articles perhaps could be a bit easier to read, and sometimes give more "hard" examples of complicated formulas. I don't believe my rejected edit helped very much there, but I though it might get some reader more interested. (and if so, necessaraly early in the article) However I agree it could be better fleshed-out. Boeing720 (talk) 23:25, 25 September 2014 (UTC)
 * There is no prohibition on the lead including material covered later on, but I don't think your addition belonged for two reasons. First it was not encyclopaedic. It is not for us to say what we find interesting, or to comment on facts. We should simply state the facts and let them speak for themselves. There is nothing to stop you making them interesting, with good writing, presentation, layout. In fact that's of the main things distinguishes our best articles from the merely good ones. But you should not say so. Second the lead section is the place to say what something is (so a definition), say why it's important (so put it in context), and say how it is used/what it relates to. It's not the place for more detailed properties and examples, which can go in their own section and expand on the definition. The lead is the wrong place for this.-- JohnBlackburne wordsdeeds 23:48, 25 September 2014 (UTC)


 * (ec) But this is exactly the content of the first paragraph of the calculation section, is it not? That $$1+1/\phi=\phi$$ is just a quadratic equation.  This has two roots: one is the golden ratio $$\phi$$ and the other is $$-1/\phi$$.  So there are exactly two numbers with that property.   Sławomir Biały  (talk) 23:54, 25 September 2014 (UTC)
 * Thanks ! Boeing720 (talk) 20:34, 26 September 2014 (UTC)

Section on Irrationality
The picture and caption under "irrationality" does not prove that the golden ratio is irrational. It states that: "If φ were rational, then it would be the ratio of sides of a rectangle with integer sides. But it is also a ratio of sides, which are also integers, of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely, so φ cannot be rational."

But I fail to see how the fact that the side lengths stop being integers implies irrationality. — Preceding unsigned comment added by 130.245.221.218 (talk • contribs) 11:13, 23 November 2014
 * Because it's not possible to have an infinite descending sequence of positive integers. If the first number in a sequence is n, then any descending sequence starting from that number can run at most n steps. —David Eppstein (talk) 18:57, 23 November 2014 (UTC)
 * I think the OP has a point here - that simply means that the sides won't be integers; it doesn't entail that their ratio cannot be rational (i.e. that the ratio cannot be expressed as a ratio of integers). The proof seems to be missing something. -- Scray (talk) 19:23, 23 November 2014 (UTC)
 * Huh? Read the first line the OP quoted: if the ratio were rational, there would exist a rectangle with integer sides. For instance, one side can be the numerator and the other the denominator of this supposed rational number. And if this rectangle existed, there would exist a smaller rectangle that still had integer sides. And so on into an impossible infinite descent. There is nothing missing from a careful reading of the text that's already there. —David Eppstein (talk) 20:22, 23 November 2014 (UTC)
 * I'm sure I'm missing something, so let me explain where I'm coming from and you can tell me what I'm missing (and note that I'm not arguing that φ is rational - I'm just asking if the proof is compelling): let's take a classic 3-4-5 (side length) right triangle. The ratio of any pair of side lengths is rational. If one takes similar triangles of progressively smaller area, their side lengths are not integers but their side length ratios remain rational (and constant, of course). That the side length of the triangle (or in the case of this proof, the rectangle) are not integers does not mean the ratio is not rational. I think the proof lacks evidence that the side length ratios cannot remain rational, which has to do with the way the side lengths shrink (rather than their resepective absolute lengths). What am I missing? -- Scray (talk) 21:08, 23 November 2014 (UTC)
 * What you are missing is that in this case the side lengths remain integers, as the caption already states — see the phrase "which are also integers" in the quote given by the OP. Specifically, when you delete a square, one of the sides of the new rectangle is also a side of the old rectangle (an integer) and the other side is the difference of the old rectangle's sides (a difference of two integers is always an integer). —David Eppstein (talk) 22:11, 23 November 2014 (UTC)
 * Ok, thanks. -- Scray (talk) 22:29, 23 November 2014 (UTC)


 * Ah, I see. What I was missing was the "which are also integers" bit.  The placement of that clause in the sentence is a bit awkward and breaks up the sentence in an unexpected way that threw me off.  But you're right--the sentence as written is perfectly correct and a careful reading reveals no problem in the proof..

First paragraph needs to be improved and simplified
I speak various languages and I find the English version unnecessarily vague in the first paragraph exactly where it shouldn't be. For Wikipedia, I consider the current first-paragraph delivery a disservice.

In the introduction ordinary persons capable of thinking for themselves need to find out in one paragraph what the golden ratio is. Right now, that relatively simple task is way off mark; it is written by specialists and they are using specialists' language only. For an introduction on something this beautiful - that is bad.

I checked the Dutch site on the same matter and found they do have what it takes:

Here is the translation made by Google that I improved some (but not to perfection): "The golden ratio, also called the division in extreme and mean ratio, is the division of a line into two parts that shows a special relationship. To find the golden dissection (ratio/cut), the comparison of the largest of the two parts to the smallest is identical to the comparison of the whole to the largest segment. When indicating the largest segment as a and the smallest part as b, then the ratio of the two is such that a: b = (a + b): a."

In my words: the relationship that the larger segment has to the whole is identical in measurement (but not in position) to the measurement the smaller has compared to the larger segment.

Thank you for whoever is guarding this page. Please, will you make it a more wiki-worthy page in the first most vital paragraph? It's a rule of thumb that all interested readers should be able to index the idea right then and there.

P.S. I also found an image on the Dutch wiki page that is insightful:

http://commons.wikimedia.org/wiki/File:Gulden_snede_02.jpg

The image shows that when creating a circular line from location A, and -next- creating a circular line from C, starting at this position D, that we then find E. And E delivers the golden ratio on the line B - C. It is lovely. Maybe you'll also use it (though not in the first paragraph)?


 * Looking at the current lead, I presume the problem is that you see the word "ratio" as to specialist compared to the more nonspecific or vague words "relationship" and "comparison".  Is that it?  The Dutch "verhouden" or "verhoudt" for "relationship" might equally be translated as "ratio", perhaps, since Google translate shows English "ratio" going to Dutch "verhouding".  In that respect our lead is sort of like the Dutch one, except a bit more abstract in focusing on "quantities" rather than lengths of line segments.  The Dutch title "Gulden snede" is more about the "cut" than the "ratio", so maybe that's the relevant difference?  What would you suggest?  Dicklyon (talk) 05:35, 10 March 2015 (UTC)

Semi-protected edit request on 11 May 2015
Dear wiki-crew/administrators,

The mathematical analist Johan Louis Lobo from the Netherlands developed in 2010 a set of variables (21) derived from Phi, root 2, root 5, root 10 after being approached by a friend who asked him to look deeper into the dynamics of the ratio. After some time he was able to prove that varibles derived from these constants (work together) so to say. In the next demonstration he points out the dynamics of the golden ratio and its variables. He worked 5 hard years to finally convince students arround him what he had discoverd :)(obeisances to all the people who worked on this ratio)

Lets start simply with (re)defining the number 1. As i found like fifteen new formulas with this set of variables to do so.

The best example how they work together is next......>

And Eureka, because; $$\left(1/_2 +\rho\right)$$ * $$\zeta^2$$ = 1

$$\rho$$ = $$\frac{\varphi}{_4}$$...... ζ = - 1 + $$\sqrt{_5}$$

$$\frac {\zeta} {\iota}$$ = $$\psi$$ ...and... $$\frac {\iota} {\zeta} $$ = $$\nu$$..... this results in; $$\nu$$ * $$\psi$$ = 1

Maybe its good to consider; $$\frac {_1} {\zeta * {\eta}}$$ = 0.6545084971 = $$\frac {_1} {_4}$$+$$\rho$$

So in this way the Golden ratio works together with root 5 to produce 1.

Lets see the next approach to 1 with the variables, Iota, Mu and Phi now,

$$\frac {\varphi} {\mu}$$ = $$\iota$$ ....$$\frac {\mu} {\varphi}= \varphi^2$$ And if we now multiply the two products while we squere Phi, we see that we approach one aswel,

$$\varphi^2$$ * $$\iota$$ ˜ 1 . Now $$\mu$$ = -1+$$\varphi$$ and $$\iota$$ = -$$\zeta+\varphi$$ See that, $$\varphi^2$$ - $$\iota$$ = $$\gamma$$...or..$$\mu$$ + $$\varphi$$ = $$\gamma$$....... $$\gamma$$ = $$\sqrt{_5}$$

($$\gamma$$)R^>

$$\gamma$$ = $$\varphi^2$$ - $$\iota$$

$$\gamma$$ = $$\varphi^2$$ * $$\iota$$ * $$\gamma$$

$$\gamma$$ = $$\psi$$ * $$\varphi$$

$$\gamma$$ = $$\frac{\psi}{\mu}$$

$$\gamma$$ = 2$$\xi\lambda$$...where $$\lambda$$ = $$\frac{\kappa}{_4}$$.... and.... $$\xi$$ = $$\gamma\kappa$$

Lets consider we have an expanding squere.....If side a= Phi and b aswel, then the root will be 2$$\tau$$ To understand what is $$\tau$$.......$$\zeta$$ * $$\tau$$ = $$\kappa$$ which is $$\sqrt{_2}$$ If this is true then $$\kappa$$ * $$\tau$$ = $$\varphi$$

This means that we can simulate the function of $$\sqrt{_2}$$ with $$\varphi$$, because $$\frac{\varphi}{\tau}$$ = $$\sqrt{_2}$$

How its possible is as follows; $$-\kappa {_+} \frac{\varphi}{\omega}$$ = 0.033000033..... $$\omega$$ = $$\frac{_1}{_2}\varphi$$

In this way $$\omega *\kappa$$ = $$\sqrt{_2},{_5}$$

And what follows is....4 * $$\frac{\omega}{\kappa}$$ = $$\xi$$....and $$\xi$$ = $$\sqrt$$

Another approach to this argument is..... $$\kappa $$ = 4 * $$\left(\frac{\omega^2} {_1{_0}}\right)$$

$$\frac{\mu}{\tau^2}$$ = $$.{_4{_7{_2{_1{_3{_5{_9{_5{_4{_9}}}}}}}}}}$$... ...$$\frac\frac{\mu}{\tau^2}{_1{_0}}+ {_1.{_4}}-.{_0{_3{_3}}}$$ = $$\kappa$$ or in more complex ways...

4 * $$\frac\sqrt\frac{\rho}{_2}{\varphi}$$ .... 2*$$\left(\frac\frac\frac\sqrt{_2,_5}{\alpha}{\delta}{\nu^2} -3\right) $$.......$$\frac\frac{\left(\left(\sqrt{\tau}\right)* \psi\right)^2}{\nu}{_5}$$.......$${_0}.{_0}{_0}{_1}{_3}{_6}{_8}{_8}{_9}$$ + $$\zeta$$ + $$\frac{\varphi\lambda}{\xi}$$....hehehe ")

Lets see some more quallity's of these great ratio's;

$$.{_5}$$ = $$\frac{(\varphi+.5)*\mu}{\varphi^2}$$..............$$\eta$$ = $$\frac{(\varphi+.5)*\mu}{\varphi}$$

$$1$$ = $$\frac{(\varphi+.5)*\mu}{\tau^2}$$..............$$\eta$$ = $$\frac{(\varphi+.5)*\mu}{{_2}\zeta}$$..............

$${2}$$ = $$\frac{(\varphi+.5)*\mu}{\nu^2}$$.............. $${\rho}$$ = $$\frac{(\varphi+.5)*\mu}{\zeta\iota}$$..............($${_2}\tau$$) = $$\frac{(\varphi+.5)*\mu}{\varphi\kappa}$$

$${8}$$ = $$\frac{(\varphi+.5)*\mu}{\rho^2}$$..............$$\rho$$ = $$\frac{(\varphi+.5)*\mu}{\sigma}$$..............$$\sigma$$ = $$\frac{(\varphi+.5)*\mu}{{_2}\rho}$$..............

$$\sqrt{\mu^2+\varphi}$$ = $$\sqrt{_2}$$..... $$\sqrt{\mu+\varphi^2}$$ =$$\sqrt{\sigma}$$...........$$\sqrt{\mu^4+\varphi^4}$$=$$\sqrt{_7}$$........$$\sqrt{\mu^6+\varphi^6}$$=$$\sqrt{_1{_8}}$$......etc

$$\sqrt {\varphi^2+\iota}$$ = $$\sqrt {_3}$$.... $$\sqrt {\varphi^2+\psi}$$ = $$\sqrt {_4}$$....

Here we see a few ways how 16 of the (21) variables are derived from $$\varphi$$

$$\frac{({_1}{_+}{\eta)}}{\varphi\lambda}$$ = $$\xi$$.......... $$\frac{({_1}{_+}{\nu)}}{\varphi\lambda}$$ = $${_2}\tau$$.......... $$\frac{({_1}{_+}{\mu)}}{\varphi\lambda}$$ = $${_2}\kappa$$.......... $$\frac{\varphi}{\tau}$$ = $$\kappa$$.......... $${_2}\varphi$$ = $$\theta$$.......... $$\gamma$$ - $$\varphi$$ = $$\mu$$........... 4 * $$\frac\sqrt\frac{\rho}{_2}{\varphi}$$ = $$\kappa$$...........$$\frac\sqrt\frac{\rho}{_2}{\varphi}$$ = $$\lambda$$.......... $$\varphi$$ - $$\delta$$ = $$\psi$$.......... $$\varphi$$ * $$\delta$$ = $$\iota$$

$$\varphi^2$$ - $${_2},{_5}$$ = $$\alpha$$.......... $$\varphi^2$$ - $$({_1}+\psi)$$ = $$\delta$$.......... $$\frac{\varphi\lambda}{\kappa}$$ = $$\rho$$............... $$\frac{\varphi}{\kappa}$$ = $$\tau$$.............. $$\varphi$$ + $$\mu$$ = $$\gamma$$..............

$$2\theta$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}{_1}$$.......... $$\theta$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}{_2}$$.......... ..... $${_0},{_1}{_5} = -\rho+\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}\frac{\theta^2}{\psi^2}$$................$${_0},{_2}{_5} = \frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2}  \frac{\eta}{\iota}\right)^2}{\theta^2} * {\delta}$$..........$${_0},{_1}{_2}{_5} = \frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2}  \frac{\eta}{\iota}\right)^2}{\theta^2} * {\sigma}$$..........$$\eta= \frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2}  \frac{\eta}{\iota}\right)^2}{\delta^2} - {_0},{_7}{_5}...........$$$$\eta$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2}  \frac{\mu}{\iota}\right)^2}{_8}$$

$$\rho$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}{_1{_6}}$$.......... $$\frac{_1}{_2}\rho$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}{_3{_2}}$$

$$\varphi$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}\frac{\theta^2}{\eta^2}$$....of.... $$\varphi = \frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\eta}{\iota}\right)^2}{\theta^2} * {\gamma}- {_0},{_7}{_5}...........$$ $$\varphi$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}{_4}$$.

$${_1}+\nu$$ = $$\frac{\left(\left(\frac\sqrt{\zeta}{\kappa}\right) * {_2} \frac{\mu}{\iota}\right)^2}\frac{\theta^2}{\eta}$$..........

Another question was if the Golden ratio could be approached by the constants $$\pi$$..and..$${e}$$.... So i discovered that $$\frac{_2,{_8{*\pi}}}{e}$$ / 2 = 1.61801

Hanuman das (talk) 21:03, 11 May 2015 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format. --I am  k6ka  Talk to me!   See what I have done  21:25, 11 May 2015 (UTC)
 * Don't you think it's rather deceptive of you to mention “The mathematical analist Johan Louis Lobo from the Netherlands” without also mentioning that you are he?  —Mark Dominus (talk) 17:18, 12 May 2015 (UTC)

Series representations
Another infinite series for phi is $$\varphi=\frac{1}{2}+\sum_{n=0}^{\infty}\frac{(2n-1)!!}{n!\cdot 10^n}$$ which converges less quickly than the one in the current article but has a much simpler form. This is original work (from http://blog.bretmulvey.com/post/126839701187), so I'm looking for another source to include as a citation. 2001:4898:80E8:4:0:0:0:12 (talk) 15:26, 18 August 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Golden ratio. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20100106000322/http://www.matematicas.unal.edu.co/airlande/phi.html.en to http://www.matematicas.unal.edu.co/airlande/phi.html.en

When you have finished reviewing my changes, please set the checked parameter below to true to let others know.

Cheers. —cyberbot II  Talk to my owner :Online 06:08, 26 August 2015 (UTC)

Is the subsection "Symmetries" wrong?
Maybe I'm overlooking something, but I don't think the set of fractional transformation given in the subsection "Symmetries" leaves invariant/interchanges the golden ratio and its conjugate. In fact, I think the fractional transformations are related to x^2-x+1 (3rd roots of unity) and not x^2-x-1, the defining relation of the golden ratio. Also there is no reference given in this subsection. — Preceding unsigned comment added by 93.223.176.205 (talk) 13:42, 28 August 2015 (UTC)

Edit request
This article describes Vitruvius as a "near contemporary" of Euclid, which is laughable as they lived almost 300 years apart in very different times--one just after Alexander the Great and the other a contemporary of Augustus Caesar. This ought to be corrected. --CRATYLUS22
 * ✅ Thanks! Gap9551 (talk) 16:36, 30 November 2015 (UTC)

The math in the calculation paragraph needs to get simplified to understand
$$\frac{a+b}{a}=\frac{1}{a}a+\frac{1}{a}b=1+\frac{1}{a}=1+\frac{1}{\varphi}b$$

If $$a=\varphi$$ then $$b=1$$ such that $$\frac{1}{\varphi}:1$$, substitute such that $$\frac{1}{\varphi}:1 \longleftrightarrow \underbrace{a:b}_{a+b}$$ thus $$b=1$$

$$1+\frac{1}{\varphi}*1=1+\frac{1}{\varphi}$$ after multiplication with $$\varphi$$ we obtain $$\frac{1}{\varphi}:1 \longleftrightarrow 1:\varphi$$

Therefore,

$$\frac{1}{\varphi}+\frac{\varphi^2}{\varphi}=\frac{1+\varphi^2}{\varphi}=\frac{1}{\varphi}+\varphi \rightarrow ...$$

Multiplying by φ gives ... (stays unchanged in comparison with the original article) — Preceding unsigned comment added by 217.121.229.115 (talk) 07:46, 23 September 2015 (UTC)

(pardon me if I'm using this forum wrongly). What is the general consensus on wikipedia? Writing out the standard math to calculate the algebraic expression for the golden ratio doesn't seem to be encyclopedic. As there is a "simple English" wiki, what about a "simple math" wiki?


 * Well in WP it may be up to the discretion of the author to certain degree. There are several things that should be considered here one of them is the "target audience". Meaning if the main audience for an article is expected to have little math background and it is an elementary subject then a more detailed explicit algebraic manipulation or calculations maybe given on occasion. Providing such explicit details are helpful for readers and do disturb the flow of reading or excessively bloat the article. So there is no simple yes or no answer but it depends on the specific context. While it is probably fair to say that usually explicit calculations/manipulations are omitted, there is nevertheless a considerable number of cases where they may deem appropriate.
 * As far as a simple math wiki is considered, that would be a separate project, that might have its place within the wikimedia family of projects, but imho it doesn't really fit the wikipedia language structure.--Kmhkmh (talk) 12:26, 3 January 2016 (UTC)

Golden ratio, exterior division?
Wy not in english wikipedia? See this in German Wikipedia: exterior division (German) --Petrus3743 (talk) 13:16, 15 January 2016 (UTC)
 * ✅ Petrus3743 (talk) 14:07, 8 February 2016 (UTC)

The Golden ratio has the simplest of all periodic continued fractions
The Golden ratio has the simplest of all Category:Periodic continued fractions, all 1's, the period length being 1. All Category:Quadratic irrational numbers have an [eventually] periodic continued fraction, the period length could be any finite length, including 1. (Category:Rational numbers all have a finite continued fraction, which is a special case of [eventually] periodic continued fraction, i.e. all 0's, the period length being 1.) &mdash; TentaclesTalk or ✉ mailto:Tentacles 22:46, 21 April 2016 (UTC)

In other words, the set of [eventually] constant sequences is the subset of the [eventually] periodic sequences for which the period is 1. &mdash; TentaclesTalk or ✉ mailto:Tentacles 22:56, 21 April 2016 (UTC)

Quadratic Formula for the Golden Ratio
In the page, it is stated that:

>>> Using the quadratic formula, two solutions are obtained: >>> φ = 1+(sqrt(5)/2 = 1.6180339887 >>> and >>> φ = 1-(sqrt(5)/2 = -0.6180339887 >>> >>> Because φ is the ratio between positive or negative quantities φ is necessarily positive:

I believe the two solutions should have been:

φ = (sqrt(5)+1)/2 = 1.6180339887 and φ = (sqrt(5)-1)/2 = 0.6180339887.

It is common to state φ = 1.6180339887, but I think it is not incorrect to put it as 0.6180339887, depending on whether you are viewing the ratio from the angle of [ b/a = (a+b)/b ] or [ a/b = b/(a+b) ]; "a" being the shorter side of the rectangle. My humble opinion.

Best regards

Robinkklam (talk) 02:21, 6 June 2016 (UTC)
 * $$1-\varphi$$ (the negative number) is a solution of the defining quadratic equation $$x^2-x-1$$. $$\varphi-1$$ (the positive number) is not. Try using a calculator to plug these numbers into the equation and see for yourself. —David Eppstein (talk) 02:43, 6 June 2016 (UTC)

Yes. I did make a wrong turn in my calculation. Thank you for your note. — Preceding unsigned comment added by Robinkklam (talk • contribs) 07:53, 6 June 2016 (UTC)

Golden ratio
I would enjoy discussing your entry under the section "Golden triangle" as I suspect that the statement is false or at the minimum needs clarification. The entry reads: I'd love to see a proof that the triangles so described are similar.

Most literature describes a golden triangle as an isosceles triangle whose ratio of the common side to the distinct side is equal to the golden ratio φ.

Following the figure in your section the side CB is identical to the same side in the original triangle thus forcing all corresponding sides to be proportional in the ratio of 1:1. But the triangles are not congruent as Euclid's Proposition 3 in book VI asserts that XB = AX*CB/AC for any triangle whose angle is so bisected. To recapitulate: I'd love to see a proof that the triangles so described are similar. Frank Gordon (talk) 02:15, 30 November 2015 (UTC)
 * I'm not quite sure what you're asking or not clear about, but there are 2 "types" of golden rectangles (ABX-like and BCX-like) and golden triangles of different types are not similar, only golden triangles of the same type are similar. So ABX and BCX are not similar but ABC and BCX are similar.--Kmhkmh (talk) 02:43, 30 November 2015 (UTC)

ABX is a straight line! — Preceding unsigned comment added by Frank Gordon (talk • contribs) 22:03, 7 December 2015 (UTC)

"1 cubit = 7 palms and 1 palm = 4 digits. The theory is that the Great Pyramid is based on the application of a gradient of 5.5 sekeds" Pi square root matches the seked which is what they actually used, THERES NO EVIDENCE THEY KNEW ABOUT PI, the earliest recorded egyptian pi was in 1850BC and it was (16/9)2. Maybe they liked 22/7 or maybe its was magic for sight of any of the millions of ref on google for 3.14 or rather 3.16 which is what they thought? What ever baseless claim people make it was not the egyptians using Pi and this should be stated instead of baseless speculation that contradicts the evidence!--Thelawlollol (talk) 04:26, 18 June 2016 (UTC)

Semi-protected edit request on 27 June 2016-Golden Ratio-applications in nature.
Presence of Golden Ratio in Nature-

1)In a honeycomb the female honeybees always outnumber male honeybees and the ratio in which they do so is the Golden Ratio(1.618:1).

2)Sunflower seeds grow in opposing spirals and the ratio of by adjacent diameters is always The Golden Ratio.

3)The nautilus-a cephalopod mollusc pumps gas into its chambered shell to adjust its bouyancy and the ratio of each spiral's diameter to next is Golden Ratio.

Namami2011 (talk) 09:36, 27 June 2016 (UTC)
 * A typical honey bee colony includes on the order of 50 000 individuals. In winter, there may be no drones at all. In summer, there may be several hundred drones per colony. The summertime ratio is on the order of 100:1, two orders of magnitude away from the golden ratio. Just plain Bill (talk) 11:39, 27 June 2016 (UTC)
 * Also, be careful when using the word "always". You would have to prove that it is true for every flower, every individual, every species. It may be true that patterns "tend" to the golden ratio, but when you deal something that can be numbered, at best you get a fraction, which cannot be a rational number like the golden ratio. Dhrm77 (talk) 12:45, 27 June 2016 (UTC)
 * I'm closing this edit request as it is clear that there is no consensus for it. In addition to the above objections, the nautilis shell, while being shaped like a log-spiral, has a different aspect ration than the golden spiral. —David Eppstein (talk) 17:02, 27 June 2016 (UTC)
 * Right, because none of these things is true. Dicklyon (talk) 21:24, 2 July 2016 (UTC)

Edit request for end of "Decimal expansion" section.
The computation result in the last paragraph is seriously out of date. Replace the last paragraph with:

The decimal expansion of the golden ratio  has been calculated to an accuracy of two trillion ($2$ = 2,000,000,000,000) digits.

71.41.210.146 (talk) 13:57, 2 July 2016 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format.  B E C K Y S A Y L E S  14:52, 6 July 2016 (UTC)


 * I want X = the last paragraph of the "Golden Ratio" section, which currently reads:
 * The golden ratio has been calculated to an accuracy of several millions of decimal digits . Alexis Irlande performed computations and verification of the first 17,000,000,000  digits.
 * replaced by Y = the text I supplied above:
 * The decimal expansion of the golden ratio  has been calculated to an accuracy of two trillion ($2$ = 2,000,000,000,000) digits.
 * Sorry, I didn't quote the text to change, but I thought I specified it clearly enough. 71.41.210.146 (talk) 01:46, 7 July 2016 (UTC)


 * ✅ - the heading "Decimal Digits: 1,000,000,000,000" confused me initially - Arjayay (talk) 08:22, 7 July 2016 (UTC)


 * Yes, your description did not specify adequately. In the future it may be helpful to follow instructions more carefully as they are written to avoid these types of common mistakes.  In particular, the phrase "last paragraph" may be ambiguous as to its meaning given that an editor may recognize it to be the sentences after a break in the source, or some other number of sentences in the apparent text depending on the width of the screen they are using at the time and possibly settings affecting the appearance and indentation.  Additionally your identification of the section within the heading, as opposed to the body of your request, makes it more difficult than is necessary to locate the text addressed.  One of the purposes behind protecting pages is to reduce the number of inappropriate edits.  The process has evolved over time and requires specification in a particular format based on the experience that edit requests without take significantly longer to be read and are much less likely to be accepted.  The time and effort editors spend to make edit requests becomes wasted, and the progress of the encyclopedia is slower.  If you feel that there is a better way to handle edit requests or something that could make the instructions easier to follow, discussion at Wikipedia talk:Protection policy may be helpful.  B E C K Y S A Y L E S  13:33, 7 July 2016 (UTC)


 * Thank you!
 * As for "RTFM"... I have to laugh. I created that template and I wrote those instructions, so please forgive me if I thought I understood them pretty well. :-)  I think the problem was simply that putting essential information only in the section heading was a bad idea; I realize in hindsight that the natural thing to do is to ignore the boiler plate in the message box and above and only read carefully the request after the prominent message box.  (It's a lot less prominent in the edit box.)
 * I apologize for that, but I think I really did follow the letter of the instructions. I just showed that you can follow the letter and still do a really bad job.  (https://i203.photobucket.com/albums/aa310/brykoe/CalvinHobbes-1.jpg comes to mind.)  Sorry for taking up up your time. 71.41.210.146 (talk) 15:15, 7 July 2016 (UTC)

About the given series representation
Is there some particular reason for electing the series $$\frac{13}{8}-\sum_{n\geq 0}\frac{(-1)^n (2n+1)!}{n!(n+2)!4^{2n+3}}$$ as a relevant example of a series converging to $$\varphi$$? Its convergence is not so fast, and the series representation $$1-\sum_{n\geq 1}\frac{(-1)^n}{F_n F_{n+1}}$$ that comes from the continued fraction is better-looking and has a similar convergence speed. 131.114.104.188 (talk) 11:14, 11 November 2016 (UTC)Jack D'Aurizio

Semi-protected edit request on 9 February 2017
Footnote 29: Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6 should be amended: Le Corbusier, The Modulor p. 25, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 316, Taylor and Francis, ISBN 0-419-22780-6, Marcus Frings: The Golden Section in Architectural Theory, Nexus Network Journal vol. 4 no. 1 (Winter 2002), available online Mfrings (talk) 11:07, 9 February 2017 (UTC)
 * Yes check.svg Done  B E C K Y S A Y L E S  05:22, 16 February 2017 (UTC)

Semi-protected edit request on 24 February 2017
Guvengunver (talk) 13:01, 24 February 2017 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format. DRAGON BOOSTER   ★  15:02, 24 February 2017 (UTC)

Binary and hexadecimal values
It makes no sense to have the binary and hexadecimal values of the golden ratio in the article. The golden ratio is usually not handled within those bases. If someone really needs those numbers in a rare occasion, they can easily enough do the conversion themselves. For the article they are redundant information. 84.249.218.170 (talk) 06:00, 10 March 2017 (UTC)

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