Talk:Golden ratio/Archive 3

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Non-Greek Origins
Its pretty clear to me that the "Sherrifs" of this page are in VIOLENT OPPOSITION TO THE PRINICPLES OF WIKIPEDIA. They are going to stick to the idea that the Greeks were the first with Phi, no matter what the evidence. The deletion of refrences to Egypt and Hebrew use is uttlery without excuse. The Hebrew Temple deletion refered to the text itself, so assuming the reader can do fractions, then the lack of another citation is not a "reason" for deltion. It is an EXCUSE TO PROMOTE A POINT OF VIEW. —Preceding unsigned comment added by Special:Contributions/ (talk)


 * You have provided no references for your assertions. Until you do, they will not be in the article.  &mdash; Arthur Rubin |  (talk) 17:58, 24 August 2007 (UTC)


 * I'm with Arthur on this. We've spent the last year or more getting unsourced bull out of this article.  If there are facts we missed, please bring them in and share your sources.  And read up on the principles of wikipedia yourself, esp. WP:V and WP:RS. Dicklyon 04:23, 25 September 2007 (UTC)

Recursive/continuous fractions notation
Regarding this reversion. If Phi is definded recursively (1 + 1/(1+1/(etc...))) the subscript means nothing, but if defined iteratively (1 + 1/1 = 2, 1 + 1/2 = 1.5, 1 + 1/1.5= 1.66, etc..), the subscripts are absolutely necessary (as none of the subscripted Phis are actally Phi but approximations of Phi). This should be explicitly stated in the applicable section as it is a little confusing. Adam McCormick 05:08, 27 August 2007 (UTC)
 * I've made an edit which should alleviate any possible confusion. Please discuss before reverting. Adam McCormick 05:27, 27 August 2007 (UTC)


 * I don't think it helped. The old way was more clear and correct, so I restored it.  There's no need to introduce the subscripted phi notation for the convergents.  Dicklyon 06:57, 28 August 2007 (UTC)


 * But when calculating Phi iteratively, none of the calculations actually produces Phi, it may need to be put somewhere else but it does need to exist, I'm reverting back to my version, please reorganize if you don't think this applies to convergents Adam McCormick 16:38, 28 August 2007 (UTC)

I still don't see why we need to give the convergents subscripted-phi names. The old text that you changed already states that the convergents are not phi themselves, but approximate it. To me, it is confusing to see the infinite continued fraction and then to see your wording about how this formula may be "expanded iteratively" when the iterative expansion is not the same formula, but rather a sequence of approximations. I think the older wording about using the convergents of the infinite continued fractions as approximations makes this more clear. —David Eppstein 17:01, 28 August 2007 (UTC)


 * I understand that, my problem is that a non-math person trying to approximate Phi wouldn't have any idea how to implement the continued fraction, and even with some exposure to the subject I still find the convergents statement very confusing. I think the iterative form needs to exist somewhere, that's all. Adam McCormick 19:20, 28 August 2007 (UTC)


 * But it is somewhere: it's in the decimal expansion / calculation methods section, which is I think a more likely place to look for this as a calculation method. Specifically, the continued fraction section that you've been trying to edit says that the convergents are ratios of Fibonacci numbers, and the last sentence of the calculation methods section says to take the ratio of consecutive Fibonacci numbers. However as a method for approximately calculating the golden ratio this iteration is significantly slower than the Newton iteration described earlier in the calculation methods section. —David Eppstein 19:35, 28 August 2007 (UTC)

P.S. Please do not remove the periods from the ends of sentences. Displaying them as part of the formula, the way they were before you removed them, is a standard convention in mathematical typography. See Wikipedia talk:WikiProject Mathematics/Archive 28 for a recent discussino on exactly this issue. —David Eppstein 19:39, 28 August 2007 (UTC)


 * The alternative typographical convention of not putting end punctuation after displayed equations is also quite commonly used by reputable publishers. Does wikipedia have a policy on this?  Or a style guideline? Dicklyon 23:27, 24 September 2007 (UTC)


 * Examples, please? I just checked some journals by AMS, SIAM, ACM, Elsevier, Wiley, Springer, and World Scientific, and they all had punctuated displayed equations. I did find a Kluwer journal in which some sentences that ended in displayed equations had a period, others didn't, often within the same article, but that seems more like a lack of copyediting than a consistent house style. —David Eppstein 00:49, 25 September 2007 (UTC)


 * The first book I picked up off the top of my stack consistently omits end punct on displayed equations (6th ed. of Hunt's The Reproduction of Colour in case you want to check). I'm not saying it's the most popular convention these days, but in fact it's what I was taught, maybe 15 years ago or so, when I had been doing endpunct.  The argument is that it's less likely to introduce confusion into equations if you don't add dots and things in the display. Dicklyon 04:20, 25 September 2007 (UTC)

Mathematical pyramid redundancy
"The height of this pyramid is Phi times the semi-base...The slope of the face is also Phi." This is kind of redundantly redundant. Since the height of a right triangle is always the base times the slope of the hypotenuse, the statement says the same the twice. You think that we should change that, or leave it anyway? ROBO_HEN5000 ROBO 01:43, 27 September 2007 (UTC)


 * See if you like the way I fixed it. The link that was hanging on there was also irrelvant, so I took it out. Dicklyon 02:59, 27 September 2007 (UTC)

Bartok

 * copied here from my talk page: Dicklyon 22:33, 14 October 2007 (UTC)

It is pretty clear to me that whoever wrote the section on Bartok's use of Fibonacci numbers is not a musician. They use the use the terms progression and interval incorrectly. My edit corrected that. —Preceding unsigned comment added by 89.49.14.18 (talk) 22:17, 14 October 2007 (UTC)


 * Your edit also added some stuff not supported by the reference. I added a link to the ref so you can check what it says; so try again, but if what you add is not supported by, or contradicts, the reference, than you'll need to find a new reference to support it.  We have to be rather strict on this particular article as it tends to accumulate a lot of junk otherwise. Dicklyon 22:30, 14 October 2007 (UTC)
 * Sorry for posting on wrong page, I'm new here. I definately believe that this page would accumulate a lot of junk - so many misconceptions - so I understand your vigilance. You should definately consider changing the part about Bartok's use of Fibonacci numbers...how about a musical score as a reference? For us musicians, that is much better than any book. It is, in fact, not even known for a fact that Bartok used the Fibonacci numbers. Even the passage discussed, although definately suggestive of their use, is marked rubato. Bartoks musical language conceals his use of the Fibonacci numbers and the golden ratio if he in fact used them - and that should be noted here. If someone can provide a writing in which Bartok states that he used it, only then is it proven beyond any doubt. I checked out the reference; if you want to take such writing as a reference you are welcome to, but you are doing a disfavor to wikipedia. Check the score and you will see that my addition was more complete; ask educated musicians and you will see that terminology was incorrectly used. —Preceding unsigned comment added by 89.49.56.56 (talk) 21:00, 15 October 2007 (UTC)


 * If you have sources with more info, please do fix it; however, also keep in mind that this article is supposed to be about the golden ratio, so if you have expansions on the fib.no. stuff, that might be better in a different article. Or if there's reason to believe the source cited is not to be regarded as a reliable source, tell us the reason and take that info out. Also, wikipedia prefers to report what is in secondary sources, not new interpretations of primary sources; the latter are called WP:OR and are discouraged. Dicklyon 05:52, 16 October 2007 (UTC)


 * I guess I can understand that policy. Just goes to show it is impossible to have a perfect policy. Thanks for the info anyway. —Preceding unsigned comment added by 89.49.32.214 (talk) 15:04, 20 October 2007 (UTC)

penny arcade reference?
http://www.pennyarcademerch.com/pat070421.html —Preceding unsigned comment added by 71.116.131.170 (talk) 14:17, 9 November 2007 (UTC)

Explicit expression
The term explicit expression is used in its mathematical sense in about 20 Wikipedia articles, but I did not see a definition in any of them. I redlinked its use in Golden ratio. It would be helpful for Wikipedia to have a definition of this term to which usages can be linked, either as a short article devoted to the term or as a section in an appropriate article on mathematical expressions. Finell (Talk) 06:00, 14 December 2007 (UTC)


 * It's implicitly defined in implicit function; I recommend you redirect to there. Dicklyon (talk) 06:22, 14 December 2007 (UTC)

Disambiguation
Just wanted to comment that the Golden Mean isn't "the felicitous meeting of two extremes." According to the article it seems to be more about moderation and temperance, not a random meeting of two unlike things. [{Special:Contributions/71.82.122.91|71.82.122.91]] (talk) 01:13, 19 December 2007 (UTC)
 * It says "middle", not "meeting", and is accurate and sufficient for the purpose of disambiguation. Finell (Talk) 09:02, 19 December 2007 (UTC)

Controversy section needed
I read a journal article on this topic that shows that most of these interpretations of art matching golden ratios are rather unlikely. There is nothing but correlation, and the connection between the golden ratio and these masterpieces may be a simple effect of biased selection: In other words, there are many measurements in a building or a face, and to dismiss the measurements that do not conform, or the aesthetically pleasing art that doesn't conform to the golden ratio would certainly introduce bias.

Here is a link to the article: http://laptops.maine.edu/GoldenRatio.pdf by George Markowsky. Deepstratagem (talk) 17:07, 20 December 2007 (UTC)


 * We decided about 18 months ago to not have a controversy section, but just to focus on verifiable points. We've already done a pretty good job of reducing the frivolous claims, so it's not clear to me where you feel there's a problem.  The Markowsky paper is already linked in further reading; if you figure out what journal it's from, you might want to use it as a source and add a bit of what it says where needed. Dicklyon (talk) 17:27, 20 December 2007 (UTC)

Phi and the semitone
In the "Music" section, I have added "Phi and the semitone". However, I have used non-Wiki terms for mathematical expressions. Perhaps someone more familiar with Wiki math would be kind enough to revise my mathematical expressions. Thanks! Prof.rick (talk) 19:49, 22 December 2007 (UTC)


 * Since it's just unsourced numerology, I just took it out. Let us know if you have a reliable source connecting phi to the semitone, and maybe we can put it back. Dicklyon (talk) 21:25, 22 December 2007 (UTC)


 * We could use a third opinion about this now in Mathematical coincidences. Dicklyon (talk) 09:31, 28 December 2007 (UTC)

timeline
Life is too short to debate this edit; so I provide the deleted information here, for anyone who is interested:
 * The oldest known applications of the golden ratio are likely in vedic mathematics (more than 3.000 BC) and likely also in some Egyptian pyramids (2000 BC).

&#151; Xiutwel ♫☺♥♪ (talk) 23:34, 22 December 2007 (UTC)
 * It's not about debate. See WP:V and WP:RS.  Dicklyon (talk) 00:25, 23 December 2007 (UTC)

Former FA candidate
In April 2007, a well-meaning Wikipedian, who had not participated in the article and appears not to have expertise in its subject matter, nominated Golden ratio to be a featured article. This drive-by nomination was short lived: all who commented on the nomination opposed it. Since this nomination was essentially an accident, and the embarrassment of failure was not self-inflicted, I propose deleting the box concerning its failed FA candidacy from the top of this Talk page. Comments? Finell (Talk) 06:29, 14 December 2007 (UTC)


 * I would agree. I wonder somewhat what difference it makes, but I don't think its presence really matters. Adam McCormick (talk) 05:49, 18 December 2007 (UTC)


 * Done. Finell (Talk) 23:01, 2 January 2008 (UTC)

Weasel Words
"Some suggest that he.." in the bit about Mona Lisa. —Preceding unsigned comment added by Lukechumley (talk • contribs) 04:02, 27 January 2008 (UTC)

Temporal Sandbox
Here is a temporal sandbox to source, explain, and develope several items of artistic production studied because of their golden ratio proportions:

Golden ratio/sandbox

All of them are trascendental works of art and still don't have their articles. That's something that has to e changed.

The product of the sandbox could be placed either directly in this article on in a subsidiary list article (if it gets too big).

--20-dude (talk) 19:09, 8 March 2008 (UTC)


 * I notice that you have not included reliable sources for anything there. Without sources, speculation that these artworks involve the golden ratio is original research and cannot be included here. —David Eppstein (talk) 19:33, 8 March 2008 (UTC)


 * What's with editors expecting to include strong sourcing when articles are just stubs? Jeez, I'm just starting. —Preceding unsigned comment added by 20-dude (talk • contribs) -->
 * The point is WP:OR. The information going into an article should come from reliable sources in the first place, and thus it is logical to place the sources as you go. For an article such as this one (with frequent vandalism), new sections need to be referenced from the outset. I'm not saying that it's a huge deal, but you do need them, and the sooner, the better. Adam McCormick (talk) 00:57, 10 March 2008 (UTC)

On the other hand, here is to show even sourced comments can have no sense:

Interestingly, a statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is 1.34, with averages for individual artists ranging from 1.04 (Goya) to 1.46 (Bellini).

I don't doubt the study is right but basically which great painters? and are you sure we're talking about their entire work? I don't even think the entire work of the most insistive like Le Corbusier and Da Vinci is done with Golden ratio. In other words which canvases?--20-dude (talk) 00:43, 10 March 2008 (UTC)


 * That's exactly why we put sources in our articles. Because the source is there, and linked from the article, you can go to it yourself and find out whatever additional information the authors of the study included in their paper, such as which painters they considered. —David Eppstein (talk) 01:01, 10 March 2008 (UTC)


 * I've never said references aren't extremely necesary.--20-dude (talk) 01:11, 10 March 2008 (UTC) Just that I'd be ridiculous to expect users to start the stubs already with the references. That's why it is an article in development and not a featured article.
 * On another related note, I used some of the info and sources in this article to finish already with the easier examples. I'm thinking about moving it to List of art works designed with golden ratio already. Accoring to wikipedia's guidelines it's about good time, since it's not that rough as a stub but not that finished as an article (the guidelines say it's not that recomendable to start articles with too much information because editors don't react well)--20-dude (talk) 01:19, 10 March 2008 (UTC)
 * I wouldn't worry to much about that, just move it rather than copy the text. I would also recommend a more succinct title. Maybe "Golden ratio in art" or some such. Also please fix the broken ref tag before moving. Adam McCormick (talk) 04:28, 10 March 2008 (UTC)
 * Nice advise.
 * Realizing there are already a good amount of articles realated to the Golden ratio I started WikiProject Golden ratio to improve the edition of them as a whole. --20-dude (talk) 04:29, 11 March 2008 (UTC)

The Golden Ratio
The last time I checked, there was also the well-known ratio 0.61803398875:1, which is closely related to the little-more-known 1.61803398875:1. It makes no sense for the smaller greek letter to represent the larger ratio.168.103.222.216 (talk) 20:32, 13 March 2008 (UTC)


 * Nevertheless, that is the convention that is established in the literature, so please stop changing the long-standing consensus notation in the article. Actually, 0.618...:1 and 1.618...:1 are equivalent ratios; the only difference is whether you express the ratio as long:short or short:long. Again, expressing the golden ratio as long:short is the common convention in the literature, although one sometimes sees the ratio expressed as 0.618... Please consider creating a Wikipedia account for yourself, as it will make your participation here easier. And please do continue to participate. Thanks. Finell (Talk) 21:51, 13 March 2008 (UTC)


 * I agree with Finell; we have been over this here before, and settled based on a consensus of many editors. To change it, you would need to bring some pretty strong references to outweigh what we have already weighed. Dicklyon (talk) 23:26, 13 March 2008 (UTC)

To this anonymous user I say: your argument lacks cogency.
 * (1) There is no clear need to have separate letters for these two numbers.
 * (2) There is no convention of using capital letters for big number and lower-case letters for small numbers.
 * (3) Wikipedia is no place for overthrowing standard conventions, just as it is no place for original research.

Michael Hardy (talk) 00:11, 14 March 2008 (UTC)

Capitalization of Phi
Is there established consensus of $$\Phi$$ versus $$\varphi$$. I ask because they've all been changed (the intermediate edit by Jossi looks good though) and should be reverted or not based on that consensus. Adam McCormick (talk) 22:49, 12 March 2008 (UTC)


 * Yes there is. Most modern professional works on the golden ratio use lower case phi for the golden ratio. See, for example, the Mathworld link. That has been the accepted usage on Wikipedia for over a year, at least; feel free to check the edit history for yourself. Although I have great respect for Jossi and his many contributions to this article and throughout Wikipedia, there is long-standing consensus here that the lead would be incomplete without the defining equation. If Jossi wants to reopen discussion of whether the equation should be in the lead, he is welcome to do that on this Talk page. I reverted the anon edits and Jossi's to restore the consensus. Finell (Talk) 23:24, 12 March 2008 (UTC)


 * No problem. So long as Jossi's edit was considered separately, I've got no complaints. Adam McCormick (talk) 06:21, 13 March 2008 (UTC)


 * I think that the lead could do without the formula, per WP:LEAD. ≈ jossi ≈ (talk) 23:29, 13 March 2008 (UTC)


 * Dear Jossi: Welcome back! What specifically in WP:LEAD makes the formula inappropriate? Finell (Talk) 02:01, 14 March 2008 (UTC)

Disputed sightings
Dear Dicklyon: Thanks for clearing up the misinformation about the sculpture. I assumed (but should not have) that it was entitled "Golden Ratio" because that is what the text and image caption said; likewise, I should not have uncritically qualified my {Fact} tag with the hidden comment. Since you're on a roll, how about cleaning up the bee ancestry entry. For starters, integer:integer can't be irrational, so it can't be the golden ratio (although it can approach it). Finell (Talk) 19:18, 16 March 2008 (UTC)


 * I've tried, and can't find anything on the bees, or where that idea came from, or what the actual ratios might be. I'd just get rid of it.  But I did some work on the Roses of H., and concluded that it's probably just a wikipedian's original research, since there's absolutely nothing on it in books or papers that I can find. Dicklyon (talk) 20:33, 16 March 2008 (UTC)
 * I cannot find anything either. ≈ jossi ≈ (talk) 20:48, 16 March 2008 (UTC)

Miles in Kilometres
1 Mile = 1.618 Kilometres. Interesting, but may not be useful or appropriate for article?

124.148.1.217 (talk) 02:14, 27 February 2008 (UTC)


 * By my arithmetic, 1 mile = 1.609344 kilometers. We have:


 * 2.54 centimeters = 1 inch
 * 12 inches = 1 foot
 * 5280 feet = 1 mile
 * 100 000 centimeters = 1 kilometer
 * So 1 mile = 2.54 &times; 12 &times; 5280 centimeters
 * = 160 934.4 centimeters = 1.609344 kilometers. Michael Hardy (talk) 04:48, 21 March 2008 (UTC)

Title
Why do they call it golden ratio? the ratio goes from the middle pont (that's 0.5) of a side of a perfect square, to the opposing vertex. After you extent the circle to 0° or 180°, you se that the extra section is 0.618, that's 0.5 + 0.618 = 1.118, thus the golden ratio is 1.118 of a square of 1 per side. The golden section might be 1.618, but the ratio is not.

Can someone explain what's happening to me?--20-dude (talk) 03:35, 17 March 2008 (UTC)


 * It's the ratio a:b or (a+b):a as shown; or the ratio of the dimensions of a golden rectangle; or (1+sqrt(5)):2; etc. In the rectangle picture, the sum of the distances sqrt(5)/2 and 1/2 is it: 1.118 + 0.5, not as you describe.  "They" can call it whatever they want; we just report on it.  Sorry, I can't explain what's happening to you. Dicklyon (talk) 03:41, 17 March 2008 (UTC)


 * Dude: The OED traces the etymology of golden ratio and golden section. I believe that the first use of golden was in German by Ohm (mentioned in the article), which led to translations such as golden section. I hadn't heard of auric (Latin for golden?) section. Where did you find that one? Finell (Talk) 22:34, 17 March 2008 (UTC)

Suggested Addition to Mathematics
A recent discovery in mathematics, by Lin McMullin, has shown that the Golden Ratio and it's conjugate are related to the line through the points of inflection of fourth degree polynomials. Assume that the polynomial has points of inflection at $$x =j$$ and $$x = k$$ the equation


 * $$ q(x)=ax^4+bx^3+cx^2+dx+e $$


 * If j and k are the roots of the second derivative, then
 * $$q''(x)=12a(x-k)(x-j)=12ax^2-12a(j+k)x+12ajk $$

working backwards, using d and e as the constants of integration, we can find a slightly different form for q(x)
 * $$ q'(x)=4ax^3-6a(j+k)x^2+12ajkx+d $$
 * $$ q(x)=ax^4-2a(j+k)x^3+6ajkx^2+dx+e$$

Use this to write the equation of the line through the points of inflection
 * $$ y(x)=\frac {q(k)-q(j)}{k-j}(x-k)+q(a)$$, then
 * $$ y(x)=-(aj^3-3aj^2k-3ajk^2+aj^3-d)x+(aj^3k-3aj^2k^2+ajk^3+e)$$

Then solving the equation
 * $$ q(x)=y(x)$$

We find four solutions :the obvious two are $$ x=j$$ and $$x=k $$ the other two are
 * $$x=(\frac{1+\sqrt 5}{2})j+(\frac{1-\sqrt 5}{2})k=\varphi j + \Phi k $$
 * $$x=(\frac{1+\sqrt 5}{2})k+(\frac{1-\sqrt 5}{2})j=\varphi k + \Phi j $$

The line through the points of inflection of a fourth degree polynomial intersects the polynomial at two other points whose x-coordinates are linear combinations of the x-coordinates of the points of inflection with $$\Phi$$ and $$\varphi$$ as coefficients.

If there isn't major objection to putting this into the main article within two weeks from 1/7/08, I'll be putting it in.

Sources http://www.linmcmullin.net/PDF_Files/Qolden_Ratio_in_Quartics_2007.pdf ; http://www.cut-the-knot.org/Curriculum/Calculus/FourthDegree.shtml L. McMullin, A. Weeks, The Golden Ratio and Fourth Degree Polynomials, On-Math Winter 2004-05, Volume 3, Number 2 —Preceding unsigned comment added by 68.199.24.156 (talk • contribs) 01:31, 8 January 2008 (UTC) -->

Revision as of 04:19, 8 January 2008 (edit) (undo)68.199.24.156 (Talk)


 * That's very cool. Let's hope that Lin McMullin can get it published, so we can use it. Dicklyon (talk) 01:31, 8 January 2008 (UTC)


 * Actually, it is already published three times, not counting the author's own monograph. The result is independently reported in Cut-the-knot, which appears to be a WP:RS; that site and its author, Alexander Bogomolny, are cited several times on Wikipedia. Also, the Cut-the-knott article cites two publications by McMullin: one in ON-Math, which is peer reviewed; another in The North Carolina Association of Advanced Placement Mathematics Teachers Newsletter. Further, the math is fully stated and is therefore verifiable (or falsifiable) by anyone with the requisite mathematical knowledge; for that reason, the majority of equations, other mathematical expressions, proofs, and the like in Wikipedia math and science articles are not cited to a source. Therefore, there is a sufficient basis under Wikipedia guidelines to include this "very cool" discovery. Would Dicklyon or one of the other strong mathematicians here please work this into the article? Thanks. Finell (Talk) 08:48, 27 January 2008 (UTC)


 * I don't think being cited on wikipedia makes the math blogger a reliable source. I couldn't even find his name on his site, so it seemed too anonymous, and full of ads, to be taken as such.  But if the result is in a peer-reveiwed paper, I'd be willing to take a look and report what it says.  Do you have a copy, or a way to get a copy?  Maybe the author will see this and respond or email me one.  By the way, math being checkable is not at all the same concept as wikipedia's WP:V concept. Dicklyon (talk) 18:22, 27 January 2008 (UTC)


 * I have the distinct pleasure of having Lin help me write this entry, so he has no problem with it, in fact he just wanted his name attached to it (can't blame him for that, it was after all his discovery) and it being sourced correctly. If you want Lin to e-mail you I think I can arrange that for you, for a copy just look at the sources. I'll wait to add it for your response. I was planning on checking this site before today, but I forgot. As for his site are you sure you had the right one, http://www.Linmcmullin.net, his name and picture are both clearly visible, not to mention the site is his name, as for his reliablitiy check his teaching experience and his publication history. Further I think this "very cool" discovery could explain a few of the natural discoveries made by others, if you can prove that nature uses fourth degree, and third degree (there is another proof for that, but this one was made first and inspired the other), polynomials then you wouldn't be too surprised if the Golden Ratio pops it's head up here and there. —Preceding unsigned comment added by 63.99.26.3 (talk) 20:25, 4 February 2008 (UTC)


 * I don't think I referred to his site, but was commenting on cut-the-knot. I have a copy of "The Golden Ratio and Quartic Polynomials" with Lin's copyright notice on it, but not a copy of or evidence of a peer-reveiwed publication.  If you add something about it with a proper citation, I expect that will be fine.  Dicklyon (talk) 00:42, 5 February 2008 (UTC)


 * Added March 20, 2008. And removed 5 min later WTF, it's been here in discussion for months with no problem, tried to put it in and while I was checking the format it disappears. Well I tried again I hope it stays this time. —Preceding unsigned comment added by 24.219.181.187 (talk • contribs) -->


 * There are unresolved issues and unanswered questions, the section should not be re-added without community consensus and proof of peer-reviewed publication. Adam McCormick (talk) 02:42, 21 March 2008 (UTC)
 * one more source added. —Preceding unsigned comment added by 24.219.181.187 (talk) 16:15, 21 March 2008 (UTC)
 * I think the Cut-the-knot reference is a reliable source and there seem to be peer-reviewed references as well, so I see no reason not to include this result. But I think it should be cut down to just a statement of the result, without the derivation. Also, be careful to use φ and Φ in the same sense as the rest of the article. Gandalf61 (talk) 09:55, 22 March 2008 (UTC)
 * Nice catch on the reverse use of φ and Φ, and has been corrected. I agree with everything else, it could use a little shortening, since the full proof is laid out with the reference, but I don't think it should just be the result without any type of proof.

regarding the recent "OR" reverts
The material reverted is about the golden ratio's relationship to inflection points of a fourth degree polynomial. $$\phi$$ is the root of a quadratic; the quadratic is the second derivative of quartics; and inflection points are roots of second derivatives. In other words, the OR appears to be tautological. Find a reference in a peer-reviewed math paper, otherwise it just doesn't seem worth reading much less proving. Pete St.John (talk) 18:34, 25 March 2008 (UTC)


 * While it's not the deepest result ever, I don't think it is entirely trivial either. The result says that the line segment between the points of inflection intersects the quartic in the two points that divide it in the ratio φ:Φ. And that ratio is the same for any quartic with real points of inflection. That's not completely obvious. Alexander Bogomolny discusses it in some detail at Cut-the-knot, and he doesn't typically write up trivia. And a peer-reviewed paper in On Math was cited above. Gandalf61 (talk) 19:51, 25 March 2008 (UTC)
 * Well yeah that sounds nontrivial, but now I don't see my own revert in the article history. Sometimes I'm flumoxxed; I could have sworn that I had undone to Paul's version, but he doesn't have one for days back. Anyway, Gandalf, feel free to put in maybe something more brief? Pete St.John (talk) 20:39, 25 March 2008 (UTC)


 * Has anyone actually seen this paper from On Math? Dicklyon (talk) 21:34, 25 March 2008 (UTC)


 * Well, there is certainly a paper with the cited title in the index of the relevant journal issue here. I think you need a subscription to see the paper itself. But why be so suspicious ? Do you have any evidence that suggests the paper may not be legitimate ? If not, I think we should follow WP:AGF and asume that the paper is a valid reference. Gandalf61 (talk) 09:16, 26 March 2008 (UTC)


 * Is it possibly this? -- Dominus (talk) 13:37, 26 March 2008 (UTC)


 * That's a different paper, different title, apparently self-published. That one I have, but I'm wondering if there's a published peer-reviewed one that anyone has seen.  I'm not suspicious of its existence, but how can we cite it as a source if nobody has seen it?  Do we even have reason to believe that it's peer-reviewed at this point?  Someone who has it or knows of its content should just speak up and fix this; the rest of us should wait, or go try to get a copy. Dicklyon (talk) 15:34, 26 March 2008 (UTC)


 * I've requested a copy from the author. -- Dominus (talk) 15:59, 26 March 2008 (UTC)

The da Vinci face
It was just a slip that I removed this image along with the Rose of H., but now that I look at it, I see that the commentary on it is completely weasel-worded and unsourced. And it's not at all clear from the figure what distances are supposed to be in the golden ratio. Could somebody who knows about this please fix it? Dicklyon (talk) 04:57, 17 March 2008 (UTC)


 * I believe the source of the idealized face with golden ratio reference lines is Divina, which Leonardo illustrated. In my research I found a sketch by Leonardo of a real face with the same grid of lines superimposed. I wanted to track down more information before adding it to Wikipedia. Finell (Talk) 22:25, 17 March 2008 (UTC)


 * At this point, I can accept that it's from that book since the page headers appear to say so. But how does it relate to golden ratio?  No way that I can see. Dicklyon (talk) 15:37, 26 March 2008 (UTC)


 * Here is a link to an image of that page, from a copy of the book in the LoC's collection . Perhaps you or someone who speaks Italian (Jossi?) can explain in more detail, from the surrounding text, what this image represents in a book that is about or largely devoted to the golden ratio. Or, if we wait a few months, an English translation of the book is scheduled for publication in June 2008. Finell (Talk) 01:37, 27 March 2008 (UTC)


 * Thanks for that pointer. I hadn't realized the LoC had that sort of online collection.  Looks like there's no text near the picture, however, and lots of the figures appear to be about things other than the divine proportion.  So maybe we just shouldn't make any claims about it being an application of the golden ratio unless someone finds evidence that it is.  I look forward to that translation. Dicklyon (talk) 02:50, 27 March 2008 (UTC)
 * It reads, Divina Proportio, and it refers to mapping the human face to the golden ratio. Check the ratio of the six rectangles (the leftmost two are split in the middle). ≈ jossi ≈ (talk) 22:24, 27 March 2008 (UTC)
 * The full title is Divina proportione: opera a tutti glingegni perspicaci e curiosi necessaria oue ciascun studioso di philosophia, prospectiua pictura sculpura, architectura, musica, e altre mathematice, suauissima, sottile, e admirabile doctrina consequira, e delectarassi, cõ varie questione de secretissima scientia. M. Antonio Capella eruditiss. recensente. it is in ancient Italian. ≈ jossi ≈ (talk) 22:31, 27 March 2008 (UTC)


 * Yes, and the title page would be a nice image; doesn't make much sense to me, but nice. Jossi, since you uploaded the face image in the first place, what do you know about it?  I was looking for reason to suspect that the rectangles with diagonals across them were meant to be golden rectangles, but I couldn't find one. There's another face (page seventy something iirc) with various and sundry unexplained proportions marked as well.  Any clue? Dicklyon (talk) 23:01, 27 March 2008 (UTC)


 * What page number, Dick? ≈ jossi ≈ (talk) 20:38, 28 March 2008 (UTC)
 * You probably mean this . My knowledge of Italian is basic, and the olde Italian used by Pacioli does not help.... ≈ jossi ≈ (talk) 20:40, 28 March 2008 (UTC)


 * Right, that one. It's got the same top and bottom rectangles with diagonals across them, but they're divided up different ways.  I wonder still if those, or some other proportions, are said by Pacioli or Da Vinci to be approximately divine. Dicklyon (talk) 22:49, 28 March 2008 (UTC)
 * Dunno... All we have is Divina Proportio stamped above the head. We can just simply describe that without additional commentary. ≈ jossi ≈ (talk) 01:37, 29 March 2008 (UTC)
 * There's some stuff here ≈ jossi ≈ (talk) 01:40, 29 March 2008 (UTC)

General Form
I went through the page a couple of times but couldn't find a "general form" that involves deriving the golden ratio for all positive n. That is, if you take any positive n and add 1, then add the original number. Then add the previous number. Take many iterations and when you divide the new number by the one before it, it is close to the golden ratio. For example:

{7,8,15,23,38,61,99,160,259} 259/160 = 1.61875

Do this with any n. Here is another:

{5.2,6.2,11.4,17.6,29,46.6,75.6,122.2} 122.2/75.6 = 1.616402

Has anyone seen this before? InfoNation101 |  talk  | 19:06, 22 April 2008 (UTC)


 * This property of number sequences similar to the Fibonacci sequence is described in generalizations of Fibonacci numbers. Gandalf61 (talk) 19:32, 22 April 2008 (UTC)


 * Hm. Interesting page.  Thanks for the heads up.   InfoNation101  |  talk  | 19:39, 22 April 2008 (UTC)

"Approximately 1.6180339887"
That doesn't seem approximate (thesaurus: about, almost, nearly, roughly) to me, that's 11 significant figures. I've done plenty of science, and "approximate" tends to me values such as Avogadro's number being 6.022X10^23, or the specific heat capacity of water being 4.18j/kgC. On the page for pi, pi is "approximately equal to 3.14159". That seems more like an approximation to me.

Either we should:
 * Cut down on the number of significant figures (possibly to 1.618, the commonly stated value of the golden ratio), or
 * Switch "approximate" with something like "the golden ratio, to the tenth decimal place, is valued at 1.6180339887."

What are everyone else's thoughts? (Bonzai273 (talk) 04:09, 24 May 2008 (UTC))

Disputed fact tag
The following is cross-posted from my talk page to elicit wider feedback Adam McCormick (talk) 17:00, 18 May 2008 (UTC)


 * Dear Alanbly.
 * In Golden ratio, you added a fact tag to the following:
 * The secretive Leonardo seldom disclosed the bases of his art, and retrospective analysis of the proportions in his paintings can never be conclusive.
 * I think the statement is so undeniable that no tag is needed. Will yo ureconsider?--Noe (talk) 14:31, 18 May 2008 (UTC)
 * I only restored a fact tag removed by an IP. The way it's worded I think it needs to be cited. As I said in my edit summary, there are ways of rewording it that might eliminate this need maybe:
 * "Leonardo seldom disclosed the bases of his art, and, as such, retrospective analysis of the proportions in his paintings is not conclusive."
 * or
 * "Retrospective analysis of the proportions in da Vinci's paintings is not conclusive evidence of the bases of his art."
 * But I am always hesitant to make most edits to Golden ratio without discussion as it is a hot topic. Would you agree with either of my rewordings? Adam McCormick (talk) 17:00, 18 May 2008 (UTC)
 * I have no problems with the original, or with any of your versions.--Noe (talk) 20:26, 18 May 2008 (UTC)
 * Seems to me like a source might be a good idea, or some research to see if there is actual support for the idea that he was "secretive"; here's a book that seems to argue against that idea, for example. Or say nothing if there's nothing sourced that needs to be said.  But I think what really needs a citation is the previous sentence "Whether Leonardo proportioned his paintings according to the golden ratio has been the subject of intense debate."  Shouldn't we link a source for this debate if we're going to assert that it exists? Dicklyon (talk) 21:53, 18 May 2008 (UTC)
 * By the way, calling for a citation should not be interpreted as a suggestion that a fact is "disputed"; rather, that it may not be a fact (i.e. it's someone's interpretation), or that it's a fact that deserves some verification and access to sources for further info about it. I agree with what's written (except maybe the secretive part), but especially in articles like this one that tend to accumulate a lot of hearsay and weasel words, we need to stick to high standards of verifiability. Dicklyon (talk) 21:56, 18 May 2008 (UTC)
 * I think we - as a source used e.g. by primary and high school students worldwide - should, as far as possible, not just state established facts, but also debunk common myths and miscomprehensions, like those that can be found by googling for "golden ratio" etc. Of course, unless it's very trivial, sources are required for this debunking. I agree sources are required for the secretive Leonardo thing, but not for stating that observation of a ratios between two measurements close to 1.6 imply intelligent design - sorry, I mean deliberate use of golden ratio (understood as an exact mathematical construction or proportion) by the artist.--Noe (talk) 11:34, 19 May 2008 (UTC)
 * Up to a point, one can add obvious stuff without sources. But if someone calls for a source, we should work to provide one, and if we don't they may legitimately remove the material we thought was obvious.  If we try to argue for an exception based on obviousness we leave to door open to lots of interpretatin about where to draw the line.  In the case of interpretations of Leonardo, there's plenty of sourced material that can be used, for example in Livio, to debunk the silly interpretations. Dicklyon (talk) 14:29, 19 May 2008 (UTC)
 * ...which is why I brought this tag up at the talk page of the editor who added it (or as I know understand, restored it), asking the editor to reconsider. As I understand it, no-one here disputes the contents of the statement in question.--Noe (talk) 16:00, 19 May 2008 (UTC)

"Approximately 1.6180339887"
That doesn't seem approximate (thesaurus: about, almost, nearly, roughly) to me, that's 11 significant figures. I've done plenty of science, and "approximate" tends to me values such as Avogadro's number being 6.022X10^23, or the specific heat capacity of water being 4.18j/kgC. On the page for pi, pi is "approximately equal to 3.14159". That seems more like an approximation to me.

Either we should:
 * Cut down on the number of significant figures (possibly to 1.618, the commonly stated value of the golden ratio), or
 * Switch "approximate" with something like "the golden ratio, to the tenth decimal place, is valued at 1.6180339887."

What are everyone else's thoughts? (Bonzai273 (talk) 04:09, 24 May 2008 (UTC))


 * I don't see the problem. "Approximate" doesn't mean "inaccurate", it just means that it's not perfect. —David Eppstein (talk) 04:25, 24 May 2008 (UTC)

I just think that it's unnecessary to have so many significant figures. The objective of this wikipedia article isn't to give a value for the golden ratio that can be used for the construction of monuments or whatever, it's to give readers a better understanding of the golden ratio and its importance and uses. Having a long-winded value for the golden ratio just seems a waste, and doesn't add to the article. A small, concise value, such as the commonly recognized 1.618, or if you want to go a bit further, 1.61803, is definitely sufficient. It may even make it more memorable/interesting, as some people may just see the golden ratio as this abstract scientific value only used by rocket scientists, if you understand me. If we did want to give a huge value of the golden ratio, a page could be made called "extended value for golden ratio", or a link could be added to a page with it. (Bonzai273 (talk) 04:48, 24 May 2008 (UTC))


 * Sure, it's unnecessary. So what?  We've fought the creeping digit syndrome for years, and we've held the line stable here for a long time.  Why mess with it?  Dicklyon (talk) 04:55, 24 May 2008 (UTC)

Because it could be better. If we just were content with things as is, why bother editing this article? May as well just lock it down and not change it in that case... . As you said, it is unnecessary. How about this solution: shorten the digit number there to something like 1.61803, then people can see later on in the calculations extended a little. This is a good solution, as the non-mathematical people won't see all the digits as confusing or whatever, and the mathematical people, who will be interested in the calculations, will still get the warm, fuzzy feeling of seeing a precise value of the golden ratio. What do you think about that? (Bonzai273 (talk) 05:03, 24 May 2008 (UTC))


 * If you want "better" you need to justify that and get consensus. This article has a long history of compromise, so if you just jump in and make random-looking "improvements" that will typically not be accepted. Dicklyon (talk) 05:18, 24 May 2008 (UTC)
 * The expression of the number with 11 digits is approximate, as must be any expression in decimals of this irrational number. It is also verifiable. I do not think it Wikipedia's mission to dumb down facts or to be so patronising as to segregate readers according to whether they are allegedly non-mathematical or crave warm, fuzzy feelings. It is a straw man argument by Bonzai273 to confuse contentment with the expression with a wish to lock down the whole article. —Preceding unsigned comment added by Cuddlyable3 (talk • contribs) 14:50, 24 May 2008 (UTC)


 * I strongly support approximately 1.618 in the lead. Even building a monument - if you want a 10 m tall golden rectangle, this would give the width to plus/minus 2 mm. Just a few paragraphs further down, 10 decimals are given.--Noe (talk) 15:01, 24 May 2008 (UTC)


 * I would support starting off with the same precision as we do pi: 3.14159, 1.61803.  But not less than that.  Dicklyon (talk) 15:36, 24 May 2008 (UTC)

An engineer might find the statement "approximately 1.6180339887" silly (unless dealing with an extremely unusual situation where that many significant figures were significant for the job at hand). However, a mathematician would reach a completely different conclusion. In this article, mathematical usage is appropriate, and the word "approximately" simply means that the specified value is not exact. Don't change the article. --Johnuniq (talk) 00:50, 5 July 2008 (UTC)

Golden mean
I'm not sure golden ratio (1.618...) is synonym with golden mean with I believe is close to 0.618(...), as wikipedia states. I think, golden mean always means the second element in a golden section, not the whole thing, but I'm not sure.

I thing golden ratio would be synonym with golden extreme and mean, but not just one of either. That's unles there is somethng I'm not taking in consideration.--20-dude (talk) 06:44, 27 May 2008 (UTC)


 * Sources? What do you mean "as wikipedia states?" Dicklyon (talk) 06:47, 27 May 2008 (UTC)


 * He means, the Golden ratio article in "Wikipedia states" that golden ratio and golden mean are synonyms, but he thinks that may be wrong because the golden ratio is 1.618 while, he thinks, the golden mean is 0.618. He thinks that because he reads some sources that do in fact say that the golden mean is 0.618... There are also sources that say that the golden ratio is 0.618..., but perhaps he has not come across those yet. His underlying problem is that he reads literature that uses the various synonymous terms, define them, and state their values differently, and he does not get past that because he does not understand the concept. So he is, understandably, confused. To his credit, this time he is asking, not inserting erroneous information in article space. Finell (Talk) 20:14, 31 May 2008 (UTC)


 * Actually the golden ratio 1.618... is not the mean of anything. The qualifier conjugate used for 0.618... should be multiplicative inverse. Cuddlyable3 (talk) 07:56, 27 May 2008 (UTC)


 * Definitive answer: By the most common current convention, the golden ratio is 1.618... because, by the most common current convention, the ratio is stated as . If one states the golden ratio as , then the golden ratio is 0.618..., and some published sources do state it that way. The relationship between the lengths of the segments of a line sectioned extreme and mean ratio (a golden section)  or between the sides of a golden rectangle is the same, regardless of which way one states the ratio. In any event, whichever way the ratio is stated, it is always, and only, a ratio between two lengths; it is never solely the short one or the long one. And, whichever way one states the golden ratio, golden mean is sometimes uses as its synonym (although golden mean has an unrelated meaning in philosophy). Dude, please stop sowing confusion with your misinformation: there is no such term as golden extreme and mean in the reliable sources on the subject. Finell (Talk) 20:14, 31 May 2008 (UTC)


 * The golden ratio, golden mean, or divine proportion are one and the same thing (perhaps with slight differences in usage). This thing is not a number, it's a concept, a relation between quantities (numbers, lengths, times intervals, or whatever). In some sources, this concept is identifed with one of the numbers phi=1.618... or tau=0.618... (other symbols may be used), both of which make sense, but neither of which is the basic meaning of the concept. In some (older) sources, there is a distinction between a ratio (like 1:3, phi:1 or 1:phi) and the value of the fraction (1/3, phi/1=phi resp. 1/phi=tau); in other sources, there is no such distinction.--Noe (talk) 12:21, 1 June 2008 (UTC)


 * I agree completely, but the difference is not between phi and tau, either. Tau used to be the symbol of choice for the golden ratio (the initial letter of the Greek word for cut), but phi became more popular in the 20th century. This is mentioned in the article. So if you see tau = 0.618, the author is stating the golden ratio as ; some sources say phi = 0.618 for the same reason. And the unusual property that 1/φ = 1-φ (or, if you go the opposite direction, 1/φ = 1+φ), doesn't help. And then there is the rarely seen (but one place is Mathworld) term golden ratio conjugate which, according to our article and Mathworld is 0.618 and is denoted Φ (capital Phi), but is sometimes denoted φ′ (phi prime), and which is defined as the reciprocal of the golden ratio (if you state the golden ratio as  ). But some think that golden ratio conjugate should mean the conjugate root of the defining quadratic equation, which we say is -0.618. This just introduces more confusion, for anyone who gets that far. Personally, I think the world would be better off if the term golden ratio conjugate were outlawed, since 1/φ always works and expresses the concept, whether you think long or short; but golden ratio conjugate is out there, so our article can't ignore it. By the way, none of this confusion arises at all if one thinks in pure pre-analytic geometry, which is where the whole idea started and was the way Euclid codified it. A couple years ago, I was going to do a section on Terminology and notation for the article, which would go through the etymology of the terms and symbols, and the varying definitions, historically, but I never got around to it. Finell (Talk) 14:25, 1 June 2008 (UTC)


 * Noe says "This thing is not a number, it's a concept, a relation between quantities (numbers, lengths, times intervals, or whatever)." But that relation is most often expressed mathematically as a ratio, which is a number.  In Pacioli's day, divine proportion was never made into a decimal number, and I think maybe also not expressed as the irrational in terms of root 5 (not sure about that).  But certainly by the time anyone called it "golden", it was a number, since that's how modern mathematics expresses such geometric relationships algebraically, thanks to Descartes.  To try to ignore the number and express it again as a just a concept, using words alone, tends to confuse the picture, and is what has made it so easy for 20-Dude to use language that he can't explain, and that the rest of us are unable to intepret, such as "with golden extreme and mean, but not just one of either". Dicklyon (talk) 15:36, 1 June 2008 (UTC)


 * I agree, but we must accept that sources differ as to whether that number is 1.618... or 0.618..., and my post (like Finell's preceeding it) should explain why it's so, and why it's not terribly important either.--Noe (talk) 07:54, 2 June 2008 (UTC)


 * Euclid's sectioning a line in extreme and mean ratio, and everything else that Elements says on that topic, is understandable to anyone who sufficiently understands geometry taught in the Euclidean manner. Likewise, the geometry, the quadratic equations, and the solutions, whether expressed as irrational numbers or as decimal approximations, is understandable to anyone who sufficiently understands geometry taught in the Cartesian manner. (When I was in 7th grade honors math, the so-called "new math" taught exclusively Cartesian geometry, with theorems proven by algebra, which was interesting [in the Chinese sense] because algebra was first taught in 8th grade. I read selectively from Elements much later for pleasure, probably as a result of reading Einstein's Autobiographical Notes.) Anyone who understands either knows that   or   is purely a matter of convention: anyone who understands the algebra can flip one around to the other; if one thinks in Euclidean terms, there is nothing to flip. Dude's problems are, first, that he doesn't understand either (something that he shares with an alarmingly substantial majority of the American public), and second, that this does not deter him from declaiming on the subject (it is in this respect that he is unusual). Finell (Talk) 19:18, 1 June 2008 (UTC)


 * Well, I've studied geometry in the way it was taught in American high schools in the 1960s, all about proofs and such, which may be what you call the Euclidean manner or may not. So clue me in: what is the interpretation of "extreme" and "mean" in that expression?  Dicklyon (talk) 19:52, 1 June 2008 (UTC)


 * We are contemporaries. However, "it was [not] taught [the same way] in [all] American high schools in the 1960s". In my high school, the way it was taught depended on which track you were in. If your geometry was not taught as Cartesian coordinates, then it was (almost certainly) what I refer to as the Euclidean manner (straightedge and compass constructions, non-algebraic proofs). As for extreme and mean ratio, my surmise (I have not found a source) is that it refers to the property that, in the defining proportion, mean1=mean2 and extreme1/mean = mean/extreme2 = (extreme1 + mean)/mean. By the way, I have never pretended that my level in math approaches yours. Now, here's one for you, since you also love old books: What is the original source of the quotation so often attributed to Kepler, "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." Pity (for terminological consistency) that he did not reverse these comparisons. And which did he value more? The value of "a measure of gold" depends on the measure, while the value of "a precious jewel" depends on the particular gem stone, the size, and other factors (color, cut, and clarity, to be specific). Finell (Talk) 21:40, 1 June 2008 (UTC)


 * The geometry I studied was definitely constructions, non algebraic, so I guess that means Euclidean. I don't recall extreme and mean ratio coming up, but it's possible. Anyway, I take it the "mean" here means what we call "geometric mean" in algebra, the square root of the product.  So if we take the short segment and the total line as extremes, the long segment is their mean.  The ratio of the mean to either extreme is thus the same (depending on the order).  I still don't know why they call it "extreme and mean ratio"; what does the "and" designate?


 * Great question on the quote. Hambidge 1920 p.153 cites two German books for exact references to sources, but I don't have access to either of those.  I'll see what I can find. Dicklyon (talk) 00:00, 2 June 2008 (UTC)


 * Your explanation of "extreme and mean" is much cleaner than my groping, and that may be as close as we can get without finding a scholarly interpretation of the precise phrase "extreme and mean ratio". One problem may be that classical Greek is hard to translate accurately; the translation of classical Greek to a European language is less linear than translation from Latin (I learned this about 15 years ago, when I came across a widely quoted statement by Plato about about taxes that I knew had to be a mis-translation [although it is the translation used in Bartlett's] because it was contrary to the history of the income tax). And we don't even get it directly from the Greek: Heath's English edition is based on the leading German translation; other editions come to us via Arabic. So we are trying to parse 4 words in English that may not be an accurate translation of what Euclid actually wrote in Greek.


 * I agree, it would be fascinating to see how these expressions connect. Dicklyon (talk) 05:33, 2 June 2008 (UTC)


 * My interest in the original Kepler source (in addition to the fact that it would be nice to cite) is because I want to read the quotation in its context, if there is an English translation: why did he single out the golden ratio as being comparable in importance to the Pythagorean theorem? The latter seems to me to be much more fundamental. Most people think of Kepler the astronomer, but was primarily a mathematician. All of his theories about cosmology, both correct and incorrect, were based on geometry.


 * The "Kepler triangle" combines these two concepts into one triangle, and like a lot of people, he fell for the mystical power of geometry and numbers. It's the only right triangle with edge lengths in geometric progression.  He wasn't much of an astronomer, having very poor eyesight; that's why he had to kill Tycho and steal his data. Dicklyon (talk) 05:33, 2 June 2008 (UTC)


 * I've been watching Kepler triangle. As for mysticism, let's give Kepler (and Pacioli, for that matter) a break based on historical context. Mysticism was the norm then, even for mathematicians and scientists. Astrology was taught as part of medicine. Galileo was a devout Catholic, despite being a Copernican. Newton was deeply into mysticism. But despite that, the latter two were not shabby as physicists, and Newton was OK at math. As for Tycho, you obviously subscribe to Peter Schaffer's version of Mozart's death. Finell (Talk) 06:46, 2 June 2008 (UTC)


 * Here is one more. Do you have access to a bibliographic index of early 20th century math papers? If so, can you find any papers published by Mark Barr. So far as I can tell, he never published a book. The closest I have gotten to his work on the golden ratio and his suggestion of using phi for its symbol is writing by TA Cook. Ghyka mentions it too, but I think that everything in Ghyka is secondary accounts of others' work (with attribution). Finell (Talk) 05:22, 2 June 2008 (UTC)


 * No, I don't have, and I've never been able to find anything cited on Barr either. Dicklyon (talk) 05:33, 2 June 2008 (UTC)


 * I tried searching Zentralblatt (goes back earlier than MathSciNet) for his name but didn't find anything. —David Eppstein (talk) 05:41, 2 June 2008 (UTC)

"Reasons for the occurrence of phi in nature"
I have seen very elegant arguments for why it is advantageous for some plants to incorporate the golden ratio into the distribution of their leaves or seeds. It is based on the fact that phi is the "most irrational" number in the sense that it is the most difficult to approximate with a ratio of integers (an approximation up to a certain precision requires a large denominator). I am not familiar enough with the argument to write it myself, but if someone else is, I think it would definitely be worthwhile.

More generally, a precise statement (and possibly a proof) of the fact that phi is the "most irrational" is lacking from this article. —Preceding unsigned comment added by 68.89.174.144 (talk) 04:29, 6 June 2008 (UTC)


 * I'll be happy to write it if you'll tell me your source. Dicklyon (talk) 21:53, 6 June 2008 (UTC)


 * See eg Atela et al (2008) A Dynamical System for Plant Pattern Formation: A Rigorous Analysis, Journal of Nonlinear Science, Volume 12, Number 6 / March 2008, pages 	641-676.


 * This particular model and its explanation goes back to papers Stéphane Douady and Yves Couder on spiral phyllotaxis, from 1991 onwards. But see the article Phyllotaxis at Mathworld for discussion and observeration of the phenomenon going back much earlier, including papers by Coxeter.  Jheald (talk) 15:11, 21 July 2008 (UTC)

on Further Reading
I moved a number of this items to refs, since they usefully support some points in the article. It's not clear what the remaining 6 are good for, or why we'd want a further reading section in addition to all the cited reading. Some are not very accessible. I bought a copy of Doczi, and it doesn't have much useful (lots of fanciful stuff about spirals that someone might find interesting, but most of it not very connected to the golden ratio). The Walser book looks pretty, but I haven't spotted a great place to cite it.

So should we remove these, if they're not sources that support the article? Or if someone has a copy of some that arent' online, can we add something that they support? Dicklyon (talk) 05:13, 24 August 2008 (UTC)


 * Per WP:MOS, the Further Reading section is intended for additional books on a topic that are not cited in the article as sources. I think we would need a rationale to remove an entry that fits this definition. Also, I see no need to make an extra effort to cite something that is now in Further Reading. The only one that I have read is Huntley, which is a pretty good overall treatment by a mathematician (I don't have it; I got it from a local library). Finell (Talk) 03:06, 26 August 2008 (UTC)

Nature
Just an observation, the major and the minor grooves of DNA are in the golden ratio. Also, when a string on a guitar is separated in the golden ratio you get the tonic resolving tone. —Preceding unsigned comment added by Niubrad (talk • contribs) 11:23, 26 October 2008 (UTC)

Difference?
I've seen stuff that tells me 1.618 is the golden ratio, but I've seen people call the Golden ratio 0.618 as well. What is the difference? and can anyone tell me anymore natural stuff that is based on the goloden ratio? Thanks a lot.Ericrules2363 (talk) 03:04, 2 October 2008 (UTC)Ericrules2363


 * The difference? 1.618 &minus; 0.618 = 1. Fredrik Johansson 05:54, 2 October 2008 (UTC)


 * Phi, phi, pho, fum, I smell a troll... Dicklyon (talk) 05:58, 2 October 2008 (UTC)

(sin 30) / (sin 18) = Golden number
I found this out using a calculator. Maybe it could be added to the article. —Preceding unsigned comment added by 74.58.215.201 (talk) 20:47, 28 June 2008 (UTC)


 * It already says that $$\varphi = {1\over2}\csc 18^\circ$$, which is exactly the same, since $$\sin 30^\circ = {1\over2}$$. -- Dominus (talk) 21:02, 28 June 2008 (UTC)


 * More generally, things you find out using a calculator are not encyclopedic, per WP:RS and WP:NOR. Dicklyon (talk) 21:04, 28 June 2008 (UTC)
 * A calculator isn't a reliable source? KenFehling (talk) 07:13, 21 July 2008 (UTC)
 * That's like putting your own chemistry researches on here and claiming a test tube is a reliable source. Reyk  YO!  07:44, 29 July 2008 (UTC)
 * That's not really the same at all. The results of a calculator are verifiable by anyone.  In fact, this very article and many math related articles have computations that someone did with a calculator. KenFehling (talk) 19:19, 29 July 2008 (UTC)
 * If the calculations and examples are used to illustrate a known fact, then that's fine. For instance look at the example section in Engel expansion. I picked the number 1.175 because it would illustrate the algorithm concisely, and I did those calculations with the aid of a calculator. But that's not original research because I didn't invent the algorithm and I'm concerned only with explaining an already known fact. If I'd just been fiddling around with my calculator one day, found something interesting and decided to put it up on Wikipedia going, "Look what I've found out!", then that would be OR. The fact that anyone with a calculator could check to see if I'm right is immaterial. Reyk  YO!  22:16, 29 July 2008 (UTC)

A good properly functioning calculator is NOT a reliable source for something like this, but anyone can do this sort of calculation on paper in a minute and that seems like a reliable source. Calculators approximate, and knowing how and when such approximations appreciably affect the bottom line requires keeping your brain in gear. Talking students out of gullibly believing their calculators is a substantial challenge. (Talking them out of using a calculator as an anesthetic device is another challenge, but not the same one.) Michael Hardy (talk) 22:56, 29 July 2008 (UTC)
 * PS: See also exact trigonometric constants. Michael Hardy (talk) 22:58, 29 July 2008 (UTC)
 * Not all calculators always approximate their results. The TI-89 Titanium, for example, performs symbolic computation.  Given the expression $$\frac{\sin{30^\circ}}{\sin{18^\circ}}$$, it can display either an exact result of $$\frac{\sqrt{5}+1}{2}$$ or an approximate result of 1.6180339887499. –Wdfarmer (talk) 03:52, 29 October 2008 (UTC)

That the Parthenon uses Golden Ratio is now contested?
I recently saw a Nova program on the current Greek reconstruction of the Parthenon. The program implied that the Parthenon did not use the Golden ratio anywhere, and that the theory that it did was a creation of renaissance thought, not careful measurement. —Preceding unsigned comment added by 71.104.240.132 (talk) 03:56, 4 September 2008 (UTC)


 * There is no evidence the Greeks used golden ratio in art or architecture (they favoured rational proportions). I don't know who first cooked up the theory that they did, but I doubt it goes as far back as the rennaisance - I think it is after Fechner's psychophysics (which was about 1890, as far as I recall). The golden ratio myth is based on speculation and secondary popular modern sources. (Please prove me wrong, if you can!)--Noe (talk) 06:37, 4 September 2008 (UTC)


 * The earliest such claim I've seen is this guy in 1912. And he clearly doesn't know what he's talking about (see this note 2 on p.22). Dicklyon (talk) 07:18, 4 September 2008 (UTC)


 * I just noticed that he attributes some of the Parthenon proportioning to Jay Hambidge, though not necessarily the divine section aspects; here is more on that guy and the Parthenon. Dicklyon (talk) 07:31, 4 September 2008 (UTC)

This is further backed in Mario Livio's Golden Ratio, The Story of Phi. (ISBN-10: 0767908155 ISBN-13: 978-0767908153)

Livio flatly says that the Parthenon's columns do not measure to the Golden Ratio, and suggested that the Greeks knew about Phi but had never previously used it in any architecture. —Preceding unsigned comment added by 168.28.202.184 (talk) 11:47, 28 October 2008 (UTC)

Short Proofs of Irrationality
The first proof of its irrationality, using the Euclidean Algorithm, seems to me to need work. I may try something later (say, by March of 2009). Rick lightburn (talk) 21:28, 15 November 2008 (UTC)

Math images
These are great for complex fractions but do try to avoid overusing them for single letters and other simple expressions where plain text will suffice, especially as there is no reliable way to get an image and a text to scale properly with respect to each other (so don't use math tags mid-paragraph unless there is no other option). Also they will never work in section headings:

The $$\sqrt{}$$ of $$5$$ is $$\sqrt{5}$$
Now look up at the table of contents. Don't do this. — CharlotteWebb 14:36, 23 November 2008 (UTC)


 * On a related subject, please avoid using the superscript-2 character — I see you have just replaced many &lt;sup&gt;2&lt;/sup&gt; with this, and in general it's a bad idea. The reason is that it produces very different spacing of the characters than is obtained for other superscripts. See WP:MOSMATH. —David Eppstein (talk) 16:20, 23 November 2008 (UTC)


 * No doubt the issue of scaling is important, but I have to say that the fix introduced by CharlotteWebb has a downside: there is now a mixture of fonts used to represent phi, and the result is displeasing to my eye. For example, just under the "Golden ratio conjugate" heading, we see a bold phi in one font (why bold?), then we see an attractive varphi in an expression. The capital phi at the end of the sentence is particularly disappointing (BTW, shouldn't "capital Phi" just be "capital phi"?). --Johnuniq (talk) 07:55, 24 November 2008 (UTC)

Images disappearing
Over at Fibonacci number, a couple of us have recently changed the math expression used to display varphi because we found that the png image was not being delivered, so the browser was displaying "varphi". While I did lookup the docs to confirm that there was a rational basis for the change I made, I have almost no knowledge of   and no idea why it had failed.

I see that the same problem is now apparent here (I think the problem was not there yesterday, becaused I scanned the article to see if it suffered the problems I was seeing at "Fibonacci"). Just above the Geometry heading, there are two equations where the png is not delivered. They appear as:
 * -\frac{\varphi}{2}=\sin666^\circ=\cos(6\cdot 6 \cdot 6^\circ).
 * -\varphi=\sin666^\circ+\cos(6\cdot 6 \cdot 6^\circ).

The two images are each downloaded as an empty file. What's going on? --Johnuniq (talk) 07:55, 24 November 2008 (UTC)

Phi's evil twin?
The only positive solution to this


 * $$x^2 - x - 1 = 0.$$

is this


 * $$x = \frac{1 + \sqrt{5}}{2} \approx 1.61803\,39887\dots\,$$

but the negative solution is this


 * $$x = \frac{1 - \sqrt{5}}{2} \approx -0.61803\,39887\dots\,$$

This negative number seems to have all of the same properties as phi ie. the number squared is itself + 1 (0.381), and 1 / the number is itself - 1 (-1.618).

I don't know how significant this is, but I hear people say all the time that phi is the only number that does this. (Though I'm glad to see this article makes no such claim.) Aykantspel (talk) 18:26, 29 June 2009 (UTC)

I got rid of the "Euclidean algorithm" section
In a very bad edit on March 18th, 2008, user:Marc van Leeuwen claimed to be adding an even shorter proof of irrationality, this one using the Euclidean algorithm.

In fact, the proof was the SAME as the proof that followed it. But it was phrased in such a way as to make those unfamiliar with the Euclidean algorithm think that they could not follow it without learning the Euclidean algorithm.

So I got rid of it today.

The short proof that follows it is phrased in a needlessly complicated way. I will change that later. Michael Hardy (talk) 17:12, 23 November 2008 (UTC)


 * The two methods clearly are related, but are you sure they are equivalent? The "Euclidean algorithm" proof you omitted concluded with the claim that "this proof does not require any number theoretic facts, such as prime factorisation or even the fact that any fraction can be expressed in lowest terms". That claim looks likely to me.
 * I read the "Euclidean algorithm" proof a month ago and was enchanted with its elegance. The only reason it was long was that the author had attempted to first explain the algorithm. I think the proof could be offered in one line (to anyone familiar with the algorithm). I would hope there is a way to restore the essence of the Euclidean argument, probably as the second proof (because it would be pretty opaque to many general readers). --Johnuniq (talk) 08:20, 24 November 2008 (UTC)

Infinite Surds

 * $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}}}} = \varphi = \frac{1+\sqrt{5}}{2}\approx 1.61803\,39887\ldots$$ —Preceding unsigned comment added by 98.165.243.53 (talk) 21:21, 27 November 2008 (UTC)

The Golden Ratio Optimizes Cardiomelic Form and Function
Recently, the Golden Ratio was linked to cardiac and hand development, in relation to optimal form and function. The full article can be found at http://www.facebook.com/ext/share.php?sid=46890696782&h=VLLLZ&u=9dzAQ

Enjoy, and comment!

Jasoncys (talk) 11:35, 3 January 2009 (UTC)


 * That's a facebook link that leads to a warning page. Dicklyon (talk) 18:41, 3 January 2009 (UTC)


 * Ah, I see that you can get past facebook to the link you probably intended: at journals.tums.ac.ir. Dicklyon (talk) 18:44, 3 January 2009 (UTC)


 * Interesting conjecture. Too bad there's so little support for it; just speculation.  Is this journal thought to be a reliable source? Dicklyon (talk) 18:54, 3 January 2009 (UTC)