Talk:Golden ratio/Archive 8

Rotating black holes
Regarding this removal: this isn't one of those cases where there is an unresolved difference of legitimate opinion in the field (like when Physicist A uses a particular approximation that Physicist B thinks is inapplicable). Davies just screwed up. XOR&#39;easter (talk) 13:08, 26 July 2019 (UTC)
 * The link you included above is a better mathematical source than any of the three I included. It mentions the more nuanced detail that the commenter Greg Egan worked out that the golden ratio does show in black holes in a more subtle way, while "angular momentum is held constant". I personally think it would be worth including in the "Disputed claims" (not as an undisputed fact) with this nuance added for clarity. See the Azimuth Project website for attribution to John C. Baez. UpdateNerd (talk) 19:39, 26 July 2019 (UTC)
 * I removed this again anyway; sorry I didn't say anything sooner, but I didn't realize you were going to try to re-add. This isn't a purported observation of $&phi;$; it's just a random popping out of it during a theoretical physics calculation.  And considering the story around this, there doesn't actually seem to be any sort of dispute; there was just a mix-up in exactly what was being calculated. So this isn't appropriate to add, regardless of sourcing.  –Deacon Vorbis (carbon &bull; videos) 20:20, 26 July 2019 (UTC)
 * Fair enough (though I actually disagree; it's an observation even if it's done through equations instead of geometry). If better secondary sources explained the nuances of this, it would be completely appropriate to add. UpdateNerd (talk) 20:24, 26 July 2019 (UTC)
 * Well, it would be an observation if someone actually measured this on a real black hole, rather than some sort of derivation from an idealized one. And in this case, unlike the real iffy things (like nautilus shells), there would be a theoretical basis for it, too.  Moreover, it still wouldn't be interesting enough to include  there was also some sort of explanation linking a defining property of $&phi;$ to the result, rather than just something akin to the Strong Law of Small Numbers rearing its head.  –Deacon Vorbis (carbon &bull; videos) 20:33, 26 July 2019 (UTC)
 * Good point about detecting it on a real black hole, or further connecting the data to them somehow. UpdateNerd (talk) 20:36, 26 July 2019 (UTC)
 * Yeah, it's just a number falling out of an equation (a wrong equation, as it happens). And once the calculation is corrected, the only way to get $$\phi$$ to pop out is to compute something that's physically irrelevant. Davies' original paper didn't make a big deal out of a threshold working out to be $$(\sqrt{5}-1)/2$$; I'd call it a curiosity, except that it turns out to be not very curious. XOR&#39;easter (talk) 23:18, 26 July 2019 (UTC)
 * Oh, and The Golden Ratio: The Divine Beauty of Mathematics is not a reliable source. XOR&#39;easter (talk) 23:28, 26 July 2019 (UTC)
 * One other thing: the source in The Fountain is just a typical regurgitation of golden-ratio fannishness. It repeats as fact an explanation for phyllotaxis which is highly dubious (and can't apply to all plants). And it garbles the claim about black holes (I don't even know what it's trying to mean by spinning parameter). It's not reliable either. XOR&#39;easter (talk) 00:07, 27 July 2019 (UTC)
 * Thanks, I'll disregard those as sources, although I was mostly using them for general details not found in the Livio article—not the accuracy of the claims or mathematics. UpdateNerd (talk) 00:48, 27 July 2019 (UTC)
 * For posterity: except to point out where Davies went wrong (as on the rotating black hole article), the Azimuth source isn't much good either. It claims that the golden ratio surfaces when the gravitational constant and speed of light equal one, but how can these two velocities be equal? I wish I'd noticed this irregularity before, but I'm also not an astrophysicist. It appears that Egan only found what he was looking for, which shouldn't be too surprising. UpdateNerd (talk) 18:49, 27 July 2019 (UTC)
 * Setting $$G = c = 1$$ is commonplace. See Planck units. XOR&#39;easter (talk) 19:13, 27 July 2019 (UTC)
 * I see both G = 1 and c = 1 depending on the context, but nothing about setting them both to 1 in the same equation. UpdateNerd (talk) 20:14, 27 July 2019 (UTC)
 * The table in §List of physical equations does so, for starters. It is literally a thing that physicists do all the time. It is completely unremarkable. The first reference that springs to mind is Chapter 1 of Gravitation, which explains how to set $$G$$ and $$c$$ simultaneously to unity, along with Boltzmann's constant $$k$$, but any decent set of lecture notes on general relativity will cover the technique. The hoary old joke is that standard practice is to set $$G = c = k = \hbar = 1$$, and if you're really an intense theorist, you set $$\pi = 1$$ as well. XOR&#39;easter (talk) 20:42, 27 July 2019 (UTC)
 * Do you have a page number for Gravitation? UpdateNerd (talk) 21:11, 27 July 2019 (UTC)

Whoops, I must have missed this notification in my watchlist. There's a box in Gravitation dedicated to the topic on page 36. But pretty much any field theory textbook will discuss the subject at least a little in the course of establishing its notational conventions. Tony Zee's Quantum Field Theory in a Nutshell does so in the preface (p. xxv), for example. XOR&#39;easter (talk) 21:03, 8 August 2019 (UTC)
 * Thanks for the helpful page references (and I also see that we have an article on the geometrized unit system), but I'm still agnostic on this issue. As far as I can tell, c=1 and G=1 can both be utilized, but I don't see c=G=1 in the same expression. They're fundamentally different values, and therefore seemingly can't be equal. UpdateNerd (talk) 21:27, 8 August 2019 (UTC)
 * If you choose your units of mass, time and length carefully, then both the speed of light and the gravitational constant are numerically equal to 1. Nothing too hard about it. XOR&#39;easter (talk) 21:32, 8 August 2019 (UTC)
 * As I said, still remaining agnostic, as that explanation is too vague for me. Thanks for your responses though, as it provides some food for thought. UpdateNerd (talk) 21:35, 8 August 2019 (UTC)

Expressing φ in terms of a and b
I find the mathworld explanation much clearer and more intuitive. My mental picture is of a piece of paper from which you keep cutting squares from the longest size, and the ratio of the sides is always φ.

As it is, it takes several steps to get from the a+b relationship to the one just for φ whereas if you directly make the sides 1 and φ then this relationship "can immediately be seen". — Preceding unsigned comment added by 2A01:CB15:8010:2F00:ECF5:C24C:4992:6347 (talk) 07:56, 7 February 2020 (UTC)

No reference for phi in financial markets
The Reference 9, while covering the first clause of the sentence does not mention financial markets, thus a separate reference for this clause is necessary. Also, I am interested if indeed phi does show up in financial markets. — Preceding unsigned comment added by 2603:3005:BE6:C000:8DF:D9BE:2C86:6D0 (talk) 22:42, 24 February 2020 (UTC)


 * From the source:
 * It's a off, but still seems reasonable overall. –Deacon Vorbis (carbon &bull; videos) 23:06, 24 February 2020 (UTC)
 * It's a off, but still seems reasonable overall. –Deacon Vorbis (carbon &bull; videos) 23:06, 24 February 2020 (UTC)

Semi-protected edit request on 22 August 2020
I found a new property of the Golden ratio:

1 / x = 1 + x

Where (x = 1,618...(Golden ratio) - 1)

thx Ricerca Veritas (talk) 01:49, 22 August 2020 (UTC)
 * Hi. This is not a new property of the golden ratio. Your equation is equivalent to saying $$x^2 + x - 1 = 0$$, and if you plug $$x = \varphi - 1$$ into that, you get
 * $$(\varphi-1)^2 + (\varphi-1) - 1 = \varphi^2 - 2\varphi + 1 + \varphi - 1 - 1 = \varphi^2 - \varphi - 1 = 0,$$
 * which is the equation already in the article. Keep at it &mdash; the first step to discovering something new is to rediscover old things for yourself! XOR&#39;easter (talk) 02:21, 22 August 2020 (UTC)

Flag of Togo
Should it be mentioned that the flag of Togo has a proportion of 1:φ? –Hydrogenation (talk) 18:55, 16 January 2021 (UTC)

"The figure on the right" should change.
It's meaningless and a bit confusing when viewed in a mobile context. Bunglero (talk) 00:23, 14 February 2021 (UTC)
 * Agreed; this violates MOS:SEEIMAGE. —David Eppstein (talk) 01:19, 14 February 2021 (UTC)
 * I cut the sentence; I wouldn't object to some rephrased version, but I'm not sure a remark pointing to the image is necessary when the image itself is right there. XOR&#39;easter (talk) 14:14, 14 February 2021 (UTC)

Equation *
For what n,m is this claimed? It fails for n=3 and m=2 (and most others) [Why not say: sqrt 5 is irrational hence phi is irrational?] 74.81.88.122 (talk) 15:40, 25 March 2020 (UTC)


 * This seems to be reasonably clearly explained in the section. As for irrationality of $&phi;$ following from that of the square root of 5, that's in the very next subsection as well.  –Deacon Vorbis (carbon &bull; videos) 15:57, 25 March 2020 (UTC)

Equation * is said to be an identity. It is certainly not true for all m an n. What is the claim? In my view Eq * does not prove phi is irrational. If you feel that it does please supply some detail to support you view. If not just go with your next paragraph which, as you note, says that it is irrational because sqrt 5 is irrational. 74.81.88.122 (talk) 01:03, 26 March 2020 (UTC)
 * It is clearly stated that in the text above (*) that n represents the whole and m the longer part of a partition of a length into smaller parts in the golden ratio. The n and m to which (*) applies are the n and m for which this is true. It is not stated that this equality applies to other values of m and n, so I don't understand why you are complaining that it doesn't. —David Eppstein (talk) 01:13, 26 March 2020 (UTC)

The most beautiful equation for golden ratio was discovered by the Indian Mathematician Srinivasa Ramanujan. The equation is as follows: Golden ratio = [2*3/1*4]*[7*8/6*9]*[12*13/11*14]* ... — Preceding unsigned comment added by Ashutoshpise (talk • contribs) 08:56, 2 March 2021 (UTC)

Music "formal boundary"
in the music section, it says that: "The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section." what is a formal boundary of a music composition? how can it be a number or a ratio? Please tell me if I should put this question somewhere else. Thank you! Bumpf (talk) 15:06, 24 March 2021 (UTC)

The Parthenon
I used my ruler to measure the proportions of the Parthenon face shown here: https://www.shutterstock.com/editorial/image-editorial/art-places-various-5850841gq

using the large version shown here, I got 22.2 mm wide measured across the top frame, and 13.7 mm deep from the tip of the triangle to the foundation. The proportion is 16.2. Using the smaller version of this image I saw on google images, I measured 10.5 mm divided by 6.5, or 1.615. So it seems you can find the golden proportion in the Parthenon front face if you include what appears to be the foundation.Eameece (talk) 23:06, 11 April 2021 (UTC)

This author Gary Meisner (2013) finds golden proportion ratios in The Parthenon as well. https://www.goldennumber.net/parthenon-phi-golden-ratio/ Meisner's article points out that the fact that the Greeks in the 5th century BC did not know about this proportion and thus did not intentionally design the building in accordance with it, does not disprove that the proportion appears. Meisner says that the proportions may be present since the appearance of the Golden Ratio in nature and the human body influences human aesthetics. Eameece (talk) 23:18, 11 April 2021 (UTC)
 * You can find the golden ratio in anything if you pick and choose which points you measure between carefully enough and are sloppy enough in what range of numbers you think are all equal to the golden ratio. See https://xkcd.com/spiral/. Your measurements are original research, and cannot be used directly here. Meanwhile, we lack sources attesting to intentional use of this ratio by classical architects, in contrast to say the Vitruvian system of rational proportions. —David Eppstein (talk) 23:11, 11 April 2021 (UTC)
 * You added the paragraph about Meisner after I replied; please do not edit earlier messages to make it appear that other editors were replying to different things than what they replied to. As for Meisner, you can find a lot of dubious sources making dubious claims for the golden ratio. And you can find reviews of his book, specifically, saying things like "a lot of the math is wrong" and "I suspect that if other proportions were searched for they might also be found". So I'm skeptical that his book can be taken as a reliable source here. —David Eppstein (talk) 00:10, 12 April 2021 (UTC)

The ratio of the width of 101′ 3.75″ to revised height of 62.491′ is 1.621, close to the golden ratio of 1.618. Any measure of the Parthenon should include it's base, since that's what observers see as "The Parthenon" Gary Meisner, 2020. Perhaps this represents another view on this subject. Eameece (talk) 06:41, 12 April 2021 (UTC)


 * The personal website of a man whose book elicited reviews saying "a lot of the math is wrong" is hardly a reliable source, and assertions about "what observers see" are speculation. XOR&#39;easter (talk) 15:13, 12 April 2021 (UTC)

OK. Thanks for reading my suggestion. I think this Meisner website is well documented and accurate, despite that one review. There may be others who hold this opinion, but that's all for now 2600:1700:BEF0:1BB0:5D6F:5F58:45F8:E304 (talk) 20:23, 12 April 2021 (UTC)
 * If you want something that looks more like a reliable source about the numerical ratios of the Parthenon, please try Markowsky, "Misconceptions about the Golden Ratio", already used as a reference for other material in our article. Markowsky observes that "None of these authors is bothered by the fact that parts of the Parthenon are outside the golden rectangle" and "The dimensions of the Parthenon vary from source to source probably because different authors are measuring between different points. With so many numbers available a golden ratio enthusiast could choose whatever numbers gave the best result." He also cites several other authors in suggesting that a 4:9 ratio is more evident in the design of this structure. —David Eppstein (talk) 22:30, 12 April 2021 (UTC)

Golden ratio in nature, magnetic resonance of spins
I went and checked the actual journal article and "the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals" is not what the article found.

Specifically, the journal article states:

"''Figure 4D shows how the ratio of the energies of those peaks varies with increasing field and approaches closely (near 5 T just below the 3D critical field of 5.5 T) the golden ratio m2/m1 = (1 + sqrt(5))/2 = 1.618 predicted for the E8 masses.

I'm not sure about admissible evidence (it is locked behind a paywall). And the source the wiki article links is quoted correctly. The source is just pop science and doesn't understand the word "approaches". I mean its nifty but so would approaching pi or sqrt(2) be. Mediocracy (talk) 19:54, 1 July 2021 (UTC)

Semi-protected edit request on 13 August 2021
Please add this subsection to the "Applications and observations" section of the current Golden Ratio wiki page

Human Physical Beauty

Some believe that the golden ratio may serve as the underlying foundation of an attractive human physical appearance. American plastic surgeon Andrew J. Hayduke, M.D. proposed that human physical beauty can be objectively tested via analysis for golden ratio anatomic relationships within the physical anatomic features of human faces and breasts. Hayduke describes a series of golden ratio based anatomic test grids within his treatise entitled The Golden Ratio Within the Human Face and Breast: A Plastic Surgeon's Method of Analyzing Beauty.

JohnEdit45 (talk) 14:01, 13 August 2021 (UTC)


 * ❌. I strongly suspect this is self-published; searching for "Ivory Crown Press" finds only this book. In any case, brand-new publications purporting to use the golden ratio to analyze something are not even worth including in the "disputed sightings" section. —David Eppstein (talk) 16:21, 13 August 2021 (UTC)

Inconsistencies in mathematical notation
I spent like 2 hours cleaning up this article’s 5 different inconsistent ways of writing inline math (sans serif italics, sans serif Roman, serif italics, serif Roman, LaTeX serif italics). Apparently David Eppstein really preferred the extremely inconsistent version / is too busy to bother with the details, so insists on trashing that effort. To be honest I don’t understand what he is trying to say. The explanation given in reverts were:

(1) Math tags should always be preferred to math templates


 * Response: clearly Wikipedia math articles in general and this article in particular do not follow that advice. I deliberately left all of the block math tags and anything that looked tricky; the rest are simple formulas well within the capabilities of math templates to display adequately. If you want, maybe we should convert all math and numbers in the article to math tags? That seems fine to me, though it doesn’t match this article or prevailing practice.

(2) Blackboard bold should be left alone


 * Response: previously the article used a mix of "blackboard" vs. regular bold for sets like $Z$ and $Q$ and Wikipedia uses regular bold versions of these extensively throughout Math articles. Just picking one of the two and sticking to it seems like an improvement to me.

(3) Supposedly $$\varphi$$ looks impossibly different from, $φ$, or $φ$, so the latter forms should be avoided/eliminated


 * Response: First of all, the latter form was previously used constantly throughout this article, alongside also $φ$, φ, φ, and ϕ. If readers were going to be confused it was already going to happen. But more to the point, in the 3 Wikipedia skins I tried across 3 browsers, $$\varphi$$ and $φ$ look approximately the same.

Cheers. –jacobolus (t) 23:33, 21 September 2021 (UTC)


 * I agree that the inconsistencies should be cleaned up. What I disagree with jacobolus on is how they should be cleaned up. Jacobolus favors templates for all inline formulas, and &lt;math&gt;-formatted only for display math. That does not work for formulas like $$\mathbb{Q}[\sqrt 5]$$ (where MOS:MATH explicitly says not to use unicode replacements for blackboard bold characters). It does not work for the extremely frequent use of varphi in the article, where (at least on my browser) $φ$ $φ$ (favored by Jacobolus) looks much narrower than $$\varphi$$ $$\varphi$$, likely confusing readers who might think that among multiple variant phi letters, these are two different ones rather than intended to be the same. The only way to get inline mathematics to look like display mathematics is to use &lt;math&gt; consistently for both, so my position is that in articles where display mathematics is used, we must use &lt;math&gt; inline as well in order to match it. And it does not work for square roots, where $\sqrt{5}$ $\sqrt{5}$ (used in many formulas including some introduced by Jacobolus) doesn't match up the top bar and radical part of the radical sign and just looks glitchy compared to $$\sqrt 5$$ $$\sqrt 5$$. For all of these reasons I think that this is not one of the articles with mathematics simple enough to be handled adequately by templates, and that we should use &lt;math&gt; throughout. —David Eppstein (talk) 23:49, 21 September 2021 (UTC)


 * Do other editors agree that all math throughout the article should be &lt;math&gt; tags? –jacobolus (t) 23:55, 21 September 2021 (UTC)


 * As an aside, what browser/skin are you using? In all the browsers I have here, $\sqrt{5}$ etc. look just fine. Maybe someone should go fix the radic template if it is glitching out. That sounds like a serious problem affecting thousands of articles –jacobolus (t) 23:55, 21 September 2021 (UTC)
 * Chrome, Monobook. These are not obscure choices of how to view articles. The radic issue is that, in $\sqrt{5}$ $\sqrt{5}$, the diagonal bar of the radical sign extends for a couple of pixels above the top bar, creating an extra lump where there should not be one. Wrapping it in a math template like $\sqrt{5}$ $\sqrt{5}$ (which you did not consistently do) fixes that issue but introduces new issues with vertical spacing: the top bar is a bit too high, and we also have in our article instances of $\sqrt{φ}$ $\sqrt{φ}$ where the phi is far too low below the top bar of the radical compared to $$\sqrt\varphi$$ $$\sqrt\varphi$$. —David Eppstein (talk) 00:41, 22 September 2021 (UTC)


 * Okay how is that looking then? I switched the article to (unless I missed some) uniformly use math tags, with the exception of the colorful letters in a caption. –jacobolus (t) 01:39, 22 September 2021 (UTC)
 * There's still an unformatted square root in the Tschichold quote, and I'm not sure we need math formatting for some but not all of the numeric values in the "Music" section. There's an unclosed tag that needs fixing in "Relationship to Fibonacci sequence". In "Symmetries", I'm not sure what the notation used to describe the identity is supposed to mean. And in "Mathematical pyramids", I would prefer not to use inline mathematics for a square root of an exponent; it makes the line too high and messes up the line spacing. But these are quibbles; I think it looks good now. —David Eppstein (talk) 01:53, 22 September 2021 (UTC)


 * I have no idea what the 'symmetries' notation is about. Maybe we can track down its original introduction and ping the editor who added it. Update: I am guessing it is supposed to be Permutation –jacobolus (t) 02:33, 22 September 2021 (UTC)


 * Is there a way to get $$\sqrt\varphi$$ not drop the square root sign quite so low? It also messes up the spacing of the line below. If it were bumped slightly up I think it would still look fine. But Wikipedia LaTeX doesn't seem to support most spacing tricks, e.g. vphantom. –jacobolus (t) 02:24, 22 September 2021 (UTC)
 * Wikipedia talk:WikiProject Mathematics/Archive/2020/Nov discusses this but the subscript-backspace trick from there doesn't seem to work with varphi. $$\sqrt{\varphi_{\!}}$$ $$\sqrt{\varphi_{\!}}$$. Superscript-backspace is no better: $$\sqrt{\varphi^{\!}}$$ $$\sqrt{\varphi^{\!}}$$. Instead, the overset-thinspace trick produces $$\sqrt{\overset{\,}{\varphi}}$$ $$\sqrt{\overset{\,}{\varphi}}$$ which looks better above but also messes up the line spacing above, and I'm not sure it fixes the spacing below. —David Eppstein (talk) 07:40, 22 September 2021 (UTC)
 * Probably okay to leave it slightly disrupting the following line then. –jacobolus (t) 08:39, 22 September 2021 (UTC)


 * Does my take on clarifying the cycle notation seem reasonable? –jacobolus (t) 08:38, 22 September 2021 (UTC)
 * Sure. —David Eppstein (talk) 16:19, 22 September 2021 (UTC)

Silveren Ratio
Why 1:φ ratio is so special? I mean, of course φ has its properties such as 1,618.. & 0,618...; but in everyday life only one of them is used, the one that allows to create a paper page (A format) with the same ratio upon dividing it in half on the longer side. But, you know what? a 1:√2 ratio has exactly the same property because √2/2:1 |*2/√2| = 1:2/√2 |*√2/√2| = 1:2√2/2 = 1:√2. — Preceding unsigned comment added by 31.182.229.99 (talk) 20:43, 30 December 2021 (UTC)


 * Yes, you are correct that if you cut the a piece of paper in half then $√2$ is the relevant value. The golden ratio comes into play if you instead cut off a square (that is width-by-width in size) from one end of a piece of paper.  — Q uantling (talk &#124; contribs) 21:27, 30 December 2021 (UTC)


 * Additionally there is the silver ratio, $1 + √2$. It is the relevant quantity when you cut off a rectangle (that is width by double width in size) from one end of the piece of paper.— Q uantling (talk &#124; contribs) 22:06, 30 December 2021 (UTC)

indent method
Even though display="block" adds too much white space, block indent completely cuts off text at the edge of the screen with no scrolling capabilities. Especially a problem on mobile screens. Text gets wrapped, but since is rendered as an image anything to the right of a boundary is lost Hellacioussatyr (talk) 23:25, 28 July 2022 (UTC)


 * Oh wait, I was mis-remembering. It seems I was the one who preferred the bi templates. Context: First I switched the article to consistent use of math templates (previously it was an inconsistent mishmash of ~4 different styles of math markup), but found the results ugly, so we instead agreed to switch to  tags. I used : indentation, but apprently that doesn’t play well with accessibility tools, so David switched them to  . I found the gobs of extra whitespace a bit too much (is there some other way to avoid those?), so switched to using the bi template instead for indentation. I have one other main issue with  (across Wikipedia): It often it adds boxes where some mistake in the markup/CSS results in mismatched box sizes creating unnecessary scroll bars, and then these end up hijacking the mouse scroll to scroll that individual box up and down (by a few pixels) instead of scrolling the whole page (not to mention adding unsightly useless scroll bars all over the place). I’m not sure who we should talk to to get that one fixed. –jacobolus (t) 23:51, 28 July 2022 (UTC)

Lucas numbers vs Fibonacci numbers
If you test for convergence for sequences 1,1 1,2 2,1 2,2; 1,2 will converge faster, as you are already one step along the Fibonacci sequence. 1,1 converges faster than 2,1. Therefore, bringing the concept of Lucas numbers, and giving a specific example starting 2,1 could be misleading as to suggest that Lucas series beginning with numbers other than 1,1 possess special properties not found in the Fibonacci number when it comes to dividing consecutive numbers and the golden ratio.

I suggest the section on Lucas numbers should be slimmed down, Lucas examples removed, and a note that: In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence 1,1. As you move along the Fibonacci sequence, dividing two consecutive numbers, the closer you move to the golden ratio. Lucas numbers begin with a base sequence other than 1,1 and will similarly converge towards the golden ratio. Nick Hill (talk) 11:42, 8 June 2022 (UTC)


 * The purpose of that section is to show that both the Fibonacci sequence and sequence of Lucas numbers together have a kin relationship with the golden ratio, particularly as evidenced by: their convergence to $$\varphi$$ of successive terms, the limit of their quotient for $$\sqrt5$$ that makes the golden ratio a fundamental unit of the algebraic number field of $$\sqrt5$$, their ability to produce close golden spirals with different square sizes, and their definition of $$\varphi$$ under $$({{L_n + F_n \sqrt{5}})/2}\, .$$ So I find the section appropriately lengthened and partitioned, personally, and I'm always open to any changes. In fact, I was going to add a tiny bit more Lucas-$$\varphi$$ related content. They also share independent properties with the golden ratio as mentioned - i.e. Lucas numbers round to powers of $$\varphi$$. I think it's more encyclopedic to include a wholesome amount of information about both sequences, and in relative equal amounts, with maybe slightly more oriented toward Fibonacci numbers. I'll try and see how I can make that more apparent. Thank you for your input Nick. Radlrb (talk) 08:30, 17 June 2022 (UTC)


 * To Nick: You have it backwards. The Lucas numbers are ahead of the Fibonacci numbers. Where Fibonacci has 1,1, Lucas has 1,3. And $$ L_n \approx \sqrt{5} F_n > 2 F_n > \phi F_n \approx F_{n+1} $$. JRSpriggs (talk) 23:45, 5 August 2022 (UTC)

Semi-protected edit request on 4 August 2022
In the section labeled Music, the third paragraph says, “ The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.[119] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[120]”

This is incorrect. In fact, Trezise’s actual quote is, “ The precise significance of these findings is hard to assess, but the intrinsic evidence is remarkable; there is, however, no material proof (letters, sketches, anecdotes, for example)  other than the finished works that Debussy consciously contrived such proportional schemes.

There is, in fact, a quote from Debussy himself, in Debussy’s published letters, titled ‘Debussy’s Letters’, in a letter to his publisher, referencing “the divine number”. I’m quoting Debussy word for word. This is taken from letter #108, to Jacques Durand: “I was just getting ready to send you the proofs…. You’ll see, on page 8 of ‘Jardins sous la pluie’, there’s a bar missing; my fault, in fact, as it’s not in the manuscript. Even so, it’s necessary from the point of view of number; the divine number,…”

I would suggest you amend the third paragraph, by quoting Debussy and adding a reference to the ‘letters’ book. There’s speculation that Debussy may have been influenced by a particular group of artists and poets in France he was spending time with. Regardless, that is a direct quote from his letters, and the piece of music mentioned has been analyzed for use of the golden ratio and checks out. If you’d like I’ll send you the findings in that score. I believe Roy Howat may have also discussed this in his book, ‘Debussy in Proportion.’ I would need to get a new copy of that book to verify, which I’d be happy to do.

You may contact me directly at email@stevesteele.com.

Regards, Steve Steele Andromeda Expat (talk) 05:37, 4 August 2022 (UTC)
 * Is it your contention that "the divine number" is intended as a reference for the golden ratio, rather than to the general concept of numbers? That seems a stretch to me. Far too little to base your proposed edit on. —David Eppstein (talk) 06:28, 4 August 2022 (UTC)
 * "... the divine number, as Plato and Mlle Liane de Pougy would say, though each for a different reason, admittedly." That sounds like he's talking about something more general. XOR&#39;easter (talk) 16:33, 4 August 2022 (UTC)
 * I would at least include it as a possible allusion to the golden ratio. The correct translation of "elle est nécessaire, quant au nombre; le divin nombre" in French, actually makes an allusion to a number. It should really be translated as "it’s necessary from the point of view of the number; the divine number". That is because in French, au is a contraction of à le which translates in English to a singular "the". To write in French "number" as in the general concept of all numbers, would actually be written in French as "aux nombres," with an x, which makes it plural - saying "au nombre", in high French, or colloquial French, would not be attributed to "number" as it would be in English - that would need to be written as "aux nombres". I am fluent in French and can distinguish that at least in this letter, Debussy is likely speaking of a particular number, whether that is the golden ratio or not, one cannot tell. However, from a linguistic perspective, and since he mentions Plato in particular, I would be inclined to believe that it is in fact an allusion to the divine proportion. Really, in French, to allude to numbers in general, would not be followed also by a singular "divine number". It makes more sense, to allude to a number being spoken of, which is then specified directly as the divine number, as this flows together from a singular grammatical article that alludes to a singular object. This is my take. Great find, thank you for sharing. Radlrb (talk) 18:16, 11 August 2022 (UTC)
 * If we do not have reliable sources EXPLICITLY saying that Debussy meant the golden ratio, and not some other number, we cannot use this quote. —David Eppstein (talk) 18:35, 11 August 2022 (UTC)
 * Not true, we can state that it is ambiguous. Plus, other authors have used this quote as alluding to the fact that it could very well be what was meant. Eric Frederick Jensen's Debussy on page 200 mentions this letter specifically, and says it is unclear. However, from a linguistic point of view, it's quite clear that the possibility is very real. And proper sourcing would demand that we remain impartial and at least relay that information as it was presented, especially since it is a point of contention. It makes sense to be fair and include both possibilities, instead of blankly rejecting such a valuable source altogether. Radlrb (talk) 18:43, 11 August 2022 (UTC)
 * We cannot include it at all. It is a primary quote, saying only something about a divine number. Any interpretation that it is connected to the golden ratio, without sourcing, is original research. For all we know, he is referring to the trinity and the "divine number" is 3. —David Eppstein (talk) 19:18, 11 August 2022 (UTC)
 * You do not understand - a person in a book wrote that it could mean the golden ratio. That is not original research. I'm not coming up with that. Radlrb (talk) 19:39, 11 August 2022 (UTC)
 * Which person in which page of which book analyzed this specific quote as referring to the golden ratio? It's not evident from the discussion above. —David Eppstein (talk) 19:42, 11 August 2022 (UTC)
 * Mariu Luvio writes, on page 191 in The Golden Ratio: "Since Debussy didn't say much about his compositional technique, we must maintain a clear distinction between what may be a forced interpretation imposed on the composition and the composer's real and conscious intention (which remains unknown). To support his analysis, Howat relies primarily on two pieces of circumstantial evidence: Debussy's close association with some of the symbolist painters who are known to have been interested in the Golden Ratio, and a letter Debussy wrote in August 1903 to his publisher, Jacque Durand. In that letter, which accompanied the corrected proofs of Jardins sous la Pltiie, Debussy talks about a bar missing in the composition and explains:  'However, it's necessary, as regards number; the divine number.'  The implication here is that not only was Debussy constructing his harmonic structure with numbers in general but that the "divine number" (assumed to refer to the Golden Ratio) played an important role."
 * Here Mariu Luvio is assuming that Debussy meant the golden ratio.
 * Eric Frederick Jensen's Debussy on page 200 (as I wrote on my previous comment), writes: "One Debussy scholar, Roy Howat, used bars as a basis and justified its selection by pointing to what he felt was a clue in a letter by Debussy. Looking through the proofs of Estampes, Debussy complained to his publisher that a measure --  'necessary from the point of view of number; the divine number, as Plato and Mlle Liane de Pougy would say, though each for a different reason, admittedly'  -- was missing. The tone of the letter is bantering, and the meaning unclear. But using Debussy's comment as a point of departure, Howat examined several of Debussy's compositions, including La Mer, to see if the golden section were present."
 * Here Jensen shows how what Debussy wrote in the letter is what instigated Howat to search for the golden section within his music.
 * So, naturally Steve's proposed addition would not be original research when these sources, as examples, are referenced, and would be meaningful to the section under Music in the article mentioning Debussy and his scores that have apparent golden sections. Also, it's directly related to what we are already are speaking of in the article, pertaining to Howat's analyses. Agree? Radlrb (talk) 20:01, 11 August 2022 (UTC)
 * This article by Jorge Variego actually summarizes the matter at hand eloquently, and provides the actual golden spiral Howat outlines in the music score: http://jorgevariego.com/?p=696. It also mentions the quote. It's generally a well referenced quote, it turns out. Radlrb (talk) 20:46, 11 August 2022 (UTC)
 * I don't know what kind of spiral that is supposed to depict but it is not the golden spiral. In the golden spiral the scale goes up by a factor of $$\varphi$$ every quarter-turn. In Variego's figure it is every half-turn. And Variego immediately notes that a different spiral from Hokusai is "admittedly broader than Debussy’s variety". To me this seems typical of the sort of fuzzy-headed mysticism we should be avoiding in this article: claiming that the logarithmic spiral shape is somehow synonymous with the golden ratio (they are two very different things). —David Eppstein (talk) 21:52, 11 August 2022 (UTC)
 * Okay. Obviously that is not the main point, and I in fact agree with you that the spiral does not directly look like a golden spiral, maybe it's one of the spirals incorporated, I don't know - however both are logarithmic spirals, I didn't really take a closer look at it because it does not read well on my computer.
 * Aside from that, you haven't answered my main question. I'll go ahead and make the edit, since the main point at hand is valid - it is sourced, from the two main sources I took the time to type and share with you. The other reference was to simply give background information. Anyways, it's a reference from an actual musician, clearly, who is not trying to fool anyone into anything, but rather make a point, regardless of whether that image was the correct image to incorporate, or not. If you really want to see all the golden proportions Howat found within Debussy's music then take a look at Debussy in Proportion: A Musical Analysis by him. I will rest now. Radlrb (talk) 21:55, 11 August 2022 (UTC)
 * An accurate description would be "some golden-ratio-loving scholars have picked at vague patterns in Debussy's compositions and even vaguer wording in quotes from Debussy in an attempt to find the golden ratio in his works. The results have been inconclusive". —David Eppstein (talk) 22:08, 11 August 2022 (UTC)
 * No. Radlrb (talk) 22:13, 11 August 2022 (UTC)
 * In any case six of the ten lines of the music section (for my browser width) are already devoted to Debussy, rather out of proportion. Expansion of this material seems likely to cause greater issues with balance. —David Eppstein (talk) 22:16, 11 August 2022 (UTC)
 * Then we add to the section in general, and the other sections like Nature, which is missing information. Can you think out of the box for once. Ugh. Radlrb (talk) 22:20, 11 August 2022 (UTC)
 * Your aversion and loathe toward the spiritual community is noted. You cannot fathom intelligent people taking their time and effort to unfurl truths that are against your paradigm. Mind you, all the greats in the world, for the most part, believed in the divine and in mathematical absolutism. Go figure, you call yourself an impartial editor. I feel sadness that you're so full of scorn and apathy. If Galileo didn't have the guts to look into the stars, we'd maybe be behind by an easy 500 years if someone like him didn't come around soon after, given that time compounds itself in proportion to its advancements; and no, Galileo was a genius, not just anyone was meant to look into the stars, which also included Hans Lipperhey, the first patenting inventor, amongst other contributors, all with the fortitude and fearlessness to go against the Christian status quo... for what a telescope could reveal. Radlrb (talk) 22:19, 11 August 2022 (UTC)
 * Mentioning Plato is not much of anything, since he had no evident direct connect with the golden ratio, according to Livio's book – his "Platonic solids" and his interest in noncommensurate numbers were precursors, but that's about it. Dicklyon (talk) 21:36, 13 August 2022 (UTC)
 * Can elaboration about this (including the quotation from Debussy’s letter and scholars’ speculations) go in a footnote? I agree that more than a couple of sentences of main text about this one piece of music starts to derail the article. –jacobolus (t) 16:03, 12 August 2022 (UTC)
 * Yes, that's one solution that would appease me and the original author of this thread, I believe. As far as for D.Eppstein, I don't know, and frankly, I don't care.
 * Do as you please jacobolus. That's already 3 people who approve the edit, making his wishes against the matter moot. I was thinking on rewriting that entire section on Debussy and include just 2 sentences. That's always an option, instead of adding sentences, much of the time if new information is added, the rest can be re-written accordingly, for flow and to not-reiterate points. Radlrb (talk) 17:58, 12 August 2022 (UTC)
 * All circles are golden spirals, and time is a type of width. DeBussy said that. Jar of room temp urine (talk) 05:21, 13 August 2022 (UTC)
 * Did he really? What a joker. Dicklyon (talk) 05:39, 13 August 2022 (UTC)
 * A footnote might be OK. But let's make sure it says what's actually supported by sources.  I sympathize a lot with Eppstein's point that this is just another random vague GR comment, not anything supportable. Dicklyon (talk) 05:39, 13 August 2022 (UTC)
 * Passing mentions of possible appearances of the golden ratio are not worth including. Frankly, given what's been offered, I find even a footnote unwarranted. XOR&#39;easter (talk) 16:25, 13 August 2022 (UTC)
 * You're probably right, but I'd review a specific draft proposal if someone makes one. Dicklyon (talk) 21:36, 13 August 2022 (UTC)
 * I'm not surprised this article is long but I am surprised it seems to be looked after pretty well. There's lots of mysticism about the Golden ratio and I'd expected either they'd have taken over completely and produced something extremely long and disjointed, or the fact brigade against fringe would be removing everything like that and just have maths and a few classical references. But it seems to cater for both sides reasonably by just requiring good references and not just passing mentions.
 * In that light I don't think Debussy's letters can be included unless someone in a reliable source puts forward the possibility that when he said divine number he meant the golden ratio. Without that it would fail WP:OR I believe. NadVolum (talk) 10:55, 14 August 2022 (UTC)
 * In fact I think some of the maths bits seem uncited and probably should be removed unless good citations can be found. NadVolum (talk) 11:01, 14 August 2022 (UTC)
 * True, though we've tried to keep it sourced and balanced (you should have seen it 15 years ago!), we may have gone easier on math things than on mystical/art/etc things. Put a cn tag on anything you think needs a source, and we can find one or remove it. Dicklyon (talk) 00:23, 15 August 2022 (UTC)
 * Please do tag them rather than just immediately removing. Some of the unsourced material may be original research (and therefore should be removed) but this is a topic where most of the basic mathematical information should be sourceable with some searching. —David Eppstein (talk) 00:48, 15 August 2022 (UTC)
 * Almost everything here is pretty straightforward to prove, and undoubtedly can be sourced to some published paper (or realistically, several). Before removing something, consider hunting for a source yourself. If something seems nontrivial and you can’t find a source, add a tag or bring it up on this talk page. –jacobolus (t) 07:38, 18 August 2022 (UTC)

Works Cited vs. Further Reading
Is there a reason for these two as separate categories? –jacobolus (t) 07:38, 18 August 2022 (UTC)
 * The further reading ones should not be listed as references because they are not used as references. So they need to be in a different section. —David Eppstein (talk) 16:24, 18 August 2022 (UTC)
 * Is there a reason the two categories can’t be consolidated though, under some other name? –jacobolus (t) 22:35, 18 August 2022 (UTC)
 * The standard section ordering in Manual of Style/Layout keeps them separate, under those names. Also, it is useful to distinguish works that are cited as references from other works that are not yet cited but could be. —David Eppstein (talk) 22:40, 18 August 2022 (UTC)
 * To my reading this is a (slight) misadoption of MOS:FNNR and MOS:FURTHER, and seems like an arbitrary separation between apparently similar books to someone who is just looking at the bottom of the page. The MOS says the references section(s) may contain any or all of the following: Explanatory footnotes ... Citation footnotes ... Full citations to sources, ... General references (full bibliographic citations to sources that were consulted in writing the article but that are not explicitly connected to any specific material in the article). This seems to perfectly well include everything currently in the "further reading" section of this article. To be honest I don’t get the point of a "further reading" section, for which it recommends a reasonable number of publications that would help interested readers learn more about the article subject.... This section is not intended as a repository for general references or full citations that were used to create the article content.... To me the separation into two sections seems not that helpful. It creates extra work for anyone adding or removing footnotes (need to check which section the source is in and move it if necessary) and potential confusion for readers (is one of the two types being more recommended than the other? Are there any model articles where both of these sections appear and seem to have particularly clear/helpful content and structure? –jacobolus (t) 05:43, 19 August 2022 (UTC)
 * As for model articles that do it this way, Leonhard Euler (a featured article, and recently reconfirmed as featured) seems as good an example as any. There are huge numbers of publications about Euler in some way (in that respect, similar to the golden ratio); the "Further reading" section is limited to book-length works directly about Euler that were not already cited in the article. —David Eppstein (talk) 06:29, 19 August 2022 (UTC)

Add EXACT connection with e, pi & logarithm!
$$\phi=e^{-0.2i\pi}+e^{0.2i\pi}=e^{-0.2\ln -1}+e^{0.2\ln -1}=-1^{-0.2}+-1^{0.2}=\frac{1}\sqrt[5]{-1}+\sqrt[5]{-1}$$

— Preceding unsigned comment added by 46.39.54.40 (talk) 18:44, 28 May 2022 (UTC)


 * You may have something there. However, beware that raising to a non-integer power anything other than a non-negative real number is generally not well defined.  In particular, there are five complex values $x$ for which $x^{5} = −1$. Yes, you can specifically indicate which one you mean in a number of ways, but $(−1)^{0.2}$ is not one of the ways to indicate that.  Similarly, $ln −1$ is not sufficient to uniquely indicate the value that I believe you mean to indicate.  — Q uantling (talk &#124; contribs) 18:32, 29 May 2022 (UTC)


 * Here we operate with the principal root, so these examples are obvious:
 * $$\varphi = 2\mathcal{Re}({-1}^{1/5})$$
 * $$\varphi = 2\mathfrak{R}(\sqrt[5]{-1})$$
 * $$\varphi = \frac{1}\sqrt[5]{-1}+\sqrt[5]{-1}$$
 * $$\varphi = e^{-{i\pi/5}}+e^{i\pi/5}$$
 * Moreover, the last form has the only one root due to its irrational power:
 * $$\varphi = e^{-{i\pi/5}}+e^{i\pi/5}\approx e^{-0.62831853i}+e^{0.62831853i}$$ 46.39.54.78 (talk) 18:50, 2 June 2022 (UTC)
 * Very nice. Is there a source we could use, I think we could add this to the article, maybe under a heading for other forms for phi, since we have several that might not fit nicely inside the headings we have. Radlrb (talk) 22:06, 4 June 2022 (UTC)
 * Well, these calculations based upon sophisticated mathematical equations. The source we could use is Wolfram|Alpha:
 * $$\varphi -e^{-{i\pi/5}} = e^{i\pi/5} = \sqrt[5]{-1}$$ (simplification via Euler's formula)
 * see "Alternate forms" section for proof:
 * https://www.wolframalpha.com/input?i=%CF%86-%28e%29%5E-%28ipi%2F5%29
 * or the same with prepended Euler's formula:
 * https://www.wolframalpha.com/input?i=%CF%86-1%2F%28%28-1%29%5E%281%2F5%29%29
 * And see "Result" section for proof of $$\varphi = 2*\mathcal{Re}(\sqrt[5]{-1})$$:
 * https://www.wolframalpha.com/input?i=%CF%86%2FRe%28%28-1%29%5E%281%2F5%29%29
 * or
 * https://www.wolframalpha.com/input?i=%CF%86%2FRe%28e%5E-%28i%CF%80%2F5%29%29
 * or
 * https://www.wolframalpha.com/input?i=%CF%86%2FRe%28e%5E%28i%CF%80%2F5%29%29
 * Finally for $$e^{-{i\pi/5}}+e^{i\pi/5} = \frac{1+\sqrt5}{2} = \varphi$$:
 * https://www.wolframalpha.com/input?i=e%5E%28-%28i%CF%80%2F5%29%29+%2B+e%5E%28%28i%CF%80%2F5%29%29
 * or
 * https://www.wolframalpha.com/input?i=%CF%86%2F%28e%5E%28-i%CF%80%2F5%29%2Be%5E%28i%CF%80%2F5%29%29
 * or
 * https://www.wolframalpha.com/input?i=%CF%86%2F%28%28-1%29%5E%28-1%2F5%29%2B%28-1%29%5E%281%2F5%29%29 46.39.54.78 (talk) 16:50, 11 June 2022 (UTC)
 * Wolfram Alpha is not usable as a reference. See Reliable sources/Noticeboard/Archive 264. You need to find an actual publication (presumably in a journal or a book) by an actual human saying something about the significance of these formulas. —David Eppstein (talk) 18:26, 11 June 2022 (UTC)
 * Indeed, somewhere we can see how this can be applied, plugged in for another value, or something else? Maybe you came up with it, which is still great, but to use it here it needs to be able to be cited, at least if it isn't well-understood knowledge. Radlrb (talk) 08:06, 17 June 2022 (UTC)
 * maybe no connection with logarhythm (and to messing with i), but apparently √e is colose to Φ. Φ÷√(e-0,1) is the closest, google can calculate, and equals to 0,99995267018... (which means that √e is slightly bigger than Φ) — Preceding unsigned comment added by 31.182.229.99 (talk) 22:02, 9 July 2022 (UTC)
 * Wolfram Alpha is not usable as a reference. See Reliable sources/Noticeboard/Archive 264. You need to find an actual publication (presumably in a journal or a book) by an actual human saying something about the significance of these formulas. —David Eppstein (talk) 18:26, 11 June 2022 (UTC)
 * Indeed, somewhere we can see how this can be applied, plugged in for another value, or something else? Maybe you came up with it, which is still great, but to use it here it needs to be able to be cited, at least if it isn't well-understood knowledge. Radlrb (talk) 08:06, 17 June 2022 (UTC)
 * maybe no connection with logarhythm (and to messing with i), but apparently √e is colose to Φ. Φ÷√(e-0,1) is the closest, google can calculate, and equals to 0,99995267018... (which means that √e is slightly bigger than Φ) — Preceding unsigned comment added by 31.182.229.99 (talk) 22:02, 9 July 2022 (UTC)

I made the bit about 5th roots of unity a bit more explicit in golden ratio. Does that cover what folks here were looking for? –jacobolus (t) 05:45, 23 August 2022 (UTC)

Alternate Expression
The formula for phi can be reduced to a positive exponent, multiplication, and addition:

This is clear and easy to remember, demonstrates interesting relationships, and is less obtuse than the formula as shown in the article. This can also improve computational efficiency (of admittedly marginal importance). Here is an example for javascript: I placed this under computation, but ‎Ovinus reverted it. Placing it here in the event someone else sees it as useful. Myndex talk  00:45, 23 August 2022 (UTC)


 * YMMV, but this doesn’t seem mathematically interesting to me or any easier to remember. You should generally use  in preference to   for practical computation, though it doesn’t really matter in this case because Javascript JIT compilers are smart enough to evaluate the constant expression one time and save the result then not bother repeating the computation. The definitional “formulas” in the article are $$\varphi^2 = \varphi + 1$$ or $$\varphi = \tfrac12(1 + \sqrt5)$$ and they do not seem “obtuse”. –jacobolus (t) 01:35, 23 August 2022 (UTC)
 * Maybe "obtuse" is the wrong word... this is in part something I am working on relating to "plain language" descriptions of math, wherein the math is laid out in as simple a manner as possible, and minimizing the need for symbols. This is not a simplification for math people, it is a simplification towards plain language.
 * What I felt was interesting is this is described verbally as "five to the power of a tenth of five times a tenth of five plus a tenth of five." Myndex  talk  03:46, 23 August 2022 (UTC)
 * If you want “plain language”, then the definition you are looking for is “the ratio of the diagonal to the side of a regular pentagon”. –jacobolus (t) 04:54, 23 August 2022 (UTC)
 * And for what it’s worth, the lead section could do a better job of explicitly mentioning this one. –jacobolus (t) 05:30, 23 August 2022 (UTC)
 * Also, pow runs slightly faster for me, (OS X) ... I see that it's close and sqrt may be faster on some environments, according to: https://www.measurethat.net/Benchmarks/ListResults/9063  Myndex  talk  03:56, 23 August 2022 (UTC)
 * No way pow runs faster than sqrt. ( / is a single instruction;   is usually a complex beast to get faithful rounding (e.g., ).) In any case, that's still OR, and I agree with jacobolus that it's no easier to remember. $$\tfrac{1+\sqrt{5}}{2}$$ recalls the beloved quadratic formula, while this formula uses unsightly decimals. Ovinus (talk) 04:45, 23 August 2022 (UTC)
 * Also, the speed of a machine-precision formula for phi is totally pointless because if you ever need this in a context where speed is relevant it will be a hardcoded constant or computed once rather than computed repeatedly. —David Eppstein (talk) 05:27, 23 August 2022 (UTC)
 * I linked to a page that runs hundreds of trials of both. It appears to be compiler and environment dependent, but is too close to consider either the preferred.
 * Is interesting because phi approximates the Fibonacci ratio starting with the 5th Fibonacci number pair. So I suppose only I find that interesting. Myndex  talk  08:46, 23 August 2022 (UTC)
 * Here is a benchmark with four variations: https://www.measurethat.net/Benchmarks/Show/20563/1/calculatephi splitting hairs for performance.... though on my machine, depends on browser, but pow version is slightly paster that sqrt   Myndex  talk  09:33, 23 August 2022 (UTC)
 * ...or not. Hmmm inconsistent... Myndex  talk  09:36, 23 August 2022 (UTC)
 * This benchmarking tool is entirely inadequate to answer your question, because the JavaScript JIT compilers are all smart enough to perform this computation once and save the result as a constant. So what you are measuring is some overhead of function calls and looping and so on, rather than your explicit arithmetic. If you want to perform JavaScript performance benchmarking you need to be a lot more careful about how you set up your test. (And please don’t waste your time on it; it will ultimately just show you that sqrt is faster or possibly the two will be the same speed if pow(x, 0.5) gets compiled down to sqrt(x) by a smart enough compiler.) –jacobolus (t) 19:07, 23 August 2022 (UTC)
 * This all seems like WP:Original research to me unless there is a reliable source with that formula in it. NadVolum (talk) 12:58, 23 August 2022 (UTC)
 * @NadVolum It is only a different expression. It is a restatement of the SAME formula.  Myndex  talk  15:14, 23 August 2022 (UTC)
 * I guess so but it did seem to be presented as something new and interesting of itself. NadVolum (talk) 17:03, 23 August 2022 (UTC)
 * phi approximates the Fibonacci ratio starting with the 5th Fibonacci number pair – this is starting to sound suspiciously like numerology. –jacobolus (t) 19:07, 23 August 2022 (UTC)
 * Oh now now, you know I am not going that way at all.  Myndex  talk  14:14, 25 August 2022 (UTC)
 * Oh now now, you know I am not going that way at all.  Myndex  talk  14:14, 25 August 2022 (UTC)

Adding new content to Applications and observations
Hello!

I hope my edits have been pleasant (and thank you to those who have pitched in to mine and prior), if you need me to add/remove or change anything I've done in a similar style, please let me know or feel free to further improve the article yourself. Since the mathematics content was expanded by a relatively large amount, I am going to also help update Applications and observations and Disputed observations to balance the length of Mathematics (maybe expand History, too). If you'd like to join me in this effort, please feel free, there is plenty more that can be added, almost everywhere. If you also have ideas on minor unmentioned golden ratio topics for the article under Mathematics that could be added inside a larger heading of Other properties let me know too, I can work on them if you don't feel inclined. I am going to expand Other properties with sections detailing:
 * $$\varphi$$ as a Pisot–Vijayaraghavan number and relationship to other metallic means and similar properties between them
 * $$\varphi$$ regarding the algebraic number field $$\mathbb{Q}(\sqrt5)$$
 * Golden ratio base mathematical properties for $$\varphi$$
 * Decimal expansion of $$\varphi$$
 * $$\varphi$$ in other geometric examples
 * Move Optimization under applications to Other properties and list as many such properties that can be added as tidbits, such as its low-discrepancy

That way we have a proper Other properties section that flows into Applications and observations. Thank you for reading and/or helping the article! Radlrb (talk) 13:11, 24 June 2022 (UTC)


 * "Other properties" should probably be broken into multiple sections. For more about golden integers/rationals, perhaps https://archive.org/details/numbertheoryinqu0000dodd/ would be a useful reference. –jacobolus (t) 05:49, 23 August 2022 (UTC)
 * That's what I meant when I listed these, as possible individual sections within "Other properties". That seems like a great source. Radlrb (talk) 21:24, 25 August 2022 (UTC)

Straight phi vs. Curly phi
Is there any reason why this article shouldn't be changed from using the mathematically-rare curly phi $$\varphi$$ (LaTeX \varphi) to the historically-accurate straight phi $$\phi$$ (LaTeX \phi)? D.keenan (talk) 04:46, 14 September 2022 (UTC)


 * Not that I know of. Someone changed a bunch of phis to curly and so I just followed along. If you want to change all of them to straight phis, that is fine with me. JRSpriggs (talk) 17:47, 14 September 2022 (UTC)
 * There's not really a semantic difference here; it's more like the question of whether you are using a lowercase "a" with only a small tail on the bottom right like $$a$$ or with a big handle over the top like (at least in my Wikipedia rendering preferences) a. Straight seems to be a little more popular in publications but curly is the form preferred by the MathVault Compendium of Mathematical Symbols for what that's worth. I have a slight preference for curly because it's more visually distinctive from a capital phi but not a strong opinion. The vertical and heavyweight html &phi; should be avoided; the article currently uses it in the caption for the dodecahedron coordinate image. —David Eppstein (talk) 19:25, 14 September 2022 (UTC)
 * Both forms are common (for the golden ratio and other purposes) and neither is more or less "historically accurate". Some sources prefer one or another, and occasionally sources use the two symbols for separate variables. –jacobolus (t) 00:55, 15 September 2022 (UTC)

Semi-protected edit request on 14 October 2022
Please allow me to edit this page.I am currently a college student and i have spotted a gramatical error SoopBruv (talk) 08:22, 14 October 2022 (UTC)


 * Where? Please include more details here and I'll fix it. UNITE TOGETHER, STRIVE FOR SURVIVAL! 08:27, 14 October 2022 (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 and provide a reliable source if appropriate. ScottishFinnishRadish (talk) 09:19, 14 October 2022 (UTC)

Semi-protected edit request on 27 October 2022
Justification for change: There is no reason, to my understanding, that content that is not sky-is-blue obvious should be allowed to appear, unsourced, without a reader warning. To do so violates WP:VERIFY and/or WP:ORIGINAL RESEARCH. (As a former professor of the physical sciences, I state without reservation, the content of this section is not obvious, rather, it needs to have been taught to the editor placing it, and therefore is derived, ultimately, from some authoritative source.) The "Calculation" section certainly must appear so-derived in a wide variety of reputable sources. Find an autoritative source, edit the content to that source, and place the citation. Until then, I ask the following reader warning edit, in keeping with clear WP policies and guidelines.

Change from:
 * ==Calculation==

Change to:
 * ==Calculation==

If rejecting this request, please state the applicable WP policies and guidelines that trump the clear statements in WPVERIFY and WP:ORIGINAL RESEARCH. Thank you. 2601:246:C700:2:8D90:7012:872:558C (talk) 20:01, 27 October 2022 (UTC)


 * The citation to WP policies and guidelines is WP:CALC, part of policy. This section is purely elementary-school-level algebraic manipulation + quadratic formula. —David Eppstein (talk) 20:33, 27 October 2022 (UTC)
 * I fully anticipate that that the local populace here will reject this request, given the general status of this and other articles with regard to WP:VER and WP:OR. (Because the early work on Maths articles went so far in the direction of personal knowledge and OR over compliance with WP:VER, the articles are beyond the pale with regard to being verifiable content.) Just understand that as long as there are no sources, or no reader warnings until sources appear, we find the material presented as completely unusable for beginning maths students. (They may consult it on their own, but they may not cite it, because they cannot source their arguments and calculations from the WP article.)
 * Then, to begin a rejection of this request relying on the WP:CALC subsection, which states "[b]asic arithmetic, such as adding numbers, converting units, or calculating a person's age, is almost always permissible" appears disingenuous, and further, irrelevant to the content of this irrational number-containing calculation describing positive and negative roots of a quadratic. Granted, "[m]athematical literacy may be necessary to follow a 'routine' calculation, particularly for articles on mathematics or in the hard sciences", but providing a derivation that applies a simple array of algebraic steps, even just requiring that level of literacy, does not mitigate the need to establish who with authority has presented this derivation, as it is now persented. You know as well as I, identifying whose derivation is presented is a scholarly expectation, broadly speaking.
 * No, material here is either sky-is-blue (which this is not), or it sourced, or it is plagiarised, or it is original. That fact that you, as a Prof of CS, and I, as a prof in another phys sci, can follow it, and are sure it is true, is not the test given us. The test is that it appears published somewhere, so a reader or other (non-expert) editor can verify it. (And the suggestion that this is "purely elementary-school-level" suggests that you have never been involved very generally in such a level's curricula or teaching — meaning, not with a gifted student, but with a classroom of all-comers — and that's simply not an accurate basis to allow the status quo to continue.)
 * The "As a former professor..." remains my evaluation of the actual content and its level — my teaching spans from inner city fourth-tier institutions, to Ivy Leagues, with the bulk at a Big Ten — even if it makes more work for WP:WikiProject Mathematics. That is, my request remains. (As esteemed is the editor, the foregoing rapid argument does not address the relevant audiences, or needs, of this article.) 2601:246:C700:2:8D90:7012:872:558C (talk) 21:17, 27 October 2022 (UTC)
 * I've added some citations. However, note as you did the following text in WP:CALC: Mathematical literacy may be necessary to follow a "routine" calculation, particularly for articles on mathematics or in the hard sciences. In some cases, editors may show their work in a footnote. This calculation is decidedly "routine", in the context of this sentence of CALC (assuming mathematical "literacy", a higher standard than what you suggest). What is certainly not routine: medium-sized proofs, rarely used or opaque techniques, judgments on the nature of proofs (simplicity, beauty). To be honest, I've never seen people using footnotes for work, although I think it's a nice idea.
 * I understand where you're coming from. Citations are always nice to have, yet they are of limited use to the subset of people who are reading this article but also do not understand elementary-to-high-school algebra; after all, few people have access to JSTOR. It might help educators. Next time, I'd suggest you just provide some sources which support the content, and they can be incorporated smoothly. And/or, make an account! Ovinus (talk) 22:04, 27 October 2022 (UTC)
 * articles are beyond the pale with regard to being verifiable content – You can certainly find examples that push (or exceed) the boundary of WP:CALC, but this is not one of them.
 * you ... and I ... can follow it, and are sure it is true, is not the test given us – Yes it is, more or less. More concretely, anyone with a reasonable background (introductory middle school / high school algebra, and a reasonably careful effort) can follow the steps here; if your undergraduate technical students are having a problem following this line of reasoning something is going very wrong with their background preparation. Adding a source is not going to make basic algebraic manipulations easier to follow for someone who has not yet been through a year or two of introductory algebra courses, but teaching elementary algebra can’t be expected of every technical article, even those aimed at a broad audience. You can trivially find some sources for these specific manipulations if you need to (e.g. by skimming through a few of the cited books about the golden ratio), but even if you couldn’t, these are well within the scope of WP:CALC for this article. –jacobolus (t) 02:09, 28 October 2022 (UTC)
 * I should add here, the specific ratio manipulations here long pre-date algebraic notation. You can find them in Euclid and many (many!) other sources. For a comprehensive history, you can look at Herz-Fischler (1998) https://archive.org/details/mathematicalhist0000herz/ jacobolus (t) 02:30, 28 October 2022 (UTC)
 * Red information icon with gradient background.svg Not done for now: please establish a consensus for this alteration before using the template.  —  Paper9oll  (🔔 • 📝)  14:25, 29 October 2022 (UTC)