Talk:Golden ratio

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The value of φ[edit]

The value of φ with 2000 digits

Section d'Or[edit]

In the article it is mentioned: "However, despite this general interest in mathematical harmony, the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition did not use the golden ratio itself in their compositions." (Livio 2003, p. 169)

However, upon close examination, the painting reproduced below by Albert Gleizes, exhibited at the 1912 Salon de la Section d'Or (catalogue no. 40), most conspicuously did employ the golden ratio. The proportions of the canvas correspond exactly to the golden rectangle (a ratio of 1 to 1.618 ± 0.01). The probability this is a chance occurrence would be exceedingly small. This work has a rare dimension of 105 x 171 cm. Gleizes, as most artists at the time, generally used standard format chassis (stretchers), which are not golden rectangles:

Albert Gleizes, 1912, Les Baigneuses (The Bathers), oil on canvas, 105 x 171 cm, Musée d'Art Moderne de la Ville de Paris, golden rectangle (painting, 1 to 1.618 ± 0.07), golden ratio grid (golden/yellow grid, so1 - so4), and 4 : 6 ratio grid overlay

Unfortunately, observers have long attributed the golden ratio to works where nothing of the kind was used (arbitrarily placing the grid or spiral in the preferred section of the composition), and when it was used, it's not alway easy to identify. Here however, the fact is, the canvas itself corresponds to the golden rectangle. I have sent an email to a Gleizes expert (Peter Brooke)[1] on this topic who responded: "it wouldn't surprise me if Gleizes would have experimented with the section d'Or, indeed it would rather surprise me if he didn't." So too, "Gris and Villon". Because Liovio's statement is likely inaccurate (and he provides no verifiable source to back his claim), it should be removed from the article, and replace with something more inline with, "the extent to which these artists engaged in the such mathematical formalism is not known, or difficult to discern..." If need be, I can search for a another source that states something similar. Coldcreation (talk) 08:32, 30 November 2018 (UTC)

The golden rectangle frame size in that one is convincing enough. And it's easy to believe that some artists of that time started deliberately using golden ratios in their work. But throwing a spiral on top and saying "see how the features line up" (when they don't) is unhelpful pareidolia. See XKCD for more examples that are equally (not) convincing. —David Eppstein (talk) 18:58, 30 November 2018 (UTC)
It fits quite nicely flipped horizontally too. Coldcreation (talk) 19:33, 30 November 2018 (UTC)
Yes, the spiral fits nicely into the rectangle, but that has nothing to do with the painting. And if you can't even tell which way up the spiral goes, you might consider that to be evidence that the artist did not use the spiral as a compositional principle. —David Eppstein (talk) 20:54, 30 November 2018 (UTC)
A new version will be rolling out shortly. Coldcreation (talk) 21:25, 30 November 2018 (UTC)
The 1912 compositions of Juan gris, according to art historian Christopher Green, were often "modular and regular... easily fitted to the demands of Golden Section composing in the pictures of the summer, such as Man in a Café and The Watch." The "synthetic and analytic were visibly fused. In the Golden Section paintings... he laid the grids like systems of fault-lines across things, faut-lines on either side of which view-points switch.[2][3][4]
Juan Gris, 1912, Man in a Café, oil on canvas, 127.6 x 88.3 cm, Philadelphia Museum of Art. Exhibited at the 1912 Salon de la Section d'Or
While Gris' Man in a Café canvas/stretcher format and ratio do not correspond to a golden rectangle, an overlay clearly does match multiple golden ratio points, lines and intersections of the composition. I have examined, but not yet uploaded the Golden ratio overlay version of this painting. I will if anyone is interested in seeing the close relationship. Evidently, several works at the Section d'Or exhibit were based on the ratio; incorporating it within the standard stretcher formats at their disposition, and personalized golden section formats. Gleizes, as seen above, made a concerted effort to make sure his canvas was equal in proportion to the golden ratio. (See references below) Coldcreation (talk) 10:55, 30 November 2018 (UTC)
Juan Gris, 1912, Still Life with Flowers, oil on canvas, 112.1 x 70.2 cm, Museum of Modern Art, New York

This early 1912 painting by Juan Gris, Still Life with Flowers (above), also exhibited at the salon, corresponds precisely to the golden rectangle (a ratio of 1 to 1.618 ± 0.01), measuring 112.1 x 70.2 cm. Not only that, there are elements of at least two grids that line up with parts of the structural composition, one is a 6/4 grid, the other varieties of golden sections (from the the outer stretcher, and from the inner painted frame). I can upload the golden ratio overlay version if need be. This stretcher was likely custom built to the golden rectangle ratio, since, again, it was not a standard format.

These observations, and those referenced above and below, disprove Livio's claim that "the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition did not use the golden ratio itself in their compositions." (Livio 2003, p. 169) Coldcreation (talk) 15:38, 30 November 2018 (UTC)


  1. ^ Peter Brooke, private communication
  2. ^ Christopher Green, ‎Juan Gris, Whitechapel Art Gallery, London, 18 September - 29 November 1992 ; Staatsgalerie Stuttgart 18 December 1992-14 February 1993 ; Rijksmuseum Kröller-Müller, Otterlo, 6 March - 2 May 1993, Yale University Press, 1992, pp. 37-38, ISBN 0300053746
  3. ^ William A. Camfield, Juan Gris and the Golden Section, Art Bulletin, 47, no. 1, March 1965, 128-34. 68
  4. ^ David Cottington, Cubism and Its Histories, Barber Institute's critical perspectives in art history series, Critical Perspectives in Art History, Manchester University Press, 2004, pp. 112, 142, ISBN 0719050049

This entire topic would by now be better served by writing an analysis for another venue that could, eventually, in turn benefit Wikipedia. We spill over into WP:OR every which way. Of the cases mentioned so far, the most interesting (in terms of print sources that can actually be evaluated) may be Juan Gris. The references appear to be originally dependent upon Camfield (1965), who is at least cautious and careful to distinguish mathematics from metaphor. His placement of the golden section at La Section d'Or is rather tentative:

A few of these paintings were moreover based on simple geometrical compositions. Yet, not a single artist there displayed a serious commitment to geometrical proportions with the one exception of Juan Gris. [...] Although all of the Puteaux artists were interested in mathematics (Marcel Duchamp, interview with the author, April 4, 1961), neither Marcel Duchamp (ibid.) nor Jacques Villon, who suggested the title for "La Section d'Or" (Dora Vallier, Jacques Villon, Paris, 1957, pp. 60-62), believes that the golden section was actually used in their paintings. And in Du Cubisme (1st English translation), London, 1913, p. 64) Gleizes and Metzinger chastise those painters who would rely on mathematics for certitude.

Camfield says that neither of the paintings he analyses "can be definitively identified with "La Section d'Or" paintings", only that the "stylistic evidence" places them in that period and they "would almost certainly have been shown" there. So, it is really a question of what later observers have claimed about the art, and we may have to content ourselves with the "not known, or difficult to discern" kind of language. XOR'easter (talk) 18:22, 30 November 2018 (UTC)

Good points. I'm in the process of gathering more sourced information. I'll be back. Coldcreation (talk) 18:35, 30 November 2018 (UTC)
In Du "Cubisme" it was argued that Cubism itself was not based on any geometrical theory, but that non-Euclidean geometry corresponded better than classical, or Euclidean geometry, to what the Cubists were doing: "If we wished to relate the space of the [Cubist] painters to geometry, we should have to refer it to the non-Euclidean mathematicians; we should have to study, at some length, certain of Riemann's theorems." In fact, they were free do overlap multiple methods including the golden section (or non at all). That is what some of them they did. Coldcreation (talk) 18:46, 30 November 2018 (UTC)
Quoting the Cottington book (p. 112),
It will be remembered that Du "Cubisme", written probably as these paintings were being made, gestured somewhat obscurely to non-Euclidean concepts, and Riemann's theorems; as Linda Henderson has shown, these references betray not an informed understanding of modern mathematics but a shaky hold on some of their principles, culled (indeed plagiarised) from Henri Poincaré's La Science et l'Hypothèse. The authors themselves had little clear idea of how such mathematics related to their art, except as a vague synecdoche for "modern science".
As mathematicians, they were great artists. XOR'easter (talk) 21:23, 30 November 2018 (UTC)
Today their 'equations' are on the expensive side. Coldcreation (talk) 23:18, 30 November 2018 (UTC)

───────────────────────── Stepping back a bit for perspective, I think it's important to record what Livio has to say about this topic, since his book is one of the standard references — possibly the treatment of the subject at the popularized or semi-popularized level. The fact that such a work takes a position itself makes that position worth noting. We're outlining the history of an idea, not merely giving the "final word" on it. XOR'easter (talk) 00:44, 1 December 2018 (UTC)

Fair points, but Livio apparently did not bother to measure the paintings, or analyze their geometric structural elements. (Instead, he seems to have relied on what others had written on the topic, without citing his sources). Had he done so, he would have seen that some of the paintings featured in the 1912 Salon de la Section d'Or exhibition did in fact use the golden ratio itself in their compositions. Theses artists were more than just interested in the golden ratio, some actually experimented with it to varying degrees. From the literature, it is evident that, notwithstanding, the Groupe de Puteaux felt no need to limit themselves to any such formula. Mentioning Livio's view in the article will only serve, at best, to show how capable the Cubists were at dissimulating their strategies; at worst, to highlight Livio's mistake. That said, Wikipedia readers might be well served to have access to this information. And the best place for elaborating on it would likely be in the article dedicated to it: Section d'Or. As for this article, the reader now has enough information, and more if needed. Coldcreation (talk) 06:54, 1 December 2018 (UTC)
Another point about Livio's inability to 'definitively identify' the works shown at the 1912 show (thus leading to his erroneous claim). From the titles, dates and previous exhibitions listed in the 1912 Salon de la Section d'Or catalogue, many are easily identifiable, e.g., Les Baigneuses (The Bathers) and Le goûter (Tea Time), and many more works by Gleizes, Metzinger and others. The main exception is for the works of Juan Gris, since no titles are given for his submissions in the catalogue. However, it is now known, from published correspondence between the artist and the dealer Léonce Rosenberg that 13 paintings by Gris were shown (Green 1992), most of which have been identified by their titles, dates, and dimensions. Coldcreation (talk) 07:16, 1 December 2018 (UTC)


I realize that there are many supposed golden ratio attributions that lack rigor or are simply false. Is it possible to identify the golden ratio in a figure that is not, say, a golden rectangle, e.g., a square, as shown here, or some rectangle between the two? Coldcreation (talk) 07:43, 1 December 2018 (UTC)

The obvious answer is that it's the ratio between side and diagonal of a regular pentagon (the main reason the Greeks studied it). —David Eppstein (talk) 07:45, 1 December 2018 (UTC)
My objective here is to be able to determine if a painting has been constructed with the golden ratio in mind, or not, regardless of its format, e.g., Juan Gris' Man in a Café (shown above). Coldcreation (talk) 09:06, 1 December 2018 (UTC)
The better question is, once you overlay a grid of some sort on an artwork of some sort, what is your criterion for determining how good a fit it is? And where is your null hypothesis? —David Eppstein (talk) 18:15, 1 December 2018 (UTC)
Here is an example:

Here, shown side by side rather than overlaid (I have not uploaded the overlay version) it is clear that Braque used the Golden rhombus in the design of this 1917 painting. The match is exact (assuming the Commons image is correct). The angles and sides should line up to within a margin of error (± 0.01): equal to (a close approximation) the width of the edges of the painting hidden by the frame, and/or cropped off in the photo. The stretcher of Braque's painting is a golden rectangle (ratio of 1 to 1.618 ± 0.02). The ± 0.02 is attributed to frame overlap/cropping. That is the criteria. I found plenty of Null hypotheses (I tested about 50 paintings today, not just from the 1912 exhibition), until I tested the 1917 Braque. It actually surprised me that Braque used rhombus, as he was not a member of the Section d'Or (though from 1919 they exhibited together at Rosenberg's gallery l'Effort moderne). Coldcreation (talk) 20:34, 1 December 2018 (UTC)

Georges Seurat's, 1887–88, Parade de Cirque (Circus Sideshow) is an interesting case study:
The golden section (golden/yellow grid, so1 - so4) does not govern Parade's geometric structure. Modern consensus is that Seurat never used the 'divine proportion'. Parade is shown here divided horizontally into fourths and vertically into sixths. The 4 : 6 ratio corresponds to the dimensions of the canvas (one-half times wider than its vertical dimension). The ratio of Seurat's painting/stretcher corresponded to a ratio of 1 to 1.502, ± 0.002 (as opposed to the golden ratio of 1 to 1.618). The compositional axes in the painting correspond to basic mathematical divisions (simple ratios that appear to approximate the golden section). Coldcreation (talk) 09:47, 2 December 2018 (UTC)

I don't understand the point of this section, nor what you have done. See WP:NOTFORUM, item 4, which says "bear in mind that article talk pages exist solely to discuss how to improve articles; they are not for general discussion about the subject of the article, nor are they a help desk for obtaining instructions or technical assistance". Dicklyon (talk) 16:22, 2 December 2018 (UTC)

Actually, it's a subsection of the previous. In the article it was mentioned: "...despite this general interest in mathematical harmony, the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition did not use the golden ratio itself in their compositions." (Livio 2003, p. 169) Several sources contradicted that statement. While examining some of the works exhibited, mentioned in the latter references, it was confirmed that indeed Livio's claim was erroneous. This subsection was set up to test more works (those that were not constructed in a golden rectangle format). The question has been resolved. The Art section of the article has now been improved, with a statement that reflects the actual 1912 developments. Coldcreation (talk) 16:31, 2 December 2018 (UTC)
Thanks for making it a subsection to clarify. I still don't understand what the section is for. You seem to be presenting your original research, while may be interesting, but not really relevant here. Why don't you get it published in a journal, and then let us know? Dicklyon (talk) 19:33, 2 December 2018 (UTC)
We had several sources with conflicting statements regarding the use (or not) of the golden ratio in 1911-12 works. Livio, nor Camfield, could identify the paintings exhibited. Others have identified the works. They are well-known works. I posted a couple of them above. The dimensions correspond to golden rectangles. We now know, and have conveyed in the article, the correct statement. Before, we had no way of knowing which source was correct. While some original research to verify the sources was involved here at Talk, reliable sources back the current claims in the article. The Seurat info above is well documented. So all is well and good. Coldcreation (talk) 19:44, 2 December 2018 (UTC)
Got it. I'm all for representing both sides in such disagreements, as long as we don't do our own OR and editorializing about which source is correct. I'll try to catch up on what's happening... Dicklyon (talk) 19:55, 2 December 2018 (UTC)

This section got wiped out in the massive rollback to the Nov 1 version. I have a moderately copy-edited version on my draft under 'Disputed claims'. Before it was just under Art with no subsection. UpdateNerd (talk) 10:05, 27 January 2019 (UTC)

Wholesale changes[edit]

@UpdateNerd: – With no discussion, this editor has made about a hundred edits, reinstating a lot of the kind of POV that we took years to weed out. For example, he writes "The Great Pyramid of Giza (c. 2560 BC) has properties which closely approximate the golden ratio, and the Babylonian Tablet of Shamash (c. 888–855 BC) can be superimposed with two orders of golden rectangles", referencing a golden ratio hype book. This is unnecessarily misleading, suggesting as it does that a few observations of ratios near phi might mean something. I reverted one day's changes, but he put it all back and carried on with more. How can we deal with this rate of flux? Is anyone up for giving these changes the critical review they need, or should we roll back and take it slowly as I had suggested? Dicklyon (talk) 00:26, 12 January 2019 (UTC)

I haven't studied the changes but the lead should not define the golden ratio as a number then detour with observations about self similarity. I'm inclined to roll everything back and proceed slowly, one section at a time, with the lead last. Slowly means that edits to one section should occur, then wait three days for responses before starting on another section. Bold editing is great, but rolling everything back is also a bold edit and will happen if it has consensus. Johnuniq (talk) 00:57, 12 January 2019 (UTC)
Well, I'm happy to walk away from working on the article while other editors vet it, but my only request is that this be done constructively, by bringing up new/existing reliable sources and reflecting what the majority of them say; not rolling back a lot of sourced work indiscriminately. Look at the actual references; many of them are linked. If you want something in particular removed, e.g. the Tablet of Shamash, why not start a discussion about that one point? Rolling everything back is unproductive. I used Livio as my main source and the other refs bring up points he never addresses, but can easily be verified by the reader. Please let's discuss this one topic at a time, and make consensus changes, rather than making wholesale accusations and disruptive rollbacks. UpdateNerd (talk) 01:37, 12 January 2019 (UTC)
I'm inclined to roll everything back and proceed slowly, one section at a time, with the lead last. Sounds sensible to me. (I find that intros and leads and abstracts are, often, good things to do last. One has to establish the content to be summarized before attempting the summary. I know this isn't how everyone writes, but it seems the appropriate method to follow here.) Cheers, XOR'easter (talk) 18:11, 12 January 2019 (UTC)
Per WP:STATUSQUO: "Reverting is appropriate mostly for vandalism or other disruptive edits. The Wikipedia edit warring policy forbids repetitive reverting. If you see a good-faith edit which you feel does not improve the article, make a good-faith effort to reword instead of reverting it." If you wish to dispute material recently added to the article you should do so with consensus. Or, if you see material added by User:UpdateNerd that could be improved, be bold, and improve it. Coldcreation (talk) 18:35, 12 January 2019 (UTC)
"reinstating a lot of the kind of POV that we took years to weed out" sounds like disruptive editing to me. And your link is an essay describing some editors' editing philosophy; it is not policy or even a guideline. I think a more appropriate editing philosophy is to roll back edits that are clear disimprovements, regardless of the good faith of the editor, and not put the onus on cleaning up someone's mess on someone else. I think the number of edits involved is significantly greater than 100; the last version prior to UpdateNerd's revisions appears to be this one. —David Eppstein (talk) 20:05, 12 January 2019 (UTC)
If my edits had been "reinstating a lot of the kind of POV that we took years to weed out" I would agree with reverting them! But nothing I added was copied from an old version, it was directly sourced. For now, I've removed the controversial information from the lead and History section; I think this way the improvements that I've made can be compared to old versions without appearing POV. Implications that the golden ratio was intentionally included in pre-Pythagorean objects gives the perception of POV, but that was not my intent. It's fair not to include them since enough RS question their intentional use. If anyone's looking for the validity of ancient objects with golden ratio properties, they can look for the information in the more detailed section or article without having it emblazened in the golden ratio's lead. Please, before reinstating any old work, compare how the individual references are being used. Making claims with missing page numbers, or misrepresenting what they say is WP:SYNTH and another form of POV. UpdateNerd (talk) 20:59, 12 January 2019 (UTC)

─────────────────────────@David Eppstein: This is Wikipedia:Editing policy: Instead of removing content from an article, consider: WP:FIXTHEPROBLEM. Coldcreation (talk) 00:05, 13 January 2019 (UTC)

The material you link is about material that is appropriate but merely badly written. But in this case "the problem" is reinstatement of debunked claims about pyramids and off-topic material about whether the Egyptians knew the Pythagorean theorem, and the like, and the correct solution to the problem is bulk removal. The simplest way of doing that would be to revert to the last clean version. —David Eppstein (talk) 00:32, 13 January 2019 (UTC)
Another aspect of the problem is that it will be a monumental effort to find all the errors, with edits done in a way that the diff tool and undo are hard to use. Going back to his very first edit, the very first word he inserts is wrong. Should I fix that and move to the next one? Maybe it's not even there in the present version. If I work from the present instead, there nothing to guide the inspection of this huge complicated article, so that's not likely to take care of the problem either. It's best if we proceed incrementally, vetting changes made at a moderate pace, rather than having to contend with this sort of wholesale rewrite via hundreds of ad-hoc steps. Dicklyon (talk) 00:36, 13 January 2019 (UTC)
At some point the quoted phrase "extreme and mean ratio" was removed; now it appears only in a quote and in the lead in a way that's disconnected and meaningless. Dicklyon (talk) 00:52, 13 January 2019 (UTC)
It appears in the Euclid quote; in what other context is it more relevant? The Kepler triangle? UpdateNerd (talk) 01:36, 13 January 2019 (UTC)

Here is the diff that I propose we revert. It starts with an in appropriate hatnote, continues through an ill-advised lead rewrite ("the golden ratio is an irrational number represented by the Greek letter ..." doesn't tell you anything useful or interesting), includes a wrong definition of golden spiral in a figure caption, and then gets harder to understand from there as not much lines up. If there are improvements in it, we can work on finding them and putting them back. Perhaps some of the ref detailing. It looks like 4 us of agree this is the best course of action. Dicklyon (talk) 00:58, 13 January 2019 (UTC)

He's also got Kepler connected to art in the lead, but not in the body, and I don't find anything in Livio or elsewhere supporting that. And did we really need a second image of Kepler's solar system model? Dicklyon (talk) 01:13, 13 January 2019 (UTC)

I don't see why we need even the first image. The connection to the golden ratio is indirect and unexplained. —David Eppstein (talk) 01:19, 13 January 2019 (UTC)
I fixed the Kepler & golden spiral misphrasing in the lead & replaced the first solar system image with a simple dodecahedron, which produces less ambiguity. I really think that this process of pointing out the issues and working to fix them is more productive than a rollback. Please continue to raise concerns and I'll provide what info I can to address them. UpdateNerd (talk) 02:11, 13 January 2019 (UTC)

The pyramid info sourced to Maor and referencing the Rhind papyrus also got dropped and replaced by speculation. Dicklyon (talk) 01:21, 13 January 2019 (UTC)

The Maor ref never had a page number, and I couldn't find the info in the source that it was supposed to represent. I don't see how the Rhine Papyrus was ever relevant, but the Pythagorean theorem notes certainly are in the context of the Kepler triangle/related pyramid. Everything else is sourced, not speculation. UpdateNerd (talk) 01:34, 13 January 2019 (UTC)
It is not difficult to find pirated copies of Maor. And with a copy in hand it is also not difficult to determine that the appropriate page number range is 7–9. It is relevant because it provides evidence against the use of the golden ratio, by explaining the Egyptian pyramid shape in a different way that fits much better with what we know of Egyptian mathematics. —David Eppstein (talk) 02:14, 13 January 2019 (UTC)
According to my email records, I bought a copy in 2006, which is when we started a long hard de-bullshitting phase on this article. I went to the NY Public Library to look at books about the Rhind papyrus, too. Sorry I didn't get the page numbers in at that time, but UpdateNerd, you can ask; don't remove refs for lack of understanding their details. Dicklyon (talk) 04:54, 13 January 2019 (UTC)
Page numbers help. The most relevant information I can find in the article citing Maor is that the 3:4:5 triangle is described in the Rhind papyrus... but which source claims that the Great Pyramid of Giza might be based on that triangle, or that it is a "nearly similar pyramid shape", as asserted in the Nov. 1 version of the article? The reinstated version of the article synthesizes a bunch of poor citations such as this offline personal site, while ignoring the very informative Bartlett. UpdateNerd (talk) 06:15, 13 January 2019 (UTC)

The proposal is to revert all edits back to this permalink: 00:17, 2 November 2018. The diff of edits since then was given by Dicklyon but it is too complex to digest. Instead, I put the two versions (then and now) side-by-side and scanned them. The original is superior in several ways, starting with the lead and Calculation section which tell the reader what the golden ratio is. The current version moved Calculation into the Mathematics section which has a consistency appeal but which fails to define the topic somewhere near the top of the article. Other adjustments of that nature weakened the article IMHO. I support a revert. People wanting the new version need to justify it, section-by-section. WP:FIXTHEPROBLEM is not relevant as explained by David Eppstein above. Johnuniq (talk) 01:58, 13 January 2019 (UTC)

I've restored the calculation to the lead (minus the footnote of the derivation of -0.618, which I feel is overly detailed and should remain only in the body). Please feel free to improve it if you can. UpdateNerd (talk) 02:17, 13 January 2019 (UTC)

I've done the revert. UpdateNerd, I hope you'll help us identify and retrieve the good parts of your edits. Dicklyon (talk) 04:48, 13 January 2019 (UTC)

I don't think so, as I attempted to improve the article in the public eye for anyone to individually revert/revise/add to. I'm happy to continue discussing which part of my revisions were in error, perhaps in a draft, and reinstating it from the status-quo revision you rolled back to once common ground can be found. Thanks UpdateNerd (talk) 05:15, 13 January 2019 (UTC)

Working through the changes[edit]

I added back the Livio refs to the history from you first edit. At second edit, I'm stumped already. It's not clear why Plato's divided line analogy is worth a mention, as it works for arbitrary ratios; nothing relevant to golden ratio there. Also not clear why so much name unlinking in that edit, or why you go to the trouble of asking for nowrap on hyphenated terms. Mixing so many issues in an edit makes it hard on us. I'll skip that one and move forward. Dicklyon (talk) 17:28, 13 January 2019 (UTC)

I know of no other ratio Plato's analogy is true for. It's relevant because his definition precedes Euclid's. UpdateNerd (talk) 20:11, 13 January 2019 (UTC)
This reply makes no sense to me, and seems symptomatic of the kind of fuzzy thinking that got us into this mess. First, what mathematical fact is expressed in Plato's analogy that might be true for some ratios and false for others? And second, in what way is it more true for the golden ratio than for simple rational (but unequal) ratios like 1:2? —David Eppstein (talk) 20:21, 13 January 2019 (UTC)
It's a simple but not "fuzzy" concept. A line with the golden cut has parts proportional to each other in the same ratio as the larger is to the whole; it is self-similar. That is not true of the 1:2 ratio. It is not self-similar in any way that I can think of. UpdateNerd (talk) 20:27, 13 January 2019 (UTC)
There is nothing in analogy of the divided line about comparing the part to the whole. It is only about comparing one subdivided part to another subdivided part with the same proportions. And your attempt to view this as an example of self-similarity is extremely anachronistic; that's a 20th-century concept. —David Eppstein (talk) 20:34, 13 January 2019 (UTC)
You're correct on the misattribution. I think Plato describes a similar concept in the Republic about geometrical self-similarity, but not in his Analogy, which is a metaphysical model. I'll see if I can find the other quote, and sorry for the confusion (I should have caught this lapse in citations). UpdateNerd (talk) 20:57, 13 January 2019 (UTC)
The self-similarity quote is from Timaeus; if I can find a source linking it to the history of the golden ratio, I will share it. UpdateNerd (talk) 21:40, 13 January 2019 (UTC)
It doesn't look like there's anything there either. Sounds more like the usual fan hype of trying to connect golden ratio to everything (which is always possible, of course). Dicklyon (talk) 22:31, 13 January 2019 (UTC)
I agree with the StackExchange user who posted that question — it's hard to see the golden ratio anywhere in that passage. To me, it looks like a statement about the conditions for arithmetic and geometric means being equal, maybe garbled a bit by a translator not knowing enough math (though I can't be sure about that). XOR'easter (talk) 23:57, 13 January 2019 (UTC)

Your third edit suggests that the name tau went out as the twentieth century came in. This is not so; it just gradually became less common. So I'll skip that one, too. In this edit you remove all the &dq= info from book-search links, which removes highlighting that can be very useful for readers looking for the cited content; is there consensus that this is a good direction to go? Let me know. Dicklyon (talk) 17:37, 13 January 2019 (UTC)

I tend to prefer only the &pg= parameter in Google Books links (so that the preview goes directly to the chosen page), omitting &dq and other parameters. But I don't know of official guidelines on that issue. The {{Google books}} template does have a parameter corresponding to dq, suggesting that it's ok to continue including it. —David Eppstein (talk) 18:51, 13 January 2019 (UTC)

This edit added the sentence, The ratio of Lucas numbers to Fibonacci numbers appears to converge to , which is strongly related to the golden ratio. There are two problems with this. First, we shouldn't say "appears to": either , or it doesn't. There are oodles of relations between Lucas and Fibonacci numbers, and this statement can actually be proven, so if it's worth including, we need to make clear that it is not an unresolved conjecture. Second, "strongly related" is vague. Are all numbers that can be obtained from by an integer addition and a multiplication "strongly related" to it? Or, are we reserving "strongly related" for something more specific yet esoteric, e.g., is "strongly related" to if is a fundamental unit in the field ? I'm not convinced it's significant enough to warrant inclusion, but I wouldn't strongly object, provided it were rewritten for clarity.

To be honest, the preceding sentence isn't that great, either (I don't know how long it has been in the article). It doesn't make clear why working in the field is meaningful, as opposed to working in . XOR'easter (talk) 19:59, 13 January 2019 (UTC)

That reflects the reference, but by "appears to" it means that it does so increasingly, beyond doubt, but how exactly this would be 'proven' isn't in the ref. UpdateNerd (talk) 20:11, 13 January 2019 (UTC)
Indeed, that ref has little to say, and the "strongly related" is a jump. At your next edit you elided a digit, change pp. 86–87 to pp. 86–7. This is really not OK. Dicklyon (talk) 22:11, 13 January 2019 (UTC)


I've worked on a draft which incorporates various feedback. You can see what's changed in the edit history. Notably, I moved the Egyptian pyramids to 'disputed claims' and removed some of the irrelevant history. I'm happy to address any other feedback for a potential "goal" article. Also, I think it's simpler to look at than stepping through individual edits, particularly after weeding out some false/disputed claims. I'm happy to slowly reintroduce changes for public scrutiny, but if you wish to help in the planning process, please take a look. Thanks UpdateNerd (talk) 08:22, 18 January 2019 (UTC)

UN, that 2014 Chistopher Bartlett article is a sham. In getting from "most reliable" measurements to exact golden ratio he says "I began by taking the averages of the most reliable published data from three highly respected surveyors/researchers, Cole (1925), Dorner (1981) and Lehner (1997)." Here the Dorner and Lehner numbers are modern and identical, and the old Cole numbers, off from these by more than a part per thousand, are just about what's needed to make the part per 2000 to 3000 (one to two minutes of arc) correction to the modern data needed to arrive at the remarkable result. The Herz book is a more more objective/unbiased analysis of actual data and theories instead of cooked averages. I suggest we don't use Bartlett at all (do we even have reason to trust a word he says? who is he? Affiliation is listed as "Hunt ValleyUSA"), or if we use it, presenting it as this guy's claims, not as you have with a statement of factoid like "Based on the most reliable measurements of the Great Pyramid of Giza (c. 2560 BC), the slant height of one of its faces divided by half its base width (semi-base) is φ". That's all I've looked at so far Dicklyon (talk) 19:04, 19 January 2019 (UTC)
And your draft is based off the version we all agreed to get away from, so looking at diffs there is not going to be useful. We need to work forward from where we are, slowly. It's hard to see what changes you think are most important. If it's page numbers, I'l happy to work with you on that; I have Herz and others. Since the entire Herz book is what we're summarizing, it's not clear page numbers are really needed, but I can find some if you like. I'm not very familiar with how to convert the ref style to accommodate page numbers, though; if you can stub in new-style refs with dummy numbers to show me how, I'll put in actual numbers. Dicklyon (talk) 19:11, 19 January 2019 (UTC)
Ah, here's how you did it. I can copy that. Dicklyon (talk) 19:15, 19 January 2019 (UTC)
Yeah, that was pretty much my take-away from the Bartlett (2014) article, too. XOR'easter (talk) 20:25, 19 January 2019 (UTC)
I agree that any inclusion of Bartlett should attribute the claims more specifically, probably using direct quotes. But Bartlett does have some useful information even if the claims of the first paragraph are generally disputed (he argues for alternatives further into the article). Since it's now under disputed claims, it's not placing Bartlett as more important than the discreditation of related claims; we should simply present all the views with the appropriate balance.
The draft has many other improvements, like focusing on sourced info, not synthesized claims, and has a well-sourced History section instead of a flimsy list. It's also more logically organized into parent sections. I'm listening to all the feedback and eliminating problems to make it better than the version that was reverted and fix problems on the live article. You can also edit it yourself if you wish. UpdateNerd (talk) 21:30, 19 January 2019 (UTC)
The live version of the article is plagued with reference issues, which now include stray "</ref>" tags and missing refs that were removed. This type of editing should be avoided, but I also insist that more be done to verify that the functional refs actually support what is being claimed. Not including page numbers because "the whole book says that" is WP:SYNTH, as I've already said, and I hope my draft can be useful in fixing that. If there are issues as to its arrangement, I'd be happy to discuss how it could be improved. Per feedback, I made the Calculation section more prominent again. UpdateNerd (talk) 07:54, 20 January 2019 (UTC)
I apologize if I messed up on trying to improve the refs. I'd be happy to have your help. But we need to proceed from where we are. It's hard to see how to leverage your separate branch to do that. Dicklyon (talk) 17:16, 20 January 2019 (UTC)
OK, I fixed the ref errors. Got suggestions for what's a next good step or two? Dicklyon (talk) 18:01, 20 January 2019 (UTC)
I suggest merging the redundant Timeline section with History so we can start seeing what's missing from the prose content. Lists like the timeline should be avoided when equally good or better prose can be written. UpdateNerd (talk) 22:23, 20 January 2019 (UTC)

The same pattern of edits that got us into this mess is continuing, with UpdateNerd going back to the same pattern of making dozens and dozens of consecutive edits to this article, many of them innocuous, others that turn something that used to make sense into a garbled mess of disconnected half-thoughts. UpdateNerd: Can you explain clearly here in talk what you are trying to accomplish? Because to me it looks like you have no plan. I think you should stop making these edits until you have one. —David Eppstein (talk) 08:42, 27 January 2019 (UTC)

Well, it's fine that you reverted my recent edits, because the entire point of the recent rollback was so that edits could be reviewed on a case-by-case basis. I'm doing my best to make the minimum number of edits that are able to be easily compared. This has nothing to do with what "got us into this mess", because you intervened this time. If I make all the changes I envision in a single edit, then it's harder to see what changed. UpdateNerd (talk) 08:47, 27 January 2019 (UTC)
It would help if you rather than making many changes all over the article or providing a new draft for the complete article, simply were to post the new content and sources you want to have added as short sections. Then merit of those can be discussed in a focussed fashion. But a large number of changes who overall improve little and contain occasional problematic stuff, will at some point just trigger categorical resets by other editors and increasingly hostile attitudes (as such edits cause a lot of work for others maintaining the article), in other words you won't be able to contribute that way.--Kmhkmh (talk) 09:43, 27 January 2019 (UTC)
Based on the feedback, I will make a greater effort to keep my BOLD edits innocuous, and bring anything more controversial up for discussion. It might also generally help if others focus on fixing problems; you can use my draft to grab some of the reference page numbers which are still missing, or information which clears up some of the WP:SYNTH issues. UpdateNerd (talk) 09:49, 27 January 2019 (UTC)

Egyptian pyramids[edit]

For anyone interested, I've incorporated some of my more recent finds concerning Röber's discussion of Egyptian pyramids to the draft, and copyedited the rest for simplicity. I think this helps put Bartlett et al. into context, and clears up some of the misinformation. Incidentally, I haven't seen any modern historian sources saying that the Great Pyramid doesn't have golden ratio properties, only that it's the result of chance. Hopefully we can build up a quality argument and address any outstanding issues on the draft so we can replace the sketchy version that is currently being synthesized. UpdateNerd (talk) 11:33, 27 January 2019 (UTC)

I was able to find that Taylor indeed misrepresented Herodotus; Livio traces this error. I also reworded my version so that it's not actually saying that φ or the Kepler triangle appear in the Great Pyramid, only that they are closely approximated—according to the cited estimates. I also added a Livio quote that affirms that the Egyptians had no knowledge of the golden ratio, and even re-added the Rhind Mathematical Papyrus as the source of pi's fractional approximation. Please let me know your thoughts, as I think it's now much clearer to a first-time reader... without misrepresenting sources! UpdateNerd (talk) 09:08, 28 January 2019 (UTC)

I don't like that section at all to be honest, imho the whole pyramid thing only deserves a bullet point under disputed claims.--Kmhkmh (talk) 19:11, 11 February 2019 (UTC)
I like the bullet point idea since it doesn't merit anything but a disputed claim, and those are about a paragraph in length. I'm now just using the draft as a repository and skipping stuff not explicitly related to the golden ratio—I've updated the paragraph on Egyptian pyramids if you care to see a condensed approach. UpdateNerd (talk) 20:53, 11 February 2019 (UTC)
I'm not so sure. There's been so much writing on this speculation that some discussion may be in order. Dicklyon (talk) 04:31, 12 February 2019 (UTC)
I'm all for discussion. One thing I eliminated from the draft is the mention of mathematical phrases which would be unclear to the common reader. E.g. "secant of the angle θ" or "face angle 52° 20". (What does the second number in an angle represent?)
I've also updated the draft to sample what listing the disputed claims as bullet points would look like. UpdateNerd (talk) 06:28, 12 February 2019 (UTC)


In response to this revert and request for explanation: all I did was make Section D'or a bullet point for consistency with the rest of the section as well as remove some unclear/irrelevant statements which seem to be copy-pasted from the subpage—having nothing to do with the golden ratio. The changes related to the Parthenon were also to clean up some of the off-topic nonsense and put it in its due place under 'Disputed observations', as throwing it under Architecture (on the golden ratio article) is misleading. Let me know if that clears it up. UpdateNerd (talk) 06:59, 21 February 2019 (UTC)

THe golden ratio is also 2×sin(54°) or 2×cos(36°). — Preceding unsigned comment added by (talk) 08:47, 5 March 2019 (UTC)


Albrecht Dürer - Melencolia I - Golden Ratio.jpg

@Deacon Vorbis: The source clearly states that the vanishing point divides the diameter of the rainbow in the golden proportion, and that could also not be more apparent from the actual measurement. Since the illustration in question is quite literally full of riddles/mathematical puzzles, it would seem highly unlikely that the vanishing point underlying the entire composition would be placed at this location by accident. The article is chock-full of other reliable information, so at least with attribution, it would be appropriate to include. And as far as I'm aware Dürer didn't leave notebooks of his intentions lying around (as Da Vinci did). UpdateNerd (talk) 13:43, 2 May 2019 (UTC)

(edit conflict) The source says "divine proption", not "golden proportion" or even "golden ratio". While I realize that this term is sometimes used instead, the author never did so anywhere else in the source, despite using "golden ratio" five times explicitly. In any case, I'm very hesitant to put in a single "this one person says this one painter did something in the golden ratio this one time" kind of thing, even if the source is otherwise more or less reliable. It starts to get kind of WP:CRUFTy for one thing. But also, if you're specifically looking for φ, and you make enough measurements, you're going to find it all over the place – a point which is made elsewhere in the article even. Without some kind of corroboration or quote from the artist or something providing some context, we can't just WP:INDISCRIMINATEly include these. –Deacon Vorbis (carbon • videos) 14:15, 2 May 2019 (UTC)
Considering there was an edit conflict, did you see the image I mocked up? Also, "divine proportion" explicitly refers to the golden ratio in pretty much all contexts, expecially during the Renaissance when Pacioli (whom the article is primarily about) advocated for calling it that. Yes, it is a single source making an extraordinary claim, so if we include it we should attribute it to "one source" or something like that. There isn't likely to be a bunch of articles repeating such esoterica on the interweb, but if there was, that would make the guideline-stringest case for including this much stronger... but likely without adding any insight. But I don't see how your cruft/indiscriminate arguments apply. This is particularly discriminate on what to include (a rare strong-case use of the golden ratio during the Renaissance, of which there are only 1 or 2 other highly significant examples—currently missing from the article). I agree it would help to find corroboration, but making the point so obvious might not have been the artist's intent. :) UpdateNerd (talk) 14:42, 2 May 2019 (UTC)
Granted about the terminology, but that was the lesser of my concerns really. As for the image, using lines so close together is going to introduce extra uncertainty in determining the vanishing point. Even in the author's diagram, where he used a couple more, one of them didn't intersect the others, yet there was no indication of how much uncertainty there was in his measurements, just a blanket "omgz golden ratio!". Even in your diagram, I see that you've used the outside of the rainbow, rather than the inside. Why? Why not the center? Using one of those will doubtless give a better match than the others. How much other subtle little fudging is made in these sorts of measurements?
Could the artist have done this intentionally? Of course. Did he? Who knows; the author certainly doesn't make any sort of case that he did. And if it was unintentional, then it shouldn't be included. We need to stick to cases where there's some sort of corroboration about intentional use, or at the very least some sort of robust discussion about it, rather than including any questionable case of finding something that was specifically being sought, too obscure to have attracted any other attention. –Deacon Vorbis (carbon • videos) 15:17, 2 May 2019 (UTC)

Ahmes papyrus[edit]

While I agree that a letter to editor is not the most reliable source per se, it replicates the quoted portion of the papyrus included in Alger's Mathematics for Science and Engineering, which doesn't make any other outlandish claims related to the golden ratio. It mentions in the same section that the golden ratio was not known until Euclid, which could be added as a quote in the reference (I'll post it when I have time tomorrow). Including the information in the article gives insight as to the more scholarly views that have existed about the golden ratio and the pyramids, without saying that they're true. UpdateNerd (talk) 04:01, 6 June 2019 (UTC)

Please refrain from using bad sources. This article has been infested with woo too much already, you don't need to make it worse. Seked is apparently just a word for the slope of a pyramid; different pyramids had different slopes, and the imputation that the slope is the golden ratio is modern. As the more respectable sources already say. Including this information does not give more insight; it merely makes it look like Wikipedia editors are credulous and unscholarly. —David Eppstein (talk) 05:10, 6 June 2019 (UTC)
I didn't realize they were referring to seked. I figured the translation could have been a weasly one (although slopes derived from two side measurements aren't totally unrelated from ratios). There is nothing wrong with pointing out different views—perhaps mentioning that that's the word transliterated seqt—but I don't see a need for that now. UpdateNerd (talk) 05:59, 6 June 2019 (UTC)

Rotating black holes[edit]

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'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 φ; 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 • 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 unless there was also some sort of explanation linking a defining property of φ to the result, rather than just something akin to the Strong Law of Small Numbers rearing its head. –Deacon Vorbis (carbon • 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 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 ; I'd call it a curiosity, except that it turns out to be not very curious. XOR'easter (talk) 23:18, 26 July 2019 (UTC)
Oh, and The Golden Ratio: The Divine Beauty of Mathematics is not a reliable source. XOR'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'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 is commonplace. See Planck units. XOR'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 and simultaneously to unity, along with Boltzmann's constant , 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 , and if you're really an intense theorist, you set as well. XOR'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'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'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)