# Talk:Golden ratio

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## Straight phi vs. Curly phi

Is there any reason why this article shouldn't be changed from using the mathematically-rare curly phi ${\displaystyle \varphi }$ (LaTeX \varphi) to the historically-accurate straight phi ${\displaystyle \phi }$ (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 ${\displaystyle a}$ 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 [1] 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 φ 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)
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==
{{unsourced section|date = October 2022}}

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)
Not done for now: please establish a consensus for this alteration before using the {{Edit semi-protected}} template. Paper9oll (🔔📝) 14:25, 29 October 2022 (UTC)

## Le Corbusier Modulor quotation

I can't find anything very closely matching the quotation in this article in Le Corbusier's The Modulor, a translation of which can be found at the internet archive (along with Modulor II) here: https://archive.org/details/moduloriii00leco/ even if I try pretty loose keyword searches under the assumption that the French original was being translated differently. Can anyone find the relevant passage (in English or French) or make the citation more precise? –jacobolus (t) 21:56, 29 November 2022 (UTC)

## modular group

I think the section about the modular group has the kernel of something interesting/meaningful to it, but seems like it was previously incorrect, had no sources, and I'm not quite sure what it should say so I am temporarily removing it.

Here was the previous text:

The golden ratio and its conjugate ${\displaystyle \varphi _{\pm }={\tfrac {1}{2}}{\bigl (}1\pm {\sqrt {5}}{\bigr )}}$ have a set of symmetries that preserve and interrelate them. They are both preserved by the fractional linear transformations ${\displaystyle x,1/(1-x),(x-1)/x}$ – this fact corresponds to the identity and the definition quadratic equation. Further, they are interchanged by the three maps ${\displaystyle 1/x,1-x,x/(x-1)}$ – they are reciprocals, symmetric about ${\displaystyle {\tfrac {1}{2}},}$ and (projectively) symmetric about ${\displaystyle 2.}$ More deeply, these maps form a subgroup of the modular group ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$ isomorphic to the symmetric group on ${\displaystyle 3}$ letters, ${\displaystyle S_{3},}$ corresponding to the stabilizer of the set ${\displaystyle \{0,1,\infty \}}$ of ${\displaystyle 3}$ standard points on the projective line, and the symmetries correspond to the quotient map ${\displaystyle S_{3}\to S_{2}}$ – the subgroup ${\displaystyle C_{3} consisting of the identity and the ${\displaystyle 3}$-cycles, in cycle notation ${\displaystyle \{(1),(0\,1\,\infty ),(0\,\infty \,1)\},}$ fixes the two numbers, while the ${\displaystyle 2}$-cycles ${\displaystyle \{(0\,1),(0\,\infty ),(1\,\infty )\}}$ interchange these, thus realizing the map.

This seems incorrect to me. Specifically, while ${\displaystyle \varphi }$ and ${\displaystyle -\varphi ^{-1}}$ are fixed by the identity and exchanged by the map ${\displaystyle x\mapsto 1-x,}$ they don’t seem to be fixed or exchanged by the other maps listed there. If we call these maps

${\displaystyle a:x\mapsto {\frac {1}{x}},\quad b:x\mapsto 1-x,\quad c:x\mapsto {\frac {x}{x-1}}}$

then we have ${\displaystyle a(\varphi )=\varphi ^{2},b(\varphi )=-\varphi ^{-1},c(\varphi )=\varphi ^{-1}.}$

I’d like to get someone who is an expert here to explain what part of this is meaningful, interesting, relevant, etc., and ideally link some sources. –jacobolus (t) 02:48, 14 January 2023 (UTC)

This was added in February 2010 by Nbarth, and then has persisted since then without anyone ever really modifying it. Nbarth, maybe you can explain where you got this, what you were getting at, etc.? –jacobolus (t) 03:05, 14 January 2023 (UTC)
Overall if we apply these three maps to ${\displaystyle \varphi }$ we get 6 elements in total (as expected for something isomorphic to the dihedral group ${\displaystyle D_{3}}$). Arranging these elements in order alternating with the (projectively) equally spaced elements ${\displaystyle 0,{\tfrac {1}{2}},1,2,\infty ,-1}$ for context, we have:
${\displaystyle 0,\,\varphi ^{-2},\,{\tfrac {1}{2}},\,\varphi ^{-1},\,1,\,\varphi ,\,2,\,\varphi ^{2},\,\infty ,\,-\varphi ,\,-1,\,-\varphi ^{-1},\,0,\,\ldots }$
This doesn’t seem especially noteworthy unless it can be related to other subjects, ideas, or theorems. If anyone cares about this we can try to make a figure showing the relation of these values projected onto a circle with ${\displaystyle 0,1,\infty }$ equally spaced. I'm sure we could make it pretty to look at anyway.
From looking at modular group a bit, it does seem like the Hecke group ${\displaystyle H_{5}}$ is based on ${\displaystyle \varphi .}$ Maybe someone who is an expert can write something about that here? –jacobolus (t) 03:55, 14 January 2023 (UTC)
Sorry, I made a sign error when writing this! The points that are preserved by the 3-cycles of the anharmonic group are ${\displaystyle e^{\pm i\pi /3}=(1\pm {\sqrt {3}}i)/2}$, the solutions to ${\displaystyle x^{2}-x+1}$ (the primitive sixth roots of unity), not the golden ratio ${\displaystyle (1\pm {\sqrt {5}})/2}$, which are the solutions to ${\displaystyle x^{2}-x-1}$. I made this mistake in Projective linear group [2], then copied it to Golden ratio in [3]. I have fixed the original error now too ([4]).
There's no relation to the golden ratio; this is just about the cross ratio and projective linear group; see those for interesting (and correct!) details.
Thanks for catching this and asking me!
—Nils von Barth (nbarth) (talk) 05:39, 14 January 2023 (UTC)
Thanks for clearing that up. I'm amazed nobody else ever looked carefully at this paragraph after almost 13 years, despite millions of page views.
There is something at least a little bit interesting about the 6 elements generated from ${\displaystyle \varphi }$ by the maps ${\displaystyle x\mapsto 1-x,\,x\mapsto x^{-1}}$ being all powers of ${\displaystyle \varphi }$ or their negatives,
${\displaystyle \varphi ^{-2},\,\varphi ^{-1},\,\varphi ,\,\varphi ^{2},\,-\varphi ,\,-\varphi ^{-1}.}$
If instead we apply the maps generated by (square dihedron symmetry, corners at ${\displaystyle 0,1,\infty ,-1}$) ${\displaystyle x\mapsto -x}$ and ${\displaystyle x\mapsto (1-x)/(1+x)}$ to starting element ${\displaystyle \varphi ,}$ we get the set:
${\displaystyle \varphi ^{-3},\,\varphi ^{-1},\,\varphi ,\,\varphi ^{3},\,-\varphi ^{3},\,-\varphi ,\,-\varphi ^{-1},\,-\varphi ^{-3}.}$
I didn’t really try to figure out what Hecke groups are, but that probably is probably more relevant still, meriting some discussion on this page.
@Nbarth when looking at this I found that it was helpful to explicitly draw the projection of the projectively extended real line onto a circle (the inverse stereographic projection centered at ${\displaystyle {\tfrac {1}{2}}}$). I think that makes it easier to see what is going on than File:PGL2_stabilizer_of_3_points_on_line.svg which draws them in a straight line (P.S. you may want to edit that image description).
That other page may also benefit from a reference to a source or two if you can track them down. –jacobolus (t) 07:17, 14 January 2023 (UTC)

jacobolus Oops, good catch about the image description! Fixed in [5]; thanks!

The pattern you pointed out seems interesting; the numbers ${\displaystyle \pm \varphi ^{k}}$ are the units in the ring of quadratic integers with the golden ratio, ${\displaystyle \mathbf {Z} [\varphi ]}$ (see Quadratic integer § Examples of real quadratic integer rings, Golden ratio base, and Golden ratio § Other properties), so maybe there's something going on with these symmetries of them?

I tried seeing if there was anything obviously interesting for the Hecke group ${\displaystyle H_{5}}$, or the corresponding triangle group (2, 5, ∞), but this isn't my expertise, and Google didn't return anything promising.

A better diagram would be welcome, but I don't have the time (or probably artistic skill ;) – I was just writing a quick schematic, which hopefully gives some geometric insight to a mostly algebraic point. I suspect that a circle might be bulky (due to needing to label the point in the middle, as well as several points on the circle), and take up a lot of space on the page. Good for a book on complex geometry, but a bit distracting for a small point in these articles. —Nils von Barth (nbarth) (talk) 20:37, 14 January 2023 (UTC)

If cross-ratios of the six permutations of four points are mapped by a Möbius transformation onto a circle, they have the geometric symmetries of a regular triangular dihedron.
@Nbarth: I mean something like the image shown here to the right. (You’ll have to figure out the clearest / most accurate way to write this caption though [e.g. pointing out that these reflections also swap the interior and exterior if the circle]. And feel free to edit the wiki commons description page if you want to use this image.) –jacobolus (t) 21:36, 14 January 2023 (UTC)
jacobolus Thanks, that's very helpful! I added your diagram (and my explanation) to cross-ratio in [6], and updated the file description in [7] (initially as triangular bipyramid, but you're right that trigonal dihedron is more correct; also added a note about the point at infinity).
It's a bit big, but appropriate to clarify the geometry, thanks!
—Nils von Barth (nbarth) (talk) 17:01, 16 January 2023 (UTC)
Looks good. Let me know if you want any changes to the diagram. –jacobolus (t) 22:51, 16 January 2023 (UTC)
Regarding the diagram, there's one serious problem: the involutions are not reflections - they are rotations! Rotations by 180 degrees, most obviously because they are complex maps, so orientation-preserving, but also because they don't fix ${\displaystyle e^{i\pi /3}}$ (as the reflections would, but instead switch it with ${\displaystyle e^{-i\pi /3}}$! Thus I think it's important to replace the double-headed arrow with a "rotation symbol" like ↺. (I only noticed this when looking at it closely.)
Perhaps useful too would be to putt a point for ${\displaystyle e^{-i\pi /3}}$ in the corner "at infinity", but I leave that to your discretion (it's a bit confusing because it should be on all the lines of symmetry). We could also mention it in the caption instead? —Nils von Barth (nbarth) (talk) 03:37, 27 January 2023 (UTC)
The reason I drew them as double-headed straight arrows instead of more obvious rotation-y looking symbols was that I made the picture with Desmos and that was a lot easier to draw. https://www.desmos.com/calculator/knthhtclga But I could probably open it up in Adobe Illustrator to make a symbol that looks more properly like rotation. I’ll see what I can do tomorrow. –jacobolus (t) 04:39, 27 January 2023 (UTC)
@Jacobolus Thanks for explaining! Understood that it's easier, but it's pretty misleading; hopefully easy to just add a semicircle arrow, which should convey it? —Nils von Barth (nbarth) (talk) 04:09, 29 January 2023 (UTC)

## Octagrammum Mysticum and the Golden Cross-Ratio

(Split to other section.) —Nils von Barth (nbarth) (talk) 03:41, 27 January 2023 (UTC)

@Nbarth: While we are here, you may be interested in (and we may want to mention in this article): Evans, L. S., & Rigby, J. F. (2002). Octagrammum Mysticum and the Golden Cross-Ratio. The Mathematical Gazette, 86(505), 35. doi:10.2307/3621571, jstor:3621571. –jacobolus (t) 05:05, 26 January 2023 (UTC)

That's a pretty result, but a bit "tangential" to the content of golden ratio or cross-ratio. Maybe worth adding to Pascal's theorem, as a higher-order analog? —Nils von Barth (nbarth) (talk) 03:47, 27 January 2023 (UTC)
@Jacobolus —Nils von Barth (nbarth) (talk) 04:06, 29 January 2023 (UTC)