Talk:Golden ratio

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The value of φ

The value of φ with 2000 digits

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)

References

 Not done. 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, 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 <math>-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) {{math|''φ''}} φ (favored by Jacobolus) looks much narrower than <math>\varphi</math> $\varphi$, 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 <math> consistently for both, so my position is that in articles where display mathematics is used, we must use <math> inline as well in order to match it. And it does not work for square roots, where {{radic|5}} √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 <math>\sqrt 5</math> ${\sqrt {5}}$. 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 <math> throughout. —David Eppstein (talk) 23:49, 21 September 2021 (UTC)

Do other editors agree that all math throughout the article should be <math> 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, {{radic|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 {{radic|5}} √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 {{math|{{radic|5}}}} √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 {{math|{{radic|''φ''}}}} √φ where the phi is far too low below the top bar of the radical compared to <math>\sqrt\varphi</math> ${\sqrt {\varphi }}$. —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#Cycle notation –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#Missing LaTeX macros: mathstrut, vphantom, vrule, and smash discusses this but the subscript-backspace trick from there doesn't seem to work with varphi. <math>\sqrt{\varphi_{\!}}</math> ${\sqrt {\varphi _{\!}}}$. Superscript-backspace is no better: <math>\sqrt{\varphi^{\!}}</math> ${\sqrt {\varphi ^{\!}}}$. Instead, the overset-thinspace trick produces <math>\sqrt{\overset{\,}{\varphi}}</math> ${\sqrt {\overset {\,}{\varphi }}}$ 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.

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. —Quantling (talk | 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.—Quantling (talk | contribs) 22:06, 30 December 2021 (UTC)

Add EXACT connection with e, pi & logarithm!

$\phi =e^{-0.2i\pi }+e^{0.2i\pi }=e^{-0.2\ln -1}+e^{0.2\ln -1}=-1^{-0.2}+-1^{0.2}={\frac {1}{\sqrt[{5}]{-1}}}+{\sqrt[{5}]{-1}}$

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 x⁵ = −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. —Quantling (talk | 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)

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)