Talk:Golden ratio

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[edit] "Approximately 1.6180339887"

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

Either we should:

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

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

[edit] Suggested Addition to Mathematics

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

q(x) = ax4 + bx3 + cx2 + dx + e
If j and k are the roots of the second derivative, then
q''(x) = 12a(x − k)(x − j) = 12ax2 − 12a(j + k)x + 12ajk

working backwards, using d and e as the constants of integration, we can find a slightly different form for q(x)

q'(x) = 4ax3 − 6a(j + k)x2 + 12ajkx + d
q(x) = ax4 − 2a(j + k)x3 + 6ajkx2 + dx + e

Use this to write the equation of the line through the points of inflection

y(x)=\frac {q(k)-q(j)}{k-j}(x-k)+q(a), then
y(x) = − (aj3 − 3aj2k − 3ajk2 + aj3 − d)x + (aj3k − 3aj2k2 + ajk3 + e)

Then solving the equation

q(x) = y(x)

We find four solutions :the obvious two are x = j and x = k the other two are

x=(\frac{1+\sqrt 5}{2})j+(\frac{1-\sqrt 5}{2})k=\varphi j + \Phi k
x=(\frac{1+\sqrt 5}{2})k+(\frac{1-\sqrt 5}{2})j=\varphi k + \Phi j

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

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

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

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

That's very cool. Let's hope that Lin McMullin can get it published, so we can use it. Dicklyon (talk) 01:31, 8 January 2008 (UTC)
Actually, it is already published three times, not counting the author's own monograph. The result is independently reported in Cut-the-knot, which appears to be a WP:RS; that site and its author, Alexander Bogomolny, are cited several times on Wikipedia. Also, the Cut-the-knott article cites two publications by McMullin: one in ON-Math, which is peer reviewed; another in The North Carolina Association of Advanced Placement Mathematics Teachers Newsletter. Further, the math is fully stated and is therefore verifiable (or falsifiable) by anyone with the requisite mathematical knowledge; for that reason, the majority of equations, other mathematical expressions, proofs, and the like in Wikipedia math and science articles are not cited to a source. Therefore, there is a sufficient basis under Wikipedia guidelines to include this "very cool" discovery. Would Dicklyon or one of the other strong mathematicians here please work this into the article? Thanks. Finell (Talk) 08:48, 27 January 2008 (UTC)
I don't think being cited on wikipedia makes the math blogger a reliable source. I couldn't even find his name on his site, so it seemed too anonymous, and full of ads, to be taken as such. But if the result is in a peer-reveiwed paper, I'd be willing to take a look and report what it says. Do you have a copy, or a way to get a copy? Maybe the author will see this and respond or email me one. By the way, math being checkable is not at all the same concept as wikipedia's WP:V concept. Dicklyon (talk) 18:22, 27 January 2008 (UTC)
I have the distinct pleasure of having Lin help me write this entry, so he has no problem with it, in fact he just wanted his name attached to it (can't blame him for that, it was after all his discovery) and it being sourced correctly. If you want Lin to e-mail you I think I can arrange that for you, for a copy just look at the sources. I'll wait to add it for your response. I was planning on checking this site before today, but I forgot. As for his site are you sure you had the right one, http://www.Linmcmullin.net , his name and picture are both clearly visible, not to mention the site is his name, as for his reliablitiy check his teaching experience and his publication history. Further I think this "very cool" discovery could explain a few of the natural discoveries made by others, if you can prove that nature uses fourth degree, and third degree (there is another proof for that, but this one was made first and inspired the other), polynomials then you wouldn't be too surprised if the Golden Ratio pops it's head up here and there. —Preceding unsigned comment added by 63.99.26.3 (talk) 20:25, 4 February 2008 (UTC)
I don't think I referred to his site, but was commenting on cut-the-knot. I have a copy of "The Golden Ratio and Quartic Polynomials" with Lin's copyright notice on it, but not a copy of or evidence of a peer-reveiwed publication. If you add something about it with a proper citation, I expect that will be fine. Dicklyon (talk) 00:42, 5 February 2008 (UTC)
Added March 20, 2008. And removed 5 min later WTF, it's been here in discussion for months with no problem, tried to put it in and while I was checking the format it disappears. Well I tried again I hope it stays this time. —Preceding unsigned comment added by 24.219.181.187 (talk • contribs) -->
There are unresolved issues and unanswered questions, the section should not be re-added without community consensus and proof of peer-reviewed publication. Adam McCormick (talk) 02:42, 21 March 2008 (UTC)
one more source added. —Preceding unsigned comment added by 24.219.181.187 (talk) 16:15, 21 March 2008 (UTC)
I think the Cut-the-knot reference is a reliable source and there seem to be peer-reviewed references as well, so I see no reason not to include this result. But I think it should be cut down to just a statement of the result, without the derivation. Also, be careful to use φ and Φ in the same sense as the rest of the article. Gandalf61 (talk) 09:55, 22 March 2008 (UTC)
Nice catch on the reverse use of φ and Φ, and has been corrected. I agree with everything else, it could use a little shortening, since the full proof is laid out with the reference, but I don't think it should just be the result without any type of proof.

[edit] Miles in Kilometres

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

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

By my arithmetic, 1 mile = 1.609344 kilometers. We have:
    • 2.54 centimeters = 1 inch
    • 12 inches = 1 foot
    • 5280 feet = 1 mile
    • 100 000 centimeters = 1 kilometer
So 1 mile = 2.54 × 12 × 5280 centimeters
= 160 934.4 centimeters = 1.609344 kilometers. Michael Hardy (talk) 04:48, 21 March 2008 (UTC)

[edit] Capitalization of Phi

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

Yes there is. Most modern professional works on the golden ratio use lower case phi for the golden ratio. See, for example, the Mathworld link. That has been the accepted usage on Wikipedia for over a year, at least; feel free to check the edit history for yourself. Although I have great respect for Jossi and his many contributions to this article and throughout Wikipedia, there is long-standing consensus here that the lead would be incomplete without the defining equation. If Jossi wants to reopen discussion of whether the equation should be in the lead, he is welcome to do that on this Talk page. I reverted the anon edits and Jossi's to restore the consensus. Finell (Talk) 23:24, 12 March 2008 (UTC)
No problem. So long as Jossi's edit was considered separately, I've got no complaints. Adam McCormick (talk) 06:21, 13 March 2008 (UTC)
I think that the lead could do without the formula, per WP:LEAD. ≈ jossi ≈ (talk) 23:29, 13 March 2008 (UTC)
Dear Jossi: Welcome back! What specifically in WP:LEAD makes the formula inappropriate? Finell (Talk) 02:01, 14 March 2008 (UTC)

[edit] Disputed sightings

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

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

[edit] Title

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

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

It's the ratio a:b or (a+b):a as shown; or the ratio of the dimensions of a golden rectangle; or (1+sqrt(5)):2; etc. In the rectangle picture, the sum of the distances sqrt(5)/2 and 1/2 is it: 1.118 + 0.5, not as you describe. "They" can call it whatever they want; we just report on it. Sorry, I can't explain what's happening to you. Dicklyon (talk) 03:41, 17 March 2008 (UTC)
Dude: The OED traces the etymology of golden ratio and golden section. I believe that the first use of golden was in German by Ohm (mentioned in the article), which led to translations such as golden section. I hadn't heard of auric (Latin for golden?) section. Where did you find that one? Finell (Talk) 22:34, 17 March 2008 (UTC)

[edit] The da Vinci face

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

I believe the source of the idealized face with golden ratio reference lines is Divina, which Leonardo illustrated. In my research I found a sketch by Leonardo of a real face with the same grid of lines superimposed. I wanted to track down more information before adding it to Wikipedia. Finell (Talk) 22:25, 17 March 2008 (UTC)
At this point, I can accept that it's from that book since the page headers appear to say so. But how does it relate to golden ratio? No way that I can see. Dicklyon (talk) 15:37, 26 March 2008 (UTC)
Here is a link to an image of that page, from a copy of the book in the LoC's collection [1]. Perhaps you or someone who speaks Italian (Jossi?) can explain in more detail, from the surrounding text, what this image represents in a book that is about or largely devoted to the golden ratio. Or, if we wait a few months, an English translation of the book is scheduled for publication in June 2008. Finell (Talk) 01:37, 27 March 2008 (UTC)
Thanks for that pointer. I hadn't realized the LoC had that sort of online collection. Looks like there's no text near the picture, however, and lots of the figures appear to be about things other than the divine proportion. So maybe we just shouldn't make any claims about it being an application of the golden ratio unless someone finds evidence that it is. I look forward to that translation. Dicklyon (talk) 02:50, 27 March 2008 (UTC)
It reads, Divina Proportio, and it refers to mapping the human face to the golden ratio. Check the ratio of the six rectangles (the leftmost two are split in the middle). ≈ jossi ≈ (talk) 22:24, 27 March 2008 (UTC)
The full title is Divina proportione: opera a tutti glingegni perspicaci e curiosi necessaria oue ciascun studioso di philosophia, prospectiua pictura sculpura, architectura, musica, e altre mathematice, suauissima, sottile, e admirabile doctrina consequira, e delectarassi, cõ varie questione de secretissima scientia. M. Antonio Capella eruditiss. recensente. it is in ancient Italian. ≈ jossi ≈ (talk) 22:31, 27 March 2008 (UTC)
Yes, and the title page would be a nice image; doesn't make much sense to me, but nice. Jossi, since you uploaded the face image in the first place, what do you know about it? I was looking for reason to suspect that the rectangles with diagonals across them were meant to be golden rectangles, but I couldn't find one. There's another face (page seventy something iirc) with various and sundry unexplained proportions marked as well. Any clue? Dicklyon (talk) 23:01, 27 March 2008 (UTC)
What page number, Dick? ≈ jossi ≈ (talk) 20:38, 28 March 2008 (UTC)
You probably mean this [2]. My knowledge of Italian is basic, and the olde Italian used by Pacioli does not help.... ≈ jossi ≈ (talk) 20:40, 28 March 2008 (UTC)
Right, that one. It's got the same top and bottom rectangles with diagonals across them, but they're divided up different ways. I wonder still if those, or some other proportions, are said by Pacioli or Da Vinci to be approximately divine. Dicklyon (talk) 22:49, 28 March 2008 (UTC)
Dunno... All we have is Divina Proportio stamped above the head. We can just simply describe that without additional commentary. ≈ jossi ≈ (talk) 01:37, 29 March 2008 (UTC)
There's some stuff here [3] ≈ jossi ≈ (talk) 01:40, 29 March 2008 (UTC)

[edit] regarding the recent "OR" reverts

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

While it's not the deepest result ever, I don't think it is entirely trivial either. The result says that the line segment between the points of inflection intersects the quartic in the two points that divide it in the ratio φ:Φ. And that ratio is the same for any quartic with real points of inflection. That's not completely obvious. Alexander Bogomolny discusses it in some detail at Cut-the-knot, and he doesn't typically write up trivia. And a peer-reviewed paper in On Math was cited above. Gandalf61 (talk) 19:51, 25 March 2008 (UTC)
Well yeah that sounds nontrivial, but now I don't see my own revert in the article history. Sometimes I'm flumoxxed; I could have sworn that I had undone to Paul's version, but he doesn't have one for days back. Anyway, Gandalf, feel free to put in maybe something more brief? Pete St.John (talk) 20:39, 25 March 2008 (UTC)
Has anyone actually seen this paper from On Math? Dicklyon (talk) 21:34, 25 March 2008 (UTC)
Well, there is certainly a paper with the cited title in the index of the relevant journal issue here. I think you need a subscription to see the paper itself. But why be so suspicious ? Do you have any evidence that suggests the paper may not be legitimate ? If not, I think we should follow WP:AGF and asume that the paper is a valid reference. Gandalf61 (talk) 09:16, 26 March 2008 (UTC)
Is it possibly this? -- Dominus (talk) 13:37, 26 March 2008 (UTC)
That's a different paper, different title, apparently self-published. That one I have, but I'm wondering if there's a published peer-reviewed one that anyone has seen. I'm not suspicious of its existence, but how can we cite it as a source if nobody has seen it? Do we even have reason to believe that it's peer-reviewed at this point? Someone who has it or knows of its content should just speak up and fix this; the rest of us should wait, or go try to get a copy. Dicklyon (talk) 15:34, 26 March 2008 (UTC)
I've requested a copy from the author. -- Dominus (talk) 15:59, 26 March 2008 (UTC)

[edit] General Form

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

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

Do this with any n. Here is another:

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

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

This property of number sequences similar to the Fibonacci sequence is described in generalizations of Fibonacci numbers#Fibonacci integer sequences. Gandalf61 (talk) 19:32, 22 April 2008 (UTC)
Hm. Interesting page. Thanks for the heads up. InfoNation101 | talk | 19:39, 22 April 2008 (UTC)

[edit] Disputed fact tag

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

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

[edit] "Approximately 1.6180339887"

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

Either we should:

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

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

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

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

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

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

If you want "better" you need to justify that and get consensus. This article has a long history of compromise, so if you just jump in and make random-looking "improvements" that will typically not be accepted. Dicklyon (talk) 05:18, 24 May 2008 (UTC)
The expression of the number with 11 digits is approximate, as must be any expression in decimals of this irrational number. It is also verifiable. I do not think it Wikipedia's mission to dumb down facts or to be so patronising as to segregate readers according to whether they are allegedly non-mathematical or crave warm, fuzzy feelings. It is a straw man argument by Bonzai273 to confuse contentment with the expression with a wish to lock down the whole article. —Preceding unsigned comment added by Cuddlyable3 (talk • contribs) 14:50, 24 May 2008 (UTC)
I strongly support approximately 1.618 in the lead. Even building a monument - if you want a 10 m tall golden rectangle, this would give the width to plus/minus 2 mm. Just a few paragraphs further down, 10 decimals are given.--Noe (talk) 15:01, 24 May 2008 (UTC)
I would support starting off with the same precision as we do pi: 3.14159, 1.61803. But not less than that. Dicklyon (talk) 15:36, 24 May 2008 (UTC)

[edit] Golden mean

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

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

Sources? What do you mean "as wikipedia states?" Dicklyon (talk) 06:47, 27 May 2008 (UTC)
He means, the Golden ratio article in "Wikipedia states" that golden ratio and golden mean are synonyms, but he thinks that may be wrong because the golden ratio is 1.618 while, he thinks, the golden mean is 0.618. He thinks that because he reads some sources that do in fact say that the golden mean is 0.618... There are also sources that say that the golden ratio is 0.618..., but perhaps he has not come across those yet. His underlying problem is that he reads literature that uses the various synonymous terms, define them, and state their values differently, and he does not get past that because he does not understand the concept. So he is, understandably, confused. To his credit, this time he is asking, not inserting erroneous information in article space. Finell (Talk) 20:14, 31 May 2008 (UTC)
Actually the golden ratio 1.618... is not the mean of anything. The qualifier conjugate used for 0.618... should be multiplicative inverse. Cuddlyable3 (talk) 07:56, 27 May 2008 (UTC)
Definitive answer: By the most common current convention, the golden ratio is 1.618... because, by the most common current convention, the ratio is stated as long:short. If one states the golden ratio as short:long, then the golden ratio is 0.618..., and some published sources do state it that way. The relationship between the lengths of the segments of a line sectioned extreme and mean ratio (a golden section) or between the sides of a golden rectangle is the same, regardless of which way one states the ratio. In any event, whichever way the ratio is stated, it is always, and only, a ratio between two lengths; it is never solely the short one or the long one. And, whichever way one states the golden ratio, golden mean is sometimes uses as its synonym (although golden mean has an unrelated meaning in philosophy). Dude, please stop sowing confusion with your misinformation: there is no such term as golden extreme and mean in the reliable sources on the subject. Finell (Talk) 20:14, 31 May 2008 (UTC)
The golden ratio, golden mean, or divine proportion are one and the same thing (perhaps with slight differences in usage). This thing is not a number, it's a concept, a relation between quantities (numbers, lengths, times intervals, or whatever). In some sources, this concept is identifed with one of the numbers phi=1.618... or tau=0.618... (other symbols may be used), both of which make sense, but neither of which is the basic meaning of the concept. In some (older) sources, there is a distinction between a ratio (like 1:3, phi:1 or 1:phi) and the value of the fraction (1/3, phi/1=phi resp. 1/phi=tau); in other sources, there is no such distinction.--Noe (talk) 12:21, 1 June 2008 (UTC)
I agree completely, but the difference is not between phi and tau, either. Tau used to be the symbol of choice for the golden ratio (the initial letter of the Greek word for cut), but phi became more popular in the 20th century. This is mentioned in the article. So if you see tau = 0.618, the author is stating the golden ratio as short:long; some sources say phi = 0.618 for the same reason. And the unusual property that 1/φ = 1-φ (or, if you go the opposite direction, 1/φ = 1+φ), doesn't help. And then there is the rarely seen (but one place is Mathworld) term golden ratio conjugate which, according to our article and Mathworld is 0.618 and is denoted Φ (capital Phi), but is sometimes denoted φ′ (phi prime), and which is defined as the reciprocal of the golden ratio (if you state the golden ratio as long:short). But some think that golden ratio conjugate should mean the conjugate root of the defining quadratic equation, which we say is -0.618. This just introduces more confusion, for anyone who gets that far. Personally, I think the world would be better off if the term golden ratio conjugate were outlawed, since 1/φ always works and expresses the concept, whether you think long or short; but golden ratio conjugate is out there, so our article can't ignore it. By the way, none of this confusion arises at all if one thinks in pure pre-analytic geometry, which is where the whole idea started and was the way Euclid codified it. A couple years ago, I was going to do a section on Terminology and notation for the article, which would go through the etymology of the terms and symbols, and the varying definitions, historically, but I never got around to it. Finell (Talk) 14:25, 1 June 2008 (UTC)
Noe says "This thing is not a number, it's a concept, a relation between quantities (numbers, lengths, times intervals, or whatever)." But that relation is most often expressed mathematically as a ratio, which is a number. In Pacioli's day, divine proportion was never made into a decimal number, and I think maybe also not expressed as the irrational in terms of root 5 (not sure about that). But certainly by the time anyone called it "golden", it was a number, since that's how modern mathematics expresses such geometric relationships algebraically, thanks to Descartes. To try to ignore the number and express it again as a just a concept, using words alone, tends to confuse the picture, and is what has made it so easy for 20-Dude to use language that he can't explain, and that the rest of us are unable to intepret, such as "with golden extreme and mean, but not just one of either". Dicklyon (talk) 15:36, 1 June 2008 (UTC)
I agree, but we must accept that sources differ as to whether that number is 1.618... or 0.618..., and my post (like Finell's preceeding it) should explain why it's so, and why it's not terribly important either.--Noe (talk) 07:54, 2 June 2008 (UTC)
Euclid's sectioning a line in extreme and mean ratio, and everything else that Elements says on that topic, is understandable to anyone who sufficiently understands geometry taught in the Euclidean manner. Likewise, the geometry, the quadratic equations, and the solutions, whether expressed as irrational numbers or as decimal approximations, is understandable to anyone who sufficiently understands geometry taught in the Cartesian manner. (When I was in 7th grade honors math, the so-called "new math" taught exclusively Cartesian geometry, with theorems proven by algebra, which was interesting [in the Chinese sense] because algebra was first taught in 8th grade. I read selectively from Elements much later for pleasure, probably as a result of reading Einstein's Autobiographical Notes.) Anyone who understands either knows that long:short or short:long is purely a matter of convention: anyone who understands the algebra can flip one around to the other; if one thinks in Euclidean terms, there is nothing to flip. Dude's problems are, first, that he doesn't understand either (something that he shares with an alarmingly substantial majority of the American public), and second, that this does not deter him from declaiming on the subject (it is in this respect that he is unusual). Finell (Talk) 19:18, 1 June 2008 (UTC)
Well, I've studied geometry in the way it was taught in American high schools in the 1960s, all about proofs and such, which may be what you call the Euclidean manner or may not. So clue me in: what is the interpretation of "extreme" and "mean" in that expression? Dicklyon (talk) 19:52, 1 June 2008 (UTC)
We are contemporaries. However, "it was [not] taught [the same way] in [all] American high schools in the 1960s". In my high school, the way it was taught depended on which track you were in. If your geometry was not taught as Cartesian coordinates, then it was (almost certainly) what I refer to as the Euclidean manner (straightedge and compass constructions, non-algebraic proofs). As for extreme and mean ratio, my surmise (I have not found a source) is that it refers to the property that, in the defining proportion, mean1=mean2 and extreme1/mean = mean/extreme2 = (extreme1 + mean)/mean. By the way, I have never pretended that my level in math approaches yours. Now, here's one for you, since you also love old books: What is the original source of the quotation so often attributed to Kepler, "Geometry has two great treasures: one is the Theorem of Pythagoras, and the other the division of a line into extreme and mean ratio; the first we may compare to a measure of gold, the second we may name a precious jewel." Pity (for terminological consistency) that he did not reverse these comparisons. And which did he value more? The value of "a measure of gold" depends on the measure, while the value of "a precious jewel" depends on the particular gem stone, the size, and other factors (color, cut, and clarity, to be specific). Finell (Talk) 21:40, 1 June 2008 (UTC)
The geometry I studied was definitely constructions, non algebraic, so I guess that means Euclidean. I don't recall extreme and mean ratio coming up, but it's possible. Anyway, I take it the "mean" here means what we call "geometric mean" in algebra, the square root of the product. So if we take the short segment and the total line as extremes, the long segment is their mean. The ratio of the mean to either extreme is thus the same (depending on the order). I still don't know why they call it "extreme and mean ratio"; what does the "and" designate?
Great question on the quote. Hambidge 1920 p.153 cites two German books for exact references to sources, but I don't have access to either of those. I'll see what I can find. Dicklyon (talk) 00:00, 2 June 2008 (UTC)
Your explanation of "extreme and mean" is much cleaner than my groping, and that may be as close as we can get without finding a scholarly interpretation of the precise phrase "extreme and mean ratio". One problem may be that classical Greek is hard to translate accurately; the translation of classical Greek to a European language is less linear than translation from Latin (I learned this about 15 years ago, when I came across a widely quoted statement by Plato about about taxes that I knew had to be a mis-translation [although it is the translation used in Bartlett's] because it was contrary to the history of the income tax). And we don't even get it directly from the Greek: Heath's English edition is based on the leading German translation; other editions come to us via Arabic. So we are trying to parse 4 words in English that may not be an accurate translation of what Euclid actually wrote in Greek.
I agree, it would be fascinating to see how these expressions connect. Dicklyon (talk) 05:33, 2 June 2008 (UTC)
My interest in the original Kepler source (in addition to the fact that it would be nice to cite) is because I want to read the quotation in its context, if there is an English translation: why did he single out the golden ratio as being comparable in importance to the Pythagorean theorem? The latter seems to me to be much more fundamental. Most people think of Kepler the astronomer, but was primarily a mathematician. All of his theories about cosmology, both correct and incorrect, were based on geometry.
The "Kepler triangle" combines these two concepts into one triangle, and like a lot of people, he fell for the mystical power of geometry and numbers. It's the only right triangle with edge lengths in geometric progression. He wasn't much of an astronomer, having very poor eyesight; that's why he had to kill Tycho and steal his data. Dicklyon (talk) 05:33, 2 June 2008 (UTC)
I've been watching Kepler triangle. As for mysticism, let's give Kepler (and Pacioli, for that matter) a break based on historical context. Mysticism was the norm then, even for mathematicians and scientists. Astrology was taught as part of medicine. Galileo was a devout Catholic, despite being a Copernican. Newton was deeply into mysticism. But despite that, the latter two were not shabby as physicists, and Newton was OK at math. As for Tycho, you obviously subscribe to Peter Schaffer's version of Mozart's death. Finell (Talk) 06:46, 2 June 2008 (UTC)
Here is one more. Do you have access to a bibliographic index of early 20th century math papers? If so, can you find any papers published by Mark Barr. So far as I can tell, he never published a book. The closest I have gotten to his work on the golden ratio and his suggestion of using phi for its symbol is writing by TA Cook. Ghyka mentions it too, but I think that everything in Ghyka is secondary accounts of others' work (with attribution). Finell (Talk) 05:22, 2 June 2008 (UTC)
No, I don't have, and I've never been able to find anything cited on Barr either. Dicklyon (talk) 05:33, 2 June 2008 (UTC)
I tried searching Zentralblatt (goes back earlier than MathSciNet) for his name but didn't find anything. —David Eppstein (talk) 05:41, 2 June 2008 (UTC)

[edit] "Reasons for the occurrence of phi in nature"

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

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

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