Talk:Golden ratio/Archive 1

why quadratic equation?

can anybody please suggest what should be the correct name for a quadratic equation since quad means four and is not approriately used in this context

[Quadratic in this case refers to 'square' (ie, foursided) rather than 4th power, and so is used correctly for an equation where x^2 is the dominant term.]

And if you wondered, the term for an equation with four roots is "quartic". Quadratic, cubic, quartic, etc. Japhy 14:38, 16 October 2005 (UTC)

ax²+bx+c=y is a quadratic equation, that simply is the name. Wolfmankurd 16:32, 25 May 2006 (UTC)

Good page

Good page. Lots of info. Surely you don't need to keep the "Needs Attention" mention in place whilst you resolve final wordings about the validity of aesthetic claims. Take credit for the work done. 134.244.154.182

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Errors in article

The following is in the article: I am not good at edits so could someone find a place to put in this the simplest equasion for "Phi is 5^.5 x .5 + .5" it is as universal as the fingers on your hands and the toes on your feet and uses only positive symbols thanks greg

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But more interestingly, it is usually found in natural shapes:

• Leaves length / width
• On faces, it's everywhere! Ratio mouth width / nose width, etc.
• More examples welcome!

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The first is obviously wrong - leaves come in all sorts of proportions. The second doesn't make much sense - mouth width divided by nose width may be close to the golden ratio for most people, but it's also close to π/2, 1.6, √e, etc. --Zundark, 2001 Nov 6

I put a discussion of the relationship of φ to phyllotaxis at Talk:Fibonacci number/Phyllotaxis. It might be appropriate to incorporate some of that discussion here. -- Dominus 15:42, 11 Mar 2004 (UTC)

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Why?

Is there any known (or speculated) reason why humans find the golden rectangle beautiful? - Stuart Presnell

For me the most beautiful is its continued fraction representation, which can't be more simple φ = [1; 1, 1, 1, ...]. Perhaps golden rectangle is beautiful because it is so simple. But in fact it is not so simple. Just beautiful. Some human nature obviously can't be described with pure math. The same thing is with a devine being in Islam. Arabesques are beautiful and nobody knows why. Western Church built complex churches along the centuries. Simple orthodox Ethiopian church had built cubic ones as Lalibela's churches are. These are some of my views on a subject. Another question. Why is chaos nowadays so beautiful. Just because of a fashion? Best regard. -- XJamRastafire 15:08 Sep 5, 2002 (PDT)

I'm not convinced this ratio is special. I confess I can't easily tell the golden ratio from a simple 3:2 ratio. A lot of things are near 3:2 (A4 = 1:1.4, 35mm = 1:1.5, golden ratio = 1:1.6). -anon

I believe that Edward Tufte says much the same thing; the purported 'attractiveness' of rectangles with a 1:1.618 side ratio actually attaches to all rectangles with ratios between about 1.5 and 1.75. -- Dominus 13:19, 14 Sep 2004 (UTC)

The "attraction" or "specialness" would come from a series of the ratio. One single rectangle is, in my opinion, never special or attractive. But a rectangle with the golden ratio, part of a larger self-referential design that uses the golden ratio is special. Hyacinth 20:47, 17 Oct 2004 (UTC)

Golden rectangles are CLAIMED to be beautiful, because Fechner said so, and because people started repeating this nonsense. And why did they? I think some math people have an inferiority complex about their subject, so if they can link math to beauty, that's a bonus. Showing the BBC programme on Wiles to highschool students, to begin with the just laugh at him. (The program starts by Wiles crying over his eight years long fight with Fermat's last theorem.) Later, they may glimpse that math just possibly MIGHT be beautiful in itself - but you can't really convince them of that. But what if beauty - no, Beauty - can be connected to a mathematical construction...? - The other way round, some arts people may have an inferiority complex towards the Sciences and Math - so if beauty and proportion can be based on a solid mathematical construction, it's not just a matter of taste; then it's got the legitimity of Scinece to back it up. --Niels Ø 18:29, Dec 13, 2004 (UTC)

Quite simply if you ask some one to draw a rectangle/ pick the 'nicest' rectangle they will usually do somethign which is close to a 1:1.6 ratio. be this the golden ratio, pi/2 root e whatever but thats why it's called beautiful.Wolfmankurd 16:37, 25 May 2006 (UTC)

If that's true, you must have a reference you can give us, yes? Dicklyon 17:42, 25 May 2006 (UTC)

I have found this article on the internet by a man called Mario Livio. He make reference to a study undertaken by German physicist and psychologist Gustav Theodor Fechner that concluded that most people find the Golden rectangle most beautiful. Mr. Livio draws interesting conclusions from that. If you go to this site, this section is halfway down. --Canadian Joeldude 02:52, 2 July 2006 (UTC)

Actually, if you continue to read that article, or if you read Livio's book, you'll find that results for these sort of studies (regarding the claimed beauty of golden rectangles) are at best inconclusive. Livio quotes British psychologist Chris McManus: "whether the Golden Section [another name for the Golden Ratio] per se is important, as opposed to similar ratios (e.g. 1.5, 1.6 or even 1.75), is very unclear." It may be that people tend to pick or draw rectangles close to the 1:1.6 ratio, or it may not. Any relation between this and the golden ratio may be true or it may be number juggling. So in answer to the original question, yes, people have speculated that the golden rectangle is somehow "inherantly" beautiful, but no, there are no scientific reasons why this perception exists (if indeed it even exists). At least, that's how I read the evidence I've seen. Although I can't help but think that editing Wikipedia woud be a great deal more pleasant if the editing textbox had a 1:1.6 ratio...can't imagine why... :-) --puzzleMeister 16:14, 18 July 2006 (UTC)

I would speculate that it has to do with the human field of vision. can anyone give an approximate dimension for the human field of vision? it is hardly a rectangle, more of an oval or a rectangle with rounded corners, and is wider than it is tall. peripheral vision varies among sex and geographical descent, but if the golden rectangle and the rectangular approximation of the human field of vision are roughly equivalent, that would give my speculation some weight, no? and assuming that it varied with the same consistency as the "aesthetically pleasing rectangle" (pi/2, 1.75, 1.4, etc) then we might be onto something.

24.60.182.118 01:35, 11 August 2006 (UTC)

"Everyday people" explanation

Work remains in this article to describe the golden mean better to the everyday person. Kingturtle 21:56 Apr 19, 2003 (UTC)

I have attempted to better describe the golden ratio to laypeople (since I basically consider myself one) by adding a blue sidebar with an illustration of the golden ratio represented as a line divided into two segments (along with a description). When I first read about the golden ratio, what helped me understand it was an example of a line that had been divided into segments according to the golden ratio. So I attempted to capture that with the illustration I put on the page (Image:Golden ratio line.png). I'd be interested to know if people find that helpful. Please give a shout out if it's too big or factually incorrect (don't try measuring it, it's not exact). - Eisnel 08:28, 22 Jun 2004 (UTC)

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Digits

Putting the first few thousand digits doesn't seem encyclopedic to me. Just putting a link to a webpage containing the first few million would be enough. —seav 01:39 4 Jul 2003 (UTC)

I agree. I can't think why anybody would need more than 10 digits of this, yet the top of the page has 30 digits, and the bottom has 1024 digits. It seems so very "I figured out how to do arbitrary-precision math on my computer, I need to show it off" to me.  :-)

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Statistics?

Should add its uses in statistics or optimization. Wshun

I agree fully with this, however I am nowhere near qualified to write it. Anybody care to try? Joblio 10:38, 21 Feb 2005 (UTC)

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One version, one page

We shouldn't have multiple versions of the same page, or we might end up with a nightmare of variant editions. "Golden ratio" is the most common name. Gene Ward Smith 08:10, Feb 3, 2004

I merged the page histories, but I don't have any opinion on whether it should stay at this title or at Golden mean. Golden mean is just a redirect now, so anyone can move the page back there if they want to. I have not yet changed any of the links to Golden mean as I didn't know if the page is going to stay here or move back there. Angela. 17:20, Mar 5, 2004 (UTC)

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All over and fractals

Maybe something should be added about the multitude of seemingly unrelated places that the golden ratio appears (Stock market, pyramids of egypt, etc.). Also maybe a mention of fractals? --Starx 00:05, 8 May 2004 (UTC)

introductory paragraph and the golden ratio conjugate

OK, this "golden ratio conjugate" is interesting and belongs in the article, but does it really belong at the very beginning, in the introductory area? Other people have mentioned that this article seems to jump into complicated math pretty quickly, and doesn't explain the golden ratio to laypeople. I think that immediately jumping into a complicated aside about the golden ratio conjugate in the introductory area (as if it's central to understanding the golden ratio) isn't appropriate. Thing is, I'm afraid that I don't know enough about the math involved to know where in the article this should be moved. But we desperately need some sort of descriptive opening paragraph that's accessable to laypeople. - Eisnel 04:11, 23 Aug 2004 (UTC)

This article currently defines the golden ratio conjugate as

${\hat {\phi }}={\frac {1}{\phi }}=\phi -1={\frac {1}{2}}\left(-1+{\sqrt {5}}\right)$

This is not the field theoretic conjugate however, which is the other root of the minimal polynomial $x^{2}-x-1=0$ given by

${\hat {\phi }}=-{\frac {1}{\phi }}=1-\phi ={\frac {1}{2}}\left(1-{\sqrt {5}}\right)$.

Is the article mistaken or is this just one of those screwed up definitions? -- Fropuff 16:19, 2004 Nov 17 (UTC)

I think it's correct. Taking the reciprocal ratio is more natural in certain contexts, and historically, it has played an important role. As for this definition being "screwed up", one might as well argue that the field theoretic definition is screwed up. Conjugate is a word with many (fairly related) meanings, and it's no surprise that technical definitions in two fields such as geometry and algebra have taken divergent meanings. The use of "conjugate" to indicate a reciprocal relation is common in geometry, going back several centuries, so I expect its usage predates that in algebra, e.g. complex conjugation. The term is also common in other ways in mathematics. --Chan-Ho Suh 06:43, Nov 18, 2004 (UTC)

The definition would be screwed up if it didn't agree with the algebraic definition. Besides which, I'm not aware the word conjugate being used to mean reciprocal in geometry. Can you point me to a reference (other than MathWorld, which I don't trust, and from which I believe this statement was copied) that uses golden ratio conjugate to mean 1/φ? -- Fropuff 16:37, 2004 Nov 18 (UTC)

Conjugate hyperbolas, diameters, etc. Conjugate is used with two geometric objects that have a reciprocal relation. For example, see the OED entry, under the math and physics related definition. It's an old word that doesn't seem to be used as often nowadays. What I meant by my previous comments is that I consider it plausible that the word conjugate could be used with respect to the reciprocal of the golden ratio, in some kind of geometric situation. But I can't be certain that any such usage was widespread at any time. It's appearance in the Wikipedia page seems mysterious. --Chan-Ho Suh 08:42, Nov 20, 2004 (UTC)

Yes, I agree that it's plausible; which is why I didn't just edit the article outright. But if referring to φ⁻¹ as the conjugate is not standard or widespread I think we should revert to the algebraic definition which certainly is standard and applies in the present context. As it stands, I think the article is bound to cause confusion (it at least confused me). -- Fropuff 15:10, 2004 Nov 20 (UTC)

Nonsense should/will get scoured from this article

There's a lot of nonsensical mumbo-jumbo about phi in the article. This includes the long-refuted claims that the golden ratio is aesthetically pleasing (in shapes like rectangles, proportions of parts of the body, etc.). Also, the claims that the Greeks purposefully used the golden ratio in their architecture is unsubstantiated and has been refuted in particular cases, such as the Parthenon. The actual fact of the matter is that, contrary to the opening paragraph, all this nonsense about the golden ratio is actually fairly recent, last half-millennium or so.

What's funny is that some of the external links point this out (notably the Livio book). I'm also disturbed by the linking to websites that are obviously of a very mystical nature. I'm fine with having a section on the history of the mysticism around the number, but as it currently is, it's a confusing mix of fact and mysticism.

I encourage the regular editors of this page to fix these erroneous statements in the article. I myself will try and fix them when I get more time. --Chan-Ho Suh 09:01, Oct 17, 2004 (UTC)

I agree. However, I think one of the most interesting things about the golden ratio is the fact that all this nonsense is perpetuated by so many authors, including some quite serious authors. Removing all reference to unsubstantiated claims is not a godd policy; instead, it should be discussed and refuted.

As I said, a history of the mysticism around Phi is worthy of being in the article; however, before I edited the page, it was a confusing mix of fact and unsubstantiated claims. Unsubstantiated claims are still in the article, but now it is mentioned if there is evidence of it or not. If anybody finds evidence for such a claim, they can insert it. --Chan-Ho Suh 09:25, Dec 14, 2004 (UTC)

I have added a warning about unreliable internet sites to the main page. Some of the links in the reference section are in that category, especially [Golden Number], related to the hilarious [Evolution of Truth]. Should the nature of the site be flagged on the main page? If it is simply removed, I guess someone else will just add it again...--Niels Ø 18:40, Dec 13, 2004 (UTC)

The warning doesn't tell anything new. Of course most sites will be unreliable; that's how the Internet is. Rather than a warning, I think your suggestion about flagging the nature of website is a better idea. Since the Phi mystics will undoubtedly add and re-add their links, and also in interest of objectivity and NPOV, upon reconsideration, I think there's a place for these links, but they should clearly be marked non-mathematical and/or non-historical.

So here's my proposal. Math links go into a math links section. Historical links go into a historical link section. Mystical stuff gets put into an "Other" section. Also, links should be of high-quality or of high information content. So a math link must be high-quality, according to math standards, e.g. contain substantive math content, not just a bunch of mystical stuff and extremely simple math. Same with history. Vague stories about Pythagoras do not qualify a site as a high-quality history link. As for "Other" links, they should be very popular sites, so only the "important" ones. I'll set this up now. --Chan-Ho Suh 09:25, Dec 14, 2004 (UTC)

Nautilus nonsense

The claim that the shape of the Nautilus shell is related to the golden ratio seems unsubstantiated - what is the connection?

I believe there is no connection, and I believe this is what has happened:

Someone invented the whirling-golden-rectangle-pattern shown in the article, and discovered a logarithmic spiral in that pattern. Someone else discovered that the Nautilius shell is remarkably close to being a logarithmic spiral. Then, someone connected those two facts. And then, since usually clear minds like those of Martin Gardner and Ron Knott have perpetuated this nonsense, serious as well as cranky authors have repeated it, without documentation.

However, not all logarithmic spirals are similar. They form a family of curves that can be characterized by a parameter, which can be chosen in a number of ways.

(i) One possible parametrization is to measure the angle between the line from a point on the spiral to the centre, and a tangent line drawn at the same points. For a given logarithmic spiral, this angle is constant along the curve (hence it is also called an equiangular spiral).

(ii) Another parametrization is this: Draw a half line starting at the centre. Measure the distances from the centre of two succesive intersections of the spiral with the line, and find the ratio between these two distances. This, again, is a constant along the curve.

(iii) Parametrization (ii) involves two points separated by a complete turn of the spiral. Instead, one could consider points separated by some other angle, e.g. 1 radian (180/pi degrees).

I have never seen a sensible argument connecting the particular logarithmic spiral exhibited by the Nautilus to the golden ratio.

Niels Østergård [[1]]

I'm new to this article, but I have heard of the connection with the nautilus shell made several times in connection with phi. For example, the book The Golden Ratio by Mario Livio has a picture of a nautilus shell on the cover. I dunno if this helps, but the concepts are definitely connected, at least in a representative fashion. Cheers, DropDeadGorgias (talk) 20:02, Dec 13, 2004 (UTC)

I have not checked this particular reference, but I have checked perhaps 50 other references making this claim - never finding valid support. - My reference from the main article to this discussion has been removed by someone else; I suppose main articles in general should not reference discussions. However, I have therefore modified the text in the main article to indicate that the Nautilus claim is unsubstantiated.--Niels Ø 08:26, Feb 21, 2005 (UTC)

In "The Golden Ratio" by Mario Livio, there is a picture of a nautilus on the cover. However, When he deals with the stucture of the nautilus shell in the book (along with some other naturally ocuring phenomena), he moves seemlessly from talking about the Golden Ratio right into talking about logarithmic spirals without making any real distinction. This un-neccesarily complicates the issue and I had to re-read this section to catch it. I found this disapointing since Livio somewhat plays the debunker of golden ratio mumbo jumbo through most of the book. And for the record this book is fully indexed and has an extensive bibliography, and I would consider it a fairly reliable reference, at least to start with. Danimal 09:03, 5 September 2005 (UTC)

I'm not too strong on the math, but I tidied up the bit about how the golden rectangle relates to the logarithmic spiral, and removed the "In nature" section of the article:

The golden ratio turns up in nature as a result of the dynamics of some systems - for instance, in the angular spacing of tree limbs around a trunk, or sunflower seeds. In both cases, the problem is "wedge this next one into the biggest available space".

You can draw a nice sunflower by plotting the points $(\theta ={{2\pi } \over {\phi }}i,r={\sqrt {i}}),i=1..N$

In the popular literature, the shape of the shell of the chambered nautilus (Nautilus pompilius) is often claimed to be related to the golden ratio. However, this claim appears to be unsubstantiated.

I don't think the nautilus comparisons are nonsense, since the golden rectangle can be used to produce a curve that very closely resembles a logarithmic spiral (and the dividing lines actually do lie on a logarithmic spiral). I doubt an actual nautilus shell will match the golden spiral any less accurately than it matches a true logarithmic spiral. If comparisons to nature are to be made at all, we should clarify whether we're talking about the logarithmic spiral in general or the one with a pitch of ~17.03239 degrees in particular, and what kind of precision we're expecting nature to have. -- Wapcaplet 20:16, 26 Feb 2005 (UTC)

I'm fairly happy with the article as it stands, with no mention of the Nautilus. However, I'd like to comment on the remarks above. If someone (like I do) wants to tell the whole World about the beauty of Math and how it pops up everywhere around us, the Nautilus is a beautiful example of an alomost perfect logarithmic spiral, and hence, it could be mentioned in an article on logarithmic spirals. But it has nothing to do with the particular logarithmic spiral discussed in the golden ratio article, and hence does not belong here - unless it is made clear that the Nautilus spiral is not golden.--Niels Ø 08:56, Feb 28, 2005 (UTC)

I believe the Golden spiral and the spiral in the nautilus shell are related due to the efficiency at which things are packed. As the article points out, it is most often seen in seed and petal arrangement:

"The amazing thing is that a single fixed angle can produce the optimal design no matter how big the plant grows. So, once an angle is fixed for a leaf, say, that leaf will least obscure the leaves below and be least obscured by any future leaves above it. Similarly, once a seed is positioned on a seedhead, the seed continues out in a straight line pushed out by other new seeds, but retaining the original angle on the seedhead. No matter how large the seedhead, the seeds will always be packed uniformly on the seedhead.

And all this can be done with a single fixed angle of rotation between new cells?

Yes! This was suspected by people as early as the last century. The principle that a single angle produces uniform packings no matter how much growth appears after it was only proved mathematically in 1993 by Douady and Couder, two french mathematicians.

You will have already guessed what the fixed angle of turn is - it is Phi cells per turn or phi turns per new cell."[2] -- MacAddct1984 04:14, August 8, 2005 (UTC)

I have a rather detailed explanation of how φ truly appears in the structure of sunflowers, pine cones, and so forth at Talk:Fibonacci_number/Phyllotaxis. And of course it's quite clear that the nautilus shell is a logarithmic spiral. But if there is any connection between the nautilus and the golden ratio, it has never been made clear to me. In fact, it seems to me that since all logarithmic spirals are similar, the assertion that the nautilus is a special φ-related logarithmic spiral is probably devoid of content. -- Dominus 14:38, 9 August 2005 (UTC)

The following addition has been moved here from the middle of my original contribution to this "Nautilus nonsense" discussion:

ADDITION: The original author of this articles goes on to say that Nautilus shell does not represent a golden spiral. All one need do is to measure it. The following links show the phi relationships in the curve of the Nautilus shell: http://www.phimatrix.com/examples.htm, specifically the image at http://www.phimatrix.com/images/s-nautilus.jpg. The lines of the grid provided by the PhiMatrix software are all in phi proportion to the ones next to the it, so the Nautilus spiral expands at every OTHER phi line.

THE ORIGINAL AUTHOR CONTINUES ON WITH THE FOLLOWING INACCURACIES ABOUT THIS RELATIONSHIP BEING "INVENTED": -- Phi1618

When Dominus states that all log spirals are similar, that is incorrect. All Arithmetic spirals (e.g.) are similar, which means that they only differ by scale. If reflected, rotated and magnified or dimished appropriately, all Arithmetic spirals are identical. Log spirals are self-similar, meaning that any log spiral can be magnified by any factor, and then rotated to cover itself exactly. But that does not mean that all log spirals are similar to each other; see my original contribution.

I have checked out the nautilus image linked by Phi1618 in the addition above. I have two comments on this:

• The attempts to find golden ratios in such measurements suffer from the same weaknesses as those for finding golden ratios (or $\pi$) in the pyramids in Egypt, viz.: There are so many ratios one can find that no meaning can be assigned to the fact that some of them hapen to be near 1.618, unless it is extremely close, and/or unless it is accompanied by a plausible theory why it should mean something. In this case, I have conscientiously tried to measure the distances in the image that the superimposed grid suggests should be related to the golden ratio. I find ratios like 2.78 and 2.80, where they "should" be $\phi ^{2}=2.618$.

• In the log spiral discussed in the main article, the equivalent ratios are not $\phi ^{2}$, but $\phi ^{4}=6.854$.

A final comment: Some articles a particularly prone to addition of incorrect claims, because many people believe those claims are correct, and because many references can be cited for them. I think a serious encyclopedia should mention such claims, and debunk them. If they are just ignored, they will be added again and again, and many readers (yes, we do have readers too) may not get any wiser from reading our articles. So not mentioning the Nautilius in the main article is imho not the best choice.

--Niels Ø 08:26, 20 January 2006 (UTC)

Edit by User:Jacquerie27

I do not like this edit at all. First, it moved the nice, friendly introduction into multiple sections, which are unnecessary. The history and other names sections are pretty pointless. Second, that edit starts off the article with some math, which I don't think is a good idea. Next, the nice, flowing derivation of the math equations is interrupted, and part of it is just diverted to a "Mathematical Properties" section. --Chan-Ho Suh 22:22, Dec 14, 2004 (UTC)

I thought the introduction was a bit vague and unfocused and some math would be better at the beginning. But no problem. Jacquerie27 22:36, 14 Dec 2004 (UTC)

University of Cambridge Is Not a Dubious University

Note that in a new book recently published, modern day Egyptologist and architect Corinna Rossi (Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2004, pp. 23-56) presents fascinating and exorbitant evidence indicating ancient Egyptian knowledge of the golden ratio as demonstrated by a modern and comprehensive architectural analysis of ancient Egyptian structures.

Corinna Rossi PhD was a Junior Research Fellow (extended post doc because she couldn't get a proper job) in Egyptology at Churchill College, Cambridge, at the time that the text was originally published (last year). Over 300 references are cited.

This is not dubious scholarship by any measure.

Sorry, it seems our edits crossed. Anyway, the section right below makes clear my reasons. I suggest you've done a severe misreading of her text. At the least, your summary is misleading. Continue this in the section below please, so that it's clear to everyone what your response to my reasons are.

As a side note, your idea of what makes non-dubious scholarship is pretty amusing. The number of references is not important there. 300? So what? Rossi's work may or may not be dubious, but the number of references has little to do with that. Neither does the fact that Cambridge University Press published it. --Chan-Ho 02:18, Feb 11, 2005 (UTC)

300 is certainly a greater number than the number of references you quote. Churchill College is a "Scientific and Technological based college" inside of the University of Cambridge.

Still you don't get it. The number of references means nothing. There are books about all kinds of garbage that have hundreds of references. What does that mean? The fact that you brought it up, as if it were relevant, and the fact that you persist in thinking this is important shows you have a mistaken notion of what makes good scholarship. And yes, 300>1. What is the point of that? I'd be interested to know since it just seems to show an interesting kind of thinking. --Chan-Ho 04:04, Feb 11, 2005 (UTC)

As for the info on Churchill, thanks. Relevance? --Chan-Ho 04:06, Feb 11, 2005 (UTC)

Stay in mathematics Chan-Ho. :) You're not adept at Psychology. --Roylee

Rossi's book and the revert of a summary of her work

I reverted the following:

Note that in a new book recently published, modern day Egyptologist and architect Corinna Rossi (Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2004, pp. 23-56) presents fascinating and exorbitant evidence indicating ancient Egyptian knowledge of the golden ratio as demonstrated by a modern and comprehensive architectural analysis of ancient Egyptian structures.

My first instinct was to revert this because there's been a lot of misconceptions and frankly, bunk, and works of what I would call "dubious scholarship" by those who see the golden ratio in everything including pryamids and so forth. There's actually no evidence that ancient Egyptians had any "knowledge of the golden ratio", where by "knowledge" I mean knowledge of the golden ratio as an irrational number with certain properties.

However, after some investigation, I find that Rossi's book is probably not of the "dubious scholarship" variety. In fact, most of the book consists of her criticizing those who have perpetrated this kind of slopping thinking, e.g. see this book review. Rossi, it appears, would not argue that Egyptians had real mathematical knowledge of the golden ratio, although she says that they did use many shapes and ratios, of which the golden ratio was one (which is not surprising really). She emphatically says that they did not think of it as a preferred ratio or see it as special. Consequently, I feel describing Rossi's position as saying she believes in "ancient Egyptian knowledge of the golden ratio" to be deceptive, and at best, a sensationalized phrasing of what she actually believes. --Chan-Ho 02:09, Feb 11, 2005 (UTC)

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I don't know where you get that dubious information from, because I hold the very book in my hand as I type this now. I'd challenge you to support your statements with some direct quotes, such as:

page 32: In 1965, Alexander Badawy, the Egyptian architect and Egyptologist, suggested the most convincing theory based on the Golden Section.

page 35: Badawy suggested that the Egyptians achieved the Golden Section by means of the Fibonacci Series 1, 2, 3, 5, 8, 13, 21, 34, 55.... According to his theory, they adopted the ratio 8:5 (in which 8 and 5 are numbers of the Fibonacci Series), which gives 1.6 as a result, as a good approximation for Φ (that is, 1.618033989...).

pages 43,46 (illustrations on pages 44-45): With this system, Badawy successfully analysed more than fifty-five plans and a few elevations of Egyptian monuments from the Predynastic to the Ptolemaic Period, including civil, funerary and religious architecture (figs. 29, 30, 32 and 34-7).

page 46: His theory seems able to explain many factors. It suggests that a single set of rules was used throughout the entire history of Egyptian architecture (and beyond), that all of these rules were related to one another, that the Golden Section was among them, and that the Egyptians could have achieved these results using their own mathematical system and practical tools.

page 54: [Rossi explains at length criticisms of Badawy's scheme, but then settles on...] Despite these criticisms, Badawy's schemes seem to work [like] no one else['s]... and his method has been followed by other scholars.

page 56: He thought that the 8:5 triangle could have been a simple and practical device to approximate the convergence of the Fibonacci Series to Φ, thus implying that the Egyptian [sic] knew Φ and performed this calculation.

Quite obviously, your source (book review) has misinformed you.

Try actually reading the book some time. --User:Roylee

Thanks, I may read the book. However, your assertion that my reference is misinformed, or rather, that it has misinformed me is mistaken. For example, this excerpt:

Rossi is rather skeptical about theories that have tried to see the Golden Section in Egyptian architectural design as the preferred ratio, even though she does acknowledge that it was one of the proportions used by Egyptian architects, along with the proportions of the triangles already mentioned. Even though the author seems to have made good use of nineteenth-century primary sources that deal with such theories in her research, their presentation in the book is over-synoptic, without clarification for the reader as to what exactly these theories entailed. Furthermore, Rossi herself seems somewhat torn between the existence and the absence of a set of clear rules in Egyptian architectural design, as she mentions the 1965 theory of Badawy as "able to explain many factors," a theory which suggests that a number of triangles including the 8:5 and the Golden Section were used by the Egyptians in the design of their monuments among other geometric forms and ratios. Despite many points of criticism directed toward Badawy, Rossi acknowledges that "Badawy's schemes seem to work," and goes into a greater detail than she does for others in explaining his theory of how certain triangles seem to have been used in laying out the ground plans of certain Egyptian temples

is perfectly consistent with your quotes from the book. In fact, it's a pretty good summary of them. What's interesting is that the reviewer feels Rossi is not giving enough credit to the Egyptian builders (cf several remarks later in the review).

I don't see anything in your quotes that shows that Rossi believes Badawy's theory, just that it's the best theory that utilizes the golden ratio. My assertion that your summary is misleading stands. Your quotes have not refuted any of the remarks I made at the beginning of this section. --Chan-Ho 04:19, Feb 11, 2005 (UTC)

Your choice of words is very clever. At the time that this text was published, Rossi wrote, "...most convincing theory.... With this system, Badawy successfully analysed...." Though Rossi may not believe this today (and I would be interested to know why), at the time of printing, Badawy's ideas apparently "convinced" Rossi of a "successful" analysis of over 55 structures. Statistically, all we need is approximately 30 to prove our hypothesis to be sufficiently reliable. As a fellow mathematician, you know this already. But, yes, the sample is not randomly drawn from a large population, and perhaps the conclusion may even be biased one way or another. --Roylee

Deleted section "Fun with the ratio"

I have deleted the following recently added section:

Fun with the ratio

If you want to see how the golden ratio applies to your own body follow these steps:

Measure the distance from the tip of your head to the floor. Then divide that by the distance from your belly button to the floor. What do you get? 1.618

Measure the distance from your shoulder to your fingertips, and then divide it by the distance from your elbow to your fingertips. What do you get? 1.618

My reasons are:

1. It is not encyclopedic.
2. It is not accurate. The chances of getting 1.618 (correct to 4 significant digits) is remote. I don't have accurate statistics on this (and the results probably depend on age, sex and race), but I'd think typical results are anywhere from 1.5 to 1.7, say.
3. If these claims are to be mentioned in the article, the status of the claims must be stated too. They are obviously (to me) unrelated to the exact mathematical golden ratio.

By the way, would it be useful - and possible - to divide this page into one on fairly well established facts involving the exact golden ratio (or possibly the Fibonacci sequence), and one about the more mythical stuff?--Niels Ø 12:17, Mar 12, 2005 (UTC)

I'm not sure, but I'm guessing that the "Fun with the ratio" items came from someone who had recently read "The Da Vinci Code." I seem to remember seeing them there. I'm surprised that there isn't an entry on this page addressing the use (and misuse) of phi in that book, since it's apparently quite popular. fsufezzik 00:43, Mar 26, 2005 (UTC)

My interest in the Golden Ratio sparked from reading/hearing about how universal the number is supposed to be in nature, like those 'body ratios' that have been mentioned. I think the claims/rumors/whatevers should definitely be mentioned in the article, even if they are totally debunked. If it's untrue or supported scientifically/mathematically only to a limited extent, but commonly attributed to the number, shouldn't this article set things straight? Thus I agree that "the status of the claims must be stated too."

Re recent revision of introduction

The introduction to the article has been rewritten by user:Bill Cannon. Among other changes is this addition: The golden ratio seems to have been understood and used by the Egyptians. Can this claim be substantiated by primary sources? Otherwise, it should be moved down from the introduction, perhaps to a section dedicated to unsubstantiated historical claims.--Niels Ø 18:00, Apr 8, 2005 (UTC)

I have added a disambiguation message to the top of the page because "golden mean" redirects to this page. This seemed like the simplist and clearest manner to go about clarifying where Aristotle's theory is to be found. If anyone feels that this is inappropiate, or that there should be a seperate disambiguation page, feel free to discuss or create. ~CS 05:09, 15 Apr 2005 (UTC)

Artistic use

I would be grateful if some examples of the use of the golden ratio in architecture and painting could be added. Simonpockley 06:20, 29 Apr 2005 (UTC)

Verifiable examples are few and recent (end of 19th century or later). Unsubstantiated claims are abundant. Among the verifiable ones, le Corbusier is notable.--Niels Ø 06:31, Apr 29, 2005 (UTC)

The Roses of Heliogabalus

Is the ratio mentioned in the article The Roses of Heliogabalus correct? The 'golden mean' page would suggest so, or am I missing something? I figured I check here first instead of changing the article.

• Yup, it looks fine -- MacAddct1984 01:52, July 14, 2005 (UTC)

I looked at this article, and was confused because no mention was made of the Golden Ratio. It turns out that someone deleted it. He mentions the reason on the discussion page, and I don't know enough about it to agree or disagree with that move Bunthorne 04:13, 5 August 2005 (UTC)

Duplicate article

Perhaps the Golden Mean article should be merged with this article? Note that this is improperly named, with large M, not "Golden mean" with small m which redirects to this article. PAR 02:57, 14 July 2005 (UTC)

Perhaps a good idea. I dare to suggest, however, that whomever decides to attempt has their work cut out for them. I estimate it will take a long time and I hope the daring individual has some experience in such a task. GeneMosher 03:44, 15 July 2005 (UTC)

Merge 'em! They describe the same notion; retain as much information as possible and, where there are significant differences, treat them appropriately in a single article – Golden ratio – under different sections and with notes. E Pluribus Anthony 16:07, 22 September 2005 (UTC)

On the other hand, this article is pretty good, while the article on Golden Mean is not as well written. Also, Golden Mean mostly concentrates on the philosophical meaning of the term (not too much, not too little), although it also, confusingly, touches on the mathematics of the golden ratio and other "harmonious proportions". Perhaps the solution is a disambiguation page for golden mean, distinguishing its meaning as a synonym for golden ratio (pointing to this page), and its philosophical meaning (pointing to a version of Golden Mean that is limited to that meaning and substantially edited to improve it [unless it gets deleted first!]). Finell 10:12, 4 October 2005 (UTC)

Please see Wikipedia:Articles for deletion/Golden Mean. Michael Hardy 00:11, 3 October 2005 (UTC)

I'd like to understand this

Why is it that the number .618 is not acceptable as representing the golden ration while 1.618 is acceptable.

In other words, why it is not acceptable to represent the golden ratio this way

${\frac {x}{1}}={\frac {1}{x+1}},$

or, equivalently, the quadratic equation

$x^{2}+x-1=0.\,$

while it is acceptable to represent it this way

${\frac {x}{1}}={\frac {1}{x-1}},$

or, equivalently, the quadratic equation

$x^{2}-x-1=0.\,$

I think both are equivalent. To avoid confusion, 1.618... is chosen for historical reasons. PAR 14:54, 24 July 2005 (UTC)

When I entered the former expression it was removed and replaced by the latter expression. If both expressions are not wrong, they should both be there. I don't see the point of removing an expression that is correct. I think if they're both correct then they both should be shown. GeneMosher 18:11, 24 July 2005 (UTC)

1.618... is the golden ratio. Mathematically, theres no reason to prefer 1.618... over 0.618... but the golden ratio has been chosen to be 1.618... rather than 0.618. By chosen, I mean its the majority view of humanity, not just a few editors on Wikipedia. At least this is my present understanding. PAR 18:02, 1 August 2005 (UTC)

It's my opinion that this number's very special fascination to us derives precisely from the fact that it satisfies BOTH equations. The idea that removing an expression of a number is justified because it 'reduces confusion' makes no sense. There's no logic in such an idea. How can censoring a mathematical fact do anything but obscure the truth? It's a shame that, in yet another area, Wikipedia is a showcase of the triumph of editorial swagger over truth. GeneMosher 14:07, 6 August 2005 (UTC)

I agree completely with your first sentence. But nothing has been censored. I have seen some mathematical expressions expressed in terms of ρ=1/φ, the inverse of the golden ratio. See Polylogarithm#Polylogarithm ladders for example. Why not express what you are trying to say in terms of ρ? Why do you want to redefine the golden ratio to be maybe 1.618... and maybe 0.618... when the rest of the world defines it as 1.618...? Its not editorial swagger to follow the conventions of the rest of humanity when doing so does not suppress any truth. Any truth that could be expressed by redefining φ to be 0.618... can be expressed by using ρ=0.618... PAR 18:10, 6 August 2005 (UTC)

I regret that I'm not a mathematician. I do understand, thanks to M. Hardy, that this number, the Golden Ratio, is not a rational number! It cannot be expressed as a simple fraction, using only whole numbers, but although some prefer to express it as an indeterminate decimal, I'm uncomfortable with that. I prefer to express it as above because it makes sense to me that way. It's not a rational number but it is famously recognized as The Golden Ratio. It's an intellectual rush for me because I discovered this number for myself expressed as ${\frac {x}{1}}={\frac {1}{x+1}},$ So, perhaps someone who is comfortable with the language of mathematics can do what you suggest. What is, by the way, the term for a number that can be expressed as a ratio which uses not just whole numbers but also simple expressions?GeneMosher

Well, "simple" is hard to define mathematically, but there is such a thing as computable numbers. These are numbers which can be calculated to any precision you wish. They include every number which is "known", like any integer, any fraction, any square root, π, the golden ratio, etc. Any number for which there is a "recipe" which allows you to write it down to as many digits as you wish, is a computable number. Strangely enough there are infinitely more non-computable numbers than there are computable numbers. PAR 03:30, 8 August 2005 (UTC)

Removal of 'properties' section

I suggest the 'properties' section be removed since it is mostly redundant. Possibly the formulae for negative powers of rho should be added in the 'mathematical uses' section in the list of powers of rho. Any objections?

Acoustics

Maybe someone more numerate than me can evaluate the possibility of including a mention of the use of the ratio in design of loudspeaker enclosures? This seems like a very widespread use of phi (almost everyone reading this article either has loudspeakers or at least is familiar with them).--demonburrito 19:50, 5 August 2005 (UTC)

Startlingly Confounding Proof

Would anyone be able to expand on the proof of irrationality. I find it (as a fairly mathematical person) to be cryptic.

"The relation

${\frac {a}{b}}={\frac {b}{a-b}}$

gives a startlingly quick proof that this number is irrational: If a/b is a fraction in lowest terms, then b/(a − b) is in even lower terms — a contradiction."

A few more steps to explain how it works would be helpful. I'd do it, but I haven't the foggiest idea of what the proof is trying to say. 10/2/2005.

Your question seems quite un-specific about which part you don't understand. The definition of "irrational number" is: a number that cannot be written as a fraction j/k, where j and k are integers, i.e., "whole" numbers. The way to prove a number is irrational is by contradiction: assume that it's rational, i.e. that there is such a fraction, and show that that assumption entails a contradiction. If there is such a fraction, it can be put into lowest terms. If a/b is a fraction representing the golden ratio, and is in lowest terms, then no fraction in "lower terms", i.e. numerator smaller than a and denominator smaller than b, can also represent the golden ratio. But b/(a − b) is such a fraction in lower terms. That is the contradiction. For example, if it were claimed that the golden ratio is the rational number 21/13, we would have a = 21 and b = 13, and b/(a − b) = 13/(21 − 13) = 13/8. But since 21/13 is in lowest terms, it cannot be equal to 13/8, since that's in "lower terms".

Sorry to drag this out in such long-winded fashion, but you were not specific about what you didn't understand. Michael Hardy 23:45, 2 October 2005 (UTC)

Was this a request to expand on this argument within the article? If so, please be more specific. The argument seems crystal-clear to me. Michael Hardy 19:30, 4 October 2005 (UTC)

The argument in the proof (tacitly) assumes that b > a-b or 2b > a. In the case of a (hypothetical) rational representation of the number 1.618033... < 2 (the positive golden ratio) the argument is in fact valid. One might also argue as follows (a bit sketchy): if the representation is in lowest terms the numbers a and b are coprime. Then the denominators b and a-b of the two irreducible fractions are coprime as well. But then the fractions must represent integers and then b = 1, a = 2, a contradiction.

--212.18.24.11 08:35, 13 October 2005 (UTC)

Material from the former Golden Mean article

... which is now a redirect to a disambiguation page. Some of what is below should be considered for incorporation into this article:

The two parts of the Golden Mean can be seen in the golden rectangle. These major and minor parts are unequal opposites united in a harmonious proportion. This is a pattern that repeats itself throughout nature. This pattern-forming process is the union of opposites. György Doczi coins the term dinergy for this.

Pythagoras

According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart.(Modern music theory calls a 5:4 ratio a "major third" and an 8:5 ratio a "major sixth".) In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers.

Astonished by this discovery and awed by it, the Pythagoreans endeavored to keep this a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.

The symbol of the Pythagorean brotherhood was the pentagram, in itself embodying several Golden Means.

Golden mean in art

In architecture, the golden mean is the ideal relationships of mass and line which the Greeks perfected over time. Moreover, they found that architecture and art that incorporate this feature are more pleasing to people. This finds its perfection in the Parthenon. This can be compared to one of the first examples of Greek temple building, the temple of Poseidon at Paestum, Italy as it is squat and unelegant. The front of the Parthenon with its triangular pediment fits inside a golden rectangle. The divine proportion and its related figures were incorporated into every piece and detail of the Parthenon.

The Triumphal Arch of Constantine and the Colosseum, both in Rome, are great examples ancient use of golden relationships in architecture.

Phidias, a famous ancient Greek sculptor, incorporated the Golden Mean in all his work.

Golden mean in the Renaissance

Woodcut from Luca Pacioli's Divina Proportione depicting the golden mean as it applies to the human face.

Kepler was fascinated by the mystery of the Golden Mean and also coined it as the "divine proportion". Luca Pacioli, in 1509, wrote a dissertation called De Divina Proportione which was illustrated by Leonardo da Vinci.

Golden mean in psychology

The famous German psychologist, Gustav Fechner, inspired by Adolf Zeising’s book, Der goldene Schnitt, began a serious inquiry to see if the golden rectangle had psychological aesthetic impact. It was published in 1876. With German zeal of thoroughness, Fechner made thousands of measurements of commonly seen rectangles, such as writing pads, books, playing cards, windows, and found that most were close to Phi. He also tested people’s preferences and found most people prefer the shape of the golden rectangle. His experiments were repeated by Witmar (1894), Lalo (1908) and Thorndike (1917).

Quotations

• "Geometry has two great treasures: one is the theorem of Pythagoras; 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."
  — Johannes Kepler

References

• The Divine Proportion, p. 75

See also

• Philosophy of mathematics
• Mathematics and God
• Mathematical beauty

Bibliography

• The Divine Proportion, A Study in Mathematical Beauty, H. E. Huntley, Dover Publications, Inc., NY, l970.
• The Power of Limits, Proportional Harmonies in Nature, Art, and Architecture, György Doczi, Shambhala Publications, Inc., Boston & London, l981.
• Der goldene Schnitt, Adolf Zeising, (1884)

External links

• The Golden Mean

Leonardo Fibonacci (filius Bonacci), alias Leonardo of Pisa,

My Apologies

I'm sorry for perceivably messing up the article; I didn't mean to. Initially I was trying to make some simple clerical errors, then I began to eliminate redundant or extraneous information, and finally I got flustered by the seemingly random equation (which has now been asterisked) that seemed to imply manipulating two equations to equal one another was a valid proof or equivalency.

I'm glad that things have been sorted out, although I think there should be some note included to address the conjugate golden number.

--24.126.30.46 01:07, 16 October 2005 (UTC)

That's OK. Thanks for contributing! -- Dominus 02:07, 16 October 2005 (UTC)

Remove dab legend?

I believe that the disambiguation legend at the top of this article should be deleted, but I wanted to solicit others' views before doing this myself. No one who is specifically looking for information about golden ratio is trying to learn about the philosophical concept of the golden mean (i.e., the middle course between the extremes excess and insufficiency), so the dab legend is unnecessary. On the other hand, the dab legend at the top of Golden mean (philosophy) is appropriate because some people use golden mean as a synonym for golden ratio—but not vice versa! By the way, I am familiar with the history of the proposal to merge this article with the article that is now called Golden mean (philosophy), which was defeated in favor of editing both articles to eliminate overlap.

Does any one disagree or agree? Finell (Talk) 11:13, 9 November 2005 (UTC)

Agree. --R.Koot 17:13, 9 November 2005 (UTC)

In the absense of disagreement, I did it. Finell (Talk) 02:44, 17 November 2005 (UTC)

Phi (Golden Ratio)

A new entry was created at Phi (Golden Ratio). It seems an obvious merge, unless someone explains why. Agree? -- ( drini's _vandalproof page ☎ ) 04:22, 16 November 2005 (UTC)

I agree, but suggest that we keep all discussion at Talk:Phi (Golden Ratio), to avoid having duplicate discussions of that duplicate article. Finell (Talk) 02:31, 17 November 2005 (UTC)

overlaps that a clear on the code won't fix, nor a clear on the offending image.

screen shot of a layout problem in the article. I've tried clear:right on the code and clear:left on the image that overlaps it; neither solves the problem. I'm left puzzled, but the problem needs solving regardless.

The overlap occurs between the sections "Aesthetic uses" and "Decimal expansion," with the image "Divina proportione" and the code starting "6180339887 4989484820." Koyaanis Qatsi 04:33, 30 November 2005 (UTC)

Recurrance

Who found the φ^n = F(n)φ + F(n − 1) relationship? I discovered this a few years ago and was curious where it came from since several mathematicians (who were familiar with φ) I consulted hadn't seen this before.

Simpler

phi can be written as

$\varphi ={1+{\sqrt {5}} \over 2}$

$\varphi =.5+{{\sqrt {5}} \over 2}$

$\varphi =.5+{{\sqrt {4}}{\sqrt {1.25}} \over 2}$

$\varphi =.5+{2{\sqrt {1.25}} \over 2}$

$\varphi =.5+{\sqrt {1.25}}$

I find this simpler. JedG 00:15, 7 February 2006 (UTC)

The last form does save one operation, but it loses some transparency. For algebra (as distinct from computation) it's usually convenient to express rationals as rationals (1/4, 5/4). —Tamfang 18:06, 17 February 2006 (UTC)

And if you're doing it on a hand calculator, your way takes one key more than the algebraic approach:

1 . 2 5 √ + . 5 =

vs

5 √ + 1 = / 2 =

—Tamfang 18:41, 17 February 2006 (UTC)

JedG, the rational number you're putting under the radical is not square-free. And there is such a thing as the purposes of algebra as opposed to those of computation. Michael Hardy 22:40, 28 March 2006 (UTC)

Venus

It is also believed that after tracing the path of Venus in the sky, they found that the ratio of the length of the long arm of the pentagon shape to the length of the shorter arm was 1.618 ... ...

I'll bite: what has Venus to do with pentagons? —Tamfang 17:56, 17 February 2006 (UTC)

I agree with Tamfang's reaction. I think this sentence ought to be removed. The apparent non-sequitur is confusing to readers (such as myself and Tamfang); but even if the word "pentagon" is corrected to "pentagram," and the astronomical connection between Venus and the pentagram is explained, the resulting assertion is still unhistorical nonsense. There is no reason to believe that "the [unspecified] ancients" first encountered the pentagram and/or pentagon through 8-year observations of Venus. The Phi ratios in the regular pentagram are inherent, and can be deduced independently of the means by which the shape comes to the geometer's attention.

If we're going to have a passage describing discovery of the Phi ratio in the pentagram among "the ancients," it would be better to specify a particular person, or at least a culture, that observed it.

-- Vogelfrei 22:56, 25 April 2006 (UTC)

So-called Myth

It is historical fact that shapes according to the G.R. have long been considered pleasing. If you have a different personal aesthetic that may consider them non-pleasing, it doesn't alter the historical fact. Writers since the time of the ancient Greeks have written about how the shapes have been pleasing. To call it a myth that it has been considered pleasing is to introduce your non-neutral point of view. I suspect that what you are trying to say is that the pleasingness is a myth. That is closer to the truth, but because the pleasingness of the aesthetic is debatable does not make it a myth, it just makes it debatable. All the same, it is important to distinguish between "considered pleasing" and "pleasing". Hu 01:09, 1 March 2006 (UTC)

The issue isn't the difference between "considered pleasing" and "pleasing." Nobody's going to argue that ratios around 1.6 are pleasing in certain contexts, and that this has been experimentally established. The myths are that (a) the exact golden ratio is more pleasing than 1.6, pi/2, etc., and that (b) the ancients knew this.

Also, the statement that "Writers since the time of the ancient Greeks have written about how the shapes have been pleasing" is blatantly wrong. There are no (surviving) Greek writings that say this, or even hint at it. If there were, why would people spend centuries trying to prove Greek knowledge indirectly? Wouldn't they just cite those writings? 69.107.70.159 17:18, 5 March 2006 (UTC)

Law of Small Numbers

This sentence has been removed due to bad wording and debatable truth: "It crops up frequently as a simple consequence of the law of small numbers and therefore mathematical investigations involving this number often arise even when the golden ratio is the farthest thing from the investigator's mind."

I think the writer is trying to say that in their opinion, the coincidence of the golden ratio with natural phenomena is just chance. That means the sentence is in the wrong section, i.e. "History" is not the place to discuss the significance. And the sentence has nothing to with Egyptians, astronomers or ancient Greeks. Further, the law of small numbers has nothing to do with this. The appearance of the golden ratio is not a numerical bias, nor is it a consequence of the pigeonhole principle referenced by the LoSN article. Then the sentence talks about mathematical investigations arising and links that to the investigator being lead to the GR unexpectedly, which contradicts the unspoken thesis of bias behind the first part of the sentence.

If an editor wants to clarify the point attempted by the sentence, I invite them to do so here, and then if it can be written in a sensible neutral PoV way, we can find an appropriate section to put it in. Hu 01:26, 1 March 2006 (UTC)

PHI Pentagram and the number of the Beast 666

PHI ratio can be found in the pentagram, which is traced out in the sky by the planet venus every 8 years, which was first observed by the grooved ware people, who then through trade transmited this knowledge using religion to other cultures, such as the greeks, the name of the greek goddess of venus was aphroditie and her number was 666, the church decreed that any worship other than that of god was the devil, all western cultures have historcal licks to venus worship, thus the number of the beast is 666 which is related to PHI and Pi —The preceding unsigned comment was added by 81.157.75.166 (talk • contribs) .

World Flags

Does any country or province use the golden ratio for its official flag? Finland seems to come close, but maybe there's closer?--Sonjaaa 06:42, 19 March 2006 (UTC)

Wrong lead?

The lead claims that the Pyhthagorians appraised the Golden Ratio. This is unlikely to be correct as the golden ratio is an irrational number not expressible in terms of whole numbers, and the pythagorians refused even to admit the existence of irrational numbers. It is well known that they murdered to keep the irrationality of the square-root of two unknown. Loom91 12:32, 28 March 2006 (UTC)

If they didn't know this was irrational, why would its irrationality prevent them from studying it, just as they did √2? And if they did know that √2 is irrational, why could they not have known the same thing about this number? Michael Hardy 22:22, 28 March 2006 (UTC)

I was just asking someone to verify this doubtful factoid. They were unlikely to be fascinated by an irrational number. Only possible scenario is that they thought it to be a rational number, but this needs to be verified. Loom91 10:35, 29 March 2006 (UTC)

More about 666

I recently removed the following paragraph:

Additionally, the equation $-\varphi =\sin 666^{\circ }+\cos(6^{\circ }*6^{\circ }*6^{\circ })$ draws an interesting (albeit somewhat forced) connection between φ and 666, the Number of the Beast

I did this for two reasons. First, the statement is false. The correspondence is close, but not exact. Second, it appears to be a mathematically uninteresting coincidence. Many numbers happen to be close to other numbers, and this one is lent special "interest" only through a connection with Christian theology and numerology. However, numerology is not mathematics; it is superstition. Discussion of numerological relationships is not appropriate material for a serious encyclopedia article about mathematics.

If there were some cult or sect that ascribed importance to this particular numerical coincidence, that might be both interesting and encyclopedic. The fact might also be worth citing in an article about numerology, or an article about strange coincidences. This article, however, is neither.

The paragraph had been removed before, and was put back, with a citation to a book. The citation is irrelevant here, because, citation or no citation, the "equation" is still false, and the "fact" is only one of an infinite collection of such "facts" that certain numbers are close to certain other numbers, that are of no theoretical or practical interest. -- Dominus 02:28, 15 April 2006 (UTC)

• Hey. It was me who added the information about the relationship between the Golden Ratio and the Number of the Beast. You took it out saying that it was false and uninteresting. I beg to differ. First of all, it is not false. Type that equation into any calculator or draw it out on the unit circle and you will quickly see that you get $-\varphi ={-(1+{\sqrt {5}}) \over 2}$ which is exactly correct and not in any way "false."

In regards to your outlining this equation as simply numerology, I feel that that is false. The equation provides an interesting connection between two famous numbers (as outlined in an article in the Journal of Recreational Mathematics, which, unfortunately, I can not find to cite itself). The equation was placed in the section including alternate forms of the original φ equation -- essentially different things to plug and chug in a calculator and get the same result. Appended to the very bottom, with an innocuous "additionally," the information about this final and interesting form was both subtle and accurate.

As for the citation added, if you were to read the book (which is, admittedly, a fascinating read in its entirety) you would find the exact equation found in a book devoted entirely to the Golden Ratio. Significant portions of this chapter are published (and apparently plagiarized) on this website if you're interested.

It is also probably important to point out that I am a math major, and neither a numerologist nor some clod who skimmed a book about a number.

Anyway, it's not my place to re-revert the article without a consensus, so if you have any responses, please be so kind. ---Dana 03:06, 15 April 2006 (UTC)

I apologize. I was wrong. I did calculate it numerically, but because of roundoff error I got slightly different results, and thought it was a coincidence. I now realize that both 666 and 216 are multiples of 18° and that the equation is in fact true.

I still think it's silly, but I no longer object to including it. Thanks for taking the time to point out my error. -- Dominus 03:50, 15 April 2006 (UTC)

The equation is still wrong as it stands because of the extra degree signs. However, I now realize that I made an error in my previous calculation (when I first removed it) and it would be correct, if one said: $-\varphi =\sin 666^{\circ }+\cos((6*6*6)^{\circ })$. However, it still seems too loaded with superstition to put into the article. 66.44.0.202 07:27, 15 April 2006 (UTC)

I was going to mention that, then decided it was not a substantive complaint. It is easily corrected and doesn't need to be discussed. -- Dominus 12:53, 16 April 2006 (UTC)

Thank you guys for reconsidering. I recognise that you think the equation is silly, but I personally really like the equation and I often find it easier to remember than $\varphi ={1+{\sqrt {5}} \over 2}$ itself. I think that it should be included for the same reason the trio of triginometric equations above it are included:

$\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }$

$\varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }$

$\varphi =2\cos(\pi /5)=2\cos 36^{\circ }$

And you're right, I did make a stupid mistake in the original equation (and thanks for noticing, thats what Wikipedia is for!), so I will correct it to $-\varphi =\sin 666^{\circ }+\cos(6*6*6^{\circ })$, which is more elegant, less cufty, and actually correct.

Again, thank you guys for your reconsideration and your help. ---Dana 12:21, 15 April 2006 (UTC)

Something missing?

From the article "The early digits can be found fairly easily on a calculator, using the formula" What formula?? --Dumarest 22:31, 10 May 2006 (UTC)

Between the word formula and the period is a math-mode formula using the text {1+\sqrt{5} \over 2}. Is it not showing up in your browser? You may need to change your math preferences. Or if your calculator doesn't have a square root, use a ratio of successive fibonacci numbers to get pretty close. Dicklyon 22:58, 10 May 2006 (UTC)

ERROR in "the successive powers of φ obey the Fibonacci recurrence" ?

In the section of "Mathematical Uses" , in a third paragraph "..Furthermore, the successive powers of φ obey the Fibonacci recurrence:..." I believe there is an error in the column section starting at

"φ3 = 2φ + 1, > should read φ3 = 3φ + 3?

φ4 = 3φ + 2, > should read φ4 = 4φ + 4?

φ5 = 5φ + 3," > should read φ5 = 5φ + 5?

Can someone who understands golden ratio professionally, make sure the successive powers formula is correct? Sorry, if i'm wrong...

Thanks —Preceding unsigned comment added by MaestroMuzon (talk • contribs) 10:29, 13 June 2006

I'm afraid you are wrong! The formulas are correct as given, and you can check them with a calculator: for example, if you evaluate [(1+sqrt(5))/2]^5 and 5(1+sqrt(5))/2 + 3. you should get the same answer. Madmath789 18:39, 13 June 2006 (UTC)

Muzon, consider the ratio between successive terms of the sequence. Your version, ${{\phi ^{n+1}} \over {\phi ^{n}}}={{(n+1)(\phi +1)} \over {n(\phi +1)}}={{n+1} \over {n}}$ which is not equal to φ! —Tamfang 21:22, 13 June 2006 (UTC)

How did you get that ratio? The general term is $\varphi ^{n}=F(n)\varphi +F(n-1)$. Dicklyon 01:17, 14 June 2006 (UTC)

I gave the ratio between successive powers if MaestroMuzon's correction were accurate. —Tamfang 02:24, 14 June 2006 (UTC)

Oops, my misunderstanding. Dicklyon 00:07, 18 June 2006 (UTC)

Various names

After I added "golden cut" to the various names, user Dicklyon removed it. From the first few hits on Google I select here the two most authoritative looking ones:

• thinkquest.org
• mathcentral.uregina.ca

They use the name in one breath with several of the others in the article. Additonally of course the name "golden section" comes from the latin word secare, which means to cut. So can you explain why you think it is not a valid alternative name? −Woodstone 15:58, 17 June 2006 (UTC)

I used google, too, but I used books.google.com to check in books, which tend to be more authoritative than random web pages. I found some hits on golden cut in books on Fibonacci and golden ratio, and read how they used it, which was generally for the POINT that cuts a segments into two segments in the golden ratio. One book did say the Golden ratio "had been called...golden cut", but I don't think that should be taken as an indication that golden cut is a synonum or alternate name, if it has a more common meaning in this context. Dicklyon 20:42, 17 June 2006 (UTC)

Also, "golden section" is sometimes used like "golden cut," but it is also usually mentioned as an alternative name for golden ratio, when golden cut is not. Dicklyon 20:46, 17 June 2006 (UTC)

I cannot see any difference between "golden secion" and "golden cut". They are direct translations of each other. Also "sectio divina" has the same root (of the "cut" part). These and "golden mean" are all about dividing a line segment in special way.

The other names "golden ratio", "golden proportion", "divine proportion" and "golden number", talk about the actual ratio that this division has.

None of these talk about the dividing point itself, but about the segments created and their ratio.

So all together, I see no base for your assessment. If, as you admit, "golden section" is sometimes used like "golden cut", that is enough reason to include that name in the article. −Woodstone 21:34, 17 June 2006 (UTC)

I don't follow your logic; especially that last bit. As I said, my assessment was based on my reading of books on this subject. The only one that suggested the possibility of "golden cut" as another name for "golden mean" couched it as "has been called", which I take as only one too-weak reason to include "golden cut" among the other more accepted terms. If you disagree, find something more authoritative than a couple of amateur web pages as evidence that the golden mean is sometimes called the golden cut. Conversely if you think that "golden section" should be removed, check the literature. Our purpose to not to apply logic to decide what it SHOULD be called, but rather to document what it IS called. Dicklyon 22:47, 17 June 2006 (UTC)

ps. Your two "most authoritative" references have severe problems. In the first, the guy admits to not caring about the question of what it is called: "I could keep this up for hours. There are probably as many different names for the Golden Ratio as there are occurances and applications. It really doesn't matter what you want to call it, because at the end of the day we're all talking about the same thing: the most irrational number ever." The second references a book, and that book does NOT use golden cut as a term for the golden ratio, but rather as a cut point. Dicklyon 22:50, 17 June 2006 (UTC)

Woodstone, I see you're persistent in wanting to apply your logic to what the golden ratio might be called. Can you provide a source that would help us verify either the grouping of the terms as you've now done them, or any source where the golden ratio is called the golden cut? We're supposed to be accumulating verifiable information, not making it up or imposing our own logic on it. Dicklyon 16:23, 18 June 2006 (UTC)

I took the liberty of removing your insulting choice of words. I already gave you references. They show that golden cut is used much the same as golden ratio, with a slight difference of focus. Here are some more:

• uga.edu
• gvsu.edu math

There are aslo several that make the same grouping as I included in the article. They invariably recognise that golden section is identical to golden cut (just translated).

−Woodstone 17:22, 18 June 2006 (UTC)

My term "waffle words" to characterize your use of "focusing more towards the division of the whole into parts" as a way to justify non-names for the the golden mean was not meant to insult, but to characterize. I apologize if I caused offense. I'll look at your references... Dicklyon 17:46, 18 June 2006 (UTC)

Re the two new refs, very similar to the first pair. The first simply lists golden cut among things that the golden ratio is called but doesn't either actually call it that or provide any reference to anyone who calls it that. The second specifically uses the term "golden cut" in the way other way, as Point C is the "golden cut" of line AB.

Usage seems to be pretty much against you. While authoritative books on the golden ratio do support the inclusion of "golden section" and the latin that it is derived from, they do NOT support putting "golden cut" into that same category. I'm not sure what you mean when you say "They invariably recognize..." Have you tried looking in books? Try http://books.google.com Dicklyon 17:53, 18 June 2006 (UTC)

Of course my refs are similar: they all prove the same point. What do you expect? You say usage is against me? Which references have you shown that explicitly state that golden "cut" is something really different than "section".

Let's look at the positive side. I suppose we agree there are two (or three) aspects:

• dividing something into parts in a special way
• the special ratio between the size of those parts
• the boundary between the special parts

Both are already described and named in the article. In my view they are traditionally all at least called "golden section", and your approved version seems to agree. We should be able to work out a mutually agreeable way to formulate this in the header section.

−Woodstone 18:53, 18 June 2006 (UTC)

OK, since you're not going to check the books I'll do the work for you. Consider the set of indexed books on Google that use both "golden segment" and "golden cut" [3]. I see eight of them. Let's look at each.

1 The Golden Ratio and Fibonacci Numbers - Page 1 by Richard A Dunlap - Mathematics - 1997 - 162 pages

"It has been called the golden mean, the golden section, the golden cut, the divine proportion, the Fibonacci number and the mean of Phidias and has a value ..." -- this one's on your side.

2 Fibonacci Applications and Strategies for Traders - Page 7 by Robert Fischer - Business & Economics - 1993 - 192 pages

"C ... b H L Golden cut Figure 1—4 Golden section of a line. The Golden Section of a Rectangle In the Great Pyramid, the rectangular floor of the King's ..." -- this one uses "golden cut" in the figure, whose caption uses golden section, and on the next page refers to "point E, called the golden cut". It does makes golden section a synomym for golden cut, but does NOT make golden cut a synonym for golden ratio.

3 The Divine Proportion - Page 25 by H E Huntley - Mathematics - 1970 - 185 pages

"Golden cut respectively. If C is a point such that 1: a as a: b, C is the “golden cut” or the golden section of AB. The ratio i/a or a/b is called the golden ratio..." -- this one totally distinguishes the terms and supports my point

4 The Dynamics of Delight - Page 80 by Peter F Smith - Architecture - 2003 - 208 pages

"And when the fractions are reversed the result ix 0.618, the ‘golden cut', Altogether it is evident that the golden section ratio is one of the main ..." -- this one is unusual in that applies the term "golden cut" to phi-1, 0.618.

5 The Fine Art of Decoupage - Page 28 by Lyn Cochrane - Crafts & Hobbies - 2001 - 144 pages

"dealing with rectangles, for example, is the so-called ‘golden cut' or ‘golden section', giving you a rectangle whose sides are in the approximate ratio of 3 to 2..." -- does not apply the term to the ratio, and doesn't even seem to have the concept of their being an exact interesting ratio; it's way off topic for this book

6 Symmetry: A Unifying Concept - Page 161 by Istvan Hargittai - Science - 2003 - 240 pages

-- sorry, the page is restricted, so we can't see what it says after "There is a special rectangle with proportions corresponding to the golden ratio. It is called also golden ????..."

7 Elementary Experiments in Psychology - Page 198 by Carl E. (Carl Emil) Seashore - 1908 - 218 pages

"Thus, the most pleasing ratio centers around 1:1.6, which is known as the golden section or golden cut.* * The golden section for the rectangle is that in ..." -- this one is on your side, but again it merly says it's known as, and then never actually calls it that. Pretty weak support.

8 The New Fibonacci Trader: Tools and Strategies for Trading Success - Page 260 by Robert Fischer, Jens Fischer - Business & Economics - 2001 - 368 pages

on page 11: "through point E, also called the golden cut of AB..." -- supports the meaning that is not phi.

So I see the question this way: should the wikipedia repeat the assertion from a 1908 psychology book that says the golden ratio is "known as the...golden cut", or should we respect that term for the way it is more widely used in authoritative books on the subject? If we decide the latter, then we can ask the same question about "golden section" and others, and see what decision is best supported by history.

Notice that NOBODY actaully uses the term "golden cut" for the golden ratio (unless you find such an example). Instead, all we have is a "has been called" and a "is known as" with no further mention.

I justed checked Mario Livio's excellent book, which I have. It doesn't mention golden cut at all, but has a whole page or so on the origin of "golden section", which comes from the German "goldene Schnitt", which could have equally well been translated some place as "golden cut", but as far as I can find is not ever used that way; perhaps the 1908 book got it by translating the German. The original meaning of golden section was also not for the ratio, but probably came to be closely associated via the 1895 Scientific American article "The Golden Section", which was the first English occurence. Livio's is a great book, by the way. Check it out.

Dicklyon 20:31, 18 June 2006 (UTC)

I redid the intro. See if we agree. Dicklyon 21:14, 18 June 2006 (UTC)

Good work, but I'm a bit confused about your conclusion. First, nr 8 contains on p260 ... golden cut (also called golden section) .... For nr 4 you sould not say φ-1, but 1/φ: the inverse ratio (just looking from the other side).

No problem, whether you call it phi-1 or 1/phi, it's 0.618, not phi. The point is that it is not one of the names of 1.618. And this usage is unique to this book, as far as we know.

So number 1 equates all terms, nrs 2,3,5,7 equate cut&section, 6 cannot be counted, 4 compounds the name to "golden section ratio".

I don't don't totally agree with that simplification, and it partly misses the point about which are synonyms for golden ratio.

So the conclusion should be that indeed a grouping is warranted:

• dividing a segment into pieces in a special way
  • "golden secion", "sectio divina", "golden cut"
• the ratio between the size of the pieces
  • "golden ratio", "golden proportion", "divine proportion", "golden number"

Since the article is named "golden ratio" that list should go first.

−Woodstone 21:40, 18 June 2006 (UTC)

But since golden section really is a common name for phi, and golden cut is not, I would be against pretending that they are used interchangably or only for the cut. Dicklyon 22:28, 18 June 2006 (UTC)

How about a "google fight"? Google books shows 273 hits for "golden ratio" and "golden section" together, but only 5 for "golden ratio" and "golden cut". Google web search numbers are 43,500 versus 258. I think that's pretty compelling empirical evidence that these terms are not used interchangably. Dicklyon 22:43, 18 June 2006 (UTC)

I'm still confused by your reasoning. Many of the refs above make clear that golden section and golden cut are synonyms, as you acknowledge explicitly at nr 2. Furthermore you insist that golden ratio and golden section are synonyms. Doesn't that make all three synonyms?

Phi is the ratio between two line segments a and b. The abstract definition does not really say if the golden number is a/b or b/a; it is just much more common to take it as larger over smaller, but the other way does occur in some texts. It never has a separate name.

The google fight is misinterpreted. The term golden cut is just much less frequently used than golden ratio; so automatically the combination would score even lower. Many texts wil not use more than one word. Even the combination golden section and golden ratio scores only 39000, almost less than 10 % of them each separately. Sorted by single term, the frequencies are golden ratio (529000), golden section (344000), golden number (93000), golden cut (51000), sectio divina (800), extreme and mean ratio (700), while phi cannot be meaningfully searched.

Would it be an idea to just name the ones over 1000 first (in sequence) and follow with the less common names.?

−Woodstone 19:01, 19 June 2006 (UTC)

Since I disagree with too much of what you say to continue a friendly banter, I'll just drop out of it and let you and others decide what to do. Dicklyon 19:31, 19 June 2006 (UTC)

Organization

Wouldn't it be advisable to merge "History" and "asthetic uses", as they both deal with uses of the Golden ratio? Also, it seems that it would be best to put the article either in the order of all sections made up of text, then mathematical sections, or vice versa, rather than having the two mixed helter-skelter throughout the article. Phi*n!x 23:06, 20 June 2006 (UTC)

Done, except I kept the (shortened) calculation right after the header. −Woodstone 08:51, 4 July 2006 (UTC)

Wide-Screen

Did anyone else notice that the golden ratio is very close to wide-screen computer monitors? (16:10) or (1.6:1)--God Ω War 04:30, 9 July 2006 (UTC)

No, wow, that's an awesome discovery! I'm sitting in front of one right now, and damned if I didn't miss that! Dicklyon 04:42, 9 July 2006 (UTC)

put another way, 8:5 is a Fibonacci ratio, a member of the sequence of rational approximations to φ —Tamfang 05:03, 9 July 2006 (UTC)

exactly, just like 5:3, 3:2, 2:1, and 1:1 are. Dicklyon 06:08, 9 July 2006 (UTC)

though, is there such a thing as a 2:1 or 5:3 monitor? (PLATO IV was 512×512, and the early Mac was 512×342.) —Tamfang 06:35, 9 July 2006 (UTC)

All of those numbers are approximations. The closer you get to 0 the farther off the approximation is. However at 8:5 you are only 1% away from the true golden ratio. Numbers like 5:3 and 3:2 are way off.--God Ω War 06:53, 9 July 2006 (UTC)

Indeed, due to the continued fraction with all 1s, each step cuts the error just more than in half. 5:3 = 1.6667 is a little over 2% off, and 13:8 = 1.625 is less than a half percent off. The 5:3 is an excellent rational approximation to some wide-screen formats. Invoking the golden ratio to explain that is, however, just another false sighting. Dicklyon 15:54, 9 July 2006 (UTC)

...or not. ≈ jossi ≈ t • @ 00:17, 23 August 2006 (UTC)

Futbol (soccer)

Did anyone else notice that the football has 20 hexagons and 12 pentagons? Beside the golden ratios inherent in the pentagons, there's also that ratio of 20:12, or 5:3, one of the fibonacci approximants to the golden ratio. Besides that, the flags of both Italy and France, the world cup finalists, are composed of three rectangles of ratio 2:1 arranged into one rectangle of 3:2. So we have today a spotting of ALL successive ratios from the sequence 1, 1, 2, 3, 5, 8 if you watch it on a 8:5 TV. And I'll give 13:8 odds that France wins. Dicklyon 18:06, 9 July 2006 (UTC)

Richard K. Guy points out: "There are not enough small integers available for the many tasks assigned to them." —Tamfang 21:16, 12 July 2006 (UTC)

Leaden ratio / Leaden number

I wonder if there exists a leaden ratio. Such a ratio may be informally defined as 1:x, where:

• 1:(x±δ) is more aesthetic than 1:x
• x>0

If this exists, I wonder how one may go about determining the value of x. If not, why not? --Amit 22:46, 12 July 2006 (UTC)

The golden ratio is the most irrational; the leaden ratio could be the most rational, viz 1:1. —Tamfang 06:05, 13 July 2006 (UTC)

Upon some thought, I think no such leaden number exists, for if it did, there would then have to probably exist another golden number to counter it, which would therefore imply an entire series of golden and leaden numbers, and such a series is just implausible.

Golden names again

I think the opening of what it's called needs to be well documented, so I found a set of three books that cover the alternate names, and added the other names that I encountered along the way. I had to throw out some comments that were hard to find a reliable source for. For example, there seems to be considerable difference of opinion about what da Vinci first called it; a sentence with a reliable source for that would be useful. Whatever we add up front about terms, let's make sure it has a reliable reference (I prefer older ones, since there seems to be an unneeded explosion of terms in very recent books, but I haven't found old books for all the terms). Dicklyon 17:47, 13 August 2006 (UTC)

Luca Pacioli used divine proportione and sectio divina in his Divine Proportione (1509). We can use that as a source for these two terms. As for Da Vinci use of sectio aurea, The source I have is the same as the one I used for the "Timeline" subsection: Hemenway's Divinie Proportion. ≈ jossi ≈ t • @ 19:22, 13 August 2006 (UTC)

Formula deleted?

Why was Euclides proof deleted? ≈ jossi ≈ t • @ 22:35, 13 August 2006 (UTC)

It appears to have moved up to the section called Calculation. Is there some part of it missing? Dicklyon 00:27, 14 August 2006 (UTC)

Yes, I realized that another formula was already there that was quite similar, so I just added some words about Euclides to the existing formula. ≈ jossi ≈ t • @

Timeline

Doesn't the timeline need dates? It looks sort of uninformative, or hard to relate to anything I know, without dates. Dicklyon 00:29, 14 August 2006 (UTC)

I will add the dates. ≈ jossi ≈ t • @ 02:20, 14 August 2006 (UTC)

Rectangle image

I think that a grey rectangle on its own does not make this article better. The rectangle construction diagram, shows both the rectangle itself, and a way to construct it. ≈ jossi ≈ t • @ 19:09, 15 August 2006 (UTC)

The point of the rectangle image was to make the so-called most pleasing shape as apparent as possible, without distractors, and as accurate as a square-pixel screen could do in a limited space. It also introduces the notion of "golden rectangle". Does it make the article better? Certainly better than it was when its caption was transferred to the construction image. Opinions? Dicklyon 19:55, 15 August 2006 (UTC)

What about moving these two images to Golden rectangle? I have many nore images in the works for this article. ≈ jossi ≈ t • @ 20:01, 15 August 2006 (UTC)

If there are no objections I will move these two images to Golden rectangle in a couple of days. ≈ jossi ≈ t • @ 20:53, 16 August 2006 (UTC)

You can go ahead and add them to the other article, but we should still illustrate a golden rectangle on this page, since it has a lot of words about how pleasing the same is due to its ratio. The simple gray rectangle illustrates the point well, and only uses space that would be blank otherwise. Dicklyon 01:17, 17 August 2006 (UTC)

Sorry to nit-pick, but if we are going to have a golden rectangle illistration, and I think we should, it should be a real one, not a Fib rectangle. Yes, I realize that the difference is imperceptible, but accuracy is accuracy. Also, the very long explanation in the caption that use of the Fib rectangle requires detracts from the article. Also, while I appreciate the Bauhaus simplicity of the unadorned gray rectangle, for the purpose of this article it would be more illustrative of the golden ratio to label the a and b sides and to use the blue and red color scheme of the line segment figure for the sides and the labels. Also, Golden rectangle in the caption should be linked to that article (even though it is also linked in the text). If possible, it might also be helpful for the golden rectangle illustration and the construction illistration to be positioned side-by-side and to be of the same dimensions. I can't help with any of this because I don't have the graphics skills. Finell (Talk) 08:38, 24 August 2006 (UTC)

How do you make a true golden rectangle with discrete pixels? —Tamfang 06:36, 25 August 2006 (UTC)

Easy: start with non-square pixels of aspect ratio phi or something rationally related to it. Dicklyon 06:47, 25 August 2006 (UTC)

So you are going to manufacture your own special displays with Golden Ratio pixels? −Woodstone 22:29, 25 August 2006 (UTC)

Not me. Just telling Finell and Tamfang what would be required to beat the Fibonacci rectangles that we can use as approximations when we're stuck with square pixels. One could also use a gray-scale anti-aliased fuzzy edge to make a sort of better approximation, but it will tend to look more fuzzy rather than more precise, which is why I elected to make the Fib rect. I used the next size up initially, which has a ratio that at least agrees with the digits 1.618, but someone complained it was too big. The image is still there, I think, and you should be able to infer its name from the present one and see how it looks. Dicklyon 22:54, 25 August 2006 (UTC)

Relationship to $\pi$ and $e$

The golden ratio $\varphi$ is related to the ratio of ${\pi }$ over $e$ multiplied by a constant; $\varphi \approx {\frac {7}{5}}{\frac {\pi }{e}}$ Does anyone have any idea which article this relationship should go?

None. This is just a numerical coincidence. Fredrik Johansson 11:25, 21 August 2006 (UTC)

As a numerical coincidence, it may be worth including, if there is a reliable source that describes it as such, that is. ≈ jossi ≈ t • @ 01:46, 22 August 2006 (UTC)

Don't have any reference for it but as a friend tested in Perl (I have independently verified in the R programming language)

# Perl code snippet;
use Math::Trig;
print 7*pi/5/exp(1);
print "\n";
print ((1+sqrt(5))/2);
print "\n";

Executing using /usr/bin/perl
Execution took 0.171657 usec
1.61801828970729
1.61803398874989

is accurate to the 4^th dp, which is more accurate than the well known $\pi \approx {\frac {22}{7}}$ approximation. In R, if you were to approximate the exp(1) component as a tailor series expansion to the 7th order, on 32-bit machines it approximates phi to the 7^th decimal place (phi - approximation= 8.8451503788000707e-07). Any higher order approximation (which tends to the numerical accuracy of 32-bit machines) converges to the 5^th decimal place (phi-approximation=-1.5699042604566671e-05). That is a fairly accurate numerical coincidence.

#R Code snippet;
options(digits=22)
phi <- (1+sqrt(5))/2
phi

phi2 <- (7 * pi)/(5 * exp(1))
phi2

phiCalc <- c()

x <- 1
e <- top <- bottom <- 1
for(i in 1:100) {
top  <- top*x
bottom <- bottom*i
e <- e + top/bottom
phiCalc[i] <- (7 * pi)/(5 * e)
}

print(phi-phi2)
print(pi)
print(exp(1))
print(phiCalc - phi)
print(phi-phi2)
[1] 1.569904260434463e-05
> print(pi)
[1] 3.141592653589793
> print(exp(1))
[1] 2.718281828459045
> print(phiCalc - phi)
[1]  5.8108086876296028e-01  1.4125789726038929e-01  3.1302154384746705e-02
[4]  5.9277521826752722e-03  9.4627444975325936e-04  1.1899787172575671e-04
[7]  8.8451503788000707e-07 -1.3878442173043126e-05 -1.5518754122645362e-05
[10] -1.5682785134529809e-05 -1.5697697043126624e-05 -1.5698939702435410e-05
[13] -1.5699035291527608e-05 -1.5699042119621254e-05 -1.5699042574590649e-05
[16] -1.5699042603234403e-05 -1.5699042604566671e-05 -1.5699042604566671e-05
[19] -1.5699042604566671e-05 -1.5699042604566671e-05 ...

--Zven 19:30, 22 August 2006 (UTC)

That is a significant and fascinating numerical "coincidence" that may explain the observed math in the Giza pyramid, but unfortunately, unless this is reported by a reliable source, we cannot include it in the article, as per our policy of WP:NOR. ≈ jossi ≈ t • @ 00:09, 23 August 2006 (UTC)

Thats fine, its good enough for discussion in the talk page at the moment as far as I'm concerned. It was mentioned by someone in a seminar, I will see if I can identify/reference the source --Zven 00:43, 23 August 2006 (UTC)

By the way, you don't need perl code or any other kind of code to evaluation 7pi/(5e) or (1+sqrt(5))/2. Just type it into your google search box and google calculate it; even the difference: ((7 * pi) / (5 * e)) - ((1 + sqrt(5)) / 2) = -1.56990426 × 10-5. But even if you find an article that has mentioned it, it's unlikely to rise to the level of useful content, since it is only coincidentally related to phi, like the so-called measurements of the pyramids that people like to quote. Dicklyon 00:50, 23 August 2006 (UTC)

True for standard operations --Zven 03:34, 23 August 2006 (UTC)

The formula is neat, but there is nothing remarkable about it. Describing the approximation requires more information than you get out of it. It is a simple fact of probability that any number can be approximated in very many ways. Here are a few more approximations, all more accurate:

$2-{\frac {6}{5\pi }}=1.618028...$

${\frac {7}{6}}\,\pi ^{2/7}=1.618041...$

$e^{e^{11/28}-1}=1.618025...$

$\zeta (3)+K-1/2=1.618022...$ (ζ = Riemann's zeta function, K = Catalan's constant)

${\frac {11}{7}}+{\frac {\zeta (3)^{2}}{\pi ^{3}}}=1.618030...$

${\frac {1}{6}}+\mu =1.618035...$ (μ = Ramanujan-Soldner constant) - error less than 2 × 10^-6

By trying other arrangements of the numbers, you could generate thousands of such approximations by brute force. You'd perhaps be interested in trying out the Inverse Symbolic Calculator or Plouffe's inverter. Fredrik Johansson 08:58, 23 August 2006 (UTC)

Those approximations are interesting, I agree that there are likely to be many approximations close by brute force. Maybe some of those could be incorporated into a Approximating $\varphi$ section? - well famous or well known approximations anyway--Zven 21:43, 23 August 2006 (UTC)

The only "famous" or "known" ones are the rational convergents. None of the others have any useful or intersting mathematical, arithmetic, or geometrical relationship beyond coincidence. Or am I wrong? Dicklyon 21:49, 23 August 2006 (UTC)

Actually, it appears that the first approximation I listed has some significance. It is equivalent to the approximation given in Squaring the circle that allows one to find an approximate circle squaring by approximating π in terms of the golden ratio. Fredrik Johansson 15:20, 25 August 2006 (UTC)

That would be a good thing to add to the squaring the circle article. Dicklyon 15:35, 25 August 2006 (UTC)

Un-attributed opinions? Original research?

This reads as an editor's opinion and in violation of WP:NOR. The text needs to sourced to reliable sources and attributed to these holding these viewpoints. Otherwise, it will be mercilessly deleted... ≈ jossi ≈ t • @ 18:27, 21 August 2006 (UTC)

The ancient Greeks knew the golden ratio from their investigations into geometry, but there is no evidence that they thought the number warranted special attention above that for numbers like $\pi$ (pi), for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception of beauty. They are, at best, inconclusive [4]. Despite this, a large corpus of beliefs about the aesthetics of the golden ratio has developed. These beliefs include the mistaken idea that the purported aesthetic properties of the ratio was known in antiquity. For instance, the Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio. This has encouraged modern artists, architects, photographers, and others, during the last 500 years, to incorporate the ratio in their work. As an example, a rule of thumb for composing a photograph is called the rule of thirds; it is said to be roughly based on the golden ratio.

Yes, there's probably a better way to summarize the results of the studies as reviewed on the referenced web page. I copied in the web reference for this short bold sentence, from a version of about a year ago; it had been dropped somewhere along the way, and some kind of reference was obviously needed, as you say. So don't be merciless; the point that results are inconclusive is hardly disputable in comparison with some of the other points made, and has been in the article for a year or so. We may want to rephrase it and provide better attribution, however. Dicklyon 19:20, 21 August 2006 (UTC)

The question is: is this a widely held viewpoint? or just the opinion of that author? We should be merciless if it is the latter. ≈ jossi ≈ t • @ 19:27, 21 August 2006 (UTC)

That's a good question. The alternatives are that the results are held to be conclusive, in one direction or the other; I've never seen any compelling evidence for that (and I did look for it a year or so ago, before I knew about wikipedia), so unless you have some, we need to leave the point as open to question at least. If you find someone who thinks it's conclusive, I'll find someone to support the opposite conclusion :) Dicklyon 20:50, 21 August 2006 (UTC)

If the vaidity of the application of the golden ratio to aesthetics is inconclusive, we can present that by describing a reliable source that describe this, but only if that is a significantly held viewpoint. See WP:NPOV. ≈ jossi ≈ t • @ 01:49, 22 August 2006 (UTC)

Well, it IS a significantly held viewpoint, but you wouldn't know that compared to the amount of non-scientific hype about the ratio. I'll work on puttin in some more refs. Dicklyon 01:58, 22 August 2006 (UTC)

You've removed a significant chuck of material that's been in there for a very long time, subject to editing by many people. It seems precipitous to take it out just because I added a reference that drew your attention to it. I'll put it back, and we can work on documenting the sources for its various statements. Is there some part of it in particular that you consider questionable? I suspect Livio is a source for most of it, and I'll look there. Dicklyon 20:55, 21 August 2006 (UTC)

It is not deleted, it is just commented out. Feel free to restore when adding sources. My concern is that there is material there that is either unattributed to a reliable source, or that it is stated as an assertion of fact without disclaiming the origin of the viewpoint or its significance. ≈ jossi ≈ t • @ 01:44, 22 August 2006 (UTC)

Indeed, a widespread problem in wikipedia. But don't throw out the baby with the bathwater. Let's fix it. If you just remove the paragraph, it leaves the clear impression that all this aesthetic stuff has some kind of scientific support. Does it? It seems safer to say inconclusive than to say nothing. Dicklyon 01:58, 22 August 2006 (UTC)

Yes. We can say that it is inconclusive, if there is a reliable source that says that, and that reliable source describes a significant viewpoint. Otherwise we can't. I am sure we can find such source. Until then, it is neatly tucked between comment tags. ≈ jossi ≈ t • @ 02:08, 22 August 2006 (UTC)

I have added some Livio refs and details for the "inconclusive" part. It's pages 189–183 if you want to read about it.

See this article: http://www-history.mcs.st-and.ac.uk/HistTopics/Golden_ratio.html. Plenty of evidence provided there that the ancient Greeks new about the ratio. I am looking for direct sources cited in that article. ≈ jossi ≈ t • @ 02:24, 22 August 2006 (UTC)

OK, I see now that your POV is on that side, which is why you didn't like the other bit. I look forward to your chasing those references, but I don't see anything there that suggests more than what Livio said, which is that they knew about the ratio via the golden section, or cutting a line into mean and extreme ratio, but not that they even considered rectangles in that proportion. You might want to try books.google.com to help. Dicklyon 02:51, 22 August 2006 (UTC)

Google books is quite limited. I prefer Questia and my local library :). And BTW, my "POV" on the subject is that I find the subject fascinating, that is it, really... ≈ jossi ≈ t • @ 03:38, 22 August 2006 (UTC)

Aesthetics: Parthenon

See these two excerpts from:

Van Mersbergen, Audrey M. , Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic Communication Quarterly, Vol. 46, 1998

The Canon of Polykleitos, his treatise on the proportions of sculpture, is lost but for two fragments preserved by Philo Mechanicus (iv 1.49,20) and Plutarch (Moralia 86a). In these fragments we read that Polykleitos understood proportion as not derived from an absolute standard of beauty, but as derived from the relativity of one part of the human body to another. Furthermore, Polykleitos is said to have incorporated the asymmetries of contrapposto(5) into his compositions (Leftwich 45) and to have built the ratios of the "Golden Ratio" into his system of proportions (Stewart 129n46).(5) Indications that other fifth-century mathematical ideas participated in the architecture of the fifth century can be seen in the asymmetries of the Parthenon, the Council House, the Assembly Hall, and the Pinakotheke.(6)

But see cautionary articles about his Canon, too: [5] or [6]

Do you mean Leftwich's challenge? We can cite that, as a challenge by that scholar. ≈ jossi ≈ t • @ 03:36, 22 August 2006 (UTC)

and

Hambidge argues that fifth-century buildings were constructed according to rectangles, the proportions of whose ends to sides is based on the square root of five (1). As he explains, the square root of five is merely a diagonal to two squares, the numerical value of which is 2.236+. The fraction is endless, or irrational. Another term, according to Hambidge, that describes the proportions of these rectangular buildings is the "Golden Ratio." The numerical value of this ratio is another endless, or "irrational" number, 1.61803+. The property of proportion that this ratio entails is that .618 equals root five minus 1 divided by 2; and 1.618 equals root five plus 1 divided by 2 (Hambidge 1). Thus, a "Golden Rectangle" has a ratio of the length of its sides equal to 1:1.61803+. The Parthenon is of these dimensions.

≈ jossi ≈ t • @ 02:43, 22 August 2006 (UTC)

and you can find more rebuttals and such in scholar.google.com by looking up Hambidge golden ratio. Here's one: [7]

I do not have access to Jstor. Care to summarize? ≈ jossi ≈ t • @

As per NPOV, we can summarize these opposing views. That will make the section quite interesting to read. ≈ jossi ≈ t • @ 03:36, 22 August 2006 (UTC)

I don't have access either, just what I can see on those pages. It's tough reading the small print sometimes. Dicklyon 03:40, 22 August 2006 (UTC)

While you're at it, see if you can find a source for this paragraph in the opening, so we don't have to remove it:

"Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry. The ancient Pythagoreans, who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence."

Dicklyon 04:12, 22 August 2006 (UTC)

The whole lead to the article needs to be rewritten from scratch, as it is pretty bad overall, including that particular sentence. I would prefer to work first on augumenting the article, and only then summarize teh article context as per WP:LEAD, including its significant historical context, its applications, major disputes, and the basic math formula. ≈ jossi ≈ t • @ 04:19, 22 August 2006 (UTC)

History

Jossi, your rework of the article really sounds like it's designed to push the agenda of ancient mysticism or something. Have you even read Livio or any of the other things you're writing about? Dicklyon 04:41, 22 August 2006 (UTC)

My rewrite of the History section's lead simply removed the editorializing and the reaching of conclusions that breached WP:NOR, and attributing the POVs. This article is not Mario Livio and the golden ratio, is it? Livio is already used to as references for four passages. That is quite enough, given that super-abundant literature on the subject. I would appreciate if you can WP:AGF. I am not into mysticism or trying to push an agenda, neither I am accusing you of pushing a skeptic agenda. I simply find the subject fascinating, to which I arrived due to my interest in typography, and the geometry involved in book design. Let's make the article better, by using good research and the editorial tools at our disposition, and the parameters defined by WP:NPOV, WP:V, and WP:NOR. ≈ jossi ≈ t • @ 14:19, 22 August 2006 (UTC)

OK, I will assume good faith. But please answer whether you have read Livio, which is probably the most thorough and unbiased piece of scholarship on this topic. And if not, please do. Also, please clarify some of the typography stuff. The book page layout drawings are unclear in exactly what is the use of golden ratio. Dicklyon 16:36, 22 August 2006 (UTC)

I have not read the specific book by Livio, but I have read various of his articles. I intend to read the book as soon as I get hold of a copy. As for the use of the golde section in book design, I have modified the wikilink to the appropriate page Golden section (page proportion). ≈ jossi ≈ t • @ 17:28, 22 August 2006 (UTC)

What I know about Livio, is that he does not accept that there is a link between aesthetics and the golden ratio, and believes that "we should abandon its application as some sort of universal standard for "beauty," either in the human face or in the arts." That statement can be used in the article, as he is an authorative source on the subject, but we can only mention it as Livio's opinion, and not as a fact. ≈ jossi ≈ t • @ 17:39, 22 August 2006 (UTC)

Yes, that's fine. But have you read his book? It will be useful to inform your writing on these topics. From my point of view, this is a mathematics and history article, and I agree with Livio that people who want to assign aesthetic concepts to this irrational number are without any solid basis for doing so. Dicklyon 18:00, 22 August 2006 (UTC)

I disagree that this article's focus is mathematics and history. There is a massive body of work going back 500 years that applies the golden ratio to many human endeavors, including architecture, art, book design, psychology, aesthetics in general, cosmology, philosophy, natural sciences, etc. Livio's viewpoint is significant but certainly not the prevalent or most significant. This article needs to be informative for our readers, and present all significant viewpoints, including these that assert that there is an aesthetic component to this ratio (or mystic, esoteric, etc.) and those that do not, as per our policy of WP:NPOV. Eventually, and if we do our work well, this article can spin out multiple articles and become a summary article as per Wikipedia:Content_forking#Article spinouts - "Summary style" articles, in which the different aspects of the golden ratio, including the dispute about its application can be fully explored. ≈ jossi ≈ t • @ 18:09, 22 August 2006 (UTC)

Value judgements

Value judgements such as "a thorough modern analysis", should not be used. We can add the date of the book, and leave it at that. Let the reader make the judgement if it is thourough or not and if it is modern or not. ≈ jossi ≈ t • @ 18:29, 22 August 2006 (UTC)

I didn't think "modern" would be a controversial way to contrast a 21st century analysis to a 19th century one, but what the heck, the date will do. As to your objection to "thorough" I'd be a lot more sympathetic if the complaint were coming from someone who had read the book. I think it is useful to point out to readers that there exists a thorough modern analysis that informs this article. It is easily available, and editors who insert non-mathematical material that conflicts with it need to be carefully controlled, in my opinion, because it's hard for them to have an NPOV while being ignorant of that latest thorough analysis of the history of this field. So, have you read it? Shall I loan you a copy? Dicklyon 18:42, 22 August 2006 (UTC)

No thanks,. I have just ordered mine... :) ≈ jossi ≈ t • @ 18:52, 22 August 2006 (UTC)

As per yourt comment that we ought to "validate" non-mathematical material against Livio's book, I strongly disagree. We need to describe 'all significant viewpoints, without engaging in any type of validation or debunking. The measuring stick is WP:V: "Verifiability, not truth".

Why do you put "validate" in quotes and attribute it to me? I said we need to careful control, not sensor or validate against Livio. But validation against authoritative sources in general should be a part of attempt to control editing of wiki articles in general. And there's a different between NPOV and AllPOVs, I think; our job is to figure out where to draw the line on nutty POVs, and to represent non-nutty ones neutrally. Dicklyon 19:05, 22 August 2006 (UTC)

If you can devise a formula that can define authoritatively what a "nutty POV" is, I am all ears, hey, we can incorporate it into the wiki software and get away without needing WP:NPOV. You may have the perception that anything that cannot be mathematically proven in regard to the golden ratio, is "nutty", fair enough. But this article is not what you or I consider nutty or not nutty. It is about what reliable sources say about the subject. So let's "discuss the article and not the subject" shall we? ≈ jossi ≈ t • @ 00:28, 23 August 2006 (UTC)

For example, we can and should include views as these:

Observers of these matters, if they are not prepared to accept a mystological explanation for what must otherwise seem strange and enigmatic, can only recognise that the genius of Egyptian architects was as it was precisely because they were working in new, untried fields of endeavour and in materials which they were the first to encounter. Even such firmly material products of the human genius as the great monuments of Egypt originated first in the mind's eye, before they found expression in three dimensions. Thus it must be with the Golden Section, whose frequency of exploitation in many cultures and as many different periods suggests that it is locked into the human unconscious, ensuring an appropriate aesthetic response to the choices which need to be made to provide an aesthetically satisfying result. If this presumption is accepted, then the implications for much, perhaps all, human creative endeavour are very great. It would be wholly consonant with the underlying theme of this book that the unique Egyptian contribution to civilisation was to be the first in which all these elements were articulated and, being so articulated, were liberated, to take their place amongst the acknowledged archetypes which determine human behaviour.

... from the same book used for my last edit, in which the dispute about the use of the golden ratio is described. ≈ jossi ≈ t • @ 18:52, 22 August 2006 (UTC)

OK, I confess, I can only categorize that as a "nutty" POV. Any pseudo scientist that tells me my only choices are "mystological" or his own theory is not leaving room for any objective POV, is he? Dicklyon 19:05, 22 August 2006 (UTC)

We do not care about "objective POVs" or "nutty POVs", we care about WP:NPOV that it is a very different proposition. ≈ jossi ≈ t • @ 19:25, 22 August 2006 (UTC)

I am makig sense to you? I think that you may need to re-read WP:V aqnd WP:NPOV, if you don't... Don't you agree that we need to describe all signficant viewpoints, even if these are "incorrect"? ≈ jossi ≈ t • @ 18:56, 22 August 2006 (UTC)

Your POV is becoming clear. Describing significant viewpoints is fine, but if they are obsolete discredited viewpoints, we have to duty to make that clear as well, don't we? Are they any current scholars that support this stuff about Egypt? Dicklyon 19:05, 22 August 2006 (UTC)

What POV? I am applying Wikipedia policies. You may need a refresher course on these? Obsolete and discredited viewpoints can only be asserted as such by referring to these that make the assertion that these are so, unless these viewpoints are so widely held that it is not necessary to asserting them as such (e.g. the earth is round and not flat). ≈ jossi ≈ t • @ 19:22, 22 August 2006 (UTC)

Can a non-combatant comment? In my opinion, individual viewpioints, even if referenced, are not encyclopedic. Where there are opposing schools of thought with a signigicant number of authoritative adherents as to matters of fact, they should be presented. On the other hand, speculations without sound evidentiary support, even with citations to multiple adherents, are not encyclopedic. We do not cite the Bible or astrologers as to matters of astronomy or other sciences, and other unsubstantiatable speculations likewise have no place in Wikipedia articles on factual matters. Finell (Talk) 08:55, 24 August 2006 (UTC)

I was beginning to think I was the only one that felt that way. I've been having a similar dispute with Jossi in Golden section (page proportion), where he seems to think that anything anyone has ever said can be included as if fact, if cited, and that to point out the mathematical inconsistencies in such things is original research and therefore inadmissable. Dicklyon 16:39, 24 August 2006 (UTC)

Deletion of material?

Can you explain why are you deleting material that is properly sourced and attributed?

Michael Rice asserts that principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the Golden ratio and that it is part of mathematics of the Pyramids<ref>S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457, '''as cited in Rice, Michael''', ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' pp.24 Routledge, 2003</ref>. Some recent historians of science, have denied that the Egyptians had any such knowledge, believing that its apperance in an Egyptian building is the result of chance. <ref>Rice, Michael, ''Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C'' pp.24 Routledge, 2003, ISBN 0-4152-6876-1</ref>

≈ jossi ≈ t • @ 19:22, 22 August 2006 (UTC)

The second sentence had a Rice reference, and the first didn't. It was unclear what statement the Giedon reference was in support of, or what the right reference for the Rice opening line was. Please just fix it. Dicklyon 19:40, 22 August 2006 (UTC)

What is going on? Why do you keep reverting. Please see WP:3RR. The text is referenced exactly as referenced in the source provided. This is the full text of page 24 of Rice, Michael, Egypt's Legacy: The Archetypes of Western Civilisation, 3000 to 30 B.C:

The second of these possibilities will be favoured here. The Egyptians, as early as the Pyramid Age, set the orientation of their buildings by the stars; the precise orientation of the sides of the Great Pyramid and its alignment with the cardinal points, with a minuscule degree of error, are examples of the application of what is surely empirically-gained knowledge raised to the level of genius. There has been much consideration of the extent to which the Egyptians had knowledge of transcendental numbers, as π (pi), which can be obtained by the distinctly arcane method of observing a drum revolving and (phi) which is also known as the Golden Section. This has been defined as 'the division of a straight line into two unequal parts, in such a way that the smaller part is in the same ratio to the larger part as the larger part is to the whole'; 22 the proportion of the greater part to the total line length is approximately 0.618. The Golden Section is found in nature, in, for example, the spiral of a shell. It was widely employed in Renaissance and later classical architecture: it is said to have been demonstrated to be present in many Egyptian buildings, obviously of much greater antiquity. The matter is, however, disputed. Several of the principal authorities on the history of Egyptian architecture have argued that the inhabitants of the Valley in antiquity were well acquainted with the Golden Section; (23) they propose, for example, that it is basic to the mathematics of the Pyramids. Others, particularly some recent historians of science, have denied that the Egyptians had any such knowledge, believing that when the Golden Section, or something like it, appears in an Egyptian building this is the result of chance. This conclusion seems less likely than that the builders did at least understand the effects of such a system of proportion.

(22) and (23) cites: (22) S. Giedon, 1957, The Beginnings of Architecture, The A.W. Mellon Lectures in the Fine Arts, 457. (23) Giedon, ibid., gives a number of examples.

I will not be subjected to such scrutinity to my edits. You will need to assume good faith, and that is about it. From now on, I will not provide with copies of the text that I am citing. You can check these yourself, by cheking by sources at your local library. Please restore the text that you deleted. Thanks. ≈ jossi ≈ t • @ 21:09, 22 August 2006 (UTC)

How can you not see the problem with that? You've got the Rice reference referring to the whole paragraph, but another reference in the middle. It makes no sense that way. Fix it to make it clear what reference goes with what statement. Dicklyon 21:31, 22 August 2006 (UTC)

Are we left brain - right brain type of people, you and I? :) The reference in the middle is a reference to Giedon made by Rice and noted as such (see the text that read "as cited by xxx in xxxx" in the middle reference. You may want to read about intermediate sources at WP:CITE#Intermediate_sources:_State_where_you_got_it ≈ jossi ≈ t • @ 21:38, 22 August 2006 (UTC)

Apparently we are. I like things logical. I understand about the ref, but the 1957 ref didn't provide any support for the opening sentence that it followed, which was a sentence about Rice's more modern statement. That ref came later. I can see how you made that mistake, but am mystified about why you can't recognize and fix it, so I fixed it for you. Dicklyon 21:45, 22 August 2006 (UTC)

Sorry, but this is bordering on the ridiculous. My edit was perfectly OK, summarizing the content rather that copying and pasting, and providing inline citations as per guidelines. Your "fix" did not fix anything, rather, aseerted your preference, after edit warring and disrupting the editing process to WP:POINT. That is unacceptable behavior and creating a toxing environment that is unnecessary. ≈ jossi ≈ t • @ 21:50, 22 August 2006 (UTC)

Apology accepted. Dicklyon 22:19, 22 August 2006 (UTC)

:). I just did that to get you off my back. If that was the only thing that bothered you (i.e. adding "citing Giedon") you could have done that yourself and avoid the whole bruhaha. ≈ jossi ≈ t • @ 23:01, 22 August 2006 (UTC)

BTW, Livio "maintains there is no evidence that the Egyptians either knew about the golden ratio or used it in the dimensions of the pyramids." but there is no evidence that they did not know about the ratio, either. So that, you would agree with me, is not a valid scientific argument, but an opinion. An interesting opinion, as interesting as the one that says that the Egyptians knew about it. ≈ jossi ≈ t • @ 21:59, 22 August 2006 (UTC)

Quite the contrary. If you read him, you'll see that he examines the so-called evidence cited by others, and finds it to be just wishful numerology. There is pretty good evidence that they did NOT know about phi if you take the body of what is known as a whole and see that we know a lot about their math, and phi is not there. Dicklyon 22:18, 22 August 2006 (UTC)

Yes, he has strong opinions about the matter, and one can come to the conlusion that the Egyptians may not have known about Phi by following his line of reasoning . But that is still an opinion, nontheless. ≈ jossi ≈ t • @ 22:24, 22 August 2006 (UTC)

For example, there are others that refer to Pi and Phi in relation to Giza, but not as one excluding the other but as one complementing the other, e.g. http://milan.milanovic.org/math/english/golden/golden3.html ≈ jossi ≈ t • @ 22:30, 22 August 2006 (UTC)

More fascinating stuff about Phi

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html

Fascinating.... ≈ jossi ≈ t • @ 00:40, 23 August 2006 (UTC)

It contains other studies that provide pro and con views on Phi and the Giza pyramids, here: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#egypt. These sources will be worth exploring. I see if I can get hold of these. ≈ jossi ≈ t • @ 00:48, 23 August 2006 (UTC)

... and as I asserted before, both pro and con are theories, both fascinating IMO, and neither of which preclude the possibility that the Egyptians knew phi:

"According to Elmer Robinson (see the reference below), using the average of eight sets of data, says that "the theory that the perimeter of the pyramid divided by twice its vertical height is the value of pi" fits the data much better than the theory above about Phi. " ≈ jossi ≈ t • @ 00:51, 23 August 2006 (UTC)

Yes theres some good stuff in there. I've noticed that there is no diagram illustrating that the Pentagons and Pentagrams ratio of areas of the larger Pentagon to the smaller pentagon equals $\varphi$. Might be a good additional picie for the article one day --Zven 03:22, 23 August 2006 (UTC)

Sure... there are many pentagons and pentagrams in commons, that we can use. See: http://commons.wikimedia.org/wiki/Pentagram and http://commons.wikimedia.org/wiki/Category:Golden_ratio ≈ jossi ≈ t • @ 03:28, 23 August 2006 (UTC)

theories

How about a new section on "theories" about the golden ratio, and then we can take the history section back to just being history? J, I should have looked at your user page before. I had no idea I was dealing with an ENFJ admin; explains a lot, esp. the WP:AcronymSoup and World View. Dicklyon 02:22, 23 August 2006 (UTC)

Just thread carefuly as I could ding you for violating WP:NPA... . (BTW, I am a strange case of ENFJ, as I am also involved in technology development and engineering)

We can maybe expand the section about "disputed sightings", re-label it, and add describe there the controversial aspects. ≈ jossi ≈ t • @ 03:00, 23 August 2006 (UTC)

You feel attacked to have your declared orientation and world view mention? Or the fact that you're a wikipedia admin? Dicklyon 03:24, 23 August 2006 (UTC)

I was just kidding, hence the smiley. What I was referring to is that WP:NPA reads that a personal attack is when "Using someone's affiliations as a means of dismissing or discrediting their views — regardless of whether said affiliations are mainstream or extreme." ≈ jossi ≈ t • @ 03:26, 23 August 2006 (UTC)

Not to worry. That's not the reason. Dicklyon 03:31, 23 August 2006 (UTC)

Standardization of $\varphi$ and $\phi$ notation

I suggest some consensus is made on the notation used, as it is not entirely consistent in the article, probably due to two ways of writing the notation, e.g. using math mode in LaTeX, or HTML encoding (which look slightly different). It seems in literature that generally lower case $\varphi$ is the golden ratio and upper case $\phi$ is the golden ratio conjugate e.g. $\phi ={\frac {1}{\varphi }}=\varphi -1$ (wolfram) What are other peoples thoughts? --Zven 00:46, 24 August 2006 (UTC)

p.s. Anyone missing the first $\varphi$ in the Table of contents for this section? maybe its a bug.

The first character in the equation above is incorrect; it is another form of lower case phi. The equation on Wolfram correctly uses an upper case Phi as the first character. The article should mention the conjugate and its notation. Because upper case Phi is used for the conjugate, the upper case Phi should be removed from the graphic. Also, I definitely agree that the notation (the phi forms) should be consistent throughout the article. Finell (Talk) 03:20, 24 August 2006 (UTC)

One of the obvious problems is that using LaTeX notation cannot link directly to the phi symbol symbol e.g. φ has to be done in HTML encoding. --Zven 06:19, 24 August 2006 (UTC)

Just noticed what you were saying, yes the Golden ratio conjugate should have been $\Phi$ not $\phi$. --Zven 10:28, 25 August 2006 (UTC)

Golden mean redux

Before anyone attacks me, let me say that I participated in the debate over merging this article with what is now Golden mean (philosophy), and very strongly supported, and still strongly agree with, the resolution of that debate that was implemented: to keep the articles separate and to differente the articles more sharply (keeping material from each out of the other). I do not wish to reargue, or to relive, that debate. Nevertheless, it is a fact (although I wish it were not so) that golden mean is used, fairly widely, as a synonym for golden ratio (but not vice versa). In fact, along with golden section, golden mean is one of the most commonly used synonyms. Dictionaries give the golden ratio as one of the definitions of golden mean, although not as the first definition (at least insofar as I have seen). Therefore, in my opinion, we cannot justify omitting golden mean from the list of synonyms in the lead. I attempted to deal with this in my 21:44, 21 August 2006 edit as follows:

Many people use golden mean as a synonym for the golden ratio, but that usage is ambiguous becasue the former term has an unrelated meaning as an admonition to follow the moderate course or position between two opposite extremes (see Golden mean (philosophy)).

I am not wedded to this language. In fact, looking at it now, I don't like it that much. Neverthelss, in my opinion, we must name golden mean as a synonym (indeed, near the beginning of the list) and we must explain the potential confusion (ambiguity) in using golden mean to mean golden ratio. The tag now at the top of the page, again in my opinion, does not get this message across. In fact, at this article now stands, the tag is a head-scratcher: why would anyone even think to confuse golden ratio with [[Golden mean (philosophy. What do others think? Finell (Talk) 04:15, 24 August 2006 (UTC)

Hello Finell, and welcome back. My view is that it would be best if we can find a source that makes the distinction between these terms, rather than come up with our own ideas about it. ≈ jossi ≈ t • @ 04:22, 24 August 2006 (UTC)

I apologize. I seem to have lost it in an Aug.03 edit. I'll put it back, and verify that it is in at least one of the referenced sources. Will the disambig be enough in that case? Dicklyon 05:33, 24 August 2006 (UTC)

I thought you lost it intentionally, but I now think that my explanation was too wordy. I am going to try another solution in the lead paragraph. Please look at it (and you are of course free to change it). Finell (Talk) 05

52, 24 August 2006 (UTC)

It's amazing how one can screw up an article and nobody points it out for weeks. Such is wiki life, sometimes. Other times it only takes 30 seconds. Even if it's not a screwup. Dicklyon 06:23, 24 August 2006 (UTC)

I tried my version, but kept the references you added. Please see what you think (if you haven't already). If my version is accepted, I think the tag can go (but I didn't delete it). Finell (Talk) 06:27, 24 August 2006 (UTC)

The trouble with the new scheme is that it is again in the domain where the ordering and positioning of terms is going to be subject to a lot of opinions. We had settled (for a while) on Livio, the most thorough modern treatment, as a source of "most common," but he inexplicably leaves out golden mean, even though searching other sources shows it to be more common than golden section and most of the others. I'm ready to punt it back to anarchy, as soon as I take golden cut out again (which there's zero support for that I can find). Dicklyon 06:30, 24 August 2006 (UTC)

Using authority to substantiate facts is good. Being confined to a single authority, especially where that authority omits a widely known fact, is not good, and is moreover an abandonment of our editorial responsibility. Anarchy would be adding material without a reputable source or insisting on following one reputable source over another, instead of attempting to combine and harmonize, or where that is impossible, to point out the differences neutrally. At least on the topic being discussed in this article (and on most other topics), I don't think any single authority trumps all others. By the way, I am sure that I have seen golden cut somewhere, but I don't have a citation at hand. Finell (Talk) 06:48, 24 August 2006 (UTC)

Finell, I understand all that. I don't have a great solution, however, and 'anarchy' is what I expect, in terms of more argument about what names to list in what order. You can review the status of citations of "golden cut" in the discussion above. Let me know if you find ANYONE actually referring to the golden ratio as golden cut. I can't find any (but see above to see exactly what I mean by this). I suspect it came up somewhere once as an alternate translation of the German goldene Schnitt, but can't be sure. Dicklyon 06:54, 24 August 2006 (UTC)

The following uses of golden cut (thanks, Google):

http://home.uchicago.edu/~jswaters/web/math/construction.html

http://jwilson.coe.uga.edu/emt669/student.folders/banker.teresa/golden/goldcut.html

http://www.beautyanalysis.com/mba_phiin1dBOTTOM_page.htm

http://www.gvsu.edu/math/students/blt/Intro.html

http://www.assiah.net/science-and-mathematics/scientests/fibonacci/fibonaccis-golden-number-and-the-golden-cut.htm

and there are more, are sufficient to warrant inclusion as an "other name". Finell (Talk) 08:18, 24 August 2006 (UTC)

Those pretty much support the alternative meaning of the cut itself, or the cut point, as opposed to the numeric value of the ratio. One says it's an alternative name for the ratio, but I don't give much weight to one random web page when hundreds of books disagree. So, I took it out, lacking a convincing reason to believe it's true. Dicklyon 00:37, 26 August 2006 (UTC)

Well, now Jossi found it in a book, so I suppose we're stuck with it, even though it's a less than 1 in 1000 thing. Dicklyon 01:50, 26 August 2006 (UTC)

There are mores sources that use "golden cut". Do we need more than one reference? ≈ jossi ≈ t • @ 01:53, 26 August 2006 (UTC)

No, one's enough. But if you find others that use it that way (not for the cut point) and say more than "has been called", I'd like to know about them. Last time I looked, google book search didn't have any, but by now maybe they do. Dicklyon 05:27, 26 August 2006 (UTC)

I was just now searching for some pre-1923 books with good info or illustrations, and found a few more with "golden cut" (not all use it for the ratio, but a couple do). Interestingly, the term "golden mean" was much more common than either "golden ratio" or "golden cut" (factor of 50 more than either of those, before 1923). Furthermore the most common term by far seems to have been "extreme and mean ratio", unless I missed something. I'm glad that mouthful has fallen out of favor. It's hard to tell how common "divine proportion" is because that term comes up in many other contexts. There are some interesting analyses of Fechner's and Witmer's phychological studies and results. And a simple construction for dividing a line segment into mean and extreme ratio. I might upload some page images and y'all can see if any would be useful for illustration. Dicklyon

As translated into English, Euclid's Elements defines and uses throughout "a straight-line ... cut in extreme and mean ratio". That (or perhaps Euclid's uncited predecessor) is probably the source of golden cut (after golden got into the terminology) and the "mouthful". The timeline quotes a usage of the "mouthful" by Kepler, so the use of that terminology spans several centuries. The wide usage of golden mean for the ratio may have something to do with some people equating the ratio with Aristotilean balance, and from there to idealization and other metaphysics and beyond. Finell (Talk) 16:54, 30 August 2006 (UTC)

Boy, these paragraphs are really getting narrow! Finell

My impression was that "golden cut" was an alternate translation from the German popular term "goldene Schnitt", common since the mid nineteenth century, which is more commonly translated as "golden section". To section and to cut are the same concept, but different words for it. The German, of course, probaby came from Euclid ultimately. I pulled your little paragraph back a bit as it was causing a formatting headache for me. Dicklyon 19:25, 30 August 2006 (UTC)

Consistent citation form

Especially since several Wikipedians are doing a lot of work to bring this article up to Featured Article quality, I believe that all of the citations should be in the same form. I suggest that we standardize on the Wikipedia citation templates—they are tedious, but they ensure consistency and full bibliographic info. Yes, it will be a lot of work. Yes, I know that I am a pedant, but I can't help that. What do others think (about my proposal, not my personality)? Finell (Talk) 06:36, 24 August 2006 (UTC)

Go for it. Dicklyon 06:39, 24 August 2006 (UTC)

Sure. ≈ jossi ≈ t • @ 00:52, 25 August 2006 (UTC)

Squishy paragraph in the lead

The lead includes this paragraph:

• "Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry. The ancient Pythagoreans, who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence."

Why is this unsubstantiated stuff still there? Should we just take it out of the lead and leave it for other sections to worry about? Or reduce it to just stuff that is supportable or has a consensus? Dicklyon 06:39, 24 August 2006 (UTC)

It isn't unsubstantiated, just unsourced. It is the premise of the aesthetics section, and the principal reason for most serious scholarly interest in the golden ratio (although the pure mathematicians may be interested just because of the mathematical propeties). More work to do citing the many sources that support these statements (although I am personally not familiar with the Greek numbers as ratios). Finell (Talk) 06:57, 24 August 2006 (UTC)

Which it are you referring to? There are several controversial points here, at least:

• have long been considered
• is still used frequently in art and design
• suggesting a natural balance between symmetry and asymmetry
• believed that reality is numerical
• that the golden ratio expressed an underlying truth about existence

most particularly that last one! By unsubstantiated I meant in this article. If there is support for them, let's have it. If they are going way beyond the kind of topic intro that belongs in the lead, let's fix that. I think it would be much better to say that phi has become a focus for aesthetic and mystical numerology since the 15th century, or something more explicitly objective about it like that. Dicklyon 16:45, 24 August 2006 (UTC)

Properties of $\varphi$

I think that;

${\frac {1}{\varphi }}=\varphi -1$ is a property of $\varphi$. Is there another real number that has this property out there? This was discussed (briefly) furthur up the talk page but was not in the article. --Zven 22:13, 24 August 2006 (UTC)

yes, because it leads to the defining quadratic equation, which has two real roots. Dicklyon 22:55, 24 August 2006 (UTC)

And isn't it wonderful that the other root is $-\Phi$. Woodstone 08:20, 25 August 2006 (UTC)

Wonderful. I prefer to avoid the case confusion and call it $1-\varphi$. Dicklyon 14:53, 25 August 2006 (UTC)

Construction picture

There are now two almost identical pictures to illustrate the construction of the golden ration. The first has a caption with good instructions, but no annotation in the picture. The second has annotation, but does not show the rectangle. Can someone merge the two to make a single complete one? −Woodstone 12:27, 26 August 2006 (UTC)

Specifically you are talking about these two pictures;

A line segment sectioned into two, to illustrate the golden ratio. The total length a+b is to the longer segment a as a is to the shorter segment b.

Nearly a golden rectangle: The ratio of adjacent numbers in the Fibonacci Series approach the golden ratio as the sequence approaches infinity. This gray Fibonacci rectangle has an aspect ratio of 144/89, or about 1.617978. The golden ratio is thought by many to be the most pleasing ratio for a rectangle.

The first is a 1-D representation of the $\varphi$ ratio and is pretty good, the second is a 2-D representation of it with respect to a reportedly asthetically pleasing rectangle. We could have the pentagon/pentogram picie too.

I think the second should still be in the article but in a place more relevant to discussing the Fibonacci Series. It could also possibly be improved with x y coordinate annotation. --219.89.171.42 11:47, 27 August 2006 (UTC)

The plain gray rectangle is intended to be the most accurate possible representation of a golden rectangle, to allow readers to appreciate the shape in isolation. Maybe the fib numbers should be removed from the caption, since they do seem to be distracting people from the fact that this is the most accurate possible way to depict a golden rect in square pixels with sharp edges. An SVG could be internally more accurate, but would display fuzzy if anti-aliased, or less accurate if sharp. Any annotations will interfere with the clean perception of the shape itself, the point of which is its beauty based on its proportion. Dicklyon 17:06, 27 August 2006 (UTC)

But there's not much room for the caption this way.

Thers is an obvious problem with that gray rectangle: it is in horizontal position, when most of its appications use a vertical position. Also, regardless of the problem of square pixels and anti-aliasing, SVG is the preferred format. I want this article to eventually be upgraded to FA status, and one of the requirements is that all diagrams be in SVG format for portability and resolution independence. I intend to convert that gray rectangle to SVG in vertical positioniong as well as convert or recreate Image:Golden ratio line.png to SVG as well. ≈ jossi ≈ t • @ 17:16, 27 August 2006 (UTC)

Per suggestion, I got InkScape and made a vertical SVG gray 89x144 rect. It doesn't leave much room for a caption. Would a bigger one be better? I had thought, based on all the buzz in the photographic community, that a horizontal rect was the usual, since it represents a sort of normal "field of view". But I've since learned of the use in book design where vertical is more relevant. The other trouble, at least on my screen when I compare to the one above, is that the pixmap image is sharp, while the scalable is fuzzy. You don't get pixel-level control very easily with scalable structures, but maybe someone knows how to force it better than I do.

I'll leave it here in case anyone wants to substitute it in or try to improve it. Its Image:FibRect89x144.svg Now I suppose I better work on the construction figure as SVG, but so far I don't see any way to make a sqrt sign. Do I need to get in and edit the xml? Advice anybody? Somebody else want to do it instead? Dicklyon 06:22, 29 August 2006 (UTC)

Sorry, I should have been more clear. I was talking about the following two:

Construction of a golden rectangle: 1. Construct a simple square. 2. Draw a line from the midpoint of one side of the square to an opposite corner 3. Use that line as the radius to draw an arc that defines the height of the rectangle.

summary of classical construction upon a unit square

This very first picture (the line cut you show above) is excellent for defining the ratio and should stay. The two construction pictures (I was talking about) should be merged into one with all elements combined. And with that, the rectangle becomes superfluous as well. −Woodstone 14:39, 27 August 2006 (UTC)

Images are not usually "merged". There are many diagrams in commons, some of which are different depictions of the same subject. ≈ jossi ≈ t • @ 14:55, 27 August 2006 (UTC)

It's just a manner of speaking. I mean create and insert a new picture containing all good elements of both pictures and then discard the old two. −Woodstone 15:10, 27 August 2006 (UTC)

New attempt by User:dicklyon

How about something like this one I just whipped up?

Excellent merge, if the construction instructions are used in the caption.

By the way, as simple rectangle doesn't even have to be an svg file, because it can be done in html:

approximate golden rectangle 144 by 233 pixels

−Woodstone 20:42, 27 August 2006 (UTC)

another try with less intense annotations

Interesting. I'm not much of an html hacker, so I wouldn't know how to position that with the figs, etc. I'll stick to wiki formatting, but if someone wants to try it that way, hopefully with a less ugly color and some border space, and no text inside, I'd like to see how it's done. Dicklyon 20:58, 27 August 2006 (UTC)

Excellent idea about the HTML version. I can place it in a wiki table aligned right, if needed. ≈ jossi ≈ t • @ 00:00, 28 August 2006 (UTC)

I'll go ahead and put the new construction figure in. Any opinion on placement relative to the present two? Feel free to move it if wherever I put it doesn't suit. Dicklyon 20:58, 27 August 2006 (UTC)

Just make sure it is an SVG and not a PNG, although I think that the image I created is superior in quality, and in design. The arrows and formulae detracts from the image, IMO. Again, you chose the unilateral way of doing things, wich is quite innapropriate. ≈ jossi ≈ t • @ 00:04, 28 August 2006 (UTC)

I might work on an SVG version if someone will recommend a good editor for Mac OS X. I don't have any SVG experience at this point, but am eager to learn. Of course, if someone else wants to make a nicer version, I'm good with that, too. Dicklyon 00:06, 28 August 2006 (UTC)

Jossi, please lay off the unfounded assertions of "unilateral" and "inappropriate." I followed up on a suggestion here, got positive feedback on my effort, invited others to improve it. I'm getting tired of your criticisms and your stonewalling of improvements to things you're involved in. Dicklyon 00:56, 28 August 2006 (UTC)

Here's one with less intense annotations. I think it looks better. Anyone want to put it in the article? Or wait for an SVG to appear? Dicklyon 03:55, 28 August 2006 (UTC)

I went ahead and put it in. With the breaks in the caption, the default thumb size like here works OK, I think. Dicklyon 04:46, 28 August 2006 (UTC)

We had two images showing construction. Not one, but two. You decided to create a third, and replace it. I am not stonewalling, I am asking only why? And if that is not unilateral, what is it? Why would you create a new image and replace two images that where already there? Care to explain why? I find it hard to understand your motivations for editing and your attitude. ≈ jossi ≈ t • @ 05:03, 28 August 2006 (UTC)

Right, two. Woodstone suggested it would be better to have one, combining the best properties of each. Nobody objected. It seemed like an obvious next move to improve the article. I tried to help. Improvements are still invited. Sorry if one of your contributions got removed in the process. Put it back if you think it's better. Or don't if you find having people mess with your contributions too "toxic". Please stop hounding me. Dicklyon 05:21, 28 August 2006 (UTC)

Bauhaus, Jung and the golden mean

There is substantial information about the application of the Golden mean by the Bauhaus and well as its by Carl Jung in his works. Are others willing to help and research the subject? ≈ jossi ≈ t • @ 16:53, 27 August 2006 (UTC)

Maybe you better help us get started. My quick search in books.google.com didn't turn up any connections. Dicklyon 17:13, 27 August 2006 (UTC)

Oh... Google books have but a very small sample of all books printed. Hopefully one day, publisher's permitting... ≈ jossi ≈ t • @ 17:17, 27 August 2006 (UTC)

Indeed, it's a modest sample, but on topics with "substatial information" it can be very helpful already. So, as I said, we need your help since I know you have books that aren't on it. Dicklyon 18:15, 27 August 2006 (UTC)

I am seriously considering my involvement in this article. It is becoming too toxic, and a waste of my time. ≈ jossi ≈ t • @ 05:04, 28 August 2006 (UTC)

Wasting my time

I have decided to remove this article from my watch list for a month or two. My reasons are explained here: Talk:Canons_of_page_construction#Merge_proposal ≈ jossi ≈ t • @ 05:24, 28 August 2006 (UTC)

That is a real shame. Over a long period of time, you have done a lot of excellent work on this article. Please reconsider. Finell (Talk) 08:24, 29 August 2006 (UTC)

Excessive textual attribution, Sources, Scope of Article, Suggestions for Improvements, and Recent Controversy

I'm a strong supporter of WP:CITE, but normally citation is enough (an exception is an historically notable source, such as Euclid or Plato). In this article, the attribution of statements to particular authors in the text in addition to the citations is excessive and detracts from readability. In the History section, we even have double attribution, direct and indirect: Rice says some other authorities said Egytians knew about the golden ratio; Rice says some other other authorities said that they didn't; Livio reviews the controversy and concludes .... It sounds too much like we are reviewing the books rather than writing about the topic and using the books as verifiable sources.

Also, what is with the repeated So-and-so "asserts"? It sounds too much like a lawyer's argument. Moreover, the use of "asserts" usually precedes refutation: "My opponent's witness asserts that [blah, blah], but the real facts are ...." My opponent's untrustworthy witnesses "assert"; my reliable witnesses "show," "prove," or "demonstrate". So, for example, is Rice "asserts" followed by Livio "reviews" and concludes an implied deprecation of Rice and promotion of Livio? If so, what place does that have in WP?

Is this the fallout of edit wars? What is going on here? Finell (Talk) 08:21, 29 August 2006 (UTC)

You take the words right out of my mouth. I have been many times on the verge of removing these unnecessary and distracting invocations of expert opinions. But in fear of edit wars (that have happened before on this article) I have desisted so far. The page looks more like a book review than an encyclopedic article. It's not the sources, but the facts that should be central. −Woodstone 08:34, 29 August 2006 (UTC)

Yes, I agree that it is a problem, but you willl need to discuss that with the editor that forced the hand in this regard. ≈ jossi ≈ t • @ 14:37, 29 August 2006 (UTC)

As I have declared, I will not be editing this article for a while, but I may comment from time to time. One obvious problem with the article is it's lead. It starts with a math equation and without context for what this is. Think of the reader. See WP:LEAD. Another problem is the order of the sections. The calculation section should not be the first section. We need to start with Pythagoras, then Euclid, then Paciolli, Kepler, etc. Provide the necessary historical context, the evolution of the concept, its application in the Renaissance, then progress to modern views, Bauhaus, Carl Jung archetypes. Then move to the math, the different proofs and numerical coincidences that have been widely reported and written about. Then a section about psychology and aesthetics, as there are several excellent studies on the subject. Then a section about the ratio and nature. Lastly, add a section about the debunking of the "mystical" properties ratio by Livio and a few others. This article can be well written, engaging and interesting without asserting any viewpoint. I was under the impression that it would be fun to edit and without controversy, such as I have experienced in political and religious articles I have edited, but I was wrong. Even articles such as these can become a minefield. Human nature, I guess, and a pity. So, I am now researching material for a section or a spinout article about the golden ratio in the Arts, that I will submit for your consideration in due time, rather than adding bit by bit and submitting myself to painful debates and specious concerns about each word. ≈ jossi ≈ t • @ 14:54, 29 August 2006 (UTC)

There are obviously some very different approaches taken to the golden ratio, by editors here and elsewhere. I had stated pretty much the antithesis of Jossi's proposal, which is that this article should be about the math, and the history of the math, which is pretty much uncontroversial. I support his current plan to build a separate article on the aesthetic/artistist applications and history, sightings in nature and in retrospective analyses, etc. Maybe we'll find a suitable way to re-integrate these aspects later. What hath Pacioli wrought? Dicklyon 18:55, 29 August 2006 (UTC)

Sorry, but I disagree with separating out aesthetics and leaving this article as pure math. That would make this article incomplete. It would be like an article on nitroglycerin that discussed its chemistry, but omitted signigicant discussion of its uses as an explosive and as a medicine. In most other treatments of the golden ratio that I have seen, the math, aesthetics, and history of both (which are difficuly to tease apart) are all discussed, and they are inherently inseparable. I do not see this being regarded as controversial anywhere else. If anything, I would rather see what Jossi proposes as a separate article to be folded into this one. If his idea of a separate article is primarily to avoid the arguments over this one, then this article and WP will suffer as a result. This article has FA potential, but it will not be realized without a more collaborative approach by all the editors.

I have taken the liberty of changing the heading to reflect the broader scope of what is being discussed (although continuing discussion of the several topics under separate headings would be preferable). Finell (Talk) 21:26, 29 August 2006 (UTC)

Finell, I agree with you. It needs to be a lot more than pure math. I just wanted to air an admission that my tendency is on the other end of the spectrum from Jossi's. Dicklyon 21:59, 29 August 2006 (UTC)

Truce proposal

Jossi, it's OK to mention my name as the editor you have this problem with. Here's how I propose we move on. I will (unilaterally) totally refrain from editing any of your contributions; however, I also suggest that you not directly edit mine nor revert recent things that have settled out. Instead, I will only comment on talk pages, and I invite you to do the same, so that changes will be made by others who support one approach or another. I urge you to agree, but I will adopt this approach unilaterally in any case.

Our issues stem from this very issue of how to attribute controversial statements, starting with your initial additons of the Rice stuff to this very article. I really only have one important issue with you, and I'm sorry that I or we allowed it to escalate; I apologize especially to the other editors here who may have been annoyed and inconvenienced by it. To summarize for others, the issue is this: statements of the form "X discovered (or describes) the use of the golden rectangle in Y", when controversial, unproven, or at odds with other material or widespread belief, should be rephrased as "X asserts that..." or "X hypothesized..." or something that indicates what X said without implying that it is accepted. I know you hate the word hypothesized for some reason, which is why we end up with asserts. Perhaps others will find a better way to avoid introducing the POV that such statements are accepted on their face.

I don't think we have any other substative issues, and I will try to avoid allowing myself to engage in any more bickering on non-substative issues.

Everyone else, feel free (as I've said many times) to undo any of my recent changes, additions, figures, etc., if you think it will help the article. Dicklyon 16:55, 29 August 2006 (UTC)

It can all be resolved if we focus on creating a great article that we can all feel proud of. I appreciate your comments and offer above, but for know I would remain unengaged for a while. It helps cool things off, and bring the common sense back. It is OK. We are all learning, an that is a good thing. Note that I have this page off my watchlist, so if you need my attention, please add a comment in my talk page. ≈ jossi ≈ t • @ 17:45, 29 August 2006 (UTC)

OK, we're good. By the way, I ordered a copy of Tshichold off ABE, but my order got cancelled because two other people ordered it, too. I guess somebody is noticing things... Dicklyon 17:50, 29 August 2006 (UTC)

Controversy

To me, one of the most interesting things about the golden ratio is the controversy over it. Like religion and politics, you find people with strongly held but opposing views on its place in art, history, and nature. We've danced around these issues by just saying who says what here and there, but I think it would be easier, and more productive to collaboration, if we could address it head on. It's pointless to argue over which view is correct; much better to address the differences, and discuss the analyses and viewpoints in the literature, than to just point at a few authors' opinions while ignoring the elephant in the room. We should be able to assemble and compare sources in an NPOV way (it would be WP:OR to use such assembly to support one point of view, so that must be avoided).

I think this would work better than sprinkling the controversy into history, aesthetic, and "disputed sightings" sections. Not that it won't be a challenge to get it right and broadly acceptable, but it's an approach that might help.

So, how about a section on "Controversies around the golden ratio" or something like that? We could juxtapose contrary viewpoints quite explicitly, so that the article could stay neutral instead of having POV items sprinkled here and there couched in quotes and weasle words.

For example, juxtapose the words of H. E. Huntley in his recent book The Divine Proportion,

• There seems to be no doubt that Greek architects and sculptors incorporated this ratio in their artifacts. Phidias, a famous Greek sculptor, made use of it.

with the words of Mario Livio, who, after noting that the mathematical theorems about the extreme and mean ratio were all developed some time AFTER the Pathenon was built, says,

• This is another example of the number-juggling opportunity afforded by claims based on measured dimensions alone. Using the numbers quoted..., I am not convinced that the Parthenon has anything to do with the Golden Ratio. ... Other researchers are also skeptical about phi's role in the Parthenon design."

Probably a better source than Huntley can be found, since he provides little reason to accept his assertion. This particular example is sort of "meta" in that it's a contrast between two authors' opinions on what is believed by others. More particular assertions on specific topics will be easy to find, too.

If this approach works, we can address the controversy without getting caught up in it.

Comments? Dicklyon 20:11, 30 August 2006 (UTC)

I would agree to this approach, if there is another source for opposing views tha nLivio. Otherwise this article will be Livio vs others and that will make the article tedious to read and innapropriate. ≈ jossi ≈ t • @ 00:51, 31 August 2006 (UTC)

I guess by "opposing" you mean on the skeptical side, like Livio and me. I'm sure we can find other sources, like Martin Gardner's 1955/57 book Fads and Fallacies in the Name of Science that has a section on Pyramidology. And others, though the more speculative works seem to be much more numerous. I suppose anything on which we don't find two sides is not properly part of this section, although not everything that I might regard as controversial is necessarily specifically addressed by the skeptical guys. Dicklyon 01:10, 31 August 2006 (UTC)

Actually, on quick look back at Gardner, he takes Taylor to task for his Pyramidology, but doesn't talk about the golden ratio claim per se, as far as I can see. Still might have some relevant quotes. Dicklyon 01:18, 31 August 2006 (UTC)

Here are some useful sources: [8] [9] Dicklyon 01:37, 31 August 2006 (UTC)

I strongly disagree with having a "controversy" section. To the extent that there is substantial disagreement among reliable sources, that can be presented in the text while maintaining WP:NPOV. To the exent that there are statements that are not supported by reliable sources, they should be eliminated in normal editing. Reliable sources, of course, are not limited by an editor's personal taste: Livio is a reliable source, but not the only reliable source. The subect of this article really is not that controversial in the world of scholarship (the editing process and discussion here is an aberration that we should not project onto the article), and it is much less controversial than many other WP articles that do not have a "controversy" section. A "controversy" section invites POV pushing, which is what we are trying to get away from (aren't we?). It also invites statements for which there are not reliable sorces, but that one might attempt to justify in a "controversy" section as another non-mainstream viewpoint. That is not encyclopedic. I am as much of a skeptic as anyone, by the way. Finell (Talk) 04:16, 31 August 2006 (UTC)

Thanks for your feedback. Personally, I think the current structure has invited more POV pushing. If we had a section where controversies could be explicitly aired, in terms of pairs of opposing views with citations, but omitting anything that doesn't fit that mold, then we could avoid the "excessive citation" problem when controversial points get embedded in other sections. Let's see who else has an opinion or counter-proposal. Dicklyon 04:23, 31 August 2006 (UTC)

What Rice said

The complicated Rice statements on both sides of the Eqyptian question leave one wondering what Rice's conclusion was. As I read him, he thinks the random ocurrence of golden ratio is unlikely, but he concludes with a balanced observation: "There are no examples from the surviving Egyptian literature which can be accepted as demonstrating the conscious or planned use of the Golden Section in any theoretical or abstract sense. It may be quite simply, that the Eqyptians' extraordinary instinct for form and balance produced the effect of the Golden Section without the need actually to define it." That sounds to me like the thing most worth quoting in the article. Dicklyon 04:47, 31 August 2006 (UTC)

Variant definitions

Here's another "definition" of sorts from Euclid (one of the early books; p. 81 in 1828 edition of the Rev. Dionysius Lardner's London edition):

• Proposition XI. Problem.

(279) To divide a given finite right line (AB) so that the rectangle under the whole line and one segment shall be equal to the square of the other segment.

A line divided, as in the proposition, is said (vide Book VI) to be cut 'in extreme and mean ratio.'

Corresponding and related definitions are found in other 19th century books.

Elements of Geometry, Theoretical and Practical: Including Constructions of the Right Line and by... By Eugenius Nulty, Philadelphia: J. Whetham 1836 p.94

• PROBLEM VIII.

57. To divide a given right line AB in extreme and mean ratio, or so that the square of the greater part may be equal to the product of the whole line and the less part.

New Plane Geometry By Webster Wells, Robert Louis Short, D.C. Heath & Co. 1909 p.132

• 272. Def. A Straight line is said to be divided by a given point in extreme and mean ratio when one of the segments is a mean proportional between the whole line and the other segment.

A Treatise on Elementary Geometry: With Appendices Containing a Collection of Exercises for Students and An Introduction to Modern Geometry By William Chauvenet, Philadelphia: J.B. Lippincott & Co. 1884 p. 120

• 73. Definition. When a straight line is divided into two segments such that one of the segments is a mean proportional between the given line and the other segment, it is said to be divided in extreme and mean ratio.

Enjoy. Dicklyon 05:18, 31 August 2006 (UTC)

Excellent. Please consider adding these definitions, maybe in their own section in the article. ≈ jossi ≈ t • @ 01:47, 10 September 2006 (UTC)

Broad preference

There's an OK plot of preferences versus aspect ratio of a rectangle in this old book: [10]. It shows a very broad preference centered near 1.6. This might be worth citing as one of the early psychological studies. Dicklyon 01:32, 4 September 2006 (UTC)

Excellent find. It is definitively worth citing. ≈ jossi ≈ t • @ 01:48, 10 September 2006 (UTC)

Philosophical considerations

Does this "Philosophical considerations" section make any sense to anyone? I recommend getting rid of it if not, and fixing it if it has some discernable value. Dicklyon 06:43, 9 September 2006 (UTC)

• Support removing this section. Never consciously noticed these empty phrases before. −Woodstone 11:36, 9 September 2006 (UTC)

• Same here. Utter twaddle! —Tamfang 00:33, 10 September 2006 (UTC)

I would argue that material that is properly sourced should either be kept, or moved to a more appropriate article. Removal of material is not an option unless the material is orginal research (i.e. not sourced to a reputable sources), or if it is the viewpoint of a tiny minority as per WP:NPOV. ≈ jossi ≈ t • @ 01:46, 10 September 2006 (UTC)

There's the problem, then isn't it? If "utter twaddle" ideas are shared by more than a few people and properly sourced, we have to include them. Such is life in the realm of the divine proportion, I guess. So we better look at each of the statements and see where they fall.

• In Timaeus, Plato wrote that two things cannot join without a third component, that being a bond which unites them; and the greatest bond is the one that makes the most complete unification where the two things and the bond have all joined to become one. He contends that such a unification is effected through the adaptation of proportion represented by the golden ratio.[16] Plato (360 BCE). Timaeus (HTML). The Internet Classics Archive. Retrieved on 2006-05-30.

The cited page does not mention golden, divine, or ratio, but has the mean-and-extreme-ratio concept in this line: "For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean-then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one." This is a line worth quoting. Where the other twattle came from is unclear. And Timaeus isn't linked right.

• In The Power of Limits, Gyorgy Doczi writes that Buddha's teachings expressed the need to avoid excess by walking a middle path between self-indulgence and self-mortification. Doczi indicates the middle path reflects the harmony of the golden ratio as it pertains to human behavior. 17 Doczi, Gyorgy [1981] (1994). The Power of Limits. Boston: Shambhala Publications, Inc., p. 128. ISBN 0-87773-193-4.

Here Doczi, a recent golden gusher, confuses the golden ratio with the golden mean of philosophy. Checking the reference, on p. 128, Doczi actual says nothing of the golden ratio, but rather says golden mean in a completely non-numeric context. So this one definitely gets flushed.

• The philosophy of Summum maintains that because it is the human mind that interprets the characteristics and qualities of the golden ratio, it should be considered in its relation to the human psyche. According to Summum, our mental states dictate our sense of harmony and disharmony, and the resulting experiences become part of our memories that then hold us captive. 18 Ra, Summum Bonum Amen [1975] (2004). “Chapter 4”, SUMMUM: Sealed Except to the Open Mind (HTML), Salt Lake City: Summum. Retrieved on 2006-05-30.

• Our observations of the world are influenced by the rule of proportion, and the golden ratio can be considered a representation of the collective consciousness of humanity. "For if we were to view nature from an altered state of consciousness, the proportion would also be altered." 19 The Divine Proportion (HTML). Summum. Retrieved on 2006-05-30.

I have no idea what Summum is. Probably come cult. How many members does it need to be considered not a tiny minority? What about the fact that the ebook says "No part of this ebook may be used or reproduced in any manner without written permission except in the case of brief quotations embodied in critical articles and reviews."? Does that mean we can't quote since we're not doing an critical article or review? Here's what the ebook's chapter 4 (html) actually has to say (this is a critical review):

7 The Divine Proportion ascribed to the collective consciousness of your state of evolution has been expressed, "For of three (3) magnitudes, if the greatest (AB) is to the mean (CB) as the mean (CB) is to the least (AC), they therefore all shall be one."

8 Modern science calculates the joining and splitting apart of "string proportions" (EVENTS) as the source of Creation. The elementary particles observed by science represent excitation modes of strings. This leaves us with the question, "What creates the strings?" It is the subjective copulation of Nothing and Possibility that produces these vibrating string proportions in all their modes of excitation. As mentioned before, Creation cannot be examined in its source objectively for it requires a philosophical examination. Any examination in a material form is limited by time based dimensional space and creation's manifestation is infinitely dimensional. At this point it is left up to the student to fill in the details of their understanding of the mathematics of the Divine Proportion, also known as the Golden Section, Phi, the Magic Ratio, the Additive Series, etc. There is available plentiful published material which, when investigated by the student, will initiate them to an understanding of the Divine Proportion.

9 The Divine Proportion can be found throughout this universe; from the swirls of galaxies to the swirls of quarks; from the harmony of music to the very physical nature of Creation. The Divine Proportion is seen as the beauty and organization in nature, the harmony and glue holding the unity of the universe. This beauty is the quality or combination of qualities which affords keen pleasure to the senses, especially that of sight or that which charms the intellectual faculties.

10 The states of discord, insanity, and chaos in the universe are observations of states seemingly opposed to the accepted Divine Proportion. Yet this is a singularly human point of view. For within chaos is found organization, within the irrational is found the rational, within ugliness is found beauty, within hate is found love, within pain is found pleasure, within discord is found harmony, within insanity is found sanity, within falsity is found truth. In reality, all discord and chaos are human judgments of their perception of a state of consciousness in opposition to the Divine Proportion. Death and dissolution may appear as states of discord, and from the point of the fixed mind seem out of harmony.

Now, I'm not sure why you'd want any of this in an article about the golden ratio, but here it is, properly sourced, so I guess we have to put it in now. And I'm sure there's more where that came from. I suppose a more appropriate article, as Jossi suggested, is a good idea. Maybe Golden gushing or Golden twaddle. I know it's not our job to stamp out nonsense, but do we have to include it just because it exists? Have editors no discretion in deciding what ideas are mainstream enough, or sensible enough, or even coherent enough, to be encyclopedic? I guess not, that would be WP:OR.

In general, I would argue that the idea of the golden mean having philosophical implications is the POV of a tiny minority and therefore must be removed. But I'm not touching it. Y'all decide.

Dicklyon 02:34, 10 September 2006 (UTC)

You can spare the sarcasm, Dicklyon. In Wikipedia we describe all significant viewpoints, without asserting them. You may need to re-read WP:NPOV. Please do. Thanks. ≈ jossi ≈ t • @ 03:03, 10 September 2006 (UTC)

Math first in lead? or "concepts"?

Jossi, since the article is mostly in mathematical categories, and the math is the well defined part that says what the golden ratio is, I restored the lead to the math-first form from which you had inverted it. Discussion is hereby invited. Dicklyon 02:53, 10 September 2006 (UTC)

If this article was called Phi (math), I would agree with you. But is not. It is called Golden ratio. If you wish, we could split this article in two: Phi that descibes the matematical properties of this irrational number, and Golden ratio that describes everything else. ≈ jossi ≈ t • @ 02:59, 10 September 2006 (UTC)

The title is pretty inclusive. But the categories suggest that it is more of a mathematical topic than an art/history/philosophy topic. Defining the golden ratio before saying why it is interesting to those other areas seems totally appropriate, for such an article. If you want another article that's primarily about the application of golden ratio in non-mathematical fields, I suppose that's OK, too. Dicklyon 06:26, 10 September 2006 (UTC)

The "Golden ratio" is not a mathematical concept. Phi is. It is your POV that this is a mathematical article, but the vast majority of all literature about this subject all include what you consider to be spurious concepts. Please do not use this article to push your POV. Wikipedia is not a place for advocacy of any kind. ≈ jossi ≈ t • @ 14:38, 10 September 2006 (UTC)

Phi is a symbol for the extreme and mean ratio that is named after a Greek sculptor who may have used it. How is that more mathematical than its other names? And if the golden ratio is not a mathematical concept, I'll be a monkey's uncle. Dicklyon 14:44, 10 September 2006 (UTC)

1. Go to the library
2. Check a few encyclopedias and see the entry for "Golden Ratio" or "Goldeb Section"
3. Ask the librarian to show you some books about the Golden section
4. If any of these books or encylopedic articles start with a math formula, give your nephew a banana. ≈ jossi ≈ t • @ 14:54, 10 September 2006 (UTC)

Jossi, I thought we were going to both stop provoking each other by changing things in this article. It's easy to see why it might be better to leave that to others. By the way, we must hang out in different libraries, because I don't find a lot of book on the golden ratio, and one I have handy, which is the most thorough study I've ever seen on it, contradicts you. The subtitle "The store of phi, the world's most astoninshing number" is followed up by the opening line "The Golden Ratio is a book about one number—a very special number. You will encounter this number, 1.61803..., in lectures on art history, and it appears in lists of "favorite numbers" compiled by mathematicians." Nobody's tring to deny the popular application of this number in the arts, but first you need to know about why the number is so special and appealing to know why artists and pseudo-scientists love to adopt it.

Here's another: The Golden Ratio and Fibonacci Numbers, by Richard A Dunlap, opens with "The golden ratio is an irrational number defined to be... It is of interest to mathematicians, physicists, philosophers, ..."

Here's one on your side, maybe: The Divine Proportion: a study in mathematical beauty, by H E Huntley, "The theme of this book is the aesthetic appreciation of mathematics". It's hard to tell from that if the book is even about the golden ratio, but it is. And it's very mathematical. That's the kind of beauty we need to represent more in the article on golden ratio.

If there are particular books that you think will help convince us that the golden ratio is less a mathematical topic and more an arts/philosophy topic, please mention them, and I will endeavor to access and read them.

The new lead paragraph is quite a floater. It doesn't really give much clue what the golden ratio is. I'm sure one could include something in the lead about the number being interesting in the arts without removing the substance of the mathematical definition.

And for someone who has done such a thorough job of excoriating me for making changes without discussion and consensus first, and accusing me of pushing a PPOV, this is pretty bold and obnoxious of you to make such a major change of POV to an article. As a long-time admin you are much more skilled and powerful about getting your way in wikipedia. All I'm trying to do is to help defend this article against it obvious long-term problem of trending toward unsupportable mumbo-jumbo (read the talk and you'll see it). Just because you sometimes cite a book to backup your factoids doesn't mean that the information is verifying or encyclopedic; I continue to discover errors in the POV expressed in your contributions by getting and checking your references. You are in serious need of watchdog, and I don't being it, but nobody else has helped, so you're stuck with me.

I hope we get at least some feedback on a couple of our issues at the request for comment that I started. So far nothing; are the other admins all afraid to cross you?

Dicklyon 15:53, 10 September 2006 (UTC)

Afraid to "cross" me?, no way... What we need, Dicklyon is a good lead to the article. But not starting with a formula. The current version, is not good as it does not explain what this subject is all about. See WP:LEAD. ≈ jossi ≈ t • @ 16:05, 10 September 2006 (UTC)

I'll see if I can fix it then, if nobody else does. What you've done to it was a big step backwards, but it may force us to a better place. Dicklyon 16:27, 10 September 2006 (UTC)

With Woodstone's help, I think we converged on a much better lead. See if you agree. Dicklyon 20:41, 10 September 2006 (UTC)

Yes. Much better. I removed the formula from the lead as it is already presented in the first section after the lead. ≈ jossi ≈ t • @ 20:55, 10 September 2006 (UTC)

How can you on the one hand agree that it's much better, and on the other hand turn around and destroy it by removing the definition of the value of the golden ratio? (this diff) I remain perplexed at what goes on in that pea brain of yours. Dicklyon 21:35, 10 September 2006 (UTC)

As for the "watchdog" thing, note that what works best on WP is when editors do check each other's contibutions. Feel free to correct me, if needed. ≈ jossi ≈ t • @ 16:07, 10 September 2006 (UTC)

Oh, I do. Here's another. I checked the modern Britannica online, and it starts with the math "Golden ration, also known as the golden section, golden mean , or divine proportion in mathematics, the irrational number (1 + Ö5)/2, often denoted by the Greek letters t or f, and approximately equal to 1.618." then the history, then I'd have to sign up to see more. Dicklyon 16:27, 10 September 2006 (UTC)

I have to agree that the article looked much better with the precise value in the opening, rather than the arbitrary approximation it now has. On a practical note, giving an approximation to an arbitrary precision at the beginning is likely to encourage the tedious sort of "digit creep" that happens so often on the Pi page. Madmath789 21:15, 10 September 2006 (UTC)

I don't care what the first sentence says, but the exact, algebraic formula for the golden ratio should absolutely be included in the lead section. The claim that "The 'Golden ratio' is not a mathematical concept." is a very strange one. The golden ratio is a perfect example of a mathematical concept, just one that happens to have applications outside pure mathematics. Fredrik Johansson 21:38, 10 September 2006 (UTC)

A phrase like "the irrational value of 1.618033989" is such nonsense that it should help you understand that Jossi has little idea what "mathematical" means. But the equation seems to scare him. Peculiar, as he claims to be an engineer, too. Dicklyon 21:45, 10 September 2006 (UTC)

I jut wanted to delete the formula as it is alreday featured below it. There is no need to have dups. Corrected lead as per Britannica's example you provided. Also note that many of the uses of the Glden ratio predate the knowledge of that formula, and the knowledge of irrational numbers. ≈ jossi ≈ t • @ 21:51, 10 September 2006 (UTC)

Remember discussion and consensus? I guess not. And what is that you believe it means when you claim that "many uses the Golden Ratio predate the knowledge of that formula, and the knowledge of irrational numbers?" How can someone use something they don't have? Maybe you have an example of some of these many? Did you get your copy of Livio and read it yet, by the way, or was it too mathematical? And why do you attribute to me and to Britannica your butchering of their perfectly good lead to justify yours? Dicklyon 21:59, 10 September 2006 (UTC)

Hey! watch your words and your tone, please. I will not engage you if you keep this tone of voice. You keep looking at this subject from a narrow matematical viewpoint, and that is why we are having such disputes. For you, if the ratio is an approximation, then it is disputed. For a mathematician, that is important. For an artist, it is not. ≈ jossi ≈ t • @ 02:20, 11 September 2006 (UTC)

Hey! You're right. I apologize for my words on the talk page. Now you apologize for what you did to the article against an obvious consensus to the contrary? Dicklyon 02:56, 11 September 2006 (UTC)

My action "against obvious consensus" resulted in a much better lead that we had before. ≈ jossi ≈ t • @ 04:38, 11 September 2006 (UTC)

Hey! That's right. Wait; no it's not. There's no more a and b to connect with the lead image. Oh, well, if it means your admonition to not change things without seeking consensus first is no longer operative, I'll just have to settle for that. Dicklyon 04:48, 11 September 2006 (UTC)

<<<Not every single edit requires consensus, oterwise we woul never make any edits. If there is a dispute about something, then we resolve it by a dialog seeking consensus. ≈ jossi ≈ t • @ 13:36, 11 September 2006 (UTC)

Right, but if your change has been contradicted and fixed by three other editors, and comments in talk, and you persist as you did in removing the mathematical definition of the main topic of the article, that's a bit outside normally acceptable procedures, I would have thought. Maybe not any more... Dicklyon 15:23, 11 September 2006 (UTC)

Gray rectangle

The gray rectagle does not provide any value to the article's lead. I propose to delete it. ≈ jossi ≈ t • @ 13:38, 11 September 2006 (UTC)

If there's nothing in the article that suggests that a golden rectangle is a particularly pleasing shape, then we have no need to illustrate such a shape. If there is, we do. I'll have to review it. Dicklyon 15:23, 11 September 2006 (UTC)

In my opinion, this particular (almost) golden rectangle doesn't add much. I think that a golden rectangle with the same labeling and color scheme as the sectioned line segment (which is great) would aid readers' understanding. Further, discussion of the aesthetics of phi and its aesthetic applications usually involve golden rectangles (occasionally golden triangles). Finell (Talk) 17:20, 11 September 2006 (UTC)

Edit summaries

Please do not use the edit summaries to express your POV. Describe your edit, and that will be more than enough. Thanks. ≈ jossi ≈ t • @ 03:05, 10 September 2006 (UTC)

Why? What better place is there to express a POV than in an edit summary? Dicklyon 06:22, 10 September 2006 (UTC)

The Talk page (i.e., here) is the better place. Finell (Talk) 17:22, 11 September 2006 (UTC)

Because edit summaries cannot be refactored. ≈ jossi ≈ t • @ 14:35, 10 September 2006 (UTC)

I guess I just don't get your point. As usual. Dicklyon 14:42, 10 September 2006 (UTC)

Edit summaries stay forever in page history. So, when you use the edit summaries, just state what you did and then discuss your reasons here. Basic etiquette. ≈ jossi ≈ t • @ 15:14, 10 September 2006 (UTC)

Sorry, if that's the etiquette of wikipedia I will try to respect it. Is it on a policy page some place? Because it seemed to me that having a POV in an edit summary would be useful to people trying to review and understand the history of edits. Dicklyon 15:55, 10 September 2006 (UTC)