Talk:Golden ratio

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)[reply]

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

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

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

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!

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)

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

"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)

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. :-)

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)

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)

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)

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:

It is not encyclopedic.
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.
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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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

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)[reply]

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

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)[reply]

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

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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

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:

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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:

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Formula deleted?

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

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]