cars - Talk:Golden ratio

Contents

1 Good page

2 Errors in article

3 Why?

4 "Everyday people" explination

5 Digits

6 Statistics?

7 One version, one page

8 All over and fractals

9 introductory paragraph and the golden ratio conjugate

10 Nonsense should/will get scoured from this article

11 Nautilus nonsense

12 Edit by User:Jacquerie27

13 University of Cambridge Is Not a Dubious University

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

15 Deleted section "Fun with the ratio"

16 Re recent revision of introduction

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:

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)

"Everyday people" explination

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)

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)

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)

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.

Professor Corinna Rossi was a Junior Research Fellow 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:

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 ot 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 sgnificant 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)

Re recent revision of introduction

The introduction to the article has been rewritten by . 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)