Talk:Golden ratio/Archive 2

Organization

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

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

Wide-Screen

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

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

I don't see the reason why they made computer monitors 16:10 whereas televisions remain 16:9. Is it to do with the taskbar top and bottom, so one can watch full HD movies and still have the taskbar on screen? Stoney3K 22:38, 24 September 2007 (UTC)

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

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

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

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

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

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

For those interested in numerical coincidences 5:8 is the ratio between the Kilometre and the British Mile! Eljeh Yendig 03:20, 27 January 2007 (UTC)

Is that an exact conversion ratio, or an approximation?

For some further trivia, current 35mm Panavision movie prints have an aspect ratio of approximately 2,6:1 (being 35mm Academy with sound with a 2:1 anamorphic lens). That's the golden ratio squared ;-)... Regarding photography though, the classic 3:2 print format for 35mm still photo may be derived from the golden ratio, as it corresponds with the book pages, and would have been easier in print purposes. Stoney3K 22:38, 24 September 2007 (UTC)

Fuβbol (soccer)

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

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

I don't think German uses the Greek letter β. In fact.. I don't even think Fuβbol is a word in any language. Did you mean Fußball? oops, forgot to sign RedAugust 06:31, 28 July 2007 (UTC)

Yes, I believe so. I have seen the Greek β often used in place of ß for spelling German words. Back in the days of IBM Selectric typewriters, you had to do this. I recall in later years, computers would render the ISO-Latin-1 character xDF ambiguously so you could use it as either β or ß. I doubt many non-Germans can tell the difference unless they see the two characters side by side. =Axlq 16:44, 28 July 2007 (UTC)

Leaden ratio / Leaden number

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

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

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

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

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

Golden names again

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

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

Formula deleted?

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

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

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

Timeline

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

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

Rectangle image

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

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

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

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

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

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

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

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

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

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

Relationship to $\pi$ and $e$

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

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

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

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

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

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

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

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

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

phiCalc <- c()

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Un-attributed opinions? Original research?

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

The ancient Greeks knew the golden ratio from their investigations into geometry, but there is no evidence that they thought the number warranted special attention above that for numbers like $\pi$ (pi), for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception of beauty. They are, at best, inconclusive [1]. Despite this, a large corpus of beliefs about the aesthetics of the golden ratio has developed. These beliefs include the mistaken idea that the purported aesthetic properties of the ratio was known in antiquity. For instance, the Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio. This has encouraged modern artists, architects, photographers, and others, during the last 500 years, to incorporate the ratio in their work. As an example, a rule of thumb for composing a photograph is called the rule of thirds; it is said to be roughly based on the golden ratio.

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

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

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

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

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

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

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

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

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

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

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

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

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

Aesthetics: Parthenon

See these two excerpts from:

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

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

But see cautionary articles about his Canon, too: [2] or [3]

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

and

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

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

and you can find more rebuttals and such in scholar.google.com by looking up Hambidge golden ratio. Here's one: [4]

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

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

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

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

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

Dicklyon 04:12, 22 August 2006 (UTC)

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

History

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

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

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

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

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

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

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

Value judgements

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Deletion of material?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

More fascinating stuff about Phi

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

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

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

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

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

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

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

theories

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

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

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

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

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

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

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

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

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

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

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

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

I have found in most sources that you are entirely backwards. I was disappointed to see phi written with a lower case letter rather than the upper case letter. In all cases I have seen it is the upper case that refers to 1.618 INCLUDING the wolfram article that you cited earlier

http://mathworld.wolfram.com/GoldenRatio.html http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibFormulae.html http://www.amazon.com/gp/reader/0767908163/ref=sib_dp_pt/102-1480055-4996158#reader-link http://goldennumber.net/neophite.htm

The lowercase phi for 0.618 makes much more sense as it is a smaller quantity. —Preceding unsigned comment added by 129.128.232.159 (talk • contribs)

I think you are getting confused between $\displaystyle \varphi$, $\displaystyle \phi$, and $\displaystyle \Phi$. Both $\displaystyle \varphi$ and $\displaystyle \phi$ are lowercase, just written in different styles, and both are used interchangeably for 1.618. The upper case is $\displaystyle \Phi$, and is not used at e.g. the MathWorld article you link. —David Eppstein 04:34, 5 April 2007 (UTC)

Ok, I see. I understand that the upper case phi is not used really anywhere, but surely the $\displaystyle \phi$ character is much more common. It is the character that is used in the articles I cite.

I don't understand why the articles are using an image instead of ф. Is this just to distinguish between the mathemitical Phi and the Greek letter ф?RedAugust 23:17, 27 July 2007 (UTC)

Golden mean redux

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

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

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

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

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

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

52, 24 August 2006 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Boy, these paragraphs are really getting narrow! Finell

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

Consistent citation form

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

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

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

Squishy paragraph in the lead

The lead includes this paragraph:

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

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

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

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

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

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

Properties of $\varphi$

I think that;

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

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

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

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

Another unique and interesting property of $\varphi$, not mentioned (that I can see) on the main page, is that $\varphi ^{2}=\varphi +1$. Perhaps this could be worked into the article somewhere? And perhaps mention the irony of the fact that "the golden ratio" isn't a ratio as such, but an algebraic irrational. 193.122.47.170 20:43, 22 April 2007 (UTC)

It is not a unique property and only a slight variant of the defining equation and it is mentioned several times in the article, e.g. under "Alternate forms", "Other interesting properties", in "Relationship to Fibonacci sequence" in a more general form. It is a ratio, but not between whole numbers. −Woodstone 21:05, 22 April 2007 (UTC)

According to my calculator's quadratic-solving program, the only solutions to $x^{2}=x+1$ are (as already mentioned in this section) $\varphi$ and $-\Phi$. As the one is the negation of the reciprocal of the other, this clearly indicates that this property is unique to $\varphi$. 193.122.47.170 14:03, 23 April 2007 (UTC)

You're not making sense. If it's true to two numbers, it's not unique to one. And it's already in the article that there are two solutions of that quadratic equation. What more can be said about it? Dicklyon 17:38, 23 April 2007 (UTC)

Construction picture

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

Specifically you are talking about these two pictures;

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

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

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

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

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

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

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

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

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

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

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

summary of classical construction upon a unit square

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

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

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

New attempt by User:dicklyon

How about something like this one I just whipped up?

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

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

approximate golden rectangle 144 by 233 pixels

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

another try with less intense annotations

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

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

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

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

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

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

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

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

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

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

Bauhaus, Jung and the golden mean

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

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

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

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

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

Wasting my time

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

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

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

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

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

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

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

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

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

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

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

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

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

Truce proposal

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

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

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

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

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

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

Controversy

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

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

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

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

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

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

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

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

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

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

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

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

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

Here are some useful sources: [5] [6] Dicklyon 01:37, 31 August 2006 (UTC)

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

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

What Rice said

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

Variant definitions

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

• Proposition XI. Problem.

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

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

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

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

• PROBLEM VIII.

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

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

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

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

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

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

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

Broad preference

There's an OK plot of preferences versus aspect ratio of a rectangle in this old book: [7]. It shows a very broad preference centered near 1.6. This might be worth citing as one of the early psychological studies. Dicklyon 01:32, 4 September 2006 (UTC)

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

Philosophical considerations

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

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

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

• Support removing this section. Shallow and weak. I don't think that Summum is a "reputable source" on this topic (see my specific comment below). If Timaeus is referring to golden (or extreme and mean) ratio (it does sound like it is), then it is doing so 2 generations before Euclid; perhaps it should be mentioned in History section. Also, I do not want to reawaken the confusion with Golden mean (philosophy). Finell (Talk) 09:40, 12 September 2006 (UTC)

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

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

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

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

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

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

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

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

• • Summum is a small cult whose founder claims to be in contact with "advanced beings" and whose members claim to have seen the founder perform miracles. I don't think that this qualifies as a reliable source or a significant viewpoint on the subject of the golden ratio, nor have I seen it cited as such in any other reliable source on the golden ratio. Finell (Talk) 09:19, 12 September 2006 (UTC)

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

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

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

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

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

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

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

Dicklyon 02:34, 10 September 2006 (UTC)

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

Math first in lead? or "concepts"?

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

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

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

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

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

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

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

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

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

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

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

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

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

Dicklyon 15:53, 10 September 2006 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Gray rectangle

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

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

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

It's unclear why you say "(almost)" here while at the same time suggesting that a real golden rectangle is possible. As has been discussed above, this is the most accurate possible golden rectangle that can be clearly delineated in square pixels. It's 144/89 = 1.61797753, off by only 0.000056. For the new one, Inkscape shows me a ratio of 1.61754492, off by 0.00049, almost 10X as much. And by the time the SVG scaling and line thickness and fuzziness are incorporated, there's no telling what it is. So we've gone from an exact-as-possible golden rectangle to an "almost" one. Dicklyon 01:23, 12 September 2006 (UTC)

That greay rectangle is in the lanscape orientation, and does not show the rectangle as it is depicted in most sources. Usually it is depicted in "portrait" mode (Narrow side on top), and not filled-in. A simple outline on a pleasant color or uncolored would be best. ≈ jossi ≈ t • @ 23:35, 11 September 2006 (UTC)

The sources I looked at always had it in landscape orientation, your book-page sources being the exception. Dicklyon 01:23, 12 September 2006 (UTC)

Replaced the rectangle as per above, for editors consideration. Ratio 1.618:1 ≈ jossi ≈ t • @ 00:05, 12 September 2006 (UTC)

I editted the caption to be consistent with it. Suggestion, Jossi: can you put more border around it, and equal border all around, so that the shape is not so much confounded by the thumbnail frame? And should we worry that the golden color will confuse the meaning in the minds of some readers? Dicklyon 01:03, 12 September 2006 (UTC)

Border fixed. ≈ jossi ≈ t • @ 02:38, 12 September 2006 (UTC)

rectangle orientation

Some book refs on landscape orientation: [8] [9]. What you got showing portrait orientation? Dicklyon 01:32, 12 September 2006 (UTC)

Yes, there are books that show it in landscape orientation. But most contruction diagrams I have show it in portrait mode. Feel free to replace with this landscpe version Image:Golden Rectangle landscape.svg. IMO, the proportion is better appreciated in portrait mode, but maybe that is just my artist's eye. ≈ jossi ≈ t • @ 02:47, 12 September 2006 (UTC)

Well, if you're not going to give us a reference, I guess I have to. Here's more in landscape format: [10] [11] Dicklyon 03:07, 12 September 2006 (UTC)

OK, changed it, and it's less ugly than the big tall one, imnsho. At this point, if anyone wants to remove it altogether, I won't object. I think that the figure has totally lost its point, since it no longer illustrates the shape very exactly, nor in sufficient isolation to allow the shape to appreciated as distinct from the frame around it. And the color and border also distract from the basic shape. It could be improved by giving it more border still, and if you gave it enough border then a portrait-mode one would even be an improvement. Whatever. Dicklyon 03:15, 12 September 2006 (UTC)

File:The Roses of Heliogabalus by Alma-Tadema (1888), oil on canvas..jpg

The Golden rectangle is thought by many to be a most pleasign proportion. The Roses of Heliogabalus, by Alma-Tadema (1888), canvas measures 52" by 84 1/8", dimensions that are almost exactly in the golden ratio.

Here's another possibility to illustrate the idea of the golden rectangle. A rectangular painting that actually was designed to be a golden rectangle (or at least is unbelievably close, if I can believe the article The Roses of Heliogabalus), and doesn't have an extra frame line. But it's still not portrait orientation. And unfortunately the JPEG is not anywhere near the quoted shape; but we could stretch it up to fix that. Dicklyon 03:42, 12 September 2006 (UTC)

Isn't it obvious that the golden rectangle is seen widely in both orientations? Most of the architectural sightings are landscape (although some are portrait), in page design it is usually portrait, and in the dimensions of paintings the examples I recall are landscape but there are probably portrait examples as well (we do borrow the terms portrait and landscape from the orientations of paintings, after all). And we could all do without the sarcasm, please. Finell (Talk) 08:47, 12 September 2006 (UTC)

Also, the new image has more accurate colors. I didn't notice it before, but the prior version was too green. Finell (Talk) 04:01, 20 September 2006 (UTC)

Version on commons with aspect ratio off in the other directin (1:1.52); I made one in the middle and linked it in the artice; don't use either of these. Dicklyon 22:38, 19 September 2006 (UTC)

This is the one now used in the article.

Edit summaries

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

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

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

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

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

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

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

No answer? I can't find any such etiquette quideline. In fact, Help:Edit summary says "An edit summary should strive to answer the question, "Why did you make this edit?". Providing an edit summary, even if the edit is minor, makes Wikipedia work better by quickly explaining to other users what your change was about." So if I put a bit of my POV into the summary to help explain why I'm making it, isn't that a good thing? Of course, if it's too big or complicated to fit, I do bring it up on the talk page. So am I missing something, or are you guys just making up new rules for some reason? Dicklyon 01:12, 12 September 2006 (UTC)

WP:ES has a section entitled Use of edit summaries in disputes that applies here:

Proper use of edit summaries is critical to resolving content disputes. Edit summaries should accurately and succinctly summarize the nature of the edit, especially if it could be controversial. If the edit involves reverting previous changes, it should be marked as a revert ("rv") in the edit summary. Avoid using edit summaries to carry on debates or negotiation over the content or to express opinions of the other users involved. Instead, place such comments, if required on the talk page. This keeps discussions and debates away from the article page itself.

Finell (Talk) 06:57, 12 September 2006 (UTC)

Cooperation and civility

These cornerstone ideals of the WP community are being flouted here, both in the editing of this article and on this talk page. The editing of this article has become a nightmare, and the article is suffering as a result. It is time for the combatants to direct their considerable knowledge and competence toward cooperation, with the objective of improving this article to the point where it can be a featured artilce candidate. It is nowhere close to that quality now, in my opinion, and the substantial work since July of adding content and sources is not bringing it closer.

Many of the primary combatants' edits are aimed alternately at pushing his own POV (often subtly, sometimes not) and cutting back the other's POV. Adding so-called factual statements selectively (even with citation) can be a form of POV pushing, as can weakening statements with unnecessary qualifiers, as can promoting or deprecating sources (e.g., "James's analysis demonstrates" versus "Jones asserts that"). I can understand, although not condone, POV disputes over controverial subjects like the War on Terrorism, for example, but what is so controversial about the golden ratio or the explicit references to it in several areas of aesthetics (e.g., architecture, painting, page design)? Although most of the the golden ratio's history is undisputed, why does our History section devote disproportionate space to the fringe areas of dispute? Why is this article the place for a dig at pyramidology; most readers who see that word would interpret it literally as the study of pyramids, not realizing that this prarticular -ology has a derogatory connotation.

This is a primarily a math topic: a ratio is a mathematical relationship. But that does not mean that the history, applications, and cultural significance of this particular ratio should be ignored or downplayed in an encyclopedic treatment.

I hope that other non-combatants who have contributed to this article will express their views on how we can move forward productively. While the combatants are of course free to comment as well, in my opinion they have already expressed themselves sufficiently. Finell (Talk) 08:33, 12 September 2006 (UTC)

Well said, Finell. It has become a real nigthmare indeed. I will make efforts not to get upset and not to bring that feeling into play. ≈ jossi ≈ t • @ 09:21, 12 September 2006 (UTC)

Same here. Dicklyon 14:26, 12 September 2006 (UTC)

Plea for help

It would be extremely helpful if others would jump in with edits or opinions, especially when Jossi and I find ourselves on opposite ends of a 1-to-1 deadlock. Opinions before we get frustrated with each other would go a long way to preventing our "sarcasm" and such. Finell, your questions above are very good ones; please do weigh in with suggested changes, or just do it. Dicklyon 14:50, 12 September 2006 (UTC)

History issues

Finell, you ask "what is so controversial about the golden ratio or the explicit references to it in several areas of aesthetics?" and "why does our History section devote disproportionate space to the fringe areas of dispute?" I think both Jossi and I have provided partial answers to these questions. We even agreed on a possible resolution, to which you strongly disagreed. The problem is that history of the golden ratio, starting from Pacioli, is the history of overloading a mathematical concept with divine, mystical, pseudoscientific, and aesthetic properties and wild theories; and the frequent uncritical repetition of wild theories by numerous authors. Jossi likes to quote them in a way that to me makes it look like what they are saying is implicitly accepted; I've pushed back, and said that if we include such things we need to make it clear that they are just the statements of particular individuals, and try to not imply any connection to objective reality when no such support exists. For example, historian John Man has written a great book about Gutenberg, but in the section where he claims that Gutenberg's page is proportioned according to the golden rectangle, at the same time that he states dimensions that are in the ratio 1:1.45, and with a footnote that makes it clear just how innumerate he is, he must just be repeating something he heard somewhere. Jossi has added Man's statements in articles Johannes Gutenberg, Golden canon of page construction, etc., so I figured it would be a good one to add here an example of false sightings. I would welcome your rewrite of the history section, in collaboration with Woodstone and others here who have less extreme POV than Jossi and I have; and I'll keep my hands off unless Jossi puts something in that I need to react to again. Dicklyon 17:35, 12 September 2006 (UTC)

My view is that as per NPOV, we need to describe widely held viewpoints, even if these are mistaken, subjective, non-scientific, etc. This specific subject has a huge variety of reliable sources that describe the viewpoints. The article needs to describe these, and that is what we are doing. I personally do not care one way or another, Dick believes that I have a "POV" in respect of this fascinating ratio. I do not. ≈ jossi ≈ t • @ 22:25, 12 September 2006 (UTC)

And this is the kind of statement that makes it hard for me to WP:AGF when I see your edits. I at least admit my POV. Dicklyon 23:59, 12 September 2006 (UTC)

My POV is that we need to apply NPOV to this and other articles, as expressed above. I really do not have a "pseudoscientific" POV about this subject as you seem to think. As an artist, typographer and book designer, I look at this subject from an artistic perspective rather than a scientific one... Maybe that is my POV... but it should not be an opposing POV to yours, just different. ≈ jossi ≈ t • @ 00:20, 13 September 2006 (UTC)

OK, then we're in total agreement. We need to apply NPOV even though we approach it with different "perspectives". Dicklyon 00:25, 13 September 2006 (UTC)

Sure. That is the spirit. We need to report signficant viewpoints, and attribute them to those that hold them. ≈ jossi ≈ t • @ 00:27, 13 September 2006 (UTC)

Pyramids

OK, I took the messy controversial distracting pyramid stuff out of the history section. If someone wants to talk about it, put it in a section on pyramids, disputed histories, controversies, or something. Dicklyon 03:15, 13 September 2006 (UTC)

No, no, no, no, no.... You cannot delete material because you think it is "distracting". If it is a controversy, let's describe it. You could just have done what you asked others to do: move the material to a different section. 04:10, 13 September 2006 (UTC)

That's a good way to test and see if anyone cares. If anyone but you had put it back, we would have learned something. To paraphrase you anyway, "My action "against obvious consensus" resulted in a much better section that we had before." Dicklyon 04:26, 13 September 2006 (UTC)

Just a quck check on the databases I have access to, I found more than 200 sources that describes the pyramids and golden section, and possible explanations. It will take time to research these, but I will start slowly with the most prominent ones. ≈ jossi ≈ t • @ 04:17, 13 September 2006 (UTC)

That's consistent with what I've been saying. The question is what to do about it, since there's no evidence that anyone's been able to find to support any of that pyramidology. Dicklyon 04:26, 13 September 2006 (UTC)

This is a weak section. 2 separate questions need to be distingushed: (1) Did Egyptians know φ and deliberately use it in designing pyramids? (2) Do pyramid proportions approximate φ? If answer (2) is yes, that is significant for 2 independent reasons. First, if approximations are widespread and very close, that may be circunstantial evidence that answer (1) is yes. Second, if proportions are reasonably close to φ even if Egyptians did not conciously use it, that is some evidence that φ is a pleasing proportion. (Similarly distinguishing between deliberate use of φ and proportions that approximate φ unintentionally applies to a lot of the aesthetics discussion, by the way).

Organization is achronological: Rice first, then back a century, then post-Rice.

Is it ONLY the pseudoscientific pyramidologists who find pyramid measurements that approximate φ? No serious scholars (regrdless of whether Livio disagrees with them)?

Section says Rice asserts A, then says Rice asserts not A. This says nothing. Plus, both Rice and Livio are secondary or tertiary soruces. If there the are no reasonably conclusive answers to the questions that the section raises, then the section becomes a short paragraph: Evidence is inconclusive, scholars disagree [citations]. We are an encyclopedia, not a soapbox for conflicting inconclusive opinions, even if we present both sides fairly. NPOV is essential, but it is not the only criterion of encyclopedic content.

Section says that proportions that are close to φ, but same proportions are also close to π. How can same proportion be close to both? Is there any history of using π as an aesthetic proportion?

The Summum winery pyramid (which I deleted from this section) belongs in Aesthetics, as exampls of a building deliberately designed with golden ration proportions.

In any event, this should not be a main level 2 section, of equal importance to Math and History, for example. It should probably go in Aesthetics, with the rest of the architecture. And Aesthetics should be subdivided, with a lead about pleasing proportions, then subsections for architecture, painting and sculpture, page design, and music (and anything else I left out). Finell (Talk) 08:22, 13 September 2006 (UTC)

I look forward to your improvements on this section. Dicklyon 15:10, 13 September 2006 (UTC)

The lead again

Finell, your changes to the lead have good and bad parts. The Livio quote is much better that what we had. But starting with the value instead of the definition is not so good. Definition, then value, then explanations and properties, like we had before, would be my preference. As it stands, the concept of cutting a line segment into extreme and mean ratio looks like an incidental property of the value; it is actually the definition. Dicklyon 15:10, 13 September 2006 (UTC)

I am not so sure about the long quote from Livio. It would be better a summary of that quote, or a summary of several references. ≈ jossi ≈ t • @ 15:58, 13 September 2006 (UTC)

That would be OK, too, if the summary is generally acceptable to us all. That's pretty much what we had before, though perhaps too brief. Would you propose to put that one back, or do you have a new one to propose? Dicklyon

I'm giving it another try now. I'll do it in 2 or 3 edits, so please be patient. Finell (Talk) 16:46, 13 September 2006 (UTC)

I tried again. Please see what you think. I restored and edited some of what we had before about aesthetics and other interests, but I prefer to keep the Livio quote as it is. I realize that it is long. However, it is an eloquent statement of the importance and widespread interest in our topic, which is an important part of a strong lead. That it is made by a skeptic, who avidly exposes claims that he believes to be unfounded and cannot be accused of being a golden ratio "enthusiast," lends credibility to his assesment. The details, with citations to other sources, are developed in the body and can be developed further. Of course, that is just my opinion. Finell (Talk) 18:40, 13 September 2006 (UTC)

I pretty much like it, except for the fatty math equation in the opening paragraph, in which the equal sign is used as a verb; this is never a good idea, and looks particularly bad here. We need to "display" the equation or remove it, in my opinion; it's not really crucial to the definition, which works fine in words, so it can be put later or left for the calculation section. The definition and value are the top items to lead with, not the derivation or explication. The new paragraph on uses is good; not too waffly, not so specific that citations are needed. Dicklyon 18:31, 13 September 2006 (UTC)

I don't like the look of my inline equation either. Perhaps it can be formatted better. However, in my opinion, equation is the essential definition; that is, the golden ratio is the irrational number, and the rest of it is descriptive, although important. If the article were entitled golden section, I would say that the sectioning of the line is the definition and the ratio is a calculated result. I have to go back to work and can't do anymore for 8-10 hours. Please play nice, everyone! Finell (Talk) 18:52, 13 September 2006 (UTC)

You could just as well take the definition to be one plus the square root of five over two. But historically, the definition is geometric, not algebraic. See some of the old definitions that I described above in talk. See if you like the way I rewrote it. Dicklyon 18:56, 13 September 2006 (UTC)

Looking good! ≈ jossi ≈ t • @ 19:17, 13 September 2006 (UTC)

I like it too. I tweaked the golden section paragraph and figure caption to coordinate better and was able to eliminate some duplicate text that way. Please see what you think. Finell (Talk) 20:55, 13 September 2006 (UTC)

I agree with the statement above tat the sequence should be: definition (geometric, then algebraic), value, explanations (in the sense of usage) and properties. But we discussed about explicitly attributed quotes before and they should be minimised. This is not an article about books about the golden ratio. Surely a lengthy quote does not belong in the intro. −Woodstone 20:37, 13 September 2006 (UTC)

Let's see if others express an opinion about the long quote. I don't disagree with Woodstone's principle, but in this case I think Livio's quote says what we want to say as well as it could be said. Finell (Talk) 21:01, 13 September 2006 (UTC)

It could be summarized, don't you think? ≈ jossi ≈ t • @ 21:49, 13 September 2006 (UTC)

We won't know until someone tries. I don't think I could express it as eloquently, but if we summarize it we can put it in a bigger font! Dicklyon 02:15, 14 September 2006 (UTC)

in nature?

Jossi put in "The golden ratio also makes surprising appearances throughout nature." and someone took it out. I'm a bit skeptical myself, because most of what I see is about Fibonacci numbers, not so much golden ratio. Can someone recommend a good book on golden ratio in nature so I can see what the thinking is on this? Dicklyon 02:22, 14 September 2006 (UTC)

I did not put that text. I just reverted an anon edit without explanation. ≈ jossi ≈ t • @ 02:27, 14 September 2006 (UTC)

Oops, I guess I misread the diffs; I see it was Finell who originally put that in. The question remains open: what is the situation on golden ratio in nature? Refs? Dicklyon 03:50, 14 September 2006 (UTC)

I put in that whole paragraph, including the nature sentence. And I thought Dicklyon approved of that paragraph. My source was Livio, which I am reading (2 chapters to go). That is one area of golden ratio appearances that Livio does not debunk, although most do involbe Fib sequences. As I recall, though, he mentions an eagle's golden spiral flight path (so it can keep a steady eye on dinner) with φ-based angles, and also φ-based angles in the spacing of leaves around the stem (to maximize leaves' exposure to sun, air, and rain), and a few others. Also, Livio stresses the close relationship between Fib sequence and φ. (NOTE: I wrote this reply before I saw Dicklyon's preceding comment, which I discovered when the edit conflict blocked my save.) Finell (Talk) 03:59, 14 September 2006 (UTC)

Well, it's true that I did not disapprove of it. Only after I saw that sentence coming and going was my attention drawn to it, and hence my question as to whether anyone knows a basis for it. And I did look in Livio already and didn't find any specific support for golden ratio in nature, just the fib stuff and the plant leaf angle stuff, which I see repeated many places but never with any data or reference that would lead to a credible primary source for it. I've found a few more books and papers that discuss golden ratio truths and fictions, and they usually put the nature stuff as true, but when you read it, it's usually just the fib numbers and the leaf angles again; other sources say the leaf angles are quite variable. I'd feel better with it if we could find a credible source, not necessarily to reference in the lead, but to follow up with later. Dicklyon 05:12, 14 September 2006 (UTC)

Perhaps you would like to take a look at the references that Livio cites for his nature chapter. And there is also the eagle. Finell (Talk) 06:37, 14 September 2006 (UTC)

The eagle constant-angle thing is again a characteristic of ANY logarithmic spiral; this defect in Livio of confusing logarithmic spirals in general with the golden spiral in particular has been discussed at length before (see [archives]). I'll take another look at what else Livio says and cites. Dicklyon 06:53, 14 September 2006 (UTC)

Do crystals of alluminum manganese alloy count as nature? I realize that the alloy is human made, but their cryatal structure is formed by laws of nature without human guidance. Finell (Talk) 20:30, 15 September 2006 (UTC)

I think they're natural enough. Do you have a reliable source that says they embody the golden ratio? Most that I've looked at jump from golden ratio in 2D Penrose tilings to 3D quasicrystals without saying whether such ratios are in them or not. Dicklyon 23:12, 15 September 2006 (UTC)

I got it from Livio. Finell (Talk) 17:42, 16 September 2006 (UTC)

OK, I looked some more and found it this time in the sizes of the steps in the aluminum-copper-iron and aluminum-palladium-manganese alloys. The earlier bit about Golden Rhombohedra was a construction, not something identified in nature. Dicklyon 17:54, 16 September 2006 (UTC)

I think the lead is too long again (still?). I think we should take out either the paragraph of questionable statements, or the long Livio quote that follows it. I'd prefer to take out the former, and shorten the intro to the latter, and leave this:

Mario Livio summarizes how this peculiar number has fascinated intellectuals of diverse interests for 2,400 years:

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

Other opinions? Dicklyon 02:00, 16 September 2006 (UTC)

Woodstone, thanks for acting on this. I like the way you moved the blockquote into the history intro. Dicklyon 17:04, 16 September 2006 (UTC)

Timaeus?

Dicklyon wrote above, about a paragraph on Timaeus that has since been deleted:

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

If this sentence is referring to the idea of extreme and mean ratio, it would belong in the History section as the first known published reference to what we now call the golden ratio. But I don't think that it is. Plato's conditions are satisfied by any 3 numbers where a/b = b/c. The numbers 27, 9, and 3, for example, satisfy the ratio requirement and the ratio is not φ. In fact, "whether cube or square," which puzzeled me for awhile, may refer to just this sort of sequence, i.e., x³, x², x. Then again, the condition is also satisfied by any numbers a, b, c where a=kb and b=kc, k being a constant. So I go back to being puzzled by "whether cube or square." Am I missing something here about Plato's statement referring to extreme and mean ratio after all? Can anyone say what Plato meant by "whether cube or square"? Finell (Talk) 06:34, 14 September 2006 (UTC)

Yes, I agree that I misinterpreted that one. There was something else from Timaeus before that made little or no sense to me, but you might want to give it another look. Dicklyon 06:49, 14 September 2006 (UTC)

I re-checked it. The old part was from the same paragraph, referring to this same misinterpretation. It has since been removed from the article, along with the rest of the rubbish in the philosophical section (Summum stuff and other junk). Let's just hope that section doesn't come back. Dicklyon 00:46, 16 September 2006 (UTC)

Line colours used on pictures

I found it hard to distinguish the red and green lines on the first picture in the Applications section. It's possible that this is a function of my colour blindness. Could the line colours be changed so that colour blind individuals may see the distinction between the lines?Jylie 01:48, 15 September 2006 (UTC)

Welcome to Wikipedia. I am sorry that the colors cause you difficulty. How many types of color blindness would need to be accomodated? Is there a single solution that would satisfy all? Can you guide us or point us to a source that can? Finell (Talk) 05:41, 15 September 2006 (UTC)

I just looked at that graphic. It is very hard to distinguish the colors on a typcal computer monitor. I see a yellow line with very faint traces of red and green in a few places, and I had to look hard for them. I have seen this same graphic in print, and it works better there. Monitor resolution may not be sufficient to render the 3 mostly overlapping lines adequately, at least at the scale of this graphic. Finell (Talk) 05:54, 15 September 2006 (UTC)

I agree that the colors are ill-suited to the graphic, I wonder what we could do to improve it. Obviously thickening the line is going to make the overlap seem greater than it should be, which isn't advisable either. According to Color_blindness#Misconceptions_and_compensations, blue and yellow should be visible clearly to almost all users, so if our overlap section was... white? - that might satisfy. -- nae'blis 18:48, 15 September 2006 (UTC)

Dick: Maybe it is time to recreate the top two diagrams in inkScape and use other colors than red and green... ≈ jossi ≈ t • @ 01:05, 16 September 2006 (UTC)

Certainly. But I still find Inkscape unbelievably obnoxious (on the Mac it uses control instead of command for everything, so none of my reflexes work -- can this be changed?) and I haven't really learned to do much with it yet. Maybe someone else feels like taking it on. Dicklyon 01:08, 16 September 2006 (UTC)

If you are speaking about the article's very first graphic, sectioning the line, in my opinion it is very well done and contributes the most to basic understanding of golden ratio/section of all the graphics (although the others do make worthwhile contributions). If someone is going to fiddle with it, please be sure that the final result is at least as good as what we already have. What color would you substitute for red or green? The one change that would help, if someone is going to redo it, would be for the labels in the graphic (a, b, a+b) to be the same color as the element that they illustrate instead of black.

Also, for the purpose of this article, as distinguished from Golden rectangle, I would replace the golden rectangle construction illustration with a simple line drawing using the same color, line width, and labeling scheme (letters, colors, font) as the line sectioning graphic. Or perhaps 2 of them, one landscape and one portrait. Don't look at me, I don't know how to do graphics.

Finell (Talk) 07:11, 16 September 2006 (UTC)

In art?

We could really use some specific examples here (2D artworks like paintings), since there's so much non-specific nonsense in this area (as the quote from Livio that Finell took out said). The "Roses" is one, though I'd love to see a reliable source for its dimensions; so far it is unverifiable.

As to the page proportion issue, this one also remains ambiguous. I've been trying to read Tschichold on this (in Dutch and German, unfortunately, which doesn't work so well for me), and Spanish snippet about Rosarivo, but I can't find any actual examples of books with pages or text areas proportioned by the golden ratio. These guys are very vague about what they say about it, too, citing medieval manuscripts nonspecifically and things like that. Does anyone have some specific support for the idea that book pages have sometimes (modern or otherwise) been proportioned to the golden ratio? I did find an article on "Octavo" with all the modern and classic German, English, and Frence paper sizes, and none of them will lead to golden-ratio page proprtions, though the French has one size that gets closest at 1.607. If you allow enough extra margin in the gutter, of course, any of them could possibly do golden-ratio-proportioned text areas, but as I said I can't find any specific statements of books that do. Dicklyon 23:26, 15 September 2006 (UTC)

Pyramids?

The section on pyramids is very weak, since the first main source, Rice, says nothing but that different people have different opinions. Since this controversy is so central to the story of the golden ratio in popular conciousness, we ought to deal with it more seriously. There are lots of things written about Taylor and Smyth and their compaign to use numerological derivations from the great pyramid as the basis of their anti-metrification campaign in Britain. That's where the ideas come from. If Rice is right that "principal authorities on the history of Egyptian architecture have argued that the Egyptians were well acquainted with the Golden ratio" we should really reference those authorities rather than Rice; his statement is really not worth quoting unless he names his sources, as scholarly writers on this topic do. Also, the linked article "Misconceptions about the golden ratio" seems to go into more depth on this topic than Livio does, so we don't have to rely on him so much. Dicklyon 00:27, 16 September 2006 (UTC)

Ther are tons of sources on this subject. My time is quite limited over the next few weeks, so I will not be able to do any research for a while. ≈ jossi ≈ t • @ 01:04, 16 September 2006 (UTC)

That's the trouble. Tons of "sources" makes it very hard to find anything authoritative or verifiable. Dicklyon 01:10, 16 September 2006 (UTC)

"Verifiability, not truth" as per WP:V. ≈ jossi ≈ t • @ 01:25, 16 September 2006 (UTC)

If we cite something credible, there's no useful distinction. If all we want to verify is that Rice said X, that's easy. If we want something verifiable about an expert opinion of use of golden ratio in the pyramids, we need to find something that cites such an expert opinion. Dicklyon 01:48, 16 September 2006 (UTC)

No, Dick. We are not here to cite anything "credible", we are here to cite something that is "verifiable" and published by a reliable source, and that, is a big difference. ≈ jossi ≈ t • @ 15:09, 16 September 2006 (UTC)

I'm familiar with the rules and guidelines. Perhaps we have a somewhat different interpretation of how to appy them. I think we're here to create a good article. Don't forget these parts of the guidelines that you seem to have forgotten:

WP:V#Sources: Articles should rely on credible, third-party sources with a reputation 
for fact-checking and accuracy. For academic subjects, the sources should 
preferably be peer-reviewed. Sources should also be appropriate to the claims
made: outlandish claims require stronger sources.

WP:V#Other_comments: Just because some information is verifiable, doesn't mean that Wikipedia is the
right place to publish it. Verifiability is a necessary but not sufficient condition for an article.

Sure. But hen I said that there are tons of sources, I was referring to that fact that there are such sources that are reputable. Scholarly books, journal articles and the like. ≈ jossi ≈ t • @ 18:15, 16 September 2006 (UTC)

WP:V in a nutshell: Information on Wikipedia must be reliable and verifiable. Facts, viewpoints, theories, and arguments may only be included in articles if they have already been published by reliable and reputable sources. Articles should cite these sources whenever possible. Any unsourced material may be challenged and removed. ≈ jossi ≈ t • @ 18:25, 16 September 2006 (UTC)

Certainly we agree on that. I look forward to you finding time to get some such sources for interpretations of golden ratio in pyramids. Dicklyon 18:42, 16 September 2006 (UTC)

Any info on this?

I found this in: Mathematical Mysteries: The Beauty and Magic of Numbers, Book by Calvin C. Clawson; Perseus Books, 1996., ISBN 0-30645-404-1

"Evidence exists that the ratio may have been known to the ancient Egyptians, for the Rhind papyrus (ca. 1650 B.C.) refers to a "sacred ratio," and the ratio of the altitude of a face of the Great Pyramid at Gizeh to half the length of the base is almost exactly 1.618" ", citing this book: David Eugene Smith, History of Mathematics ( New York: Dover Publications, 1951), p. 143.

Can we corroborate that claim about the Rhind papyrus? ≈ jossi ≈ t • @ 00:14, 17 September 2006 (UTC)

Fascinating possibility. I just ordered a copy of the book to see what else he says. A quick search turns up many quotes of that passage or something like it in articles on math and golden ratio and such. But I can't find mention of it in any articles on the Rhind Papyrus itself, but it may be just one guy's translation or interpretation or something. Dicklyon 00:29, 17 September 2006 (UTC)

Yes, maybe... Would love to know more about it. It seems that there is a translation of that papyrus in German. If anyone finds a translation in English, let us know. ≈ jossi ≈ t • @ 00:45, 17 September 2006 (UTC)

Here's a relevant PDF: [12] which says that the Rhind Papyrus has several pyramid trig problems that result in face slopes a little bigger than the 51.83 degrees that would result in the oft-quoted golden ratio: "The seked found in Problem 56, namely 18/25 (in dimension-less units) corresponds to an angle of 54.25 degrees between the base and face. The seked found in Problems 58–59, when converted back to dimensionless units, is 5 1/4:(7) or 3:4, corresponding to an angle of 53.13 degrees."

This would seem to show that the Rhind papyrus specifically indicates that the inference of golden ratio in pyramid proportions is contradicted by the main source of known Egyption math. Good find. Dicklyon 00:48, 17 September 2006 (UTC)

According to a James R. Newman, "The Rhind Papyrus," Scientific American (August 1952), the follwing is the definitive source and includes a complete translation:

• Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7

Finell (Talk) 08:01, 19 September 2006 (UTC)

Well, I found some spare time today to visit the Humanities Department, Oriental and Middle Eastern Section, of the New York Public Library. It turns out that their copy of this wonderful two-volume work is a reproduction from microfilm. Bummer, but readable enough. I found nothing in the index about "sacred" or "ratio" or "secret" (though secret is part of the subtitle: "Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the fourth month of the inundation season, under the majesty of the king of Upper and Lower Egypt...").

There is a fair amount of stuff about the problems 56–60 about run/rise ratios in pyramids, especially an analysis of the language that scholars have argued over a bit; some notes:

p. 37. Pyramids. "The Relation of the Lengths of Two Sides of a Triangle" ...Problems 56-60... two interpretations of "seked", ratio of two lengths. Eisenlohr...cosine of the angle which the lateral edge makes with the diagonal of the base, and prob. 60 tangent of the angle which the lateral faces makes with the base. Borchardt...accepted by most Egyptologists...cotangent of the latter angle. Egyptian mathematician...thinks of the seked not as a ratio, but as so many palms per cubit.

p.38 "The important point is that at the beginning of the 18th century B.C., and probably a thousand years earlier, when the greater pyramids were built, the Egyptian mathematician had some notion of referring a right triangle to a similar triangle one of whose sides was a unit of measure, as a standard. ...the actual measurements of the pyramids vary so much that we cannot tell absolutely from the measurement which is the more probable interpretation."

p.96 Problem 56. If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked? Take 1/2 of 360; it makes 180. Multiply 250 so as to get 180; it makes 1/2 1/5 1/80 of a cubit. A cubit is 7 palms. Multiply 7 by 1/2 1/5 1/50. ... The seked is 5 1/25 palms.

This value of 5+1/25 is less palms of run per cubit of rise than the 5+1/4 (5 palms and 1 finger) that corresponds to the 3:4:5 triangle (0.75*7 is 5.25). The 5.04 palms per cubit is a cotangent of 5.04/7 = 0.72, for an angle of 54.246 degrees. This is steeper than the 3:4:5, not closer to the "golden" shape. We don't have to do the Egyptian math to see that, just use Pythagorean theorem to get slant height = 308.06 = 180*1.7114 (1.7114 being not at all close to golden ratio, worse than 1.6667 of the 3:4:5 pyramid of problems 57, 58, 59, 59B. 60 is not a pyramid, but a very steep cone or something).

I wasn't able to find anything else about ratios. Several problems on dividing things in various proportions, but nothing I could find resembled anything about a sacred ratio. But I only looked for an hour or so.

Dicklyon 02:58, 28 September 2006 (UTC)

Thanks for the effort. Finell (Talk) 06:42, 28 September 2006 (UTC)

In summary, the claim seems to be totally bogus. The oft-repeated statement, variations of "Evidence exists that the ratio may have been known to the ancient Egyptians, for the Rhind papyrus (ca. 1650 B.C.) refers to a 'sacred ratio,' and the ratio of the altitude of a face of the Great Pyramid at Gizeh to half the length of the base is almost exactly 1.618," has no basis in the most complete commented translation of the Rhind papyrus. On the contrary, the ratios implied in all the pyramid problems therein are different from the "golden" one, and the notion of a "ratio" itself is strongly contradicted by the methods of Egyptian mathematics. And the closest term to "sacred" is "seked", which is the "palms of run per cubit of rise" relationship in pyramids, which for some value not in the Rhind Papyrus may lead to the "golden pyramid", but making such a connection is way more than a stretch. Dicklyon 22:05, 30 September 2006 (UTC)

By the way, a pyramid with a seked of 5 1/2 (unlike the examples of 5 1/4) would have a h/b ratio of 1.2727, pretty close to the sqrt(φ) of the golden pyramid. That's likely to be how the great pyramid was actually designed if the Rhind Papyrus is to be taken as a source of clues. Dicklyon 22:18, 30 September 2006 (UTC)

Pyramids re-org

I've started a re-organization of the pyramids section, taking it out of "disputed sightings" and including a mathematical part in it, because it has so much more to it than the other sightings. The source mentioned above provides most of the needed citations, but a few are drawn from other sources as well, so it will take me a while to rediscover and document all the fact details that I marked. Feel free to mark more or to rework it if some parts of what I wrote seem like opinions or original research instead of facts. Dicklyon 15:17, 19 September 2006 (UTC)

Summum?

A bunch of stuff about Summum, including "Further Reading" was added by User:KamiLian, a Summum advocate whose contributions were all about that. Since this is a cult-like thing, does it belong? Certainly not in further reading... Dicklyon 00:33, 16 September 2006 (UTC)

Golden ratio aesthetics: An artist's view

I mentioned to my wife that I've been interested in and reading about the golden ratio. She is a jewelry designer with formal education in both painting and design, and no interest in math or science. To my surprise, she immediately knew what I was talking about, although she knows it as the golden mean. She said that both in painting and design, it is taught as one of the most pleasing and harmonious proportions, and she personally agrees. She said that when considering a scheme of proportions, in the composition of an art work or in a design, the golden ratio is something that any artist or designer would consider. She said that 8 and 5 (that is how she said it) was an example and 8 and 13 was another (she never heard of Fibonacci or the number series). She also said that in employing the golden ratio, an artist would not make precise measurements: an artist knows what the proportion looks like at any scale, and can draw it at any scale without measuring. But that also means that being a little off one way or the other does not affect the visual impact: anything that is not perceived as different from the golden ratio is the golden ratio. She knew absolutely nothing of the math and had no idea what an irrational number even after I explained it—and changed the subject in less than a minute after I started to tell her some of the simpler math. Finell (Talk) 08:00, 16 September 2006 (UTC)

Aye Aye! ≈ jossi ≈ t • @ 15:07, 16 September 2006 (UTC)

Exactly! To artists it is just the concept of pleasing proportions. Pacioli started that. That's why so many authors are willing to see golden ratio in things that please them, whether it went into the design or not. Dicklyon 17:07, 16 September 2006 (UTC)

In my opinion, these are really 2 separate issues, but both belong in the aesthetics section: (1) Did the artist (or designer, architect, etc.) intentionally use the golden ratio to achieve an aesthetically pleasing result? (2) Do building, artworks, etc. that are considered models of excellent design embody the golden ratio (including proportions that are reasonably close) in their proportions without the designer intentionally using the ratio (or without our knowing whether he or she did)? EITHER says that the golden ratio has aesthetic qualities.

Therefore, in my opinion, Livio's critique of the aesthetics "claims" is not definitive and is only one opinion (i.e., POV): He is insisting either on direct evidence of either intentional use (we have no extant documentation of Phidias's basis for the proportions of his sculptures, likewise the Parthenon's architect(s), and they probably never wrote any in the first place), or unreasonably near-exactitude in the measurements (which is not realistic especially in the case of very large objects given the vagaries of measurement, not perfectly straight lines, buildings that settle, builders who have to accommodate topography that is not a perfect plane and may impose other constraints, etc.). To me, the superimposition of golden rectangles on the Parthenon makes a strong case regarding its overall proportions and the proportions of other elements of the facade. Livio's critique that one must choose where to place the rectangles and quibbles about measurement is not persuasive, because the Parthenon isn't a perfect rectangular box: it has stairs and overhangs and a big triangle on top and many other irregularities that contribute to its beauty and function.

On the other hand, going too far in the other direction is not sound either. There are too many other proportions and compositional schemes to say that the golden ratio is THE single most pleasing proportion. Many beautiful buildings, artworks, and book dimensions don't exhibit the golden ratio, and in those that do it is not the only proportion used (how boring that would be!). 3 x 5 index cards are pretty close, and 5 x 8 cards are really close—but 4 x 6 cards are not, and neither are standard US stationery sizes, and neither are most standard photo or picture frame sizes. These all have convenient integer dimensions (except 8.5", which at least isn't 8.618...) that aren't too narrow in either dimension and aren't too nearly square. To me, that makes the 3 x 5 and 5 x 8 cards irrelevant (or perhaps false sightings), and I also find it hard to believe that credit cards were dimensioned for aesthetics.

So in my opinion, NPOV and factual accuracy require a balanced presentation, rather that trying to push either the extreme-skeptical or cheerleader POVs.

[The speaker climbs down from the soapbox.] Finell (Talk) 18:56, 16 September 2006 (UTC)

I actually thought Livio had a very balanced and open-minded look at these questions. But I certainly agree that other scholarly opinions on the topic should also be aired. They are somewhat difficult to find, however, among the hundreds of "cheerleader" pubs in this field. We need to keep looking. Dicklyon 19:46, 16 September 2006 (UTC)

Mathematics section

I reroganized the Mathematics section and also did some copyediting there. Also, I moved the Calculation section to the beginning of Mathematics. With the defining formulas now in the lead paragraph, Calculation need not be the first section, divorced from the rest of the math. Also, logically, the prinicpal calculation should immediately precede the Alternate forms subsection. The (new) Geometry subsection should, in my opinion, be expanded to discuss, at least briefly, the various golden figures: golden rectangle, golden triangle, etc., with links to the separate articles on those topics.

I like it, except for the part where the math is buried below everything else. So I moved it up, to where the calculation section was. I'd like to hear from people whether this approach works better (to me, for a math article, it does, but I know others appreciate the golden ratio in non-mathematical ways, so it's a tough call). Dicklyon 18:45, 18 September 2006 (UTC)

In Other interesting properties, the equations in the sentence that begins "Uniquely" should probably be formated with <math>, but I am not adept at that. Could anyone help, if you agree that would be better?

Finell (Talk) 08:42, 18 September 2006 (UTC)

For the record, I didn't write the since removed sentence that begins "Uniquely"; I moved it from Applications, where it did not belong. Also, I do see now that the second equation is already in the Conjugate section (althogh it is an interesting property, so the repition here might not be so bad). However, while perhaps I am just missing it, I do not see the first equation — unless Dicklyon is saying it is obvious from transposing $\varphi ^{2}-\varphi -1\ =\ 0$ in Calculation, or is an obvious consequence of the left side of the already stated $\varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}$. IF that is what Dicklyon is saying (again, maybe I am just missing it), I would reply that these will not be obvious to the general reader, and is not nearly striking, as an interesting property, as

$\varphi ^{2}=\varphi +1$

That's the defining quadratic equation (both the form immediately above and the equivalent with negative signs and 0 on one side), so repeating it as a property was pointless. And the solution is not unique. The also equivalent statement about the conjugate, which you can get by dividing through by phi, is also already pointed out, with its uniqueness already restricted to positive numbers. Dicklyon 18:46, 18 September 2006 (UTC)

Also, I again ask Dicklyon to please tone down his edit summaries. Please refer again to Use of edit summaries in disputes, the most relevant passage from which is quoted in Edit summaries above. Thanks. Finell (Talk) 17:06, 18 September 2006 (UTC)

So you figure someone is going to be offended by my explanation that the removed sentence was "pointless"? I'm not sure I have sufficient tact to reliably avoid saying what I think in the future, but I'll try to keep your admonition in mind. Dicklyon 18:32, 18 September 2006 (UTC)

The pupose of the edit summary is not to say what you think. The purpose is to indicate succinctly what you changed. Finell (Talk) 03:35, 20 September 2006 (UTC)

Plans and proposal

I would like to add a new first section, before History, on Terminology and notation. Also, I intend to expand History substantially, at least doubling its length. I got from the library some good source materials on the history, as well as some more Aesthetics material to work in from scholarly sources.

WP already warns, when you open the edit window, that the article is long by WP standards, and it should get quite a bit longer with all the material being added. I think the best approach is to keep everything in Golden ratio for now, so those involved don't have to run all over WP to see all that is being done. However, we may want to consider, when we believe that the material is substantially complete and in good shape, to spin off some of the content to sister articles. The excellent article on Pi (does anyone to do an animated gif like that article has?) and its spinoffs may serve as model.

What does everyone else think?

Finell (Talk) 09:01, 18 September 2006 (UTC)

Sounds good. Let's see how large it gets first, and then we can spin-off sections to articles as and if needed. I can creat animated GIFs if needed. ≈ jossi ≈ t • @ 19:11, 18 September 2006 (UTC)

Structure

I have reordered some sections. The preceding edits by others moved major pieces around, leading to an unattractive article. After the short intro, a short calculation completes the mathematical definition. Then proceeding into history and aesthetics. Then follows the big math section, only for a limited audience. Concluding by the less interesting "doubtful sightings". If the big math section goes first, many people will never reach the arts section. They will most likely be able to skip over the small calculation section. (The signalled edit conflict appears to have been a vandal reversal.). −Woodstone 20:53, 18 September 2006 (UTC)

Yes. I concurr. Much better now. ≈ jossi ≈ t • @ 21:11, 18 September 2006 (UTC)

I like it, too. Putting all the math in one place is problematic. Dicklyon 21:34, 18 September 2006 (UTC)

I like it, three. Great job! Finell (Talk) 06:05, 19 September 2006 (UTC)

End punctuation in formulas?

Trying to further my education, I looked at the Wikipedia:Manual of Style (mathematics). The Punctuation section requires a period when a formula ends a sentence. By analogy, this would require other end punctuation appropriate to the formula's role in the sentence of which it is a part. I know that Dicklyon has the opposite opinion of style and eliminated the many instances of end puncuation throughout the article's formulas; for consistency, I did the same for a couple that he missed. What do others think? Would failure to follow WP style guidelines be a negative when seeking FA status? What to do? Finell (Talk) 06:42, 19 September 2006 (UTC)

I would certainly expect full and correct punctuation in all formulae, whether inline or displayed (for instance, in a list of cases, I would expect to see commas after all but the last, and a fullstop at the end). In all mathematical journals that I have anything to do with, this is always the case, and mathematics is expected to follow the same rules regarding punctuation as if it were ordinary English. I personally would expect the same to hold for Wikipedia, but it might be argued that Wikipedia is not an academic journal ... :-) Madmath789 07:22, 19 September 2006 (UTC)

OK, let's change them back; I apologize. I wasn't aware of that style page. When I used to do that, I was corrected by technical book editors who preferred it the other way, and I came to the impression that omitting punctuation after displayed equations was pretty standard (because it sometimes looks confusing out there, like in numbers with two decimal points and funny placed other punctuation). Dicklyon 13:21, 19 September 2006 (UTC)

Pyramids and tetrahedrons

Be aware that a tetrahedron is a solid with four faces, while a pyramid has five. I have corrected the misconception that they are the same, in both directions now.

As to why I said pyramids "Both Eqyptian pyramids and mathematical pyramids that resemble them are interesting with respect to the golden ratio," rather than "geometric pyramid and the Eqyptian pyramids based on that shape have relationships to the golden ratio," I think my way is much less POV. It's an "opinion" that they "are interesting", but not one that anyone is likely to dispute. The other way, however, includes geometric pyramids too generally, rather than just the ones the golden ones that resemble the Egyptian pyramids, and when it says the Egyptian pyramids are "based on" the geometric ones, it's not clear whether it means to imply based on the interesting golden ones (assuming facts not in evidence), and asserts a "relationship" that is really at the core of the controversy; this is not sufficiently NPOV for my taste.

Dicklyon 12:27, 20 September 2006 (UTC)

One of my edits deleted "remarkably", so "remarkably came back" when my edit was reverted.

Sorry for confusing the tetrahedron, but I find this whole section, including both sub-sections, confusing. It needs substantial revision, but I don't have the expertise to fix it and don't want to confuse things further by making more mistakes.

Mathematical pyramids

A geometric pyramid is not necessarily 5-sided, as erroneously stated above: it can have any number of sides; see Pyramid (geometry). It is characterized by the polygon that forms its base: e.g., triangular pyramid (tetrahedron), square pyramid, pentagonal pyramid, and so on. Also, it can either be regular or not (e.g., a rectangular pyramid). So before this sub-section starts talking about particular geometric properties of a special case (a square pyramid with unique proportions), it needs to define what it is talking about. Also, since the WP article on this subject is entitled Pyramid (geometry), I still believe that this sub-section should be headed "Geometric pyramids", both for consistency and because it is more specific than "Mathematical".

The description of the construction of this "golden pyramid" is also hard to follow. First, positing a specifically dimensioned "1 by φ" golden rectangle is unnecessary; it would be clearer and more general to begin with a (any) golden rectangle, then point out the resulting proportions, as we have been doing throughout the article.

Second, I gather that after diagonally splitting the golden rectangle, the 2 halves are joined at the longer sides of the original rectangle to form an isosceles triangle with a base 2x the length of the rectangle's short side and sides equal to the rectangle's diagonal. If that is correct, it should be spelled out, and preferably illustrated.

"Such a pyramid is very close in shape to the Egyptian pyramids." Aren't all square pyramids the same shape? Does this sentence refer to the proportions?

"The central right triangle ..." To what does this refer?

The derivation of the slopes and angles should be shown (which will obviate the {{Fact}} tags), and their significance to φ explained.

Yes, it's all true. I will add more details over time, with refs and a figure, to illustrate the notion of the central triangle that defines the differences among the otherwise similar regular square pyramids, which is the class the term "pyramid" is most often used to refer to. Dicklyon 19:42, 20 September 2006 (UTC)

Some hint of what and when the Rhind Papyrus is (the link is necessary but not sufficient) needs to be stated before its contents are discussed. This paragraph needs to be made both mathematically clear and accessible to the general reader. Also, this paragraph might fit better in the Egyptian sub-section: isn't the only reason for discussing this to illuminate the controversy over the Egyptian pyramids?

That's not the only reason for discussing it, but is one good one. I think its useful to be able to look at the math of "golden" pyramid first in isolation, to prevent confusing it with the controversial interpretation of Egyption pyramids, and then to make the transition into the remarkable similarity there. Dicklyon 19:42, 20 September 2006 (UTC)

What is the source for these details from the Rhind? Does anyone have access to the Chace translation and commentary that I cited in a prior entry on this Talk page (I don't)? If so, it would be best to be working from a translation of the primary source rather than from the conflicting statements about it in diverse secondary treatments. Also, if we see in the Rhind itself reference to a "sacred ratio" and the context, that might be the most important find yet for the purpose of this article. Personally, I'm skeptical about this claim: from the little that I've read about ancient Egyptian math, I don't think that they would have discovered or comprehended φ. On the other hand, if they thought there was something special about 5:3 or 8:5, that would be close enough to warrant mention this article.

The source of much of the material is the document we discussed above [13] which appears to be from this book [14]. Dicklyon 19:42, 20 September 2006 (UTC)

Egyptian pyramids

Is the entire φ controversy over the pyramids confined to those at Giza, or does it extend to other locales? This should be clarified.

I only have what I got in the above document. I suspect other pyramids are not too dissimilar, but I don't have the info. Dicklyon 19:42, 20 September 2006 (UTC)

The new first paragraph disorganizes this section even more than it was before. Now, measurements of and calculations concerning the Great Pyramid of Giza appear in the first, third, and fifth paragraphs. Obviously, this needs to be consolidated and harmonized, so it can be meaningful to the reader. Rather than saying that the angles are close to what the angles would be with a φ ratio of dimensions, it would be more helpful to calculate the ratios so the reader sees how close they are or aren't to φ.

In my opinion, the intervening paragraphs about Rice (which supports both sides of the controversy, unhelpfully) and Livio (which supports one) can be deleted and replaced with footnotes citing both (and possibly others) dropped from the end of this sentence: "Whether the relationship to the golden ratio in these pyramids is by design or by accident remains a topic of controversy."

Introductory sentence

Until I saw the above explanation of the intro sentence, I did not realize that it was confined only to the single specific instance of geometric pyramid described. So my changing its meaning and resulting inaccuracy in rewriting it was unintentional. However, as the sentence stands, I don't think that the intro sentence says anything meaningful. Therefore, I am taking the liberty of deleting it without replacing it. My edit summary will NOT say, "Deleted pointless sentence".

Finell (Talk) 17:34, 18:36, 20 September 2006 (UTC)

As you can see, it's a work in progress. Thanks for your attention and help with it. Dicklyon 19:42, 20 September 2006 (UTC)

Actually, I did not realize that it was a work in progress. If I had I would have waited before commenting, especially at such length. I do appreciate Dicklyon taking the time and effort to respond so specifically above.

Before posting it here, I cut and pasted my lengthy discourse above into WordPerfect to run spell check, thereby fixing my 127 spelling errors and typos, then pasted the corrected text from WordPerfect back to here. For some bizarre reason, WordPerfect changed all the φ's to f's (not Unicode compliant, I guess). I tried to fix them all now. Sorry to sow confusion.

Finell (Talk) 03:35, 21 September 2006 (UTC)

I did a bunch more work on it today, including a figure. Feel free to chip in or let me know what you think it needs next, anybody. Dicklyon 03:14, 21 September 2006 (UTC)

Dick, if you are doing a big edit, you can place the {{inuse}} template at the top of the page to alert fellow editors. ≈ jossi ≈ t • @ 04:19, 21 September 2006 (UTC)

BIG improvements in Mathematical pyramids sub-section. Good work! I still think it should be called Geometric pyramids; to the extent that Egyptian pyramids are also geometric, they are also mathematical too. I may copyedit a bit, but not tonight. Looking forward to see what you do with the Egyptian pyramids sub-section. Finell (Talk) 04:57, 21 September 2006 (UTC)

Sorry to disappoint you, but I'm not planning to work on the Egyptian part. Go for it. Dicklyon 11:45, 21 September 2006 (UTC)

Wrong conjugate?

The section Golden ratio#Conjugate golden ratio claims that (1/φ) = (φ-1) is the conjugate of φ. However, my understanding of the word conjugate is that it is the other root of the equation x^2-x-1=0 and that would be (-1/φ) = (1-φ) which is the negative of the number refered to in that section. What do you guys think? JRSpriggs 06:02, 25 September 2006 (UTC)

I think you may be right. Someone said the capital phi was used for the "conjugate" golden ratio 1/phi, but I can't find a reference that calls it that, or from which that symbol is taken. Anybody else have a relevant reference? Dicklyon 20:58, 27 September 2006 (UTC)

I was probably the someone. Here are two (or perhaps 1.5) references, but they cite other sources:

• Weisstein, Eric W. "Golden Ratio". MathWorld.
• Weisstein, Eric W. "Golden Ratio Conjugate". MathWorld.

Finell (Talk) 06:32, 28 September 2006 (UTC)

Well, Mr.Wolfram's MathWorld says "The golden ratio conjugate, also called the silver ratio, is the quantity..." and then gives the (1/φ) = (φ-1) value. But according to our article on silver ratio, "silver ratio" means something quite different. So perhaps we should not put too much faith in MathWorld. Accordingly, I will remove that part of our article. JRSpriggs 07:09, 28 September 2006 (UTC)

Did you look at the refreence(s) that the MathWorld articles cite? The MathWorld Golden Ratio Conjugate article refers to the The MathWorld article on silver ratio. The substance of the MathWorld article on silver ratio is about the same as WP's, which is not surprising since WP's silver ratio article cites the MathWorld silver ratio article as a primary reference. Generally, MathWorld is a very reliable source on math, at least as much so as WP.

It appears that "silver ratio" has 2 meanings: (1) the one in the WP and MathWorld articles on that topic; (2) as a synonym for the golden ratio conjugate. Here is another reference that uses silver ratio as a synonym for golden ratio conjugate: [15]. Here is another reference that defines the golden ratio conjugate our way (as positive) and says it is error to refer to it as the silver ratio: [16]. There are still other references that define the conjugate as the negative root. There are even a few references that define the golden ratio itself (φ) as short:long, the reciprocal of the more common definition. As mentioned an erlier entry on this Talk page, I plan to do a new section on "Terminology and notation" to deal with these and other confusions. Until then, I think the subsection can stay as it is for now; the math is right, the only issue is terminolgy, and we are in very good company on that score. Finell (Talk) 18:51, 28 September 2006 (UTC)

It seems to me that with this source verifying the term conjugate golden ratio for 1/phi, there's no problem. Of course we could also mention alternative terminology. Dicklyon 13:21, 28 September 2006 (UTC)

Well, no doubt there are many different and sometimes inconsistent terminologies for this any many other subjects. But where there is a conflict in usage, would it not be better to choose the version which is more consistent with general practice, i.e. congugate roots of an equation. Also, if you look at the rest of this article and also the article on Fibonacci number, you will see that (1-φ) is used much more often than (φ-1) and so is more deserving of its own name. JRSpriggs 03:38, 29 September 2006 (UTC)

I have no problem with you reworking it to reflect more common usage, citing refs. I just didn't think deleting it was a good step in that direction. I tried with minor changes to reflect the notion of conjugate root, but feel free to go further in rewriting it if the references support a different pitch. Dicklyon 06:16, 29 September 2006 (UTC)

Golden Ratio Trivia

For what it's worth: the Monoliths in the Stanley Kubrick films 2001: A Space Odyssey and 2010: The Year We Make Contact are Golden Ratio Rectangles.{[ID of unsigned post: 06:33, 28 September 2006 User:Basesurge}

We could do that if we had a source to cite. I will post the question on the user's talk page. Finell (Talk) 19:06, 28 September 2006 (UTC)

Decimal expansion

The article's long decimal expansion cites (sequence A001622 in the OEIS) as the source. However, unless I am missing something, I only see about 104 decimal places in that source. What is the source for the expansion that we have? Also, I consolidated the 2 sections having to do with the decimal expansion into one, subdivided in 2 subsections. The Calculation subsection here, which deals only with calculating the decimal expansion, should not be confused with the more general Calculation section at the beginning of the article. Finell (Talk) 19:55, 28 September 2006 (UTC)

It has links to lots more digits, such as these 50,000. I also had independently verified the digits in the article against my own calculation of 10000 (actually, this was more of a verificatin of my code, but in any case no discrepancies were found). Dicklyon 21:58, 30 September 2006 (UTC)

Paintings

When the unverifiable statement "Many painters used canvases proportioned at or closely approximating the golden ratio" was in there, Jossi was happy. But when I replaced by one that stated the verifiable truth, with references, that the mean painting proportion is much less than the golden ratio, that is said to be "unrelated" and summarily removed. Seems to me that putting in the truth to temper the fiction is indeed related. And that statement was in the article for a year or so already without complaint, until someone removed it a few months ago. Anybody? Dicklyon 20:19, 1 October 2006 (UTC)

Oops, I see it was I who removed it before, on Sept. 14, this diff. At the time, there was no section on Paintings. Now that there is, it needs this back as balance to the other claims that imply the use of golden ratio in paintings is relatively common. It's actually quite rare, according to the data. Finell, you made this section, and added the unverifiable line that I took out. What's your take on it? Dicklyon 20:33, 1 October 2006 (UTC)

I failed to attract Finell's attention when he dropped in to make an edit, so I've gone ahead and restored it. Then, I added the title to clarify the relevance. Here's the abstract of "Golden Section and the Art of Painting":

A statistical study on 565 works of art of different great painters was done and it was calculated the ratio of the 2 sides of a paintings. Assuming that all the painters under discussion enter in a statistics with equal weights it is shown that the average value obtained for the ratio of the sides is 1.34. This value, determined experimentally is significantly different from the value of the Golden Section F=1.618, which is a theoretical ratio, obtained from an abstract, mathematical theory, which supposedly ought to impress on a painting a supreme harmony.

Dicklyon 21:31, 1 October 2006 (UTC)

The original statement that you replaced did not have any sources. Find a source that claims that artists used canvases using the golden ratio, and you would be welcome to add the rebuttal. Without such claim, there is no need for a rebuttal. ≈ jossi ≈ t • @ 03:39, 2 October 2006 (UTC)

It's not a rebuttal. It's a report of a study by an academician who sought to determine whether the often expressed idea of golden section proportions being good was reflected in actual dimensions of paintings. The work stands on its own, in a section on golden ratio relative to paintings. It's the only work I'm aware of that addressed this question, which is certainly relevant to this section on use of golden ratio in the arts, even more so in a section in which various uses of golden section are being cited as examples; it is good to make it clear that these are exceptional, not typical cases. Dicklyon 03:58, 2 October 2006 (UTC)

Sorry, but you have this wrong. The use of the golden ratio in paintings is not related to the dimensions of the canvas, but the application of golden rectangles to the composition used by the artists. This "work" is singular, and a minority viewpoint, a rebuttal, without a claim, and thus irrelevant. Reverted yet again. ≈ jossi ≈ t • @ 05:33, 2 October 2006 (UTC)

Yes, I see that you did. And then you put it back, rephrased and expanded. Good job. Why the change of mind? Dicklyon 06:20, 2 October 2006 (UTC)

This section would benefit from more references, preferably from books on art and design rather than books on the golden ratio, many of which have a promoting or debunking POV. However, the use of averages to address this subject is, to use a word that I have only recently learned, twaddle.

• Suppose the arithmetic mean aspect ratio of some statistically significant sample of paintings was φ, but the closest individual aspect ratio was 1.9. Suppose all the aspect ratios were significantly higher or lower than φ (perhaps dual bell curves equidistant from φ), but happened to average φ. Would that support the proposition that a golden ratio was aesthetically pleasing? No, the analysis would prove the opposite, and the mean would be a misleading artifact.

• Suppose the median aspect ratio of some sample of paintings was φ, but the next closet ratio was 1.9. Same question, same answer.

• Finding a mode aspect ratio at or very near φ could support the proposition that many artists chose it, but further analysis would be required to draw any meaningful conclusion. What does the rest of the distribution look like? Is it statistically normal? Is the mode near the center of the range of values or nearer an extreme? What other aspect ratios appear with relatively high frequencies, and how are they distributed?

• Suppose (although it clearly is not the case) that Leonardo, Raphael, Rembrandt, Durer, Titian, Caravaggio, Rubens, Van Dyck, Velazques, Vermeer, Ingres, and Corot used golden rectangle canvasses for 1/3 (or alternatively 2/3) of their paintings. Would any statistical mean, no matter how far from φ, be a refutation of the statement, "Many masters chose golden rectangle canvasses for a significant percentage of their paintings"?

I don't think that anyone is saying that all or most canvasses are golden rectangles. Further, there are other compositional reasons for choosing the aspect ratio of a canvas, the natural scale of the subject depicted being a primary one. Many paintings were commissioned to be hung in a particular space, so the dimensions of that space would be relevant to the artist's choice of dimensions. Some canvases are square or round for formal reasons, so they would pull down a mean or median. (By the way, in some of the perceptual studies, the subjects' preferences showed a secondary spike for an aspect ratio of 1 [a square] with the mode at or near φ, and the distribution was not statistically normal.) So a statistical average does not prove or disprove anything in this context.

An adult of average height can easily drown in a swimming pool with an average depth of 4 feet.

Finell (Talk) 06:12, 3 October 2006 (UTC)

Twaddle? Perhaps so. I'll be the first to agree that the mean is a very poor statistic for characterizing this distribution. But did you look at the paper (PDF online)? It includes a histogram of all the values, and it's clear than the median is around 1.25 and that there's no concentration near phi. But if I state such an observation, it will probably be removed as WP:OR if history is any guide, so I stuck to more direct quotation of the stated summary results. I haven't found any ohter relevant studies, but this one certainly is relevant to the question of whether the golden ratio is commonly used to proportion paintings, at least in their overall dimensions. Dicklyon 06:44, 3 October 2006 (UTC)

If there is anything to be said about the actual use of golden ratio proportions in art, we should find it and include it. At present, we just have a couple of examples, nothing at all general. Yet you do see such broad generalities in lots of golden ratio books and papers. I didn't find any that said canvasses are often propotioned by the golden ratio, but they say things like that it's common in art. Is it? Dicklyon 07:03, 3 October 2006 (UTC)

No, not really. The use of the golden ration is studied and applied in paintings and graphic arts, mainly to composition. The idea of golden ratio canvases, was news to me. ≈ jossi ≈ t • @ 01:58, 6 October 2006 (UTC)

The idea of golden ratio in overall proportions of artworks shows up here and there. One not so reliable source, is this Reader's Digest publication: [17] which says "the dimensions of paintings frequently conform to the golden ratio." It's not clear where they get this stuff, but they person who did the quoted study also seems to have encountered it. Dicklyon 03:26, 7 October 2006 (UTC)

Reader's Digest is not a reliable publication, in my humble opinion... I would simply dismiss that without feeling too bad about it... ≈ jossi ≈ t • @ 04:34, 7 October 2006 (UTC)

Bees

Finell, sorry about the Huntley ref for the bees. I was shooting sloppy, going by a google book search hit on a page it wouldn't show me. Thanks for checking. The other ones were pages I actually did get to read, by the way, but feel free to check those. Dicklyon 22:19, 5 October 2006 (UTC)

Everyone makes mistakes. I happen to have Huntley on loan from the library. I don't have the others you cite, nor would I undertake the chore of checking all of the cited references. Finell (Talk) 22:25, 5 October 2006 (UTC)

Book

Finell, thanks for stating the actual ratio implicit in John Man's quoted Gutenberg page sizes. By the way, I have a very nice "facsimile" copy of a few of those Bible pages, and that ratio is about right for the "text area"; the overall page and its margins are unfortunately not shown in this copy. However, I was able to see an interesting "golden ratio" factoid: if you measure the rectangle that encloses not just the "text area" but also the "header", then it's about 1.62, or essentially a golden ratio. Now, can we find someone who has said so? Jossi has quoted a bunch of book-related sources, but of the ones I've been able to find so far, they don't actually say much useful about what measurements are in the golden ratio. I have a few more on order. Jossi, does anyone point out the golden rect that I've mentioned here? Or is text area sometimes defined to include the header, in contradiction to John Man's way of looking at it? Dicklyon 22:29, 5 October 2006 (UTC)

Best would to go the source of these claims: Raul Rosarivo. I will see if I can get my hand on a copy of the original booklet in Spanish Divina proporción tipográfica. My understanding of Rosarivo's measurements of the Golden ration in Gutenberg's bibles is that it applies to those bibles that use one column of text in a page, not the ones using a double-column of text. It is also my understanding that both the page size and the text block were measured to be close to the golden ratio. I will check and let you know. ≈ jossi ≈ t • @ 01:37, 6 October 2006 (UTC)

Yes, we really need that for the various articles that reference it. I assumed you had it. The San Franciso library has a German version of it (Divina proportio typographica; das Buch vom Goldenen typographischen Modul 1:1,5 in der Proportion 2:3, dem Modul von Johannes Gutenberg und seiner Zeitgenossen [von] Raúl M. Rosarivo. Übers. aus dem Spanischen von Heinz Nieth; Bearbeitung der deutschen Ausg. von Hermann Zapf.), in special collections. If you read German and can get there, go for it. I may get a chance to look, but my German is rather too poor to get much out of it, although I can tell from the title that it seems to emphasizing 1:1.5 as a "golden module". Dicklyon 02:56, 7 October 2006 (UTC)

I understand that Rosarivo's "Eureka" moment was whe he was studying the Catholicon, one of Gutenberg's first printed works. It's colofon, by Gutenberg, describes a proportion "deo trino". I have not found the full text of the colofon, only an excerpted translation in Spanish. ≈ jossi ≈ t • @ 01:54, 6 October 2006 (UTC)

OK, now I'm chronologically confused. That catholicon article says "The Catholicon is one of the first printed books. Gutenberg completed a Catholicon in 1460 before his famous printing of the Bible." My book Johannes Gutenberg 1468–1968 says that the Catholicon has never been definitively tied to Gutenberg, since there's no name in that colophon. It says the 36-line bible (was that single column?) was printed around 1459. By 1457, Gutenberg had also lost his suit with Fust, and had lost Schöffer to Fust, too. By the way, I saw one of his 42-line Bibles when I was at the NY public library; it's a thing of extraordinary beauty. But I don't seem to have any pictures of a page of the second Bible. Dicklyon 03:02, 6 October 2006 (UTC)

The text of that colophon is included in my Gutenberg book, in English translation, though they refer to is as an epilogue. The bit you're asking about is probably what they translate to "triple unity", in reference to the holy trinity. It also refers to a "relationship and harmony of letter stamps and types" but nothing specific about proportions or anything like that. Get a copy of this book; it's beautiful and inexpensive: Geck Dicklyon 02:44, 7 October 2006 (UTC)

The English version seems to be listed with the subtitle: Form Lead Letter to the Computer, from the title page, rather than the 1468-1968 that's on the front cover. Dicklyon 02:49, 7 October 2006 (UTC)

Rosarivo's translation to Spanish of the colofon reads: "Que no ha sido compuesto con la ayuda de la pluma, del punzón o del estilo, sino con formas y tipos de maravilloso ajuste, relación, concordancia y módulo al dios uno trino este libro fue terminado". What draws my attention (and from which Rosarivo, it seems, developed the golden canon) is " módulo al dios uno trino". Do you have the original colofon text in German (or was it Latin)? ≈ jossi ≈ t • @

No, sorry, just the English. The book has lots of page images but not this one. I suspect it was Latin, based on the book title Summa grammaticalis quae vocatur Catholicon. Here's a page with a rather crude translation. Dicklyon 04:58, 7 October 2006 (UTC)

This 1911 page has a relevant Latic excerpt: The colophon of the Catholicon of 1460 says that the book was printed " non calami, stili, aut pennae suffragio, sed mira patronarum formarumque concordia, proporcione, ac modulo." Dicklyon 05:06, 7 October 2006 (UTC)

Fascinating.... ≈ jossi ≈ t • @ 08:29, 7 October 2006 (UTC)

I finally got my copy of Tschichold's The Form of the Book, by slow boat from Australia. The relevant chapter is identical to the online excerpt, so I didn't really learn much new, but it gave me a chance to read it more carefully. If you read what he says about Figure 5, that's the one he refers to as the "golden canon of book page construction." He says it is identical, essentially, to what Rosarivo did, which was attributed to Gutenberg, and he points out at least twice that the text area and type area proportions are 2:3, "a condition for this canon." With this evidence, and lacking any evidence from Rosarivo that his title Divina proporción tipográfica ("Typographical Divine Proportion") should be interpreted as something about the golden ratio, I can not see at this point anything that connects to canon to the golden ratio or the golden ratio to incunabula. And why Tschichold mentions "golden section" eight times in his essay, I can't find where he actually states that some books use this ratio (I feel it's probably in there, but I'm missing it). Ah, never mind, it's in the previous essay, where he says "There was a time when deviations from the truly beautiful page proportions 2:3, 1:sqrt(3), and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimetre." That could be. So I figure, what the heck, measure some of my old books' text areas. A 1720 book on Stereometry matches golden ratio quite well. Boyle's 1670 book on "Colours" has a much more extreme text area, in excess of 1:sqrt(3) by more than a bit. And my 1697 book on the Sliding Rule is fairly close to 1:sqrt(3), though not within the half millimetre that he likes to claim. Lots of these type areas are hard to measure, because they're not "full", so I was looking at the fullest ones. Anyway, I accept Tschichold's point, since he measured a lot of books carefully, that some are laid out in the golden ratio. Not, however, applicable to incunabula (books before 1500), when the "divine proportion" was not yet known, and where Tschichold makes no claim that it was used, but rather suppports the 2:3 shape that is much closer to John Man's measurements of Gutenberg's Bible. I recommend that some articles including this one be revised accordingly. Dicklyon 05:56, 17 October 2006 (UTC)

Math questions

I have some math questions for the real mathematicians around here. The questions are not rhetorical: if I knew how to do or show these things myself, I would.

1. Deriving φ from the general case

The mathematical derivations of φ that I have seen, including ours, proceed from a special case involving φ, 1, and related terms. Is it possible to demonstrate that the general case is necessarily true:

${\frac {a+b}{a}}={\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}\,?$

2. $\varphi$ versus $\phi \,$

Why are we using \varphi rather than \phi in our formulae? Isn't \phi cleaner looking and more recognizable to more of the general public? Most everywhere else I look, plain old \phi is used.

3. Self-similarity

Can we better capture the self-similarity and self-replication that pops up wherever φ appears? For example, golden rectangles and golden triangles easily beget smaller or larger ones ad infinitum; on a line segment sectioned in extreme and mean ratio, one can readily construct 2 golden rectangles, one on the entire line segment and one on the smaller section; from a single pentagon or pentagram (all of which have φ proportions), one can easily construct larger and smaller like figures ad infinitum, and also lots of golden triangles as a by-product; any 2 non-intersecting diagonals of a pentagon golden section one another, while any 2 intersecting diagonals inscribe a golden triangle.

4. Limit of ratios of increasing, successive Fibonacci numbers

Is there a rigorous explanation of why φ is the limit of the ratios of increasing, successive Fibonacci numbers (likewise for similar recursive series), with the ratios alternatingly greater and less than φ? I have a hunch (but no more) that this is related to the self-similarity and self-replication discussed immediately above.

Finell (Talk) 09:07, 9 October 2006 (UTC)

Some answers:

1: Why do you think that is not proven in the article? (assuming that a and b are supposed to positive numbers).

You're right. Finell (Talk) 19:18, 9 October 2006 (UTC)

2: The form resulting from \phi is just quite unusual in mathematics.

Apparently usage is all over the lot. Wolfram's MathWorld and several other cites use \phi, but \varphi is also frequently used, especially on WP. However, I recognize that this is not an important issue. Finell (Talk) 19:18, 9 October 2006 (UTC)

3: no comment

4: That is shown in the article Fibonacci numbers (except for the alternating part).

I looked at the proof there. Is there a way that, for this article, the reason could be made more accessable for the general reader, perhaps by a textual explanation? Also, it would be helpful to explain the alternating part, if that is possible. Isn't it more common for successive values to approach their limit from one direction, as the curves of so many functions do? Finell (Talk) 19:18, 9 October 2006 (UTC)

−Woodstone 11:17, 9 October 2006 (UTC)

More on 3: I don't think so. You can easily get infinite self-similarity with any triangle shape, just as you do with the golden triangle. The thing that makes the golden rectangle special is that the part removed each time is a square, and that's already captured. And it's the only rectangle shape that will do this, obviously.

I do think that there is something more to the self-similarity than has been captured. There are so many instances of it. It shows up throughout 5-fold symmetry, bisecting the base angle of a golden triangle golden sections the opposite side, and many more. It is all over the non-geometric math as well: 1/φ=1-φ, which is just a striking instance of the more general property that the fractional part of any φⁿ is the same except for all n except 0. The self-similarity of this proportion may account for the widely and long held belief among artists, designers, and scholars in those fields (post hoc skepticism by mathematicians notwithstanding) that this proportion is harmonious, balanced, and aesthetically pleasing (although not the only such proportion). I doubt that it is an accident that aesthetic interest in this proportion was most intense in the Renaissance, when, more than any other era, many artists and architects did serious work in mathematics, especially projective geometry, and explicitly stated that mathematics (or more particularly geometry) was the foundation of art. The self-similarity may also account for the widespread mystical and religious interpretations (which I personally do not share) of the golden ratio, even by mathematicians Kepler and Pacioli. We can scratch our heads and wonder why, but there is no denying the well documented history that long predates recent popular literature on the topic. Finell (Talk) 19:18, 9 October 2006 (UTC)

Finell, you need to study up on the inductive method. Just because I pointed out that the fractional part is the same for exponents -1, 1, and 2 does NOT make it true for all n except 0! In fact, it's not true for any other values of n, as you could easily prove. I think you should add material if you can find books that say something interesting that we've missed, but don't go making up new stuff. If you want to consider whether it was "accident that aesthetic interest in this proportion was most intense in the Rennaisance" you need to first establish the premise, that interest was intense, and then most intense, in the Rennaisance; I think intense overstates it quite a lot, though Pacioli did certainly start something there. So document that well documented history you speak of; that would be a contribution to the solidity of the article. Dicklyon 19:26, 9 October 2006 (UTC)

Well, at least I had the sense not to put any of this stuff in the article, to address the questions "the real mathematicians around here", and to profess in advance my ignorance. And I am well aware of WP:NOR so was not proposing that we make up anything new. I was hoping is that something more regarding self-symmetry might be milked from the math. I'm sorry I've wasted your time. Finell (Talk) 21:48, 9 October 2006 (UTC)

Not a problem. And sorry if I sounded harsh. Bad habit of mine. Dicklyon 23:40, 9 October 2006 (UTC)

And as to the varphi, survey some books and let us know if a majority seem to do it the other way around. Dicklyon 15:47, 9 October 2006 (UTC)

I did a quick GBS search on "golden ratio" phi, and found 3 phi, 3 varphi, and 2 captial Phi, among the ones that used any Greek letter at all. I think that's enough evidence that the terminology in the article is at least not unusual. Dicklyon 17:34, 9 October 2006 (UTC)

Agrippa

Finell, does Agrippa actually say something about the proportions of the body in that figure? It seems very unclear what it's getting at, since the body parts don't line up with the pentagram edges whose lengths we know to be related by the golden ratio. Do you have a source for this caption, other than the wiki article it came from? Dicklyon 06:55, 13 October 2006 (UTC)

Just the WP article from which I copied it. I wouldn't want to stand in that position for very long, either (although I wouldn't mind the physique). Finell (Talk) 10:49, 15 October 2006 (UTC)

I suggest we take it out then, unless something thinks he can come up with a verifiable connection between the pentagram and some of those body parts, which pretty much don't even align. Dicklyon 16:57, 15 October 2006 (UTC)

Unless the author of the caption was misinformed, which I would not presume, the cited work by Agrippa supports the statement in the caption. The work is not in my library. Perhaps someone else can take a look. If the caption is incorrect, other WP articles where the figure and caption are used should also be corrected. Finell (Talk) 21:52, 15 October 2006 (UTC)

I'll ask for sources on those other places. Dicklyon 21:57, 15 October 2006 (UTC)

On what basis do you assume that the cited book does not support the statement, here and in the other article? Adding a {fact} tag is more than "asking" for a source when one is already cited. Finell (Talk) 06:59, 16 October 2006 (UTC)

The guy who originally added that caption now retracts it as probably unsupportable User_talk:Nixdorf#Agrippa_caption. I guess I just had a gut feeling, as it made no sense. Dicklyon 22:18, 16 October 2006 (UTC)

And here he says he "made it up": User_talk:Dicklyon#Pentagrams_and_ratios. Dicklyon 22:22, 16 October 2006 (UTC)

Your gut was correct, AND you followed through to verify what it told you. Good work! Finell (Talk) 10:30, 17 October 2006 (UTC)

Deletion of material

Why the sudden deletion of material without discussing first? ≈ jossi ≈ t • @ 17:55, 18 October 2006 (UTC)

Sorry, perhaps I jumped a bit quick after yesterday's comments (see book above). Please take a whack at it if you think the way I fixed it is not so good. Basically, I deleted material for which no verifiable support has been forthcoming, and replaced it by something that is verifiable, making the same point that golden ratio is used in old books. That is, the comments about Rosarivo seem to have been misinterpretations, and nobody has seen the original as far as I can tell; per Tschichold, Rosarivo's canon is about a 2:3 ratio, not golden section; several other articles are in need of similar corrections, so please do examine this issue carefully, as I have been trying to do. Dicklyon 18:14, 18 October 2006 (UTC)

"Verifiability, not truth" as per WP:V. We are reporting what Rosarivo said, and that is verifiable by the provided citation. ≈ jossi ≈ t • @ 18:20, 18 October 2006 (UTC)

If there are reliable sources that challenge Rosarivo's "misinterpretation", or provide a competing viewpoint, we should include it. ≈ jossi ≈ t • @ 18:22, 18 October 2006 (UTC)

Rosarivo's interpretation is not what is being challenged; we don't even have access to a copy, so we don't really know what his interpretation is. What's challenged, based on Tschichold, is YOUR interpretation of Rosarivo, which I find to be unverifiable. Please read my comments again and respond to what I'm actually saying and finding. Dicklyon 18:47, 18 October 2006 (UTC)

More specifically, I find nothing to suggest that Rosarivo ever mentioned a "golden canon of page construction"; that concept was developed later by Tschichold. And it has nothing to do with the golden ratio. Neither do I find evidence that Rosarivo's concept of Divina proporción tipográfica refers to the golden ratio. So everything you've written about Rosarivo seems to be unverifiable, at least by my attempts. Please correct me if I'm wrong, with quotes from sources. If there's a sentence in the linked Rosarivo summary that you believe helps to verify what Rosarivo did or said, please quote it there with translation, and also say who wrote it if you know. If you can find any actual Rosarivo quotes, that would be even more useful. Dicklyon 19:08, 18 October 2006 (UTC)

It is all there in Raul Rosarivo. ≈ jossi ≈ t • @ 19:17, 18 October 2006 (UTC)

You leave me little choice of action if you won't cooperate by citing the evidence you believe to be "in there". Dicklyon 19:22, 18 October 2006 (UTC)

You always have a choice. And your choice is always to shoot first and then ask questions. You deleted the material without an attempt to ask a question, and I take exception with that. ≈ jossi ≈ t • @ 19:26, 18 October 2006 (UTC)

You have the sequence wrong. I posted my long talk item last night, in the book section above; you have not yet acknowledged that you have seen it. Have you? Perhaps I didn't wait long enough, but if you won't respond to my inquiries with more than a brush off, I feel safe in assuming that more time is not what you are asking for. Dicklyon 19:55, 18 October 2006 (UTC)

One more thing: the illustration of Van de Graaf's construction is also pretty irrelevant, since there's nothing in that method that is specific to golden ratio. It's a construction for making a text area the same proportions as the page, with twice as much outer margin as inner margin, twice as much bottom margin as top margin, and text area corners on the diagonal of the spread. That's all it is, nothing related to golden ratio. Right? Dicklyon 19:14, 18 October 2006 (UTC)

Jossi, it would help to have a translation of the Spanish quotation from Rosarivo (perhaps without the Spanish for English WP), perhaps some additional substantiating quotations (in English) in footnotes (not in the body of the article), and page references to Rosarivo in the footnotes for the specific material used in this article. Also, "Ibid" cites don't work on WP because anyone can add an intervening footnote.

Also, it is my impression from what you have written that you have Rosarivo. Dick apparently thinks that what you attribute to Rosarivo comes from a secondary source's "interpretation". Could you please clarify this?

Thanks, Finell (Talk) 20:46, 18 October 2006 (UTC)

No that is a direct quote from Rosarivo, not an interpretation. I received the quote from a friend of mine in Buenos Aires. The text of the quote is "el número de oro o número clave en que Gutenberg se basó para establecer las relaciones armónicas que guardan las diversas partes de una obra". I will see if I can get the exact page number and edition. ≈ jossi ≈ t • @ 21:14, 18 October 2006 (UTC)

Any possibility of borrowing the book or obtaining a copy through a good reserach library? Having the entire context first-hand should resolve the issue difinitively. Finell (Talk) 21:53, 18 October 2006 (UTC)

Good idea. It's impossible to know from this third-hand snippet even whether "el número de oro" refers to the golden ratio, and if so, exactly what he is saying that Gutenberg did with it, since it's been pretty well established in available English-language publications and secondary verifications that Gutenberg's pages and type areas were NOT proportioned in the golden ratio. Also, Jossi's reference to WP:V is well worth paying attention to. Dicklyon 23:50, 18 October 2006 (UTC)

I have asked for a copy of Rosarivo's book. It may take a few weeks before I get my hands on it. ≈ jossi ≈ t • @ 23:58, 18 October 2006 (UTC)

Why?

Why this approach, Dick? Do you really enjoy these squimrishes? I do not. It is a bloody waste of time. ≈ jossi ≈ t • @ 19:31, 18 October 2006 (UTC)

Just trying to get the article back to verifiable information. I left your book section until I could amass the references and evidence needed to be sure it was unverifiable and incorrect. And I laid out the evidence. And you refused to discuss it. So, what next? Dicklyon 19:55, 18 October 2006 (UTC)

Dick: One day hardly seems enough time for others to see let alone to digest what you wrote, nor does it show sufficient respect for others who are working on the article. Also, you posted your long paragraph in a section that has not been active for a couple weeks and after which a lot of text intervened (one would have to scroll up a few screens to see it), so it did not call attention to itself. Perhaps a brief new item at the bottom of the Talk page announcing a new addition to an older topic posted above would help on such a lengthy Talk page. Finell (Talk) 20:25, 18 October 2006 (UTC)

I did not add anything, Finell. I just restored a deletion of text that was there for several weeks. ≈ jossi ≈ t • @ 21:11, 18 October 2006 (UTC)

My comment was addressed to Dick, not to you, which I just clarified. Dick had said in the previous item that perhaps one day was not long enough to wait for a response to his lengthy paragraph at the end of #Book [posted 05:56, 17 October 2006 (UTC)], before digging into the article. I was agreeing that one day was not enough and explaining why. Finell (Talk) 21:49, 18 October 2006 (UTC)

Which was pointless, since I had already acknowledged and apologized. Dicklyon 23:45, 18 October 2006 (UTC)

Books and Rosarivo

I've been trying to check references and figure out what was behind this line from the article (and other articles):

Rosarivo's work and assertion that Gutenberg used φ (the "golden number" or "secret number" as he called it) to establish the harmonic relationships between the diverse parts of a work, was analyzed by experts at the Gutenberg Museum and re-published in the Gutenberg Jahrbuch, its official magazine.

So I finally got a copy of the 1955 Gutenberg Jahrbuch to see what it says. I've scanned Rosarivo's article, and put it online temporarily for anyone who wants to help interpret it: here. In the process, I also took a closer look at that second reference above. I figured the statement about Rosarivo's work being "analyzed by experts" was just hype for saying they accepted it for publication. What I didn't realize was that it was pretty much a direct translation out of the Spanish reference, which I find says this (excuse my poor translation):

(p.49 Raúl Mario Rosarivo o el amor al libro by Vicente Ros)

(from German) Raúl M. Rosarivo. Divina Proporción Tipográfica. The book of golden typographic module 1:1.5, the module of Johann Gutenberg and of his contemporaries in the proportion 2:3. Translation from the Spanish by Heinz Nieth. Treatment of the German output by Hermann Zapf. Scherpe publishing house Krefeld, 1961.

(from Spanish) Study on the aesthetic proportions of the typesetter architecture of the Bibles of 42 and 36 lines, printed in Maguncia in 1455 and Bamberg in 1467-8 respectively. In it he demonstrates that Gutenberg had a module different from the well-known one of Luca Paccioli. This study, according to reference of Rosarivo himself, was analyzed by the scholars of the Museum Gutenberg of Maguncia and published in the Gutenberg Jahrbuch, official organ of the Museum, and in the magazine of the technicians of “The Graphical” book of Madrid, and simultaneously in no. 179 of “Graphical Argentina”, organ of the House of Graphic Industries of Argentina.

So, there it is: it's actually quoting Rosarivo hyping himself. And it says explicitly that his "module" (the 2:3) is not the one of Luca Paciolli (the golden ratio). On the other hand, Rosarivo's article that I link my scan of above does mention the "Goldene Schnitt" (golden section) and has a pair of "goldene Linie" (golden line) on one of the illustrations, so there may yet be something in it worth remarking on relative to the golden ratio. Someone who is more fluent in German please help me find it. Dicklyon 04:39, 27 October 2006 (UTC)

That is very interesting, Dick, but nontheless you cannot remove the material. You can provide sources that he refers to "golden ratio" to be 2:3 and not Paciolli's phi. The fact that he has been widley cited as one that refers to "goden ratio" as it relates to medieval books, is encyclopedic in itself. I don't speak German so I cannot help with the Gutenberg Jahrbuch. ≈ jossi ≈ t • @ 22:12, 28 October 2006 (UTC)

Jossi, I believe that you are in the process of obtaining Rosarivo, so that should decide the matter. However, if Rosarvio is talking only about a 2:3 aspect ratio, no matter what he calls it that has nothing to do with our Golden ratio article. Further, if Rosarivo's terminology misled others into writing that Rosarivo was using the (i.e., our) golden ratio, that doesn't make it so. I do understand the principle of "verifiability, not truth", but that can only be carried so far. Surely it is not encyclopedic to perpetuate demonstrable falsehoods even if uttered by otherwise reliable sources. No other encyclopedia does that, and I cannot imagine that WP intends to make itself a laughingstock by being the first to do so. Am I missing something here? Finell (Talk) 03:42, 29 October 2006 (UTC)

I don't think anyone is arguing for including demonstrable falsehoods, but "the material" by and about Rosarivo fits better in other articles, since, as it now clear, it is pretty much unrelated to the subject of this article except via his choice of adjectives equivalent to "golden" and "divine". Dicklyon 04:04, 29 October 2006 (UTC)

OK. ≈ jossi ≈ t • @ 15:47, 30 October 2006 (UTC)

Still speculative stuff in the lead

We still have stuff in the lead like "many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle" which has already been denied and debunked for the case of paintings; do we have something that it definitely applies to, or something we can cite? And the unsourced statement about occurrences in nature; that's been long enough looking for a citation that I think I'll take it out now. Who knows what that first part refers to? Is Le Corbusier about all? Dicklyon 23:52, 14 November 2006 (UTC)

Misleading Phrasing

"This illustrates the unique property of the golden ratio among positive numbers, that..."

Given that there are infinitely many positive numbers whose fractional parts are their multiplicative inverses, does anyone else find this misleading? Should it be specified that the golden ratio's unique property is having a multiplicative inverse one less than itself? Otherwise, what of the silver ratios? Namely, the positive roots of quadratics of the form

$x^{2}-nx-1=0$,

where n is natural.

--Johngalt7777777 05:43, 20 November 2006 (UTC)

I don't see how it's misleading. What follows the "that..." is the equation that defines the unique property. Dicklyon 06:22, 20 November 2006 (UTC)

Free use complete book on Golden Section

I found this book on the metaphysical meaning of the Golden section in Plato and Pythagorean sources, has anyone else seen it yet? http://www.attan.com/ratio.pdf -"Pythagoras, Plato, and the Golden Ratio"-

It is a complete book released for download by the author --Attasarna

I just took a peek. At 23 pages of text, it is more a pamphlet than a "complete book". It has an ISBN number, but that number is neither in the Library of Congress catalog nor in Amazon's database. The first sentence by the author (after the quoted paragraph) contains two errors that any junior copy editor would have caught; they are not the last such errors. The author confuses Phi with phi. I could not, through a few quick searches, find any information about the author; he has no publications in the Library of Congress catalog. The graphics appear to have been cut and pasted from various different sources, including some obviously scanned from printed publications, all without credit. I see nothing that would qualify the pamphlet or its author as a sufficiently reliable source to be cited on Wikipedia. While the pamphlet is available for downloading at the URL that you gave, downloading is inconsistent with the strong copyright notice on the title page; it is therefore not available for "free use". Finell (Talk) 03:11, 22 February 2007 (UTC)

Thanks, Finell. I was about to revert those links when you beat me to it. This editor likes to link that site, but has denied that it is his own. Certainly does not appear to be what he says, however. Dicklyon 03:24, 22 February 2007 (UTC)

Use in the Renaissance

Quote from lead:

At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

The above claim is easily attributable to numerous reliable-looking texts by well reputed authors; all the same other reliable sources deny it or at least point to an astounding lack of primary sources prior to the time of Theodor Fechner stating or demonstrating that the golden ratio is used in art or architecture. I here take "golden ratio" to mean a specific principle consciously applied, though not neccesarily the specific geometric construction or the resulting irrational ratio. A dynamic balance obtained by placing picture elements a bit off the centre is of course very common, as is pleasing-looking rectangular picture frames and the like. It is well documented that Renaissance architects did use the so-called "harmonic port", a rectangle of proportions one-to-squareroot-of-two, i.e. 1 : 1.41 as opposed to 1 : 1.61. The harmonic port is used today in the European A4 paper format.

Would anyone object to a more careful statement in the lead, perhaps like this:

At least since the end of the 19th century, many artists and architects have proportioned their works to approximate the golden ratio, e.g. in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio, believing this proportion to be aesthetically pleasing. Golden proportions can be found in works from earlier periods, and many sources claim that such proportions were used deliberately in the Renaissance, or even in Antiquity.

--Niels Ø (noe) 11:07, 15 March 2007 (UTC)

I expect you'll get some objections; we'll see. I would reckon that even "At least since the end of the 19th century, many artists and architects have proportioned their works..." is a stretch. I haven't seen much evidence for this "many" claim; seems more like a "few". Do you find some support for this? Dicklyon 23:29, 15 March 2007 (UTC)

Sounds ok to me. Do include a link to silver ratio or silver rectangle when you mention the harmonic port. But are you sure that's the correct phrase? I can only find one link that uses it. "Harmonic proportion" seems much more common. —David Eppstein 04:08, 16 March 2007 (UTC)

The lead

The "relationship" described in the lead is a ratio. An anonymous editor thinks it would be more clear as a "constant". I'm wondering if there are readers who have trouble with the current lead, yet would not be confused by the notion of a "constant", a somewhat abstract mathematical notion. Seems unlikely to me. And somehow he thinks it's a good idea to change quantity to number, when then requires the introduction of a nonlinearity disambiguate the solution. It doesn't need to be so hard. Opinions? Dicklyon 05:29, 3 April 2007 (UTC)

I found the changed intro perhaps slightly better and certainly not worse. Let's not discourage new editors by bluntly reverting their edits. −Woodstone 08:15, 3 April 2007 (UTC)

I agree with Dicklyon. The golden ratio is not only a relation between numbers. The illustration in the lead shows a relation between line segments for example. So we'd have an inconsistency in the lead defining it as ratio of numbers only. The term quantity should be preferred here. Furthermore the reverted edit contained factual wrong information in the proof of the irrationality of φ (numerator <-> denominator). Let's not let new editors reduce the quality of an article only to encourage them.

However, the reverted edit had good points, too. The golden ratio is actually a constant and I don't see why we shouldn't say so. — Ocolon 08:37, 3 April 2007 (UTC)

I made those anonymous edits. I'll concede the point about using the word "quantities," though I think it is an extremely trivial issue. However, I certainly think that when we switch to the algebraic definition a few lines down, a > b should be mentioned because otherwise you would end up with the golden ratio conjugate. I am slightly annoyed by the setup of the irrationality proof, but regardless I'm confident I understand it. Can someone explain to me why my edit saying that the numerator is smaller was wrong? I mean, if a is the larger quantity, then the easier contradiction to the lowest terms condition appears in the numerator, not the denominator.

If quantities means positive amounts, as it traditionally did, and does in the illustration, then the only solution of the equations has a > b, so you don't need to specifiy it. As to the proof, I didn't look carefully at what you did there; if you're sure it's right and better, so ahead and put it again. Dicklyon 03:57, 4 April 2007 (UTC)

The way it was changed back to does not define if $/phi$ is a/b or b/a, under constraint that (a+b)/a=a/b. That is precisely what the previous version (the other) proposal avoided. −Woodstone 09:29, 15 April 2007 (UTC)

That's true. But does it matter? The first sentence is correct about what it means for two quantities to be in the golden ratio. The next two sentences define the corresponding mathematical constant pin down the order that the ratio must be evaluated in. Is there a clean way to get all that into the first sentence? Do we need to? Dicklyon 16:50, 15 April 2007 (UTC)

Yes, the lead defines what it means that two quantities are in the golden ratio, but it does not textually define completely what the golden ratio itself is. I think the previous wording defined both unambiguously and remains preferable, but I will not edit-war over it. −Woodstone 18:04, 15 April 2007 (UTC)

I agree the previous one did make it unambiguous. But it felt like rather strained wording. Maybe we need another try to satisfy all these considerations. Dicklyon 18:09, 15 April 2007 (UTC)

I just checked some books, and find that most are like the current lead. That is, they define what it means for two quantities to be in the golden ratio, without introducing phi and without saying in which order to quantify the ratio. Since this is a pretty conventional approach, I'm OK keeping it. Or, if we find a clear way to do better, that's OK, too. Dicklyon 18:12, 15 April 2007 (UTC)

I missed this problem Woodstone, but I guess its still an improvement over the other sentences, agree 100% with Dicklyon here. Ricardo sandoval 18:23, 15 April 2007 (UTC)

It looks both versions suffer from something. I was thinking if it would be wise to post this problem in a math forum to get different ideas. Ricardo sandoval 01:53, 17 April 2007 (UTC)

It seems I didn't look carefully enough at the edit either, sorry; actually both numerator and denominator are smaller on the right side of the equation than on the left side. If the numerator is smaller, the denominator has to be smaller as well and vice versa. Or else we wouldn't have an equation. So, I'll change it to "numerator and denominator". Hope you agree. If not, revert it. — Ocolon 07:47, 5 April 2007 (UTC)

I find the leading description rather vague or at least confusing. On the part "is the mathematical constant that expresses the relationship that the sum of two quantities is to the larger quantity as the larger is to the smaller" its not clear what those quantities are or can be. Also "relationship" can express a proportion or a ratio, but have also a vague general meaning that misleads an untrained audience. Wouldn't it be more clear/precise if stated, for example as:

"is the mathematical constant that expresses the ratio of two quantities if the sum of those quantities is to the larger quantity as the larger is to the smaller"

The "if" making the sentence more precise.

I made another attempt: "In mathematics, two quantities are in the golden ratio if the ratio between the sum of the quantities and the larger of the quantities is the same as the ratio between the larger quantity and the smaller. This ratio can be expressed as a mathematical constant, the following algebraic irrational number with its numerical approximation: ..."

I think it's clearer, but I won't be heartbroken if you all disagree and revert it. —David Eppstein 06:38, 7 April 2007 (UTC)

I think it is way better now, good writing! I was also thinking of making a link to the calculation section, to in some way point the reader that this relation defines is possible and defines only one constant. What I think is not obvious from the definition. In the calculation section it is not said that a b are positive. The calculation demonstration although right I think looks rather strange from the perspective of a high schooler for instance. I will try to edit it to make for a more readable.

Being in the golden ratio avoids saying what the golden ration is. And this version omits the introduction of the symbol phi needed in the next line. But I agree it may be more clear to say its a ratio as opposed to a relationship. I'll work on it... Dicklyon 15:38, 7 April 2007 (UTC)

OK, I made a new one. The intro "in mathematics" had been added, but the use of this definition is so much broader that this is maybe not appropriate; I changed to "in mathematics and the arts". Certainly in mathematics the golden ratio is a mathematical constant if it is a ratio, which I presume we all agree on. And I put back the introduction of phi so the equation line makes sense. Dicklyon 15:46, 7 April 2007 (UTC)

A ratio is a relationship between two quantities. It may be expressed by a number, but it is not the same as a number. So we should say what the golden ratio is as a ratio before talking about numbers and $\varphi$. That is, we are trying to express two different concepts, the extreme and mean proportion and its expression as a number. Trying to describe them both in one sentence makes things more confusing than keeping them as two separate sentences. Your (Dicklyon's) latest rewrite muddles these points and to my mind returns to the confusing writing of the previous versions. —David Eppstein 15:50, 7 April 2007 (UTC)

OK, I'm confused. I was happy for a long time with a ratio as a relationship, but I thought the pull was to be more clear about a ratio being a number, which it is also. What do you mean by it is not the same as a number? How will we introduce that distinction? I'm not sure what the distinction is. Dicklyon 16:07, 7 April 2007 (UTC)

Here's my take on it, which may be idiosyncratic. 3:2 is a ratio. It is a relationship between pairs of things; for instance it describes the relationship between the width and height of many of my photos. 3/2 is a fraction, a number. You can add it and multiply it and do all the things you usually do with numbers. These two concepts are closely related: if you have a ratio a:b, you can form a fraction a/b, and if you have a fraction q, you can form a ratio q:1, but ratios and fractions are not the same thing as each other. Since the article is titled "golden ratio", it should be about the ratio first; of course it should also talk about the number 1.618... but it doesn't have to do so in the very first sentence. And if you talk only about the ratio first, the part about a+b:a::a:b (expressed more verbosely in English of course), and avoid trying to simultaneously talk about the corresponding number, it keeps the first sentence simple enough to be understandable. Putting in the part about a number "expressing the relationship" into the same sentence makes it much longer and more confusing. —David Eppstein 19:04, 7 April 2007 (UTC)

You've added weight to your conviction that there's a difference, but you haven't said what you think that difference it, or maybe I'm just being dense. A rational number is a ratio of integers, which is not all that relevant here except for saying number...is...ratio. Why don't you try again to modify the lead, but don't leave out introducing phi and the idea of a mathematical constant in a second sentence or somewhere this time. Then maybe I'll get a better idea what you're trying to say, or at least not be unhappy with how you say it. Dicklyon 19:46, 7 April 2007 (UTC)

(Unindenting) I would define a ratio as a binary relation of pairs of numbers. So "9 and 6 are in the ratio 3:2" could be translated into the more formal notion that the ordered pair (9,6) is a member of the binary relation {(x,y): 2x=3y}. Does that help clarify what I mean? But I think detailed axiomatics would be too distracting for the actual article. The lede as I already wrote it expresses what I would like to see: a first sentence that says what the ratio is, and a second sentence talking about the mathematical constant that expresses it. —David Eppstein 20:14, 7 April 2007 (UTC)

OK, I'll restore that, add the introduction of the name phi, and see if it sticks. Dicklyon 23:48, 7 April 2007 (UTC)

I just realized the lead would be more clear if it was stated just as in the picture, from the sentence present now one would could think (a+b)/a = b/a of course that doesn't make sense but it would increase readability if the reader didn't have to choose between those two different meanings. Something like:

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger and the smaller.

I guess there are many different ways to phrase this, this way loses a little in conciseness, but I think is more precise, don't leave place to confusion. The sequence a+b to a , a to b. Sounds better to me then the opposite, even thought in that case we can push the first reference on the ratio and increase conciseness. Ricardo sandoval 15:34, 12 April 2007 (UTC)

Although it's in the other order, I like Dicklyon's latest version, "In mathematics and the arts, the ratio between two quantities is the golden ratio if it the same as the ratio between their sum and the larger of the two quantities." I think its conciseness compared to the other order is a big plus. —David Eppstein 15:43, 12 April 2007 (UTC)

I agree that the sentence sounds great but what about the problem of possible confusion I just raised? We could say (a+b)/a=a/b or (a+b)/a=b/a by what is said on the sentence, I know the second option doesn't make sense but it adds to confusion nonetheless. Ricardo sandoval 17:03, 12 April 2007 (UTC)

We already do say (a+b)/a=a/b, at least three times: in the figure to the right of the lede, in the "expressed algebraically" displayed equation a few lines down from the lede, and at the start of the first section "Calculation". But I don't see that moving formulas into the lede is likely to make it any gentler as an introduction to the subject. —David Eppstein 17:10, 12 April 2007 (UTC)

The textual definition should be precise, independent of an accompanying picture. Therefore it should exclude an interpretation that φ could be smaller than 1. A few lines above here Ricardo states very well why I changed the wording a few days ago. −Woodstone 17:44, 12 April 2007 (UTC)

Both a/b=1.618 and a/b=-.618 satisfy the equation (a+b)/a = a/b. However, the lede as written now talks about "the ratio between their sum and the larger of the two quantities". That word larger makes the definition in the lede precise and unambiguous. —David Eppstein 18:35, 12 April 2007 (UTC)

I think you're confused. There are no values of a and b for which (a+b)/a = a/b = 0.618. Dicklyon 18:45, 12 April 2007 (UTC)

He said "-.618", not "0.618". -- Dominus 19:35, 12 April 2007 (UTC)

The lead is still not clear since when we read "ratio of two quantities" in the beginning we can mean both a/b or b/a. Ricardo sandoval 23:27, 12 April 2007 (UTC)

It is unambiguous because there are no a,b satisfying (a+b)/a = b/a. To avoid dividing by zero, a would have to be nonzero, but then the condition a+b=b could not be satisfied. —David Eppstein 23:33, 12 April 2007 (UTC)

I already noticed that in a previous point at this discussion, the problem is that it leaves the reader to make that deduction and that adds to confusion.Ricardo sandoval 23:46, 12 April 2007 (UTC)

To avoid that this discussion goes into the infinity I will try to make a resume of some of the points involved.

For reference the present lead is:

" In mathematics and the arts, the ratio between two quantities is the golden ratio if it is the same as the ratio between their sum and the larger of the two quantities. This ratio can be expressed as a mathematical constant, ... "

and one of the contestants is

" In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger and the smaller. This ratio ..."

The main points in discussion are:

1) the original lead allows one to understand a/b=(a+b)/a or b/a=(a+b)/a and the alternative lead doesn't.

2) the word "quantities" can also mean a negative number, giving a different value for the golden ratio.

3) ratio can mean a fraction (a constant, real number) or a proportion (a pair of real numbers). Ricardo sandoval 11:58, 14 April 2007 (UTC)

For 1) From the math side b/a =(a+b)/a = 1 + b/a doesn't have a solution, so anyway the original lead have only one mathematical interpretation. The problem is that it leads to confusion, it would be better if the reader doesn't have to consider these possible meanings because there is only one. The alternative lead is less concise but it is more understandable!!

For 2) in principle "quantities" can be negative but for many audiences it is understood as positive, and this can and should be clarified latter.

For 3) although ratio has this double interpretation I still think it is better then "relationship", "proportion", or just "fraction". And it is clarified as a constant.

Remark: For me the order (a+b)/a=a/b sounds much nicer then a/b=(a+b)/a, although obviously they mean the same. Ricardo sandoval 11:58, 14 April 2007 (UTC)

I agree that your contestant is better. OK with me if you change it. Dicklyon 16:17, 14 April 2007 (UTC)

Just changed it but not sure if the wording is the best one. It looks a lot like some (very) earlier versions.Ricardo sandoval 23:05, 14 April 2007 (UTC)

Slow approximation by rationals

A section was added to state that φ has one of the worst approximation speeds in the rationals. It mentions the successive quotients of Fibonacci numbers to show this. However, it is not explained why there cannot be better approximations than those quotients. −Woodstone 21:05, 22 April 2007 (UTC)

It's a well known property of continued fractions and is explained in the linked article on Convergent (continued fraction). Dicklyon 23:13, 22 April 2007 (UTC)

That the continued fraction converges slowly may be true, but that does not imply that there cannot be better rational approximations. −Woodstone 19:57, 23 April 2007 (UTC)

Yes it does. The continued fraction convergents (and some other fractions derived from them which don't affect the result for the golden ratio) are known to give the best possible rational approximations. See continued fraction, under "semiconvergents". —David Eppstein 20:04, 23 April 2007 (UTC)

Also this recent edit actually expands on a remark about rational approximation in "Other interesting properties". These should be rationalised. (abysmal pun intended) Since it is covered elsewhere I would be in favour of reverting it. The page is already a bit of a mess - far too many random factoids. Andrew Kepert 02:13, 23 April 2007 (UTC)

Go for it. If it's already mentioned, it should be integrated, at best, or just reverted. Dicklyon 03:15, 23 April 2007 (UTC)

I don't see this specific fact mentioned anywhere else(I checked the article above and some others) but agree this is not the place for it. I will move it to the continued fraction article and make a link to it.Ricardo sandoval 04:08, 23 April 2007 (UTC)

The term "most irrational number" is linked to Continued fraction#Golden ratio is the most irrational number. There is no such heading in the linked article, but there is this heading: Continued fraction#A property of the golden ratio φ. That section does not directly support the proposition that φ is the most irrational number; perhaps that statement was edited out of Continued fraction and the heading was changed accordingly (I did not dig into the edit history). Perhaps one of the strong mathematicians here would be willing to revise this part of Golden ratio or harmonize the discussion of it in Continued fraction. Finell (Talk) 05:46, 24 April 2007 (UTC)

Just changed the link. The link in the "worst case" and the link in the "approximation theorem" seem disconnected. With no reference to φ on the second. Ricardo sandoval 19:14, 24 April 2007 (UTC)

Repeated reversions of Music addition

Disclaimer: Until yesterday when I saw a reversion in this article, I had never heard of Tool (band). I have no idea what their music is like. I still don't. I can hardly be called a "fan".

Sometimes, when I see a deletion with the reasoning that the text was unsourced, I get curious enough to look for sources. So I looked. After a few google searches, it became clear to me that the removal was unjustified. I added it back, with a reference. It was reverted again. My final addition read as follows:

The progressive rock band Tool uses the golden ratio in its music. For example, the song lyrics to Lateralus last 1.617 minutes (approximately equal to the golden ratio) and are structured according to the Fibonacci sequence. Tool's drummer Danny Carey also chooses drum patterns that reflect Fibonacci sequence.

I note the following inconsistencies between what's already in the article and the reversion of my edit:

• One reason for reversion was given as "a time is not a ratio". Well neither is a title. In that spirit, I have removed another entry accordingly.
• Another reason for reversion was given as "band's use of Fibonacci sequence is not relevant to this article". In that case, should I also delete the entry on Bartok, since that also describes only Fibonacci numbers? It should go if nobody gives a compelling reason to keep it. If it should be kept, then so should my entry on Tool for the same reason - an ascending and descending sequence of Fibonacci numbers in the lyrics converges to and diverges from the golden ratio.
• The same argument holds for the entry on Debussy.

It seems to me that editors here have a knee-jerk reaction to one particular band, no matter how encyclopedic or well-referenced. =Axlq 14:13, 25 April 2007 (UTC)

No, I don't think it's anything about the band. It's that we haven't seen a source that says they use the golden ratio; if you have such a source, bring it on. As to Bartok, if the statement is correct (I don't have the ref to verify), then his music was analyzed by this guy in terms of golden ratio; so it's OK, as it's backed up by a reliable source (presumably). Similarly, the Debussy statement specifically mentions the golden ratio as coming from a reliable source. If you want to challenge these, look up the source and let us know whether they actually do analyze these guys with respect to golden ratio. As for golden ratio in a title, it's true that's not a ratio, but it is an explicit reference to the subject of the article; seems OK. As to a passage lasting 1.617 seconds, where is the reliable source that says that's about the golden ratio? If you cite it, it's OK; else not. Dicklyon 17:50, 25 April 2007 (UTC)

If you follow the link in the reference [18] its an interview with a member of the band by Christopher diCarlo, a Philosopher of Science at University of Ontario Institute of Technology. As such I would consider this a verifiable reliable source. There is an extensive section on links with the golden ratio and fib sequence. There is enough in the link to verify a connection with fib sequence and indeed tool does get a mention in Fibonacci numbers in popular culture which seems the correct place, I've now linked to this article. The link does not verify the 1.617 claim, if it could be verified it seem OK to include it to me.

It is a fuzzy line as to when references are to the fibonanci sequence or golden ratio and I'm not quite sure what the φ position in the main climax sits at the φ position means. --Salix alba (talk) 19:01, 25 April 2007 (UTC)

But there's really nothing there. All the links shows is that some guy says that MK of the tools mentioned the golden rectangle and phi to him on the phone. Hardly a reliable source for the golden ratio being represented in their music. Dicklyon 20:14, 25 April 2007 (UTC)

As to the Bartok and Debussy source, the referenced book by Smith says "In music the golden ratio is apparent in the organisation of the sections in the music of Debussy and Bartok, ..." Books are generally presumed to be reliable sources, and this one makes an explicit connect to the article's topic, so it's OK. Find it here. Dicklyon 20:43, 25 April 2007 (UTC)

OK, I'll address the points made above.

• The band themselves say they use the golden ratio. I even provided a reference. It's paraphrased from the interview, but it's a valid reference nonetheless, based on an actual interview with a band member. Claiming "there's nothing there" defies comprehension and indicates only a cursory skimming of the cited source. The beginning states that some things were paraphrased to avoid redundancy; that's the perogative of the interviewer and doesn't detract from the statements the interviewer says were made during the interview.

All I found was that the guy "mentioned" the golden rectangle and phi. Did I miss something? Dicklyon 05:46, 26 April 2007 (UTC)

• It is ridiculous to require a "reliable source" to verify an obvious statement that 1.617 is approximately the golden ratio, just like you don't need a reliable source to verify that 2+2=4. You also don't need a source to verify that the lyrics last 1:37 or 1.617 minutes; that's a verifiable fact all on its own, available to anyone to verify the same way you'd look up something in any other source cited. Others have done so, based on my searches. So could you and I, fairly trivially. It doesn't qualify as original research.

The source is required for the observation that the interval is 1.617 minutes, and the interpretation that that is a purposeful relation to golden ratio. Without that, what is the basis of mentioning the band here?Dicklyon 05:46, 26 April 2007 (UTC)

• The band has outright claimed to incorporate the golden ratio in their music, through an ascending Fibonacci sequence of lyrics (which approaches the golden ratio as the sequence ascends). Bartok and Debussy never made such claims; rather, the fact was uncovered by analysis — but for all we know the presence of the golden ratio could have been accidental. Having the author of a work claim to use the ratio is a stronger statement than some analyst discovering it after the fact.

What is the source of the claim you mention? Are you leaping again from fib numbers to phi? Did the band suggest extending the sequence infinitely and taking a limit? I must have missed that, too. Dicklyon 05:46, 26 April 2007 (UTC)

• I see Dicklyon reverted my deletion text about another band that merely titled a song "Golden Ratio", directing me to WP:POINT. I refer Dicklyon in turn to WP:AGF. A mere song title is a pretty weak criterion for inclusion, much weaker than any arguments I give here.

I don't understand your point here; you deleted it to try to tell me that something is wrong with my criteria, didn't you? Are you saying that the article should not mention a work with title Golden ratio? OK with me if you seriously want to take it out, I won't object; someone else likely will though. Dicklyon 05:46, 26 April 2007 (UTC)

• Finally, the only reason I got involved at all is because editor David Eppstein complained "This must be the 50th time some Tool fan has tried to insert this". That tells me that a lot of people feel that this factoid belongs in the article, and that this issue isn't going away until it is addressed appropriately with references. That's what I tried to do: search for references to justify inclusion of this factoid, rather than blindly reverting it out of laziness. I encourage others to help out. =Axlq 05:24, 26 April 2007 (UTC)

He was exaggerating. But I've been working on helping it go away by taking out the unsourced material from the article on the song. No reason to give up hope just because a lot of people want to put the web noise and OR into an article. Dicklyon 05:46, 26 April 2007 (UTC)

Fibonacci numbers from the golden mean (and back again)

File:Golden mean loop.png

At each step of division, the same pattern is continued, marked sequentially in red, orange, yellow, and green, and is the sum of the preceding two patterns.

Recently I was playing with Perl on the laptop and started toying with the question of what a closed loop would look like if divided into bands which have sizes according to the golden mean ratio. So starting with a loop with one black band of size φ and a white band of size 1, divide the black band into 1/φ (black), 1/(φ**2) (white), 1/φ (black), i.e. like one side of a pentagram. Repeat with for all bands of the next largest size (1), and the next (1/φ), etc. The result is a figure with all bands of one color being the same size, and all bands of the other color either longer or shorter than that by a factor of φ, and the number of those bands (long, short, long+short) are all Fibonacci numbers. What's more, the pattern of long and short bands is also the sum of the preceding two steps. Written as a code (short=0 or 10, long=1 or 110), these form a simple L-system (example 1 or 2 from that article), a string where every interval between integers not occupied by an integral multiple of the golden mean is represented by one character, and the others by the other character. Taking every kth digit of this string yields a pseudoperiodic tiling, and the closer k is to an even multiple of φ (21, for example), the longer the runs of 0s and 1s. Could someone point out a reference that leads in this direction? (not just for curiosity, but also so some of this can be put into the article without violating WP:NOR) Mike Serfas 20:34, 25 April 2007 (UTC)

Website:[19].

What do you guys have against [20] as a source of digits of golden ratio? How is this website worse than the other one with only 20000 digits.--BorisFromStockdale 19:11, 5 May 2007 (UTC)

It's someone's personal site, essentially anonymous, non-authoritative and unsourced, and contains no information that's encyclopedic or useful in extending the content of this article; we already have way more digits than anyone has a possible use for, as well as a description of how to compute more. If it was a discussion of how to compute those digits, with code, and said by whose authority we should believe it, then maybe it would be useful. Dicklyon 22:18, 5 May 2007 (UTC)

Calculation by iteration another way

IANAM, but you can calculate $1/\varphi$ with the following iteration, which is not in the article:

$x_{n+1}={\frac {1}{1+x_{n}}}$ (starting with any x != -1)

Starting with x=1, it takes around 34 iterations to converge to 1.61803398874989, while Newton's method on $x^{2}-x-1$ needs only 6 iterations!

However, given a crazy starting value like 1e10, this method still requires only 34 iterations to converge (the last digit is corrupted, note), while Newton is only at 1.66707382493846 after 34 iterations. In fact, it converges from any starting point in the same number of iterations. Uncoolbob 15:18, 14 May 2007 (UTC) (forgot to sign first time around, sorry)

Actually, it also does NOT work if you start with -2, or -1.5, or an infinite number of other negative starting values. It's also a very expensive calculation to get many digits with, since it requires a full bignum division for each step, which adds only a fraction of a digit each time. It is much easier to compute big fibonacci numbers, which is quick, and then do a single bignum division. Dicklyon 00:19, 13 May 2007 (UTC)

Good call - I had not tried -2 or -1.5. Also, I guess this iteration is equivalent to the continued fraction at the start of this section of the article, so in a sense this is superfluous. Uncoolbob 15:18, 14 May 2007 (UTC)

Investing

The reason I took out this tidbit, calling it unsourced, is that I read the refs and only found these two things about the golden ratio in the first and second refs, respectively:

• Enthusiasts for “the golden ratio” φ have claimed to see it in the proportions of classical architecture and art, and it was very consciously used by the 20th century architect Le Corbusier.
• The series has many interesting features, including the fact that the ratio of each number to the preceding one tends towards 1.618, the so-called golden ratio. You can find this ratio, and shapes and patterns derived from it, everywhere in botany, physiology, architecture and, if you believe Elliott Wave Theory, in the stock market.

And the Elliott wave principle article doesn't say much about it either, just:

• The Fibonacci sequence is also closely connected to the Golden ratio (1.618). Practitioners commonly use this ratio and related ratios to establish support and resistance levels for market waves, namely the price points which help define the parameters of a trend.

The second ref above, talking about the Batchelor guy mentioned, says:

• Professor Roy Batchelor, at the Cass Business School, is a braver man than I. Unconvinced by the idea that market movements might be determined by a sequence of numbers uncovered by a thirteenth century mathematician, he crunched all the daily movements in the Dow Jones index from 1914 to 2002 before concluding: forget Fibonacci.

This Elliot wave stuff is much more about Fibonacci numbers, and the golden ratio barely gets a mention. This snip to go back to where it came from. Or if the golden ratio really is used in investing, we should find a source that says so, and get some verifiability into what this snip says. Dicklyon 02:22, 13 May 2007 (UTC)

But Fibonacci number doesn't have a "disputed sightings" section, and this does... More to the point, on the previous page from the one where you found the bit about Le Corbusier (that is, on p.9 of the Batchelor paper), you will see "All textbooks in technical analysis devote considerable space to description and discussion of these ratios. For example, Murphy (2000) asserts that 0.5 and 0.618 are the key ratios for determining target prices in retracements." Which seems to be almost exactly the claim in the snippet, and relates much more closely to the golden ratio than to Fibonacci. And further the part about "all textbooks" (by someone who doesn't like this theory!) seems a strong claim for notability of such theories. The fact that these other guys mistakenly talk about Fibonacci when they really mean the golden ratio doesn't mean we have to follow them in doing so. —David Eppstein 04:31, 13 May 2007 (UTC)

OK, my quick search did miss that 0.618 bit. But let's be careful and not say anything beyond what the refs support. It would be better if the support for the ratio being used wasn't coming from the same guys who claim to be debunking it; that's too self-referential, isn't it? I'll re-check when you rewrite it. Dicklyon 04:36, 13 May 2007 (UTC)

Hey, at least I included some sources and skepticism! The guy who put it into Fibonacci number didn't. I'll see what I can find. —David Eppstein 04:56, 13 May 2007 (UTC)

I added the disputed paragraph back in again, with more sourcing. If you go to the source I added (Osler), you will discover that she also uses the phrase "Fibonacci number" to refer to a relationship that is more properly described as the golden ratio (the supposed 38.2% and 61.8% pricing support levels). But she does seem to support exactly the claim made in the disputed paragraph. I moved the Elliot wave stuff to a less prominent position as it appears to be related but not quite the same. Please me know what you think of it now. —David Eppstein 16:20, 13 May 2007 (UTC)

Thanks. I tweaked it a bit, too, and I think that now it's OK, since you found those sources. Dicklyon 19:33, 13 May 2007 (UTC)

You're welcome, and thanks. I think your tweaks are an improvement, so for now I guess we can leave it as is. —David Eppstein 20:21, 13 May 2007 (UTC)

Simplification

Would anyone be willing to simplify the language in the article? For those who are curious about the golden ration yet are less mathematically inclined, the article is difficult to read and hard to comprehend. I include myself, to be honest. I find the article's math to trying to read through. And at the end of it all, the golden ratio still seems rather elusive. Any help for math laypeople? --Bentonia School 17:19, 23 May 2007 (UTC)

Give us a clue where it gets hard. Do you understand the lead OK? Do you know what is meant by larger, smaller, and sum? by ratio? irrational? square root? Decimal fractions? Letters standing for quantities in algebraic expressions? At what point do you think a more elementary explanation would be helpful? Dicklyon 05:02, 24 May 2007 (UTC)

Split this article

This article is 58k, almost double the (theoretical) recommended size. A split Phi versus Golden ratio was proposed above, but I think that's a bad idea because they're synonymous terms (besides, Phi is taken). But as the Simplification subsection above suggests, there's a huge difference in audience - is the reader seeking a technical mathematical article, or a culture and history article? Nearly the entire table of contents is made up of sections split into "mathematics" and "aesthetics". I'd suggest two articles, Golden ratio (mathematics) and Golden ratio (applications), each referring to the other in summary style. The latter category would talk about pyramids, biological spirals, paintings, etc. --- any point where the math you need to know is basically that some measurement works out to roughly 1.6. Since biological applications can involve more complex mathematics if you try to work out plant spirals in terms of morphogens or packing efficiency, and they don't fall into a historical scheme, it's also possible a third article Golden ratio (biology) would turn out to be desirable. Mike Serfas 17:04, 24 May 2007 (UTC)

There are already 12 articles in Category:Golden ratio, so I don't see that one or two more would hurt. And I agree that separating the content will likely better please the users who are there for the aesthetics and put off by the technical math. But I think that, rather than just splitting into two, it might make more sense to use summary style for the main article and split out into separate articles anything long enough to require subsections. I think there's enough material to support as separate articles Golden ratio (architecture) (the main article's architecture and Egyptian pyramid subsections), Golden ratio (aesthetics) (the rest of the aesthetics subsections, with an abbreviated description of the architectural connections), Golden ratio (geometry) (the main article's geometry, Ptolemy's theorem, pentagram, and mathematical pyramid subsections, plus better linkage to Golden rectangle, Golden angle, and Golden spiral); Golden ratio (number theory) (rest of the mathematical content, including the "calculation" section, plus better linkage to Golden ratio base. —David Eppstein 17:46, 24 May 2007 (UTC)

I don't think a split into more than two is warranted at this point. It makes sense for one to be the mathematical aspects, and the other to be all the other stuff, whatever you want to call it. Separating architecture from aesthetics makes no sense. Dicklyon 18:23, 24 May 2007 (UTC)

The point of separating architecture from aesthetics is to deal with the pyramid stuff, which is certainly architecture but not really aesthetics. —David Eppstein 18:33, 24 May 2007 (UTC)

Well, that could be argued either way. There's no evidence that any ancient pyramid was ever architected with the golden ratio in mind, and plenty of evidence to the contrary. I wrote that section, mostly, and it's mostly mathematical, actually. That's an example of why it will be difficult to do the split. Either the mathematical part will be polluted by the pseudo-science stuff, or that aeastetics/architecture stuff will get pretty mathematical. Dicklyon 18:47, 24 May 2007 (UTC)

As one (of many) who has done extensive work on this article, although not recently, I would prefer no split, despite the article's length. (Contrary to the Simplification objection, I think it is a good thing that a "general reader" or aesthetics person gets some exposure to math, and that math people get some exposure to the other stuff.) It is important not to lose the breadth of the interest in and relevance of the golden ratio across many subjects. For that reason, in my opinion it is important that an article with the present title Golden ratio remain, that it convey the full breadth of the topic, and that it not be limited either to math or to aesthetics. Therefore, if the consensus is that some split is required, I propose that an article with the title Golden ratio remain, with sections for all of its current content (not necessarily all the current section headings), but that some (my preference) or all (if that is the consensus) of the content of those sections, except for a short summary, be relegated to whatever new articles result from a split, and with a linked reference (using the standard template) under each section heading to the new (split off) Main article appropriate to the particular section.

It is not practical, in my opinion, to split the several areas of aesthetics (design, architecture, painting, sculpture) from one another because that would require too much duplication in the split articles: Buildings (or design elements, paintings, or sculptures) are (or aren't), or should (or shouldn't) be, proportioned according to phi because those proportions, most commonly as golden rectangles, are at least to some degree (or aren't) aesthetically pleasing to many (or some) people, according to reliable sources in the respective fields (although disputed by other reliable sources), with some reliably sourced support in some perceptual studies (which are criticized by other reliable sources). Music doesn't involve the same considerations (it isn't clear why composers consciously used phi), although it would be a short article on its own.

Also, in my opinion, terminology and history should remain in the main Golden ratio article: terminology, because it is relevant for anyone interested in any golden ratio subject; history, because there is too much overlap of aesthetic, mathematical, and mystical (don't blame me for that one, but it cannot be denied) history (see especially Kepler and Pacioli, and arguably Leonardo), and because the indivisible history affects all the golden ratio sub-topics.

While we are at it, we also should consider consolidation of some of the prior split-offs. Putting all the golden geometric figures together (golden rectangles and triangles, and the golden ratios in pentagons, pentagrams (another math-mysticism mix), Penrose's tilings, and 5-fold symmetry generally) is one consolidation candidate.

Administratively, if a split is to be undertaken, perhaps a project page should be created. As a Golden ratio sub-page? Or maybe as an entirely separate project page? On this I defer to others who have project page experience. A project page would allow everyone to hammer out what the remaining Golden ratio article will be, and also how the new and old split-offs will wind up, without making a total public mess in article space while we are at it.

Finell (Talk) 21:57, 24 May 2007 (UTC)

I agree with Finell, in a preference to not split the article. Dicklyon 22:11, 24 May 2007 (UTC)

Bad digit

I tracked the wrong digit to this random act of vandalism. The next editor, User:Ricardo sandoval should have looked at the edit immediately before his, before doing an edit that masked it from easy discovery; but he didn't, so we missed it. Dicklyon 17:36, 8 June 2007 (UTC)

WRONG!!!!!!!!!!!!!!!!!!!!!!!!!

The golden ratio here is WRONG!!!

The golden ratio is actually

$\varphi ={\frac {16}{9}}\approx 1.77777\,777\,...$

--Onekopaka 04:27, 10 June 2007 (UTC)

Well, if by golden you mean causing people to spend lots of money on wide-screen TVs... —David Eppstein 04:46, 10 June 2007 (UTC)

Sounds like a good new article topic: golden rule ratio, 16:9, because he who has the gold makes the rules. Dicklyon 05:38, 10 June 2007 (UTC)

LOL! ≈ jossi ≈ (talk) 05:47, 10 June 2007 (UTC)

Alternate form - not math

Text says:

The formula $\varphi =1+1/\varphi$ can be expanded recursively to obtain a continued fraction for the golden ratio:

$\varphi =[1;1,1,1,\dots ]=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\cdots }}}}}}$

The intention of this is certainly correct, but it is not math, so I have no chance in the universe to understand it! A recursive formula is constructed like this (or similar):

(1): $\varphi _{1}=c$ or

(n): $\varphi _{n}=f(\varphi _{n-1})$

I can intuitively sense the formula (n) in the above b*lderd*sh (it is $\varphi _{n}=1+1/\varphi _{n-1}$), but I cannot sense (1). Said: Rursus ☺ ★ 11:59, 12 June 2007 (UTC)

You could start with c=1, in fact it does not matter what you start with as all starting values (with a few exceptions) converge to the same result. --Salix alba (talk) 12:45, 12 June 2007 (UTC)

Hmm, that's it, I think. I'm out for an image of the convergent series, and how it relates to itself. How it relates to itself, is described in the rest of the text, and $\varphi =1+1/\varphi$ is enough for me, but nothing in $\varphi =1+1/\varphi$ suggests 1.6180339887, while the series 1, 1/2, 1/4 clearly sums up to 2, because what's to remain before reaching 2 for adding a term t_n onto s_n, is clearly t_n. Said: Rursus ☺ ★ 14:32, 12 June 2007 (UTC)

A slightly nasty side comment: it is a common idea, outside mathematical and computer science cultures, that exact math is superfluous, and a hindrance to true understanding. Having math and CompSci, I reject this for the reason: understanding requires full logic down to a mechanical level – irrespective of the background of the pupil. Exact math is the sole way to succeed (but it's not a complete prerequisit, of course). Said: Rursus ☺ ★ 12:13, 12 June 2007 (UTC)

Consider this:

$\varphi =1+{\frac {1}{\varphi }}$

$\varphi =1+{\frac {1}{1+{\frac {1}{\varphi }}}}$

$\varphi =1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{\varphi }}}}}}$

...

I think that may reasonably be described as a recursive expansion, and it has the postulated limit. So what's the problem here? If there is a slight problem (involving the $\varphi$ at the "bottom" of each of these continued fractions), it can easily be fixed, but I don't think it needs fixing in a general encyclopedia.--Niels Ø (noe) 13:07, 12 June 2007 (UTC)

The problem is ... . It's not a variable. It cannot be computed, it cannot be reasoned about. Your series above is pretty OK, in fact it's much much better than using ..., for the exception of $\varphi =1+{\frac {1}{\varphi }}$, that should be solved for a starting $\varphi$ before recursing as per your series ... but solving it require us to solve the problem before solving it, and so the series explanation commits suicide by never finding a methodical start (my stated prerequisit for pedagogy)! Instead, the proposal of Salix alba is to be preferred: we simply assert that we start with any $\varphi$, but now, in order to explain anything, we have to provide an image (intuitive counterpart to proof) of that $\varphi _{n}$ converges, and then provide an image that 1.6180339887 is a likely end result. Otherwise the explanation explains nothing. Said: Rursus ☺ ★ 14:32, 12 June 2007 (UTC)

It's not a problem. The recursive expansion (substituting the original right-hand-side into where phi appears on the new right-hand-side) is an algebraic manipulation that can be done any number of times; it doesn't need any starting value or any approximation. ... Dicklyon 14:48, 12 June 2007 (UTC) ...

Ehmm, infinite recursion? The fact is that I understand what this is about. Infinite recursion doesn't exist, but might be successfully imagined, iff you already have a vast experience of formula manipulation. The trouble with this formula is not that it isn't imaginable, but that it expresses a recursion without stop value - imagine trying to span the natural numbers without a start value - it might work if we implicitly can guess a start value, but that actually requires us to know a start value of 1.6180339887 or $\varphi ={\frac {1+{\sqrt {5}}}{2}}$, which mixes an implicit algebraic method with a recursion/induction, and that makes a pedagogic mess, unless you already know all about it already (and why would you then read this article?). It would be preferrable to start from a fixed value, f.ex. 1, that is imaginable by size, and recurse outwards like:

(1): $\varphi _{1}=1$ or

(n): $\varphi _{n}=1+{\frac {1}{1+\varphi _{n-1}}}$

and from this proper (inside out) recursion derive the series (1, 2, 3/2, 5/3, 8/5, 13/8 ...), where the "image of dimension" isn't lost. (Some more thinking needed in order to get this work ... The current formulation is defect from an imaginative perspective). Said: Rursus ☺ ★ 15:45, 12 June 2007 (UTC)

The "..." represents any number of repetitions of the same pattern, without bound. Each of these infinite number of equations has the same exact sollution, which is the golden ratio. But by using the truncation method of continued fraction expansion, this form ALSO leads to the convergenets (which are ratios of fib nunmbers) as approximations. Dicklyon 14:48, 12 June 2007 (UTC)

I think I can see Rursus problem, its really a case of getting your head round Continued fraction#Infinite continued fractions

$\varphi =[1;1,1,1,\dots ]=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\cdots }}}}}}$

The continued fraction notation is tricky as it seems to require evaluating the full thing, including the ... However you can view φ as a limit of finite continued fractions

$\varphi _{1}=[1;]=1$

$\varphi _{2}=1+{\frac {1}{\varphi _{1}}}=[1;1]=1+{\frac {1}{1}}={\frac {2}{1}}$

$\varphi _{3}=1+{\frac {1}{\varphi _{2}}}=[1;1,1]=1+{\frac {1}{1+{\frac {1}{1}}}}={\frac {3}{2}}$

$\varphi _{4}=1+{\frac {1}{\varphi _{3}}}=[1;1,1,1]=1+{\frac {1}{1+{\frac {1}{1+1}}}}={\frac {5}{3}}$

Note how the numbers on the right hand side are sucessive pairs of fibonanchi numbers, which should convince your that the limit will be the golden ratio. Indeed if

$\varphi _{n}={\frac {F_{n+1}}{F_{n}}}$

then

$\varphi _{n+1}=1+{\frac {1}{\varphi _{n}}}=1+{\frac {F_{n}}{F_{n+1}}}={\frac {F_{n}+F_{n+1}}{F_{n+1}}}={\frac {F_{n+2}}{F_{n+1}}}.$

I think we are probably using the word recussion loosly here. We could possibly expand the article a bit, but it is too long at present. --Salix alba (talk) 16:50, 12 June 2007 (UTC)

I agree so far. Infinite continued fractions (of which I just had an unformulated intuitive idea) embraces the size of the series $\varphi _{n}$ and gives the student an image of the why:s and what:s of the golden ratio, I presume ... But I also agree that the article is too long, so I accept the removal of the "fact" protest template that I added, but I must stress that I believe there is a better, more pedagogical, and more compact way to write the Alternate forms section, and that method relies on proper math. The current section isn't good. Said: Rursus ☺ ★ 17:06, 12 June 2007 (UTC)

It's generally most constructive to find a ref, and then modify the text in a way that the ref supports. I just added a ref to the math concept of the infinite continued fraction expansion by recursive substitution, but it doesn't use the word "recursive", so if you'd like to edit the text to conform more to the language of the ref, that would make sense. Dicklyon 17:49, 12 June 2007 (UTC)

Sorry for reading sloppily: the intuitive image that I grumble about is in the section: Relationship to Fibonacci sequence – that is an Ancient-Greek-kind of figure that should be part of the proofs as much as possible, once again, IMHO. Said: Rursus ☺ ★ 17:29, 12 June 2007 (UTC)

I'm unclear on your point here. What figure are you referring to? Did the Greeks know about Fib numbers or the sequence of rational approximations to phi? I don't think so. Dicklyon 17:51, 12 June 2007 (UTC)