Talk:Golden ratio

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The Theorem of the Golden Mean[edit]

Add a reference to Thean'so "The Theorem of the Golden Mean" in the Pythagorean school? it would be useful to show works on the golden mean at that time. — Preceding unsigned comment added by (talk) 10:06, 20 August 2014 (UTC)

Egyptian Pyramids[edit]

"Adding fuel to controversy over the architectural authorship of the Great Pyramid, Eric Temple Bell, mathematician and historian, claimed in 1950 that Egyptian mathematics would not have supported the ability to calculate the slant height of the pyramids, or the ratio to the height, except in the case of the 3:4:5 pyramid, since the 3:4:5 triangle was the only right triangle known to the Egyptians and they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ."

"the 3:4:5 triangle was the only right triangle known to the [ancient] Egyptians" is unverifiable and illogical. If an individual can create two measured lines (sharing an endpoint) and make them perpendicular to one another, then that person can construct any right triangle.

"they did not know the Pythagorean theorem, nor any way to reason about irrationals such as π or φ." is unverifiable.

AnonymousAuthority (talk) 05:51, 15 November 2013 (UTC)

Take it up with Eric Temple Bell's ghost. We are merely reporting what he said. —David Eppstein (talk) 06:05, 15 November 2013 (UTC)
The word "since" divides what he claimed from what is implied as some known facts. The sentence either needs to be rephrased to clarify that it is not a known fact, or a direct quote needs to be provided. AnonymousAuthority (talk) 20:07, 16 November 2013 (UTC)

That the Ancient Egyptians did know about irrationals, and that they did base their entire number system and way of doing mathematics on them, as grounding principles is outlined in detail in R A Schwaller de Lubicz's The Temple of Man - 2 Vols - Trans. Lawlor & Lawlor - ISBN 978-0892815708. Your article should take the timeline back far enough to include this information. (talk) 21:18, 17 January 2014 (UTC)

This article is neither about irrational numbers in general nor about fringe theories of Egyptian mathematics published by woo astrology/tarot publishers. That material does not belong here. —David Eppstein (talk) 21:56, 17 January 2014 (UTC)

Do you need pi in order to know about the Golden Section?? I don't think so. The Timeline section of this article arguably must start with the Egyptians, in any case, as it has been fairly well established that the proportional ratio is <ital>present</ital>in the pyramids, regardless of any controversy about their mathematical prowess. see for instance; see notwithstanding the poor English, it is written by a Serbian engineer with decent credentials; see also, and, excerpts here: "The Egyptians thought that the golden ratio was sacred. Therefore, it was very important in their religion. They used the golden ratio when building temples and places for the dead. If the proportions of their buildings weren't according to the golden ratio, the deceased might not make it to the afterlife or the temple would not be pleasing to the gods. As well, the Egyptians found the golden ratio to be pleasing to the eye. They used it in their system of writing and in the arrangement of their temples. The Egyptians were aware that they were using the golden ratio, but they called it the "sacred ratio."" Cesca1910 (talk) 09:13, 3 July 2014 (UTC)

not well fleshed-out[edit]

I understand the rejection of my edit, especially given the general style of mathematical articles. First though - are there any other such numbers, whose inverted value plus one equals the value before inversion ? I'm not a mathematician, but I have though read (and used) more maths than most other people, including at university level. Although it was some years ago now. I just find the math-related articles perhaps could be a bit easier to read, and sometimes give more "hard" examples of complicated formulas. I don't believe my rejected edit helped very much there, but I though it might get some reader more interested. (and if so, necessaraly early in the article) However I agree it could be better fleshed-out. Boeing720 (talk) 23:25, 25 September 2014 (UTC)

There is no prohibition on the lead including material covered later on, but I don't think your addition belonged for two reasons. First it was not encyclopaedic. It is not for us to say what we find interesting, or to comment on facts. We should simply state the facts and let them speak for themselves. There is nothing to stop you making them interesting, with good writing, presentation, layout. In fact that's of the main things distinguishes our best articles from the merely good ones. But you should not say so. Second the lead section is the place to say what something is (so a definition), say why it's important (so put it in context), and say how it is used/what it relates to. It's not the place for more detailed properties and examples, which can go in their own section and expand on the definition. The lead is the wrong place for this.--JohnBlackburnewordsdeeds 23:48, 25 September 2014 (UTC)
(ec) But this is exactly the content of the first paragraph of the calculation section, is it not? That 1+1/\phi=\phi is just a quadratic equation. This has two roots: one is the golden ratio \phi and the other is -1/\phi. So there are exactly two numbers with that property. Sławomir Biały (talk) 23:54, 25 September 2014 (UTC)
Thanks ! Boeing720 (talk) 20:34, 26 September 2014 (UTC)

Section on Irrationality[edit]

The picture and caption under "irrationality" does not prove that the golden ratio is irrational. It states that: "If φ were rational, then it would be the ratio of sides of a rectangle with integer sides. But it is also a ratio of sides, which are also integers, of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely, so φ cannot be rational."

But I fail to see how the fact that the side lengths stop being integers implies irrationality. — Preceding unsigned comment added by (talkcontribs) 11:13, 23 November 2014

Because it's not possible to have an infinite descending sequence of positive integers. If the first number in a sequence is n, then any descending sequence starting from that number can run at most n steps. —David Eppstein (talk) 18:57, 23 November 2014 (UTC)
I think the OP has a point here - that simply means that the sides won't be integers; it doesn't entail that their ratio cannot be rational (i.e. that the ratio cannot be expressed as a ratio of integers). The proof seems to be missing something. -- Scray (talk) 19:23, 23 November 2014 (UTC)
Huh? Read the first line the OP quoted: if the ratio were rational, there would exist a rectangle with integer sides. For instance, one side can be the numerator and the other the denominator of this supposed rational number. And if this rectangle existed, there would exist a smaller rectangle that still had integer sides. And so on into an impossible infinite descent. There is nothing missing from a careful reading of the text that's already there. —David Eppstein (talk) 20:22, 23 November 2014 (UTC)
I'm sure I'm missing something, so let me explain where I'm coming from and you can tell me what I'm missing (and note that I'm not arguing that φ is rational - I'm just asking if the proof is compelling): let's take a classic 3-4-5 (side length) right triangle. The ratio of any pair of side lengths is rational. If one takes similar triangles of progressively smaller area, their side lengths are not integers but their side length ratios remain rational (and constant, of course). That the side length of the triangle (or in the case of this proof, the rectangle) are not integers does not mean the ratio is not rational. I think the proof lacks evidence that the side length ratios cannot remain rational, which has to do with the way the side lengths shrink (rather than their resepective absolute lengths). What am I missing? -- Scray (talk) 21:08, 23 November 2014 (UTC)
What you are missing is that in this case the side lengths remain integers, as the caption already states — see the phrase "which are also integers" in the quote given by the OP. Specifically, when you delete a square, one of the sides of the new rectangle is also a side of the old rectangle (an integer) and the other side is the difference of the old rectangle's sides (a difference of two integers is always an integer). —David Eppstein (talk) 22:11, 23 November 2014 (UTC)
Ok, thanks. -- Scray (talk) 22:29, 23 November 2014 (UTC)
Ah, I see. What I was missing was the "which are also integers" bit. The placement of that clause in the sentence is a bit awkward and breaks up the sentence in an unexpected way that threw me off. But you're right--the sentence as written is perfectly correct and a careful reading reveals no problem in the proof..

First paragraph needs to be improved and simplified[edit]

I speak various languages and I find the English version unnecessarily vague in the first paragraph exactly where it shouldn't be. For Wikipedia, I consider the current first-paragraph delivery a disservice.

In the introduction ordinary persons capable of thinking for themselves need to find out in one paragraph what the golden ratio is. Right now, that relatively simple task is way off mark; it is written by specialists and they are using specialists' language only. For an introduction on something this beautiful - that is bad.

I checked the Dutch site on the same matter and found they do have what it takes:

Here is the translation made by Google that I improved some (but not to perfection): "The golden ratio, also called the division in extreme and mean ratio, is the division of a line into two parts that shows a special relationship. To find the golden dissection (ratio/cut), the comparison of the largest of the two parts to the smallest is identical to the comparison of the whole to the largest segment. When indicating the largest segment as a and the smallest part as b, then the ratio of the two is such that a: b = (a + b): a."

In my words: the relationship that the larger segment has to the whole is identical in measurement (but not in position) to the measurement the smaller has compared to the larger segment.

Thank you for whoever is guarding this page. Please, will you make it a more wiki-worthy page in the first most vital paragraph? It's a rule of thumb that all interested readers should be able to index the idea right then and there.

P.S. I also found an image on the Dutch wiki page that is insightful:

The image shows that when creating a circular line from location A, and -next- creating a circular line from C, starting at this position D, that we then find E. And E delivers the golden ratio on the line B - C. It is lovely. Maybe you'll also use it (though not in the first paragraph)?

Looking at the current lead, I presume the problem is that you see the word "ratio" as to specialist compared to the more nonspecific or vague words "relationship" and "comparison". Is that it? The Dutch "verhouden" or "verhoudt" for "relationship" might equally be translated as "ratio", perhaps, since Google translate shows English "ratio" going to Dutch "verhouding". In that respect our lead is sort of like the Dutch one, except a bit more abstract in focusing on "quantities" rather than lengths of line segments. The Dutch title "Gulden snede" is more about the "cut" than the "ratio", so maybe that's the relevant difference? What would you suggest? Dicklyon (talk) 05:35, 10 March 2015 (UTC)