Talk:Golden ratio

Reference to "Zometoy" product

The reference to "Zometoy" is erroneous; the actual product is "Zometool". 2603:8080:4E05:7812:0:0:0:BA4 (talk) 12:44, 8 July 2023 (UTC)

"Zometoy" was Steve Baer's original (very low production) toy made of wooden dowels and drilled (?) nodes. Take a close look at the picture. Zometool was a later company by Marc Pelletier and Paul Hildebrandt which took the same concept and figured out how to make a better version of it out of injection molded plastic. For some about the history, see http://people.tamu.edu/~ergun/hyperseeing/2018/06/FASE/buehler2018.pdf –jacobolus (t) 15:38, 8 July 2023 (UTC)

Summing the Contained Fraction

It is probably quite, but not sufficiently, well known, that the continued fraction expression, Ø = 1+1/(1+ 1/(1+1/(1+1/(1+1/( ... infinitely, is easy to evaluate as a short algebraic expression. The part to the right of the gap has value Ø, so we can curtail the expression to Ø = 1+1/(1+Ø); multiply to a quadratic ... .

That may be worth mentioning. 94.30.84.71 (talk) 13:47, 7 November 2023 (UTC)

Or not worth mentioning. JRSpriggs (talk) 00:56, 8 November 2023 (UTC)

When the article says "The formula $\varphi =1+1/\varphi$ can be expanded recursively to obtain a continued fraction for the golden ratio" that is the converse of your statement. But notice that each step in going from one form to the other is reversible, so these are essentially the same idea. If you like you can write this explicitly as

$$\varphi =1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}\iff \varphi =1+{\frac {1}{\varphi }}.$$

(Under the assumption that $\varphi >0.$) I think the version in the article is fine though. –jacobolus (t) 01:01, 8 November 2023 (UTC)

Baffling diagram

I'm new to the golden ratio. Got onto it via the tv show Astrid. The main Wikipedia page on the subject golden ratio has a diagram of a rectangle being divided into a square and the ratio is supposed to be illustrated, but the text below the diagram is baffling to me. DMTerp (talk) 04:28, 21 December 2023 (UTC)

@DMTerp The diagram says:

A golden rectangle with long side a and short side b (shaded red, right) and a square with sides of length a (shaded blue, left) combine to form a similar golden rectangle with long side a + b and short side a. This illustrates the relationship a + b/a = a/b = φ.

Which part do you find baffling? The setup is that the rectangle with sides a and b is similar to (has the same aspect ratio as) the rectangle with sides a + b and a. If this is the case, then the ratio of a to b is the golden ratio. –jacobolus (t) 08:21, 21 December 2023 (UTC)

There is too much waffle between "A golden rectangle" and "(shaded red)". For those of us familiar with the topic, it's obvious. But the diagram would baffle a lot of people. Unfortunately, that's rather the point of an encyclopedic article. Johnuniq (talk) 09:26, 21 December 2023 (UTC)

I don't understand what you mean. What you call "too much waffle" is just directly describing which rectangle we're talking about: the one with sides a and b, which is shaded red and toward the right, as compared afterward with the rectangle of sides a + b and a. I don't think this can be meaningfully simplified, and if someone can't follow this caption they probably aren't part of the intended roughly middle school level or above audience. –jacobolus (t) 00:13, 22 December 2023 (UTC)

I see the problem I was having. The adjacent text reads “a golden rectangle . .

may be cut into a square and a smaller rectangle . .” Then I looked at the diagram and read its smaller font text and didn’t realize that it was the same explanation backwards. I’m 86 now and it seems to take me longer to “ get it”. 107.200.90.207 (talk) 17:26, 21 December 2023 (UTC)

We could swap the order in the caption as well. I don't personally think it makes any significant improvement in clarity, but it also doesn't hurt anything. It could instead say something like:

A golden rectangle with long side a + b and short side a can be cut into two pieces: a similar golden rectangle with long side a and short side b (shaded red, right) and a square with sides of length a (shaded blue, left). This illustrates the relationship a + b/a = a/b = φ.

–jacobolus (t) 00:16, 22 December 2023 (UTC)

Short description

There appears to be an idea that the "short description" is strictly limited by character count, for something to do with the technologically handicapped who can't see a full screen. In which case I can't get too worked up about it, but fwiw... I don't think a proper description could be shorter than the first sentence of the lead. The "symbolic version" ((a + b) : a :: a : b) is not very transparent to the non-mathematical, but any very condensed prose version is unlikely to be understood by anyone who couldn't understand the symbolic version. I suggest another possibility, which is something like "A ratio (approx. 1.618) which has been ascribed mystic properties." The point is that the purpose of the short description has to be (I think) to confirm to those who have heard of it that it is indeed what they are thinking of, or to hint to those who haven't why this article exists; the purpose is not to give a mathematical definition of its value.

Aside: a lot of good work has been done on these number articles, but I cannot help feeling that a problem is that while there is more than enough mathematical expertise (which is needed of course) there is a slight shortage of sensitivity to the English language. Imaginatorium (talk) 07:54, 25 December 2023 (UTC)

The "mystic properties" part is bullshit though (as the article takes pains to explain). Any such "mystic properties" are not really of special mathematical or cultural significance.

The most important and characteristic feature of this ratio is that it is the ratio of the diagonal to the side of the regular pentagon, which is why it features prominently everywhere that 5-fold symmetry appears. Because of the strong law of small numbers (in this case meaning small algebraic numbers based on a polynomial of low degree with small integer coefficients), it also appears in other places, e.g. the solution to various combinatorial problems, which don't at first glance have a direct relation to 5-fold symmetry (but can usually be explained/interpreted that way with some effort). –jacobolus (t) 08:06, 25 December 2023 (UTC)

We really shouldn't be trying to find short descriptions that take mathematical understanding to decode or that point to the most salient properties of the topic. That is not what short descriptions are for. They're mainly for things like: in mobile, you search for something, and you get multiple results that match your search. Which one do you want to read? So they should be short, and they should disambiguate the topic, without just repeating the title, but they are not intended to be a rigorous and completely unambiguous description of the topic. I think "Number, approximately 1.618" is better for this than either trying to spell out the extreme and mean ratio or trying to describe some geometric property that fits the golden ratio. —David Eppstein (talk) 08:13, 25 December 2023 (UTC)

Correct. I just used the Wikipedia app on a phone to search for "golden". It showed "Golden ratio / Number, approximately 1.618" with the image from the infobox. That is perfect for finding the correct article to read. The short description is for disambiguation. It is not a Google snippet or an alternative to reading the article. Johnuniq (talk) 08:30, 25 December 2023 (UTC)

I think "Number, approximately 1.618" is good enough for the purposes of a short description. (I also think that "short description" might not be the best short description of what a short description is supposed to do.) XOR'easter (talk) 16:20, 25 December 2023 (UTC)

Semi-protected edit request on 8 January 2024

In "Golden ratio conjugate and powers" equation ending -0.618033 needs to have an ellipsis added -0.618033... 2605:A601:A962:AC00:549:D9C7:F671:4843 (talk) 04:24, 8 January 2024 (UTC)

I did not found any such equation without an ellipsis. Possibly, a horizontal scroll was needed for seeing the ellipsis. D.Lazard (talk) 09:13, 8 January 2024 (UTC)

Semi-protected edit request on 25 January 2024

Pleas add the hexadecimal form to the infobox. More specifically, add |hexadecimal=1.9E37 79B9 7F4A 7C15..., which will fit well and produce

Hexadecimal 1.9E37 79B9 7F4A 7C15...

The digits are the result of a routine WP:CALCulation with the dc commands 16 o 50 k 1 5 v + 2 / p, which produces 1.9E3779B97F4A7C15F39CC0605CEDC8341082276BF2 (the last digit isn't trustworthy). MOS:HEX appears to prefer upper-case letters A–F.

The hexadecimal form is useful in software development because it's used by various hashing functions as the "most irrational" number. See RC5#Key_expansion, [1][2][3][4]

Providing 64 bits after the decimal place helpfully matches the largest common numeric data type. This is one more digit than is provided in decimal, compensated by saving one space due to grouping the digits in fours rather than threes. 97.102.205.224 (talk) 03:01, 25 January 2024 (UTC)

I'm looking at the articles in which the infobox is also used: Square root of 2, Apéry's constant, Square root of 3, Square root of 5, Lieb's square ice constant, etc and it seems the hexadecimal is not provided on any of these entries. – The Grid (talk) 14:31, 25 January 2024 (UTC)

It was recently removed from a bunch of these (including this article at special:diff/1190623333), because it is not considered important enough to focus attention on. There was some meta discussion at WT:WPM (2023 Dec) § Mathematical constant infobox. –jacobolus (t) 15:53, 25 January 2024 (UTC)

97.102.205.224: If you are putting this in software, you should probably write it in as (1 + sqrt(5))/2, which is much more legible than a string of hexadecimal digits and should be correctly rounded if you can trust sqrt. In most contexts your programming language will be smart enough to just do this computation once (e.g. at compile time). If you are concerned with fibonacci hashing, the appropriate place to include a hexadecimal string is there, not in the infobox here. –jacobolus (t) 16:03, 25 January 2024 (UTC)

@Jacobolus and The Grid: Er, except that the computation you suggest will generally be done in IEEE double precision (1+52 bits of mantissa) rather than in 64-bit integer math. There's a reason I did my computation in an arbitrary-precision math package.

The value is used is numerous integer arithmetic contexts (see the four links provided in the original request, or do your own web searches for "9E3779B9" and "61C88647") where it's implicitly divided by 2³² or 2⁶⁴. And such applications often require, for compatibility, a bit-exact value; rounding error is not acceptable. A hex literal plus a comment is an easier way to get the exact value desired. (This is also the reason that Hexadecimal#Hexadecimal exponential notation was added to C99, IEEE 754-2008, and POSIX.)

(Tangent: In general, adding an exact integer to an already-rounded square root risks double rounding if the addition increases the exponent and shifts lsbits off the mantissa. For φ and binary floating-point specifically, this will not happen because 2 < √5 < 1+√5 < 4, so both have the same exponent and no such shift will take place. Division by 2 is an exponent adjustment with no additional rounding.)

One simple application is low-discrepancy sequences. It turns out[5] that the additive sequence k×i mod 1, for i = 1, 2, 3, ... achieves the lowest possible discrepancy (most uniform possible distribution on the unit interval) if k = φ. This can, and often is, done in integer arithmetic by scaling by 2³² and taking advantage of the automatic modulo-2³² operation of integer arithmetic.

This is the basis of Hash function#Fibonacci hashing.

However, you have to be sure to round to an odd value when converting to integer form (so that the multiplication by k is invertible modulo-2³²; see Weyl sequence), an operation which is not easily achieved in a compile-time computable expression. If you don't allow for this, you might get φ⁻¹ = 0x0.9E37 79B9 7F4A 7C15 F... rounded to ...7C16, which wouldn't do at all. And if you are using 64-bit words, not even IEEE double will provide enough precision.

This property also makes it a good multiplier for hashing purposes. (TAOCP vol. 3 2nd ed. pp. 517–518 & Ex. 9 p. 550).

The binary form of this particular value does come up surprisingly often, which is why I thought it worth including. Ultimately it's an m:Inclusionism judgement. The nice thing about an infobox is that it's easy to skim and ignore irrelevant details; you're not reading it linearly like main article prose. I do note that Special:Search/0x9E3779B9 already shows seven existing appearances in Wikipedia (plus one I just added to Fibonacci hashing). And Knuth judges it useful enough to include a table of the binary (octal, actually) forms of numerous mathematical constants in TAOCP (vol. 3 2nd ed. pp. 748 et seq.).

(The linked debates as to whether mathematical constants should even have infoboxes is a larger issue I prefer not to get dragged into. My edit request is assuming an infobox exists. If people would like a larger edit request, I could rework the above application examples into a new subsection of § Applications and observations, as I see it's not mentioned at present. But to do a good job would be a wider-ranging edit; e.g. the name "most irrational number" is best mentioned in § Continued fraction and square root near the discussion of the Hurwitz inequality.)

97.102.205.224 (talk) 20:11, 25 January 2024 (UTC)

such applications often require, for compatibility, a bit-exact value; rounding error is not acceptable – if you have a "reliable source" claiming this, that would be a good argument for including it as part of the section Hash function § Fibonacci hashing. Some of the people chatting at the stack exchange links you posted earlier claim that the precise constant is largely irrelevant as long as it is sufficiently mixed up.

Low-discrepancy sequences like your link are calculated in floating point and are not sensitive to slight roundoff error in the 16th decimal place. I liked that blog post, tried to promote it, recommended it to many people, and corresponded with the author, but it's not a reliable source by Wikipedia standards. If you can find peer-reviewed sources about that, it would perhaps be worth adding a new subsection to low discrepancy sequence § Construction of low-discrepancy sequences (edit: it's discussed at Low-discrepancy sequence § Additive recurrence, though would benefit from a source other than a blog post). The n-dimensional generalization is out of scope for this article, but the 1-dimensional version based on the golden ratio is discussed at Golden ratio § Golden angle and Golden angle.

The nice thing about an infobox is that it's easy to skim – the bad thing about an infobox is that it's a magnet for heaps of marginally relevant trivia. –jacobolus (t) 20:26, 25 January 2024 (UTC)

@Jacobolus: Er, yes, for strictly internal hashing, the exact constant is not too critical, although there are arguments (Knuth has the most thorough treatment) that 1/φ or its negative are the best values. There have been some notable failures due to oversimplified constants chosen to be easier to multiply by. (ISTR this happened in the Linux kernel history... aha, see https://git.kernel.org/pub/scm/linux/kernel/git/torvalds/linux.git/commit/include/linux/hash.h?id=689de1d6ca95b3b5bd8ee446863bf81a4883ea25 )

However, some hashing is part of an externally-visible interface, e.g. the symbol table hash in Executable and Linkable Format or some serialization formats. The former don't happen to use φ internally, but there exist rigidly-defined hash functions using φ which could be used in an external-facing application; Boost (C++ libraries)'s hash_combine() function comes to mind, but I don't have a specific application example where that hash value is exported. It's certainly plausible that one exists.

(The crypto applications of course require bit-exact values, but they're random-looking nothing-up-my-sleeve numbers rather than caring about the numeric value.)

A semi-crypto application is the Java SplitMix64 PRNG. Is uses a Weyl sequence with 2⁶⁴/φ as the increment, and officially supports portable deterministic-seeded applications, where the same seed is expected to produce the same output on different systems.

One thing that's annoying about the somewhat deliberate pace of Wikipedia discussions is that I forget things between rounds. I know this whole thing started when I came to this page expecting to find the hex value for some reason, but I've since forgotten what it was!

I will claim that the scaled integer form of φ is used in computer software more than any other irrational mathematical constant. In floating point, of course, π and 2π win hands-down.

(Trivia found as part of my research for this discussion: https://www.guinnessworldrecords.com/world-records/100485-most-irrational-number )

97.102.205.224 (talk) 22:48, 25 January 2024 (UTC)

Fair enough, but if a bit-exact value is needed and it has to be rounded to an error in the last place because of extra constraints, then someone who needs to know this should be finding it in a specification, not copying it out of a loosely related Wikipedia article. Indeed, the latter is certain to cause an error in this instance! It seems like a decent argument for adding a more specific section about the application to hashing though, and perhaps including 1 or 2 of these hexadecimal values there. –jacobolus (t) 22:55, 25 January 2024 (UTC)

@Jacobolus: As I mentioned, I could adapt the preceding discussion into a whole new subsection (two, actually) under the applications section. It would, however, be a lot more work to cut and paste in. Also, I'd very much like to introduce the phrase "most irrational number", as it combines a fairly lay-accessible concept with a mathematically interesting property.

The issue is already referred to in § Continued fraction and square root and § Golden angle, but I'm not sure if I should expand one or the other, or pull some of it out to a separate section. Since it would come up again in any discussion of Fibonacci hashing, a separate section seems appropriate, but moving text around at much makes posing an edit-request diff to a talk page a real PITA.

Aha! Special:Diff will accept revision IDs for two separate articles! The syntax is "Special:Diff/1181263486/1194109625" (not linked because that's not a useful example). So I can come up with something in the Draft namespace. There don't appear to be any good patch/merge tools in Wikipedia, but at least the common ancestor would be clearly labelled.

Given that, do you have any suggestions for organization? I'm inclined to introduce the phrase, without references, in a summary in the lead section, and then fill in the details in the later sections, with § Continued fraction and square root containing the formal details, while § Golden angle, (not yet written) § Fibonacci hashing and § Low-discrepancy sequence will describe themselves as applications of the principle.

The big question is, should I go for it? (If you say yes, you're volunteering to be nagged to review it when it's finished.) 97.102.205.224 (talk) 00:39, 26 January 2024 (UTC)

I don't really like the "most irrational number" label, since I think it easily leads to misconceptions about what it really means. As you noticed, the article already says "The consistently small terms in its continued fraction explain why the approximants converge so slowly." This could be elaborated but is precise and not misleading, and doesn't overhype the observation. (A different way to look at the same observation is to notice that any rational number is extremely hard to approximate by rational numbers other than itself; the golden ratio is the irrational number closest to sharing this property, so in this sense it is the "least irrational number"!) @David Eppstein what do you think? –jacobolus (t) 00:45, 26 January 2024 (UTC)

As for the annoyance of making edit requests: sorry about the semi-protected status of this article. It got that way because this otherwise is a magnet for vandals and cranks. I made a page at Talk:Golden ratio/sandbox that you are welcome to use for whatever chunks of draft text you like: copy whole sections there, move them around, rewrite them, etc. Or you could also consider making an account; once you have had it for 4 days and made 10 edits, you can freely edit semi-protected pages, make new pages, etc. –jacobolus (t) 01:14, 26 January 2024 (UTC)

I think "most irrational" is misleading. Many people would think of it as some kind of qualitative distinction; that transcendental numbers are more irrational than algebraic numbers and that somehow this is the most transcendental among the transcendental numbers (obviously untrue). And "hardest to approximate" is also misleading, because there is no computational difficulty in approximating it. Instead I would prefer phrasing like "least accurately approximated by rationals". —David Eppstein (talk) 01:47, 26 January 2024 (UTC)

H'm... I agree that just by itself, the moniker "most irrational number" is easy to misinterpret (as David Eppstein has noted), but with an appropriate explanation (which obviously this article would have) it's always been a useful mnemonic for me. I think jacobolus's example comparing the best rational approximation of an irrational number with the second-best approximation of a rational number is starting out biased. Saying that "exact isn't an approximation" is basically arguing that "zero isn't a number", and I thought we'd put that to bed at least a sesquimillenium ago.

Even if you think it's misleading, it is a widely-used phrase, and should be addressed for that reason alone. I'll definitely keep the phrase "least accurately approximated by rationals" in mind!

Anyway, it's the wee hours here and I'm for bed. I'll have to pause my side of this discussion for a while. 97.102.205.224 (talk) 02:29, 26 January 2024 (UTC)

It's unfortunate that the more accurate phrasing is also significantly less catchy. —David Eppstein (talk) 07:20, 26 January 2024 (UTC)

What I'm thinking about is not "second best approximation", but instead something like: if you start plotting rational numbers, there is something like a "hole" around every integer where no other rational numbers can fit until they start to have very big denominators; to a lesser extent there is a similar "hole" around every rational number; the simpler the number, the bigger the hole (Ford circles give one visual explanation for this phenomenon). The number which creates the next biggest kind of hole around itself, besides integers and rational numbers, is the golden ratio. The denominators of rational approximations to the golden ratio at any particular level of approximation grow more quickly than for any other irrational number, while still growing less quickly than for the "second best" approximation to any rational number. So in a certain sense the golden ratio is balanced on the edge between "rational" and "irrational", just on the irrational side. This is why it might be in a certain sense called the "least irrational". A related idea: the golden ratio is the algebraic irrational number with by some definition the simplest minimal polynomial; it can't get any simpler without being rational. –jacobolus (t) 08:07, 26 January 2024 (UTC)

 Not done for now: please establish a consensus for this alteration before using the {{Edit semi-protected}} template. Clearly this is not an uncontroversial edit. PianoDan (talk) 18:22, 25 January 2024 (UTC)

@PianoDan: Yes, clearly. Didn't expect that, but it appears to be a good discussion. 97.102.205.224 (talk) 20:11, 25 January 2024 (UTC)

Does anyone else have thoughts about this? –jacobolus (t) 23:01, 25 January 2024 (UTC)

Yes, I don't think this sort of thing belongs. WP is supposed to be an encyclopedia, not a geeks' handbook; if we put in a hex approximation, why not add octal and binary? This is all stuff that can be calculated easily if required, and no sensible programmer would rely on a value in WP anyway. Imaginatorium (talk) 03:56, 26 January 2024 (UTC)

Negative Reciprocal of Golden Ratio

The golden ratio can be calculated from "(1+√5)/2." Using the positive value for √5 gives 1.6180. Using the negative value for √5 gives the negative of the reciprocal of 1.6180, -0.6180. Is this just a coincidence? Does this have any significance? 2601:18E:C700:4B7F:9592:6C05:1FCC:C6FA (talk) 16:52, 20 February 2024 (UTC)

It is not a coincidence. These are the roots of a quadratic equation. The sum of the roots is 1 which is the negative of the linear term's coefficient. The product of the roots is −1 which is the constant term. So the equation is x² −x −1 = 0. Just what we need. JRSpriggs (talk) 02:19, 21 February 2024 (UTC)

To put it another way, the golden number is the unique number with this "coincidence"; that is what makes it special. Kinda like asking whether it is a coincidence that the Equator is the one latitude where the duration of daylight never varies. —Tamfang (talk) 17:16, 27 March 2024 (UTC)

Simple Construction of a Golden Rectangle using two 2x1 Rectangles:

Constructing a golden rectangle using two 2x1 rectangles. Width = 2, Height = 1 +√5.

Align the diagonal (√5) of one of these rectangles (A) with the short side of the other rectangle (B) such that the sum of these two elements is now = 1 + √5. This line together with the long side of rectangle (B) form two sides of a Golden Rectangle.

This practical method was developed at a local MenzShed for use by woodworkers with no maths.

We have a diagram which we can't seem to upload. HoneAtHome (talk) 01:55, 14 March 2024 (UTC)

It's not new, but I agree a diagram would be appropriate. —Tamfang (talk) 16:32, 27 March 2024 (UTC)

This is roughly the same idea as this diagram from the article, except this one uses a compass to draw a circle instead of turning the whole rectangle:

–jacobolus (t) 17:28, 27 March 2024 (UTC)