Talk:Metallic mean

Lead too long?
I know that the lead is only two paragraphs, but they're somewhat long, and the article only has one content section, so maybe some of the content in the lead should be moved to the article's body. Care to differ or discuss with me? The Nth User 02:36, 28 August 2018 (UTC)

Powers of the Metallic Means and the Pascal Triangle
In equations describing the powers of the metallic means each term in the chain has a numerical coefficient and is also raised to a power of 10. It turns out that the particular coefficients are identical to terms in a (2,1)-sided generalized Pascal Triangle. In the classical Pascal Triangle terms along so-called shallow diagonals sum to give Fibonacci numbers. The same types of terms along shallow diagonals in the (2,1)-sided Pascal Triangle sum on one side to give Lucas numbers rather than Fibonacci, and Fibonacci numbers on the other (though upshifted one move). The shallow diagonal terms leading to Lucas numbers here in the (2,1) system are identical to the numerical coefficients in the terms of the equations describing the powers of the metallic means. Furthermore, in the classical Pascal Triangle the usually discussed side-parallel diagonals are associated with dimensional values. The outer 1's are 0-D, the natural numbers are 1-D, the triangulars 2-D, the tetrahedrals 3-D and so on. Equivalent dimensional labels apply to the side-parallel diagonals in the (2,1)-sided generalized Pascal Triangle as well. Interestingly, the dimensional labels belonging to the side-parallel diagonals that each numerical coefficient terms are found in are identical to the powers that each term is raised to in the equations describing the powers of the Metallic Means. 2601:89:C600:E521:6500:3D92:ABF6:9690 (talk) 21:50, 9 April 2019 (UTC)

How do you have a metallic mean between 2 numbers!?
the article states that the golden ratio is the metallic mean of 1 and 2, but both the continued fraction and the formula only have one input, n. where does this other number come from?

You've misquoted. The article says that the golden ratio is the metallic mean BETWEEN 1 and 2, not OF 1 and 2. So there is no problem. 2601:89:C701:9190:9089:70B3:33A7:F85D (talk) 17:26, 17 August 2021 (UTC)

names for each ratio
While naming the first three ratios "golden", "silver" and "bronze" is well-established or at least suggests itself, "copper" and "nickel" seem to be coined in Spinadel's "The Family of Metallic Means", and other uses of these names reference back to that paper. After that, however, I couldn't find any other name for further ratios - other than those recently added here and in my opinion they range from "kind of make sense" to "that's not even a metal". So does anyone have a source for a naming convention or with some more names to add? If not, should we come up with reasonable names for the ratios written out on the page (for N = 6 through 9) or just leave those spaces blank? 2001:A62:409:AB01:41FC:D226:8A49:4ED5 (talk) 20:56, 4 January 2022 (UTC)
 * Metals for ratios 6 to 9 were added by one-edit IPs without citation. None of the online sources use these names, though I'm unable to check Petrovich.  I've boldly removed them.  Certes (talk) 12:54, 11 March 2022 (UTC)
 * my deep apologies, it seems i was part of the miss-information, when i saw the edit that named the 9's metallic mean as waluigi i went from my memory and looked up the names from here(https://rosettacode.org/wiki/Metallic_ratios) to try and fix the mess, which for some reason i though where here before the waluigi vandal
 * that is where the 6-9(Aluminum, Iron, Tin and Lead) and 0(Platinum) names came from(i've removed platinum)... but this mistake has lead me to look a bit deeper and... it turns out that copper and nickel are the names of two different ratios than the ones expressed here, as a careful look at(https://www.mi.sanu.ac.rs/vismath/spinadel/index.html) reveals
 * the copper mean is given by the equation "(x^2)-x-2=0" which equals 2, and the nickel mean is (1+root(13))/2 as given by the equation "(x^2)-x-3=0"(https://oeis.org/A209927) and so i have removed copper and nickel as well
 * i've also changed this template to link copper with 2 and put the "etc" on its own bullet, out of the fact that i can't confidently remove it
 * a pity that i have erred so, but at least in the end i helped mend something that i didn't even know was wrong StarButterflyIsCute (talk) 12:21, 12 March 2022 (UTC)

Removal of maintenance tags
An IP user is starting an edit war for removing maintenance tag without addressing the issue. More specifically: So, the IP user did not addressed any of the issues motivating the maintenance tags, and I'll revert it again. D.Lazard (talk) 17:52, 8 April 2024 (UTC)
 * They changed into, without specifying whether "these" refers to the properties of the preceding section or to the "following properties".
 * They changed into . Again, the reader cannot know which property is referred to. The confusion between "silver means" and "metallic mean" is fixed here, but remains at several place below in the section.

A way to prove the identity e^arsinh(1/2 x)
Easy to show by solving for the inverse of (x + sqrt(x^2 + 4))/2 which yields x - 1/x. Rewriting: e^(ln x) - e^(-ln(x)), but e^t - e^(-t) is 2sinh(t) by a definition of sinh (easily found through e^(ix) and its variations), so: 2sinh(ln(x)) = y. Solving for x, we get 2 sinh(ln x) = y -> sinh(ln x) = 1/2 y -> ln x = arsinh(1/2 y) -> x = e^arsinh(1/2 y). Andrew T Porter (talk) 15:17, 12 April 2024 (UTC)

n=1
One could name 1 'Roentgenium ratio'.https://mathstodon.xyz/@rjf_berger/109830172130605119 2A04:DF80:44:4501:904B:2A91:2882:6E13 (talk) 08:01, 12 May 2024 (UTC)