Talk:Silver ratio

Copyvio
This page was an almost exact copy of the Silver Ratio info at http://encyclopedia.thefreedictionary.com/Silver+ratio and so I have deleted the entire article with the intent of rewriting it. -Voltaire 21:49, Apr 24, 2005 (UTC)

Well I at least tried to delete it, but it got put back on. -Voltaire 22:18, Apr 24, 2005 (UTC)

It is worth noting that at the bottom of http://encyclopedia.thefreedictionary.com/Silver+ratio it states:
 * This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.

So the fact that the articles are the same is not a problem! Andreww 04:35, 4 Jun 2005 (UTC)

I thought the silver ratio was the number


 * $$ \sqrt{3} + 1 = 2 + \frac{2}{2 + \frac{2}{2 + \frac{2}{2+ ...}}}\,$$

which is the limit of the ratio of the terms of the sequence


 * $$ G_n = 2G_{n-1} + 2 G_{n-2}\,$$

or am I mistaken?Scythe33 15:19, 11 Jun 2005 (UTC)

Serious errors
Hello. I do not know what actually is the definition of silver ratio, but

1.414... = Sqrt[2]

and

2.414... = 1 + Sqrt[2]

Thus NOT

1.414... = 1 + Sqrt[2] Wrong!

So, although I do not know the accurate answer, neither does this article give any such.

As a practical note, a rectangle with proportions 1:2.41 is quite elongated. So, if you asked me, I would guess that the answer is 1:1.41. However, as I've already noted, I don't know. 80.221.61.8 16:26, 13 April 2006 (UTC)

According to Mathworld, the silver ratio is defined as the sum of $$sqrt(2)$$ and 1. The corresponding continued fraction representation is correct according to the site.

http://mathworld.wolfram.com/SilverRatio.html Opinionhead 17:42, 1 May 2006 (UTC)


 * Oh for heaven's sake, it was just vandalism. See? Next time try fixing it. I'm removing the tag. Melchoir 23:38, 2 June 2006 (UTC)

So......
Why is the silver ratio important? Why does it merit being named, let alone having an article? I'm not saying there are no answers to these questions, just that it's disappointing that the article itself gives me no idea what the answers are. 84.70.26.165 11:38, 29 October 2006 (UTC)


 * I'm a bit disappointed by this article, also. I've put it on my "to do" list, and will add a good explanation later. This will have to do for now.


 * In thinking about irrational numbers, it seems natural to consider only the irrational numbers on the half open real interval (0, 1]. Any other irrational number can then be expressed in the form n + Irr, where n is an integer and Irr is the fractional irrational part. Further, when thinking about the half-open interval (0, 1], the terms of the harmonic series {1, 1/2, 1/3, 1/4, ...} spring immediately to mind. They form a sort of ladder descending from 1 toward 0 (and never quite getting there), just as the integers {1, 2, 3, 4, ...} climb from 0 toward infinity, but never quite get there, either.


 * As explained in the article about simple continued fractions in canonical form, there is a unique representation of each irrational real number as a simple continued fraction in canonical form. Now as it turns out, if the irrational number x lies on the half-open interval (1/2, 1] its canonical representation starts off like this



x = \cfrac{1}{1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\,}}}} $$


 * where a2, a3, etc. are positive integers. When real numbers are expressed in this form, the famous "golden ratio" is seen to be "half-way" between 1/2 and 1. And if you add 1 to it, you get a number that is "half-way" between 1 and 2 (in a weird sort of way, from the perspective of looking at the simple continued fraction representations of the irrational numbers).


 * Anyway, this is getting long, so I'd better wrap it up. In the representation of irrational numbers on the interval (1/3, 1/2], the first partial denominator is always the integer 2. On (1/4, 1/3] it's always a 3, and so forth. So the silver ratio isn't particularly important all by itself. But the sequence of silver means (which is not very well explained in this article, imo) is of interest, because they're "half-way" between the terms of the harmonic series. DavidCBryant 12:52, 13 December 2006 (UTC)

Separate silver means article
I think that the silver means in general deserve their own article...does anyone else agree? Carifio24 22:59, 25 December 2006 (UTC)

Silver rectangle in an octagon
Two opposite edges of a regular octagon form a silver rectangle (the long type of silver rectangle), obviously enough, and this seems interesting and relevant enough to mention. But does anyone know of a reference for this fact? I couldn't find anything searching for the terms "silver rectangle" and "octagon" together in Google or Google Scholar, and I don't want to mention this in the article if it's WP:OR. —David Eppstein 05:51, 30 January 2007 (UTC)

I verified it for myself using compass and straightedge. Because of what you mentioned, I discovered a new way to construct the silver mean. Specifically speaking, since this is an easy shape to verify with the use of simple construction tools, why can't this part of the article be written? --Opinionhead 14:14, 16 October 2007 (UTC)

In danish
Does anyone know this ratios name in danish?--83.72.7.63 18:31, 14 March 2007 (UTC)

Trigonometric Ratios
I made the connection to the article on exact trigonometric ratios because $$sqrt{2} + 1 = cot 22.5$$. Through some algebraic manipulation, I am sure other connections can be made. --Opinionhead 14:20, 16 October 2007 (UTC)

New Start, more to come
Hi

There is material in several sections (Pell number, square triangular number) that could be made more consistent. I might actually get to it. For now I put in a rough but not too rough start. If my learned fellow WIkipedians agree to let it stay (more or less) then I will move the current first line down to a new section computation and pursue further ideas. Before putting in more work I want to see how this sits with folks.--Gentlemath (talk) 08:23, 20 February 2009 (UTC)

Removal of infobox
Based upon a discussion at Wikipedia talk:WikiProject Mathematics, I've removed the infobox from the article. If anyone disagrees, could you please join the discussion there. Thanks, Paul August &#9742; 15:52, 18 October 2009 (UTC)

Interesting fact about the manitissa for this number.
I've accidentally stumbled upon the fact that the mantissa of the Silver Ratio and Silver ratio itself are reciprocals. I'm sure I'm not the first person to observe this, but it seems like it deserves mention. Is this in there somewhere that I've missed? Furthermore, there is at least one other number like this with the (to me) remarkable property of being irrational and at the same time sharing a mantissa with it's reciprocal. Square root of 5 + 2. I'm pretty sure this is not news, but I'd never seen anything like it, before I found it accidentally just puttering around. Is there an article on Wikipedia where I can read about this?Lyleq (talk) 04:43, 15 October 2010 (UTC)

Nvm. I see it is mentioned. Missed that the first time thru. Sry. Lyleq (talk) 04:48, 15 October 2010 (UTC)


 * Sharing a mantissa with its reciprocal -- Yes, that is a remarkable property. Another Wikipedia article on a number that shares its mantissa with its reciprocal is the golden ratio.
 * $$ 0+\frac{b}{a} = \frac{a}{b} = 1 =$$ one
 * $$ 1+\frac{b}{a} = \frac{a}{b} = \varphi = \frac{1+\sqrt{5}}{2} =$$ the golden ratio = 1.6180... = 1/(0.6180...)
 * $$ 2+\frac{b}{a} = \frac{a}{b} = \delta_S = 1 + \sqrt{2} =$$ the silver ratio = 2.41421... = 1/(0.41421...)
 * It sounds like you are interested in other numbers in the series
 * $$ 3+\frac{b}{a} = \frac{a}{b} = $$ ?
 * $$ 4+\frac{b}{a} = \frac{a}{b} = 2 + \sqrt{5}$$ = the number mentioned by Lyleq above = 4.2360... = 1/(0.2360...)
 * $$ 5+\frac{b}{a} = \frac{a}{b} = $$ ?
 * $$ 6+\frac{b}{a} = \frac{a}{b} = $$ ?
 * Yes, there is an article about these numbers -- metallic means.
 * --DavidCary (talk) 05:58, 12 December 2020 (UTC)
 * --DavidCary (talk) 05:58, 12 December 2020 (UTC)

PROPERTIES
As I was looking through the main page under the PROPERTIES section of Silver means I noted the following curious observation- under odd numbered powers of S_m^n in the equations, if one sums the leftmost numerical units (taking 1 as default when nothing is listed) of each contribution, the result is every other Lucas number.

That is:


 * $$ \!\ S_m^3 = S_{(1m^3 + 3m)} $$  1+3=4


 * $$ \!\ S_m^5 = S_{(1m^5 + 5m^3 + 5m)} $$  1+5+5=11


 * $$ \!\ S_m^7 = S_{(1m^7 + 7m^5 + 14m^3 + 7m)} $$  1+7+14+7=29


 * $$ \!\ S_m^9 = S_{(1m^9 + 9m^7 + 27m^5 + 30m^3 + 9m)} $$  1+9+27+30+9=76


 * $$ \!\ S_m^{11} = S_{(1m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m)}. $$  1+11+44+77+55+11=199

This matrix of equations of itself seems very similar in spirit to the Pascal Triangle, and given that a sister of Pascal with sides 2,1 generates the Lucas series, by summing samplings across every other diagonal (just as the Fibonacci series does in the classical Pascal with 1,1 sides), even more interesting. The sides of these Pascal sisters provides the 'seeds' of these Fib-like series, which may appear as contiguous numbers anywhere in the series. Thus a connection may be made between Pascal systems, and the metal means. And to think I hated math in school.... — Preceding unsigned comment added by 96.234.78.93 (talk) 05:01, 13 October 2011 (UTC)

These coefficients then turn out to come directly out of the 2,1 Pascal sister, from every other diagonal parallel to the 2'1 side, those which correspond to odd dimensions/powers- corresponding to the odd powers the above equations refer to. Very neat stuff indeed. Better than 6! — Preceding unsigned comment added by 96.234.78.93 (talk) 07:01, 13 October 2011 (UTC)

Metallic ratios
Hi all,

I think the notation for silver means should be changed to &phi;n.

$$ \begin{align} \phi_n = \frac{n + \sqrt{n^2 + 4}}{2} \\ \text{so}\quad \phi_1 = \frac{1 + \sqrt{5}}{2} = \phi \\ \end{align} $$

This could thus be generalized to all real numbers and even complex numbers.

Thanks,

The Doctahedron, 16:08, 25 December 2011 (UTC)
 * Since Vera de Spinadel has published such a sequence of silver means, this suggestion has been incorporated into the article.Rgdboer (talk) 22:26, 10 August 2013 (UTC)

Projective lines
The silver ratio √2+1 on the real projective line (or the real axis or the imaginary axis on the complex projective line a.k.a. Riemann sphere) is halfway between 1 and ∞ (infinity pole). Its reciprocal √2−1 is halfway between 0 (zero pole) and 1. Those are straightforward consequences of the stereographic projection, cot(45°/2) and cot(135°/2) respectively. — Preceding unsigned comment added by 85.65.95.79 (talk) 12:01, 3 June 2018 (UTC)