Talk:Silver ratio

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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)

Contents

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] In danish

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


[edit] Trigonometric Ratios

I made the connection to the article on exact trigonometric ratios because sqrt2 + 1 = cot22.5. Through some algebraic manipulation, I am sure other connections can be made. --Opinionhead 14:20, 16 October 2007 (UTC)