Generalizations of the golden and silver ratios, and their significance

$\Phi$, or the golden ratio, is basically $\frac{a+b}{a}=\frac{a}{b}$. The silver ratio corresponds to a similar idea of: $\frac{2a+b}{a}=\frac{a}{b}$. I've read on Wikipedia that both of these ratios are well known and have some appearance in nature (not the mystical hogwash stuff). In addition, there is the relation of the Fibonacci and Pell Numbers respectively.

Funny enough, I stumbled on these ratios originally by myself just messing around with numbers. I also noticed that the silver ratio minus 1 approximates $\sqrt{2}$ (which is how I came to be fooling around with this).

However I don't seem to find anything about further manipulation of this ratio, such as: $\frac{3a+b}{a}=\frac{a}{b}$, $\frac{4a+b}{a}=\frac{a}{b}$ and so forth. Do these ratios have any special properties that are found in nature? Perhaps called the copper ratio or something?

• The silver ratio doesn't 'approximate' $1+\sqrt{2}$; that's exactly what it is. If you rewrite your ratios non-homogeneously using $x=\frac{a}{b}$, you get $x=2+\frac1x$; multiplying by $x$ turns this into the quadratic equation $x^2-2x-1=0$ with solution $1+\sqrt{2}$. – Steven Stadnicki Apr 11 '14 at 16:55
• They have less and less importance as you increase in the height. If you look at the wikipedia page on the silver ratio (en.wikipedia.org/wiki/Silver_ratio) you can find section on "silver means" that generalizes it to "metallic means" and then specifically mentions the "bronze mean". – Foo Barrigno Apr 11 '14 at 16:57
• Ok, thanks. But this brings me to another ratio. What about (a+2b)/a=a/b? This comes out to (sqrt(3)+1)/2. Does this ratio ever appear in nature or have some use? – user2780341 Apr 11 '14 at 17:56

Answer thanks to Foo Barrigno:

They have less and less importance as you increase in the height. If you look at the wikipedia page on the silver ratio (http://en.wikipedia.org/wiki/Silver_ratio) you can find section on "silver means" that generalizes it to "metallic means" and then specifically mentions the "bronze mean".

This paper (The Generalizations of the Golden Ratio: Their Powers, Continued Fractions, and Convergents, pdf) proposes alternative way of defining silver means and related numbers and sequences.

Golden ratio can be defined as a continued fraction as follows:

$$1+\cfrac1{1 + \cfrac1{1 + \cfrac1{\cdots}}}=\cfrac{1+\sqrt5}{2}=\phi$$

Silver ratio is then:

$$2+\cfrac1{2 + \cfrac1{2 + \cfrac1{\cdots}}}={1+\sqrt2}=\delta_S$$

Now, there is a generalization of golden and silver ratio, called Silver mean of order n that is defined with following continued fraction:

$$N+\cfrac1{N + \cfrac1{N + \cfrac1{\cdots}}}=\cfrac{N+\sqrt{N^2+4}}{2}$$

Silver means are summarized in the following table: Releted to this, the authors also define silver Fibonacci sequences:

Fibonacci sequence is defined with recurrence $F_{n+2}=F_{n+1}+F_{n}$ where $F_0=0$ and $F_1=1$.

Family of silver Fibonacci sequences is defined with recurrence $F_{m,n+2}=mF_{m,n+1}+F_{m,n}$ where $F_{m,0}=0$ and $F_{m,1}=1$.

Relationship between silver means and silver Fibonacci sequences (and many other related interesting things) are explored in the paper.