I'll attempt to recapitulate what I discovered by watching this video and doing some additional investigation. The whole concept of Metallic Means is a generalisation of the Golden Mean. One way to approach matters is geometrically via the Golden Rectangle (Figure 1).
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| Figure 1: The Golden Rectangle |
This rectangle has the property that \(a \div b=(a+b) \div a \) which can be rewritten as \( a^2=ab+b^2 \). Without loss of generality, we can simplify matters by letting \(b=1\) since it is the ratio that we are interested in and not any actual values of \(a\) and \(b\). This leads to \(a^2=a+1\) which can be rewritten as \( a^2-a-1=0 \). Solving this quadratic yields two solutions, one positive and one negative. Because we are dealing with positive lengths, we can ignore the negative solution. The positive solution is the familiar \( (1+\sqrt 5)/2 \) designated using the Greek letter \( \phi \).
The equally famous Fibonacci spiral derives from the Golden Rectangle (Figure 2). If a square is removed from the rectangle, then the rectangle remaining is also a Golden Rectangle ad infinitum. The circular segments that can be inserted into each progressive square combine to form the spiral shown.
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| Figure 2: The Golden Spiral |
The Silver Mean can be derived in a similar fashion from the so-called Silver Rectangle that has the dimensions shown in Figure 3. This rectangle has the property that \(a \div b=(2a+b) \div a \) which can be rewritten as \( a^2=2ab+b^2 \). Without loss of generality, we can again simplify matters by letting \(b=1\) and this leads to \(a^2=2a+1\) which can be rewritten as \( a^2-2a-1=0 \). Solving this quadratic yields two solutions, one positive and one negative. Because we are dealing with positive lengths, we can ignore the negative solution. The positive solution is \( (2+\sqrt 8)/2 \) which can be simplified to \( 1+\sqrt 2 \) and this is the Silver Mean or Silver Ratio.
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| Figure 3: The Silver Rectangle |
In the Silver Rectangle, removing two squares at a time leaves another rectangle of the same dimensions ad infinitum. The inscribed parts of the circles in the squares join together to form the characteristic spiral of the silver variety as shown in Figure 4.
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| Figure 4: The Silver Spiral |
This process can be continued and what might be called the Bronze Rectangle is the result. Figure 5 is useful in comparing the relative dimensions of the Gold, Silver and Bronze Rectangles.
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| Figure 5: A comparison of the Golden, Silver and Bronze Rectangles |
Looking at the progression of the numbers under the square root sign, it's immediately apparent that a Fibonacci progression is in play. The dimensions of the next rectangle would be \( (1+\sqrt 21)/2 \) and so on. Let's follow up on the Fibonacci connection by looking at the ratio of progressive pairs of terms in this sequence. The terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, ... and it's well known the ratio of progressive pairs of terms approaches the Golden Mean. If we write the \(n-th\) Fibonacci term as \(F_n\) then the relationship between terms is given by:$$ F_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\F_{n-1}+F_{n-2}&\mbox{otherwise.}\end{cases} $$With the Silver Mean, the relationship is given by:$$ P_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases} $$This leads to the following sequence of terms: 0, 1, 2, 5, 12, 29, ... which is known as the Pell sequence. With the Bronze Mean, the relationship is given by: $$ T_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\3T_{n-1}+T_{n-2}&\mbox{otherwise.}\end{cases} $$Here the progression of terms is: 0, 1, 3, 10, 33, 109, ... and so, in the most general case, we have a quadratic equation of the form \(a^2-k \times a -1 \) with positive solution \( (k+ \sqrt {k^2+4}) \div 2 \) and a relationship given by:$$ T_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\k \times T_{n-1}+T_{n-2}&\mbox{otherwise.}\end{cases}$$where \(k\) is any integer greater than or equal to 1. So we've looked at the metallic means geometrically and in terms of Fibonacci-type sequences but they can also be looked at in terms of continued fractions.
For the Golden Mean: \( (1+\sqrt 5) \div 2 \), the convergents are 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55 ... and the continued fraction is:
1
1 + --------------------------------------------
1
1 + ---------------------------------------
1
1 + ----------------------------------
1
1 + -----------------------------
1
1 + ------------------------
1
1 + -------------------
1
1 + --------------
1
1 + ---------
1 + ...
For the Silver Mean: \( (2 + \sqrt 8) \div 2 \), the convergents are 5/2, 12/5, 29/12, 70/29, 169/70, 408/169, 985/408, 2378/985, 5741/2378, ... and the continued fraction is:
1
2 + -------------------------------------------------
1
2 + --------------------------------------------
1
2 + ---------------------------------------
1
2 + ----------------------------------
1
2 + -----------------------------
1
2 + ------------------------
1
2 + -------------------
1
2 + --------------
1
2 + ---------
2 + ...
For the Bronze Mean: \( (3 + \sqrt 13 ) \div 2 \), the convergents are 10/3, 33/10, 109/33, 360/109, 1189/360, 3927/1189, 12970/3927, 42837/12970, 141481/42837, ... and the continued fraction is:
1
3 + -------------------------------------------------
1
3 + --------------------------------------------
1
3 + ---------------------------------------
1
3 + ----------------------------------
1
3 + -----------------------------
1
3 + ------------------------
1
3 + -------------------
1
3 + --------------
1
3 + ---------
3 + ...
and so on and so on.
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