Metallic numbers: Beyond the golden ratio

You've heard of the golden ratio. It's portrayed in films and literature as the alternative answer to life, the Universe, and everything. And indeed, it's everywhere: from sunflower seed patterns to ammonite shells and, supposedly, the "perfect" ratio for aesthetic faces and bodies. As such, it tends to outshine its little brother you might not have heard of yet — the silver ratio.

Recall the golden ratio

Before we talk about the silver ratio, we'll recap what the golden ratio is. To paraphrase the ancient Greek mathematician Euclid:

A straight line is cut in accordance with the golden ratio when the ratio of the whole line to the longer segment is the same as the ratio of the longer segment to the shorter segment.

We can see what this looks like in the following diagram:

Here, the ratio of the length of section A to the length of section B is the same as the ratio of the length of the whole line to the length of section A. This ratio, called the golden ratio and denoted by the Greek letter $\varphi$, is approximately 1.618 in numerical value. We can find this value by first expressing Euclid's definition algebraically: $$\begin{array}{rcl}\varphi & =& A/B=(A+B)/A\\ \varphi & =& A/B=1+B/A\\ \varphi & =& 1+1/\varphi .\end{array}$$ Multiplying the last expression through by $\varphi$ gives us the quadratic equation $${\varphi }^2=\varphi +1,$$ which has solutions $$\varphi =\frac{1\pm \sqrt{5}}{2}.$$ Since we are dealing with positive lengths, we ignore the negative solution, leaving us with $$\varphi =\frac{1+\sqrt{5}}{2}=1.618033...$$

The silver ratio

Now let's cut a line into three segments, two longer segments of equal length and one smaller segment, such that the ratio of the whole line to one of the longer segments is the same as the ratio of one longer segment to the smaller segment. Then that line is cut in accordance with the silver ratio, which we will denote by the Greek letter $\sigma$. The numerical value of the silver ratio is approximately 2.414.

Once again, we can deduce this algebraically:

$$\begin{array}{rcl}\sigma & =& A/B=(2A+B)/A\\ \sigma & =& A/B=2+B/A\\ \sigma & =& 2+1/\sigma ,\end{array}$$ giving the quadratic equation $${\sigma }^2=2\sigma +1.$$ This has positive solution $$\sigma =\frac{2+\sqrt{8}}{2}=\frac{2+2\sqrt{2}}{2}=1+\sqrt{2}=2.414213...$$

To recap, the golden ratio involves dividing a line into two segments and the silver ratio involves cutting it into three segments, two being of equal length. This suggests the possibility of further generalisation...

Meet the family

Suppose we divide our line into $n$ segments of equal length, which we call $A$, and one smaller segment of length $B$. If we require the ratio between $A$ and $B$ to be the same as the ratio between the whole line and one of the segments of length $A$ we have $$A/B=(nA+B)/A.$$ Writing ${\lambda }_n$ for this ratio means that $${\lambda }_n=A/B=n+B/A,$$ so in analogy to our calculations above we have $${\lambda }_n=n+1/{\lambda }_n,$$ giving the quadratic equation $${\lambda }_n^2=n{\lambda }_n+1.$$ The positive solutions to this equation is $${\lambda }_n=\frac{n+\sqrt{n^2+4}}{2}.$$ The numbers ${\lambda }_n$, one for each value of $n>1,$ are called metallic ratios, or metallic means. For $n=1$ we get the golden ratio and for $n=2$ the silver one.

Metallic ratios share many common properties: they are linked to infinite sequences reminiscent of the famous Fibonacci sequence, to very special rectangles and to logarithmic spirals. We will explore these in the second part of this article.

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About the authors

Gokul Rajiv and Yong Zheng Yew are two former high-school level students in Singapore who happened to explore the idea of metallic means in a project and found it interesting enough to share.