Metallic Means


The golden mean occurs repeatedly in the pentagram [image Wikimedia Commons]

Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by {\phi} and is the positive root of the quadratic equation

\displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)

with the value

{\phi = (1+\sqrt{5})/2 \approx 1.618}.

There is no doubt that {\phi} is significant in many biological contexts and has also been an inspiration for artists. Called the Divine Proportion, it  was described in a book of that name by Luca Pacioli, a contemporary and friend of Leonardo da Vinci.

vitruvian-man-loresIt is claimed that {\phi} shows up in da Vinci’s Vitruvian Man, as the ratio of the man’s height to the height of his navel! This may be true; however, much of what has been written about {\phi} is nonsense. Zealots imagine that {\phi} is everywhere, and plays a central and dominant role in art, architecture and music. They find signs of {\phi} in buildings and works of art from cultures for which there is no supporting evidence of the relevant mathematical knowledge.

Expressions for \phi 

The golden mean has many amazing properties. Writing the quadratic as {x = 1 + 1/x} and repeatedly substituting {1 + 1/x} for {x} in the right hand side, we get the continued fraction expansion:

\displaystyle \phi = 1 + \cfrac{1}{ 1 + \cfrac{1}{ 1 + \cfrac{1}{ 1 + \cdots } }} = [ 1 ; 1 , 1 , 1 , \dots ]

Truncating this infinite expansion gives the sequence

\displaystyle \left\{ 1 , 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \dots \right\}

where the values are ratios of successive terms of the Fibonacci sequence.

Another beautiful expression for {\phi} is as an iterated square root

\displaystyle \phi = \sqrt{1 +\sqrt{1 + \sqrt{1 + \sqrt{ \dots }}}}

It is a simple matter to square this and get {\phi^2 = 1 + \phi}, the defining quadratic (1).

The Silver Mean

Variants and generalizations of the golden mean were introduced by de Spinadel. They are roots of quadratics that are modifications of (1) above:

\displaystyle x^2 - p x - q = 0 \ \ \ \ \ (2)

The particular case {p = 2, q = 1} yields what is called the silver mean, with value {\sigma = 1+\sqrt{2} \approx 2.424}.  Other choices of {p} and {q} give the metallic means.

We will denote the silver mean by {\sigma}. From the quadratic x^2 - 2 x - 1 = 0 it  is a simple matter to show that the continued fraction expansion for {\sigma} is

\displaystyle \sigma = 2 + \cfrac{1}{ 2 + \cfrac{1}{ 2 + \cfrac{1}{ 2 + \cdots } }} = [ 2 ; 2 , 2 , 2 , \dots ]

This follows by writing the quadratic as {x=2+1/x} and repeatedly substituting as above. The iterated square root expression is also simple to demonstrate:

\displaystyle \sigma = \sqrt{1 +2\sqrt{1 + 2\sqrt{1 + 2\sqrt{ \dots }}}}

(squaring this, the result follows immediately).

Higher metallic means are defined as positive roots of the equation {x^2 - p x - 1 = 0} and have continued fraction expansions

\displaystyle \sigma(p) = [ p ; p , p , p , \dots ]

Note that {p< \sigma(p)< p+1}.

General Metallic Means

The more general case {x^2 - p x - q = 0} is also easy to treat. Assuming {p} and {q} are positive, the positive root of this equation is

\displaystyle \mu(p,q) = \frac{p + \sqrt{p^2+4q}}{2}

The specific choice {p=1, q=1} gives the golden mean {\phi=\mu(1,1)=(1+\sqrt{5})/2} and {p=2, q=1} gives the silver mean {\sigma=\mu(2,1)=1+\sqrt{2}}. Other values are given names like copper mean, nickel mean, etc., but there does not appear to be a clear convention. Perhaps it is best to stick to {\mu(p,q)}.

The continued fraction expansion for {\mu(p,q)} is

\displaystyle \mu(p,q) = p + \cfrac{q}{ p + \cfrac{q}{ p + \cfrac{q}{ p + \cdots } }}

and the iterated square root is

\displaystyle \mu(p,q) = \sqrt{q +p\sqrt{q + p\sqrt{q + p\sqrt{ \dots }}}}


Metallic function surface. Note that the height for p = q = 1 is the golden mean.

In Conclusion

Whether the metallic numbers have any fundamental importance in mathematics or in nature is unclear, but they certainly provide an interesting case study. For example, we can treat {p} and {q} as continuous variables, and plot the “metallic function” {z = \mu(p,q)} in the first quadrant of the {(p,q)}-plane to get the figure above.


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