with the value
There is no doubt that 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.
It is claimed that 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 is nonsense. Zealots imagine that is everywhere, and plays a central and dominant role in art, architecture and music. They find signs of in buildings and works of art from cultures for which there is no supporting evidence of the relevant mathematical knowledge.
The golden mean has many amazing properties. Writing the quadratic as and repeatedly substituting for in the right hand side, we get the continued fraction expansion:
Truncating this infinite expansion gives the sequence
where the values are ratios of successive terms of the Fibonacci sequence.
Another beautiful expression for is as an iterated square root
It is a simple matter to square this and get , 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:
The particular case yields what is called the silver mean, with value . Other choices of and give the metallic means.
We will denote the silver mean by . From the quadratic it is a simple matter to show that the continued fraction expansion for is
This follows by writing the quadratic as and repeatedly substituting as above. The iterated square root expression is also simple to demonstrate:
(squaring this, the result follows immediately).
Higher metallic means are defined as positive roots of the equation and have continued fraction expansions
Note that .
General Metallic Means
The more general case is also easy to treat. Assuming and are positive, the positive root of this equation is
The specific choice gives the golden mean and gives the silver mean . 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 .
The continued fraction expansion for is
and the iterated square root is
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 and as continuous variables, and plot the “metallic function” in the first quadrant of the -plane to get the figure above.