Gold, silver, and bronze ratios

The previous post showed that if you inscribe a hexagon and a decagon in the same circle, the ratio of the sides of the two polygons is the golden ratio. After writing the post I wondered whether you could construct the silver ratio or bronze ratio in an analogous way.

Metallic ratios

To back up a bit first, what are these ratios? The golden ratio is famous, but the silver and bronze ratios are not so well known. I’ve written the silver ratio before, and hinted at the bronze ratio.

The continued fraction representation of the golden ratio contains all 1s.

 1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}

Similarly, the continued fraction for the silver ratio contains all 2s.

 2 + \cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cdots}}}

In general, the nth metallic ratio is the number whose continued fraction representation contains all ns.

n + \cfrac{1}{n+\cfrac{1}{n+\cfrac{1}{n+\cdots}}} = \frac{n + \sqrt{n^2 + 4}}{2}

The metallic ratios for n > 3 do not have standard names.

Polygon side ratios

Let’s call a number a polygon ratio if it equals the ratio of sides of polygons inscribed in the same circle. Is the silver ratio or the bronze ratio a polygon ratio?

Apparently not. I would expect that if the silver or bronze ratio were the ratio of the side of an m-gon and an n-gon, then m and n would be fairly small integers, though I have no proof of this.

I did an empirical search, letting m and n vary from 1 to 1,000. Some ratios came close to the silver and bronze ratios, but no matches. I expect you can approximate any number as closely as you like with the ratio of polygon sides, but if there’s an exact match for the silver or bronze ratio, then m and n must be larger than 1,000.

Apparently the golden ratio is the only metallic ratio which is also a polygon ratio. This is at least true for the first 100 metallic ratios and for polygon ratios with m and n less than 1,000.