[VorticesRcool]

Fascinating post. I had always assumed that the 3rd most irrational number would be the third metallic mean given by n = (n+ sqrt(n^2+4))/2, and subsequently the fourth metallic mean etc. The metallic means also pack the disks nicely. I have recently had my interest in them sparked after I came across solution to point vortex equilibria involving them.

Do you know what the metallic means are bounded by? Are they as bad as the silver ratio/(1+sqrt(2))?

These most irrational numbers, (9+sqrt(221))/10, (13+sqrt(1517))/26... how interesting that they are not just the simple generalization of the continued fraction for the golden ratio.

[extremelearning]

What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio.

As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), for some constant integer m. For example, m=1 is the golden ratio, m=2 is the silver ratio,...

But one of my other posts [2], generalizes the Golden ratio via the "Harmonious Numbers", as defined by the lagged recurrence, T(n+m) = T(n)+T(n-1), for some constant m. In this case, m=1 relates to the Golden Ratio, and m=2 relates to the Plastic Number [3].

And then finally, this post explores generalizing it via a completely different perspective, that of "Lagrange Numbers".

It seems that we need to 'think outside the box' a litte when generalizing the Golden ratio, as there is not single obvious way to generalise continued fractions.

[1] https://en.wikipedia.org/wiki/Metallic_mean

[2] http://extremelearning.com.au/unreasonable-effectiveness-of-...

[3] https://en.wikipedia.org/wiki/Plastic_number