Imaginary gold, silver, bronze, …

The previous post gave a relationship between the imaginary unit i and the golden ratio. This post highlights a comment to that post explaining that the relationship generalizes to generalizations of the golden ratio.

GlennF pointed out that taking the larger root of the equation

\phi_n = n + \frac{1}{\phi_n}

defines the golden ratio when n = 1, the silver ratio when n = 2, etc. φn is also the continued fraction made entirely of n‘s.

With this definition, we have

2 \sin \left( i \log \phi_n \right)  = ni

3 thoughts on “Imaginary gold, silver, bronze, …

  1. Did you see the cosine version? Then you get 2 cos(i log(psi_n)) = n, where psi_n is a root of x + 1/x = n. (It’s too late in the evening for me to work out the relationship between psi_n and phi_n.)

  2. The relation between \phi_n and \psi_m only seems to be a nice one for some cases. For example \psi_3 = (1 + \phi_1), which is what the cosine relation on Andrew Stacey’s page uses. There are others, e.g. \psi_6 = 1 + 2 \phi_2.

  3. I would like a ration of gold better than golden ratio, I think. (Sorry, belatedly pointing out the typo.)

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