6 thoughts on “Imaginary gold

  1. Very neat.

    I’d never heard of De Moivre’s before, and even after reading through the Wikipedia page, wondered why you reference it instead of simply referencing Euler’s identity, from which your second equality follows pretty naturally.

  2. This is a beautiful identity.. I haven’t seen it before either.

    There’s imaginary silver, too. If we introduce:

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

    So that \phi_1 is the ordinary golden ratio, \phi_2 the “silver ratio”, etc, then:

    2 \sin \imath \log \phi_n = n \imath

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