8 thoughts on “Imaginary gold

  1. Cool post! I love these identities. I think there are two typos in the proof: exp(ilog(phi)) ought to be exp(log(phi)), and the second to last line ought to be i(phi – 1/phi).

  2. 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.

  3. @Peter Fry
    Phi is a solution to the equation x^2 – x – 1 = 0, so clearly 1 – phi^2 = -phi

  4. 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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