Golden Ratio Conjugate

The quantity

$\begin{displaymath} \phi_C \equiv {1\over\phi} = \phi-1={\sqrt{5}-1 \over 2} \approx 0.6180339887, \end{displaymath}$

(1)

where $\phi$ is the Golden Ratio. The golden ratio conjugate is sometimes also called the Silver Ratio. A quantity similar to the Feigenbaum Constant can be found for the

th Continued Fraction representation

$\begin{displaymath}[a_0, a_1, a_2, \ldots]. \end{displaymath}$

(2)

Taking the limit of

$\begin{displaymath} \delta_n\equiv {\sigma_n-\sigma_{n-1}\over \sigma_n-\sigma_{n+1}} \end{displaymath}$

(3)

gives

$\begin{displaymath} \delta\equiv \lim_{n\to\infty} = 1+\phi = 2+\phi_C. \end{displaymath}$

(4)

See also Golden Ratio, Silver Ratio