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Golden Ratio Conjugate

The quantity

\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 $n$th Continued Fraction representation
\begin{displaymath}[a_0, a_1, a_2, \ldots].
\end{displaymath} (2)

Taking the limit of
\delta_n\equiv {\sigma_n-\sigma_{n-1}\over \sigma_n-\sigma_{n+1}}
\end{displaymath} (3)

\delta\equiv \lim_{n\to\infty} = 1+\phi = 2+\phi_C.
\end{displaymath} (4)

See also Golden Ratio, Silver Ratio

© 1996-9 Eric W. Weisstein