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Next: 12 Concluding remarks Up: SPINADEL Previous: 10 Golden and Silver

11 The Subtle Mean

The quadratic PV number

\begin{displaymath}
\phi^3 = \left( {\frac{{1 + \sqrt 5 }}{2}} \right)^3 = 4.236
\ldots
\end{displaymath}

appears frequently in diverse fields of knowledge. Its continued fraction expansion is purely periodic, that is,

\begin{displaymath}
\phi^3 = \left[ {\overline 4 } \right],
\end{displaymath}

and is one of the most important Metallic Means, the positive solution of the equation

\begin{displaymath}
x^2 - 4x - 1 = 0.
\end{displaymath}

It has been called the Subtle Mean by El Naschie [28] and, besides appearing in quasi periodic tiling and crystallography, it plays a significant role in the theory of Cantorian fractal-like micro-space-time

\begin{displaymath}
E^{(\infty )}
\end{displaymath}

In fact, the mean value of the Hausdorff dimension of this space-time is precisely the Subtle Mean [29]

\begin{displaymath}
\left\langle {\dim E^{(\infty )} } \right\rangle = \phi^3 .
\end{displaymath}

In addition, the Subtle Mean is involved in a fundamental way in various basic equations in knot theory, noncommutative geometry, and four manifold theory [30].


next up previous
Next: 12 Concluding remarks Up: SPINADEL Previous: 10 Golden and Silver
2003-06-05