10 Golden and Silver Mean dimensions

In 1994, El Naschie proved some important theorems connecting stability and Hausdorff dimension of a multifractal-like Cantor set at the point of global diffusion [21,22,23]. One of them is the Golden Mean theorem.

The Golden Mean theorem states a result strictly related to the notion of KAM instability and global chaos in Hamiltonian systems. Both approaches, the KAM theory and the Cantorian space-time, have the common feature that the Golden Mean winding number plays an important role with reference to the great stability of certain orbits in the $n$-dimensional phase space.

Let us recall that the Fibonacci quotients are given by $F_1 /F_0 =1; F_2 /F_1 = 0.5; F_3 /F_2 = 2/3 = 0.666\ldots; F_4 /F_3 =3/5 = 0.6; F_5 /F_4 = 5/8 = 0.625; F_6 = F_5 = 8/13; \ldots$ and that $\mathop {\lim }\limits_{n \to \infty } \frac{{F_n }}{{F_{n + 1} }} = \frac{1}{\phi } = \frac{{\sqrt 5 - 1}}{2},$ where $1/\phi$ is called by physicists the Golden Number.

The interesting point is that these quotients are the winding number ratios of the phase space orbits of numerous Hamiltonian systems. In addition, through many computer calculations, it has been proved that the orbits with the Golden winding number are the most stable. These results, known from KAM theory, agree completely with those obtained by El Naschie.

The remarkable fact that the reciprocal of the Golden Mean $1/\phi$ arises in such a natural way suggests that there must be a kind of ``universality" in the transition from quasiperiodicity to chaos.

Another main theorem is the modified Fibonacci theorem, which relates the average space dimension to a three-dimensional Cantorian space using a third order Fibonacci lattice defined by

$$G_0 = G_1 = G_2 = 1;\quad G_{n+1} = G_{n-2} + G_{n-1} + G_n \mbox{ for } n \ge 2.$$

Comparing the secondary Fibonacci sequence with this third order Fibonacci sequence, as it is stated in the following table,

Table

$n$	0	1	2	3	4	5	6	7	8	9
$F_n$	1	1	2	3	5	8	13	21	34	55
$G_n$	1	1	1	3	5	9	17	31	57	105

it is seen that for the first sequence, $F_n$ and $n$ are equal when $n = 1,2,3$, while for the second one, $G_n$ and $n$ are equal only when $n =1, 3.$ These states may be drawn into modeling some forms of ergodic behavior of Fibonacci-related systems, and they can be considered as ``ergodic-like states." The connections of this research to classical and quantum mechanical statistics, as stated by El Naschie [24], reveal that there are two types of quasi-ergodic Cantor sets:
a) an even four-dimensional that resembles the behavior of classical particles and bosons;
b) an odd five-dimensional that is related to fermions³ and the fivefold symmetry of quasicrystals.

Furthermore, this third-order Fibonacci sequence has been used in 1988 by Gumbs and Ali [25] to study the discrete Schrodinger equation with a ternary string. The experimental results indicate some unique features when compared with the second-order Fibonacci sequences. It is interesting to mention that Gumbs and Ali, using Kohmoto's technique, analyzed non-$\phi$ quasicrystals [26,27], obtaining different and characterizing results in the case of the Golden, Silver, and Bronze Means (purely periodic continued fraction expansions) and the Copper and Nickel Means (periodic continued fraction expansions).

A third main theorem is the Silver Mean theorem, where the Silver Mean $\sigma_{Ag}$ appears in calculating average Hausdorff dimensions. This number plays an important role in studying interfaces between number theory and physics, such as dynamical systems, quasicrystals, and other related topics.