The Golden Mean, Brain Waves, and the Bose-Einstein Distribution
The principle of information coding by the Brain seems to be based on the golden mean. Since decades psychologists have claimed memory span to be the missing link between psychometric intelligence and cognition. By applying Bose-Einstein-statistics to learning experiments, Pascual-Leone obtained a fit between predicted and tested span. Multiplying span by mental speed (bits processed per unit time) and using the entropy formula for bosons, we obtain the same result. If we understand span as the quantum number n of a harmonic oscillator, we obtain this result from the EEG. The metric of Brain waves can always be understood as a superposition of n harmonics times 2 F, where half of the fundamental is the golden mean F (= 1.618) as the point of resonance. Such wave packets scaled in powers of the golden mean have to be understood as numbers with directions, where bifurcations occur at the edge of Chaos, i.e. 2 F = 3+ f3. Similarities with El Naschie's theory for high energy particle's physics are also discussed.
. . . In tables of IQ test results edited by Lehrl et al. and based on concepts of information theory (see below), we read for this age and IQ 102 a short-term memory storage capacity of 84 bits. Two approaches with seemingly completely differing theoretical starting points lead on the absolute scale of information entropy to practically the same result. For Pascual-Leone's data the latter result was even obtained after applying quantum mechanics twice in series, for calculating Bose-Einstein statistics and information entropy.
The variance of the Bose-Einstein distribution equals m2 + m, where m reflects the granularity of the energy due to Einstein's photons . . . . If we set the variance 1 and m = x, we get x2 + x = 1. The solution of this equation is f (=(51/2-1)/2 = 0.618033), the golden mean. Its inverse 1/f = F (also called the golden ratio, the Golden Number, the golden section or the divine proportion) has the property 1 + F = F2. Therefore the double geometric F-series:
..., 1/F2, 1/F, 1, F, F2, F3, ... .
has the properties,
..., 1/F2 + 1/F = 1, 1/F + 1 = F, 1 + F = F2, ...
and is thus a Fibonacci series. It is the only geometric series that is also a Fibonacci series. Essential is the fact that the fractional parts .618033... of f, 1/f, and 1/f + 1 = F2 are identical. The title chosen by us refers to this golden mean in the broader sense.
Forces are now recognised as resulting from the exchange of huge numbers of discrete particles, or information patterns called vector bosons, which are exchanged between two or more particle information patterns. The absorption of a vector boson information pattern changes the internal oscillation state of a particle, and causes an impulse of motion to occur along a particular direction. This turns out to be the quantum origin of all forces. Therefore, forces can be thought of being digital rather than analogue.
In 2001 Bianconi and Barab¡si discovered that not only neural networks but all evolving networks, including the World Wide Web and business networks, can be mapped into an equilibrium Bose gas, where nodes correspond to energy levels and links represent particles. Still unaware of the research by Pascual-Leone, for these network researchers this correspondence between network dynamics and a Bose gas was highly unexpected .
”Harald Weiss and Volkmar Weiss, "The Golden Mean as Clock Cycle of Brain Waves" http://www.volkmar-weiss.de/chaos.html