Golden Mind

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The mind "knows" a moment of confirmation when the mind establishes a fractal relationship that in-dwells within the relationship. This self-embedding-acuity is the reward-generating ascension state, which generates psycho-glandular bliss.

Embedded Golden Ratio relationships exist by the pentagram inscribed within a pentagon which can repeat to have another in-dwelling pentagram/pentagon, etc., ad infinitum. So may a thought echo into itself toward a Zen moment, perhaps akin to the moment when the punch-line of a joke is realized, producing the staccato of laughter as alternatively reverberating cogitations between the initial analysis of the jocular scenario, and the synthesis of an alternative solution.  —DEM

Scientific Paper:

The golden mean as clock cycle of brain waves


The principle of information coding by the brain seems to be based on the golden mean. For decades psychologists have claimed memory span to be the missing link between psychometric intelligence and cognition. By applying Bose–Einstein-statistics[1] 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 2Φ, where half of the fundamental is the golden mean Φ (=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. 2Φ=3+φ3. Similarities with El Naschie’s theory for high energy particle's physics are also discussed.


The variance of the Bose-Einstein distribution equals m2 + m, where m reflects the granularity of the energy due to Einstein’s photons (cited from [14, p.189]). If we set the variance 1 and m = x, we get x2 + x = 1. The solution of this equation is f (=(√5-1)/2 = 0.618033), the golden mean[1]. 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,... (1)

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 Barabási [15] 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 [16].

[1] Some authors call its inverse F (=(√5+1)/2 = 1.618033) the golden mean. We hope this will cause no confusion.
[15] Bianconi G, Barabási AL. Bose-Einstein condensation in complex networks. Physical Review Letters 2001; 86:5632-5635.
[16] Albert R, Barabási AL. Statistical mechanics of complex networks. Reviews of Modern Physics 2002; 74:47-97.

Source: ( Harald Weissa and Volkmar Weiss, Rietschelstrasse 28, D-04177 Leipzig, Germany. Accepted 20 February 2003. Available online 15 May 2003)



  1. Bose-Einstein_condensation:_a_network_theory_approach (