not fit with this picture.

The observation challenging Darwinian dogma is that the moments for evolutionary breakthroughs - according to Faixat's observation - seems to come in powers of 3 for some fundamental time scale. There would be precise 3-fractality and accompanying cyclicity - something totally different from Darwinian expectations.

By looking at the diagrams demonstrating the appearance of powers of 3 as time scales, it became clear that the interpretation int terms of underlying 3-adicity could make sense. I have speculated with the possibility of small-p p-adicity. In particular, p-adic length scale hypothesis stating that primes near powers of 2 are especially important physically could reflect underlying 2-adicity. One can indeed have for each p entire hierarchy of p-adic length scales coming as powers of p^{1/2}. p=2 would give p-adic length scale hypothesis. The observations of Faixat suggest that also powers p=3 are important - at least in evolutionary time scales.

** Note**: The p-adic primes characterizing elementary particles are gigantic. For instance, Mersenne prime M_{127} = 2^{127} -1 characterizes electron. This scale could relate to the 2-adic scale L_{2}(127)= 2^{127/2}× L_{2}(1). The hierarchy of Planck constants coming as h_{eff}=n× h also predicts that the p-adic length scale hierarchy has scaled up versions obtained by scaling it by n.

The interpretation would be in terms of p-adic topology as an effective topology in some discretization defined by the scale of resolution. In short scales there would be chaos in the sense of real topology: this would correspond to Darwinian randomness. In long scales p-adic continuity would imply fractal periodicities in powers of p and possibly its square root. The reason is that in p-adic topology system's states at t and t+kp^{n}, k=0,1,...p-1, would not differ much for large values of n.

The interpretation relies on p-adic fractality . p-Adic fractals are obtained by assigning to real function its p-adic counterpart by mapping real point by canonical identification

SUM_{n} x_{n} p^{n} → SUM _{n} x_{n} p^{-n}

to p-adic number, assigning to it the value of p-adic variant of real function with a similar analytic form and mapping the value of this function to a real number by the inverse of the canonical identification, the powers of p correspond to a fractal hierarchy of discontinuities.

A possible concrete interpretation is that the moments of evolutionary breakthroughs correspond to criticality and the critical state is universal and very similar for moments which are p-adically near each other.

The amusing co-incidence was that I have been working with a model for 12-note scale based on the observation that icosahedron is a Platonic solid containing 12 vertices. The scale is represented as a closed non-self-intersecting curve - Hamiltonian cycle - connecting all 12 vertices: octave equivalence is the motivation for closedness. The cycle consists of edges connecting two neighboring vertices identified as quints - scalings of fundamental by 3/2 in Platonic scale. What is amusing that scale is obtained essentially powers of 3 are in question scaled down (octave equivalence) to the basic octave by a suitable power of 2. The faces of icosahedron are triangles and define 3-chords. Triangle can contain either 0,1, 2 edges of the cycle meaning that the 3-chords defined by faces and defining the notion of harmonic contain 0,1, or 2 quints. One obtains large number of different harmonies partially characterized by the numbers of 0-, 1-, and 2-quint icosahedral triangles.

The connection with 3-adicity comes from the fact that Pythagorean quint cycle is nothing but scaling by powers of 3 followed by suitable downwards scaling by 2 bringing the frequency to the basic octave so that 3-adicity might be realized also at the level of music!

There is also another strange co-incidence. Icosahedron has 20 faces, which is the number of amino-acids. This suggests a connection between fundamental biology and 12-note scale. This leads to a concrete geometric model for amino-acids as 3-chords and for proteins as music consisting of sequences of 3-chords. Amino-acids can be classified into 3 classes using polarity and basic - acid/neutrality character of side chain as basic criteria. DNA codons would would define the notes of this music with 3-letter codons coding for 3-chords. One ends up also to a model of genetic code relying on symmetries of icosahedron from some intriguing observations about the symmetries of the code table.

At the level of details the icosahedral model is able to predict genetic code correctly for 60 codons only, and one must extend it by fusion it with a tetrahedral code. The fusion of the two codes corresponds geometrically to the fusion of icosahedron with tetrahedron along common face identified as "empty" amino-acid and coded by 2 stopping codons in icosahedral code and 1 stopping codon in tetrahedral code. Tetrahedral code brings in 2 additional amino-acids identified as so called 21st and 22nd amino-acid discovered for few years ago and coded by stopping codons. These stopping codons certainly differ somehow from the ordinary one - it is thought that context defines somehow the difference. In TGD framework magnetic body of DNA could define the context.

The addition of tetrahedron brings one additional vertex, which correlates with the fact that rational scale does not quite closed. 12 quints gives a little bit more than 7 octaves and this forces to introduce 13 note for instance, A_{b} and G_{#} could differ slightly. Also micro-tubular geometry involves number 13 in an essential manner.

## 2 comments:

Matti, did you have time to look at http://arxiv.org/abs/math/0602610 ?

The two sets of rational numbers

Thetal := proc (chi) options operator, arrow; (1/2)*chi/(chi+1) end proc; Thetar := proc (chi) options operator, arrow; (1/2)*(chi+2)/(chi+1) end proc;

where chi ranges over the integers shows up... if you turn these into a Taylor series by interpreting them as coefficients you get

sum(Thetar(n)/factorial(n), n = 1 .. inf)=exp(1)-3/2

--PseudoAnonymousCrow

Sorry, I did not. I have been working under the big dark shadow of deadline last year and continue to do so until the end of this months. Restricts considerably my activities.

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