The self-similar numbers as a special case of cyclic numbers and their relation to the cyclic (genetic) codes

The genetic code is analyzed as one of the major natural codes, which as such can reveal the deeper principles of optimal coding. The analysis is based on a discovery that the symmetrical architecture of genetic code and the nucleon number of its constituents — the free canonical amino acids and bases, are strictly determined by the prime number with a cycling digit property −037. A detailed derivation for all types of analogues of 037 is given and shown that they represent a special case of cyclic numbers whose digits and multipliers are equidistant. The derived generic formula for analogues of 037 is correlated with cyclotomic and generalized Golden Mean polynomials, by which is shown their relation with discrete self-similarity, why author named them the Self-similar Numbers. Also it is given their relationship with the cyclic codes, which is with fact that the genetic code is arranged in multiple ways at different levels as a cyclic code, generally indicate that the hierarchically nested cyclicality is one of the principles of fractal organization of the genetic code, as well as the other organic codes in the living organisms.