Generalized DNA matrices, Silver Means, and Pythagorean Triples

Petoukhov [1], [2], [3] has studied a family of bisymmetric 2 n x 2 n matrices that code the structure of the four DNA bases, the 64 codons that make up the 20 amino acids in all living structures, and beyond that, the proteins assembled from the amino acids as building blocks [1], [2], [3]. As the result of his studies he has found that the amino acids express certain degeneracies, 8 with high degeneracy containing 4 or more codons, and 12 with low degeneracy, containing less than 4 codons. He suggests that these values may be the result of 24 hour chronocycles. These degeneracies are propagated through 17 different classes of DNA. Although different groups of codons correspond to the same amino acid in different DNA types, the quality of the degeneracy (high or low) is preserved. The first matrix of the family expresses the fact that two of the DNA bases have 3 hydrogen bonds while the other two have 2 hydrogen bonds. The rows and columns of his family of matrices reproduce the sequences of musical fifths, i.e., integer ratios of 3:2, found in a table attributed to the Roman mathematician of the first century AD, Nicomachus [4]. The integer values in this table have multiplicities given by the columns of Pascal’s triangle. The square roots of this family of matrices have entries that are all powers of the golden mean.