
ABSTRACT
Petoukhov has shown that a family of bisymmetric 2^{n} x 2^{n} matrices encode the structure of the four RNA and DNA bases and 64 codons that make up the 20 amino acids in all living structures. He discovered that the elements of the square roots of these matrices are all powers of the golden mean. We have generalized his matrices and shown that the square roots of general bisymmetric matrices are generalizations of the golden mean including a subclass that correspond to the family of silver means. Powers of these matrices are also shown to generate all Pythagorean triples. The integers in these matrices are identical to the set of integers in a table attributed to the second century Syrian mathematician, Nicomachus, who used them to describe the ancient musical scale of Pythagoras.
1. INTRODUCTION
Petoukhov (2001, 2004), (He, 2005) has studied a family of bisymmetric 2^{n} x 2^{n} matrices that code the structure of the four DNA/RNA 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. 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. These degeneracies are propagated through the 17 different genome classes of RNA/DNA. The particular class of DNA/RNA that we will be studying in this paper is the class of mitochondrial DNA. Although different groups of codons correspond to the same amino acid in different genome classes, the quality of the degeneracy (high or low) is preserved. The first matrix of the family expresses the fact that two of the RNA bases have 3 hydrogen bonds while the other two have 2 hydrogen bonds. The elements of the rows and columns of this family of matrices reproduce the sequences of musical fifths, i.e., integer ratios of 3:2, found in a table attributed to the Syrian mathematician of the first and second century AD, Nicomachus (Kappraff, 2000a). The integer values in this table have multiplicities given by the rows of Pascal’s triangle. The square roots of this family of matrices have entries that are all powers of the golden mean. A brief discussion of Petoukhov’s approach to genetic coding is given in Appendix A.
We have generalized Petoukhov’s matrices to a family of bisymmetric matrices in which the first 2x2 matrix has a pair of positive real numbers as elements, but are otherwise arbitrary. Bisymmetric matrices are matrices whose elements are symmetric with respect to both left and right leaning diagonals. We derive general formulas for the elements of the square root of this matrix. They are irrational numbers that are generalizations of the golden mean. In fact for a subclass of the matrices the elements of the square root matrix are generalizations of the golden mean known as silver means. Finally, we show that when the elements of the bisymmetric matrix are positive integers, powers of these matrices generate Pythagorean triples.
2. PETOUKHOV’S GENOMIC MATRICES
Petoukhov has shown that the four nitrogenous bases that make up RNA and DNA, adenine, cytosine, guanine, and uracil/thymine: A,C,G,U/T are equivalent in two different ways.
a. C=U and A=G according to the relation, “pyrimidine or purine.”
b. C=G and A=U/T according to the relation, “possesses three hydrogen bonds or two hydrogen bonds (Watson, 1953)”
These two properties characterize a family of matrices related to the four bases. The first of these matrices, the 2x2 RNA Matrix 1, specifies the four bases in which C is coded by 11, A by 10, U by 01 and G by 00. For relation a) C and U are pyrimidines and are assigned the value 1 in column 1 while A and G are purines and are assigned the value 0 in column 2 . For relation b) C and G have three hydrogen bonds coded by 11 and 00 respectively in Matrix 1, while A and U have two hydrogen bonds coded by 10, and 01 respectively. In this manner the bases are assigned the values 3 and 2 respectively along the two diagonals of Matrix M_{1} shown below.