Generalized Genomic Matrices, Silver Means, and Pythagorean Triples

Jay Kappraff¹* and Gary W. Adamson²

¹Mathematics Department, NJIT, University Heights, Newark, NJ 07102, U.S.A.
²P.O. Box 124571, San Diego, CA 92112-4571, U.S.A.
*E-mail address: kappraff@adm.njit.edu

(Received August 23, 2008; Accepted November 2, 2008)

Abstract. Petoukhov has shown that a family of bisymmetric 2ⁿ × 2ⁿ 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.

Keywords: dna/rna, Golden Mean, Amino Acids, Nicomachus