Symmetries, Generalized Numbers and Harmonic Laws

This paper is devoted to the presentation and analysis of matrix representations of the genetic code. Principal attention is paid to a family of the genetic matrices which are constructed on the basis of Gray code ordering of their rows and columns. This Gray code ordering reveals new connections of the genetic code to: 8dimensional bipolar algebras; Hadamard matrices; golden matrices; Pythagorean musical scale, and an integer triangle attributed to Nicomachus, a Syrian mathematician from second century A.D. All of these mathematical entities possess symmetrical features. Some questions about silver means and Pythagorean triples are also described by these genetic matrices and their generalizations.