Music, Biology & Natural Geometry (Part I)

Abstract

In an earlier article I introduced the notion of Hamiltonian cycle as a mathematical model for musical harmony and also proposed a connection with biology: motivations came from two observations. The number of icosahedral vertices is 12 and corresponds to the number of notes in 12-note system and the number of triangular faces of icosahedron is 20, the number of amino acids. This led to a group theoretical model of genetic code and replacement of icosahedron with tetraicosahedron to explain also the 21st and 22nd amino-acid and solve the problem of simplest model due to the fact that the required Hamilton's cycle does not exist. This led also to the notion of bioharmony. This article is meant to be a continuation to the mentioned article providing a proposal for a theory of harmony and detailed calculations. It however turned out that the proposed notion of bioharmony was too restricted: all isosahedral Hamilton cycles with symmetries turned out to be possible rather than only the 3 cycles forced by the assumption that the polarity characteristics of the amino-acids correlate with the properties of the Hamiltonian cycle. In particular, it turned out that the symmetries of the Hamiltonian cycles are the icosahedral symmetries needed to predict the basic numbers of the genetic code and its extension to include also 12st and 22nd amino acids. One also ends up with a proposal for what harmony is leading to non-trivial predictions both at DNA and amino-acid level.

Part I includes the main part of this article.