Pythagoras, Music, Sacred Geometry and Genetic Code

Abstract

A given triangle of icosahedron can contain 0, 1 or 2 edges of the cycle and the numbers of the triangles corresponding to these triangle types classify partially the notion of harmony characterized by the cycle. Quint cycle suggests the identification for the single edge of curve as quint interval so that triangles would represent basic 3-chords of the harmony with 0, 1, or 2 quints. Octahedron and cube which are duals of each other and have 6 and 8 vertices respectively, and dodecahedron which is dual of icosahedron having 20 vertices and 12 faces. Arabic music uses half intervals and scales with 19 and 24 notes are used. Could 20-note scale with harmony defined by 5-chords assigned to the pentagons of dodecahedron have some aesthetic appeal?  The combination of this idea with the idea of mapping 12-tone scale to a  Hamiltonian cycle at icosahedron leads to the question whether amino-acids could be assigned with the  equivalence class of Hamiltonian cycles under icosahedral group  and whether the geometric shape of cycle  could correspond to physical properties of amino-acids. The identification of 3 basic polar amino-acids with triangles containing no edges of the scale path, 7 polar and acidic polar amino-acids with those containing 2 edges of the scale path, and 10 non-polar amino-acids with triangles containing 1 edge on the scale path is what comes first in mind. The number of DNAs coding for a given amino-acid could be also seen as such a physical property. The model for dark nucleons leads to the vertebrate genetic code with correct numbers of DNAs coding for amino-acids. The treatment of the remaining 4  codons and  of the well-known 21st and 22nd amino-acids requires the fusion of icosahedral code with tetrahedral code represented  geometrically as fusion of icosahedron and tetrahedron along common  face which has empty interior and is interpreted as "empty" amino-acid coded by stopping codons.  In this manner one can satisfy the  constraints on the Hamiltonian cycles, and construct explicitly the  icosahedral  Hamiltonian cycle  as (4,8,8) cycle whose unique  modification gives (4,11,7) icosa-tetrahedral cycle.