**Authors:** Matti Pitkänen

t the 12-note scale could allow mapping to a closed path covering all vertices of icosahedron having 12 vertices and not intersecting itself is attractive. Also the idea that the triangles defining the faces of the icosahedron could have interpretation as 3-chords defining the notion of harmony for a given chord deserves study. The paths in question are known as Hamiltonian cycles and there are 1024 of them. There paths can be classified topologically by the numbers of triangles containing 0, 1, or 2 edges belonging to the cycle representing the scale. Each topology corresponds to particular notion of harmony and there are several topological equivalence classes.

I have also played with the idea that the 20 amino-acids could somehow correspond to the 20 triangles of icosahedron. 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 a topological equivalence class of Hamiltonian cycle and whether topological characteristics could correspond to physical properties of amino-acids. It turns out that 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 would be consistent with the constraints on the Hamiltonian cycles. One could of course criticize the lumping of acidic polar and polar aminoacids to same group.

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. It is however far from clear how to interpreted DNAs geometrically and the problem whether one cold understand genetic code geometrically remains open.

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[v1] 2014-09-26 07:05:54

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