About genetic code and the icosa tetrahedral tessellation of hyperbolic 3-space

The TGD based model for the genetic code (see this) relies on icosa tetrahedral hyperbolic tessellation (ITT) realized in the hyperbolic 3-space H³ representable as a light-cone proper time constant hyperboloid of light-cone of M⁴ or as a mass shell in momentum space.

1. The general idea that genetic codons as 6-bit units of ordinary "bitty" intelligence are accompanied by emotional intelligence represented in terms bio-harmonies serving as correlates for emotions. Music indeed expresses and creates emotions (for the evolution of ideas see this, this, this, this, and this). This view has far reaching implications. In particular, it means that emotions are present already at the biomolecular level. In the TGD Universe, life is universal and can appear in very many scales. This would be true also for the genetic code realized in terms of the icosa tetrahedral tessellation of H³ which can appear in arbitrary scales.
2. This interpretation of the genetic code belongs to the category of the intuitive "must-be-true" hypothesis of TGD, whose status has remained unclear. One reason for this is that I am not a specialist in the field of hyperbolic tessellations. Once again I realized that my understanding is far from perfect and decided to clarify my thoughts once again.
3. ITT involves tetrahedra (T), octahedra (O) and icosahedra (I). Genetic code would correspond to a fusion of 3 properly chosen icosahedral Hamiltonian cycles representing 12-note scale (there are many options) and one tetrahedral Hamilton cycle, which is unique. I have an intuitive geometric interpretation for this 3-1 structure: 3 I:s share 3 faces of T. This leaves one free face of T serving as an additional codon. This gives 20+20+20+1=64-3 codons and the missing 3 codons could correspond to stop codons. Also O:s are involved and the intuitive idea is that O is passive in the sense that it represents a void in the sense that the vertices, edges and faces of the octahedron can be regarded as those of octahedron or I. How to make this idea more concrete?

The ITT in the hyperbolic 3-space H³ (honeycomb) is completely unique because it includes as cells all Platonic solids, tetrahedron (T), octahedron (O) and icosahedron (I) for which the faces are equilateral triangles. One can characterize the tessellation by giving the numbers of 3-cells meeting at vertices, edges and faces.

Consider first the vertices.

1. The vertex figure of the ITT (see this) represents what an observed at a given vertex sees as intersection of a vertex-centered ball with the ITT. For instane, for cube, vertex figure is square for C(ube) and O, pentangon for I and triangle for T and D(odecahedron).

   For ITT vertex figure corresponds to an Archimedean solid known as icosadodecahedron (ID), which can be regarded as a hybrid of I and D. The 12 pentagons at the vertices of I as vertex figures of 12 I:s and the 20 triangles as vertex figures of 20 T:s correspond to vertices of D. ID has 20 triangular faces and 12 pentagonal faces, totaling 32 faces, with 30 identical vertices, at which two triangles and two pentagons touch, and 60 edges separating a triangle from a pentagon.

2. O is passive in the sense that only 20 I and 12 T but not O meet at the given vertex, hence the attribute "icosatetrahedral". One can say that O represents a void. Octahedron is a lower-dimensional example of this phenomenon: the square defining the vertex figure of O does not define a face appearing at the vertex. Only 4 triangles meet at a given vertex. This brings in mind giant voids of cosmology having galaxies at their boundaries. I have proposed that the tessellations of H³ realized as cosmological time a= constant hyperboloids in the light-cone of M⁴ could explain the observed quantization phenomena for the redshifts (see this). Could these large voids have something to do with the O:s of ITT?
3. How could one understand the 3-1 correspondence for I:s and T:s? A given T of the vertex figure is surrounded by 3 I:s. This suggests the T+3I defines a unit giving a realization of the genetic code, 4 units of this kind would meet at a given vertex.

   The proposed interpretation of T+ 3I as a unit conformas with the proposed view of the genetic code. I:s have 20 triangular faces and since the I:s have no common faces, this motivates the proposal that the 3I give rise to 20+20+20 icosahedral codons. The I:s would realize a Hamiltonian cycle with a symmetry group which is Z₆, Z₄ or Z₂. Z₂ would act as reflections or rotations. Z₆ cycle is unique, there are 2 Z₂ cycles and a large number of Z₂ cycles.

   The orbits of the symmetry group would correspond to amino-acids. Z₆ would give rise to 3 6-element orbits and 1 2-element orbit. Z₄ would give rise to 5 4-element orbits and Z₂ to 10 2-element orbits. This explains almost exactly the numbers of DNA codons coding for a given amino-acid. The 3 I:s share 3 common faces with T, which leaves one free face for T to which one can assign a tetrahedral genetic codon. The 3 missing tetrahedral faces would correspond to stop codons.

4. Interesting questions concern the interpretation of the cycles. The Hamilton cycle connects the nearest neighbor vertices of the Platonic solid. Does the cycle correspond to a closed monopole flux tube? What does it mean that one face (at least) for a given 3I+T unit is active and represents a codon: does it have protons at its vertices as the alternative realization of the genetic code in terms of the states of 3-proton triplets suggests (see this)? Can the 3I+T units of ITT contain different Hamiltonian cycles so that emotions could be local. Does DNA strand correspond to a linear structure as a substructure of ITT. Is the induction of ITT to 1-D, 2-D and even 3-D structures representing genetic code possible? Could for instance, cell membrane and microtubules represent 2-D realization of the genetic code. Could the brain and even the biological body represent a 4-D realization. Could these realizations be time dependent as the failure of strict non-determinism of the classical dynamics dictated by holography = holomorphy vision suggests: if so, even 4-D realization would be possible.

Also the numbers of cells meeting edges and faces characterize ITT.

1. Edge transitivity means that all edges are symmetry related just as the vertices are. At a given edge I, I, O, and T meet in a cyclic order IIOT.
2. 2 3-cells cells meet at a given face. Only I and T can share faces. and the shared faces correspond to O faces. O does not appear at the vertices, being realized as a "ghost" cell being analogous to the square appearing in O and having no physical realization as a face.

To sum up, the conjecture that the genetic code is realized in terms of ITT is now at a rather firm basis. During the last few years several ideas of TGD have reached a rather strong status as the understanding of the basic mathematical ideas of TGD has increased and TGD is now a mature mathematical theory and can be applied in all scales.

See the article How the genetic code is realized at the level of the magnetic body of DNA double strand? or the About honeycombs of hyperbolic 3-space and their relation to the genetic code.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.