**Bioharmony and resonance mechanism for dark photon communications**

The faces of icosahedron and tetrahedron (and also octahedron appearing in the model of genetic code as icosa-tetrahedral tessellation of hyperbolic space H^{3} (see this) are triangles. The proposal is that they somehow correspond to 3-chords made of dark photons, which in turn represent genetic codons.

Communications by dark 3-photons represent codons. 3N-photons represent in turn genes. The communications rely on cyclotron 3N-resonance so that the vertices of the faces of icosa-tetrahedron must contain charged particles coupling to a magnetic field. The magnetic field strengths at flux tubes associated with charged particles would determine the cyclotron frequencies.

Information is encoded to the frequency modulation of cyclotron frequencies. The chords serve as addresses much like in computer language LISP. If the modulations of 3N frequencies are identical and in synchrony, the outcome of the receiver consisting of 3N charged particles is a sequence of 3N-resonances giving rise to an 3N-pulse sequence. Nerve pulse patterns could emerge by this mechanism.

One can also consider 3N-signals for which only M<3N modulations are identical and in synchrony. In this manner communications to subsets of the receiver are possible. For instance, some subset of codons of dark gene or dark protein can be selected as a receiver, possibly controlled. This selection could de-entangled the receiver to de-entangled coherent pieces.

There is a direct connection with empiria. Biophotons, whose origin remains poorly understood, can be identified as ordinary photons resulting from the decay of dark 3N-photons to ordinary photons.

The realization in terms of dark nucleons looks more plausible if also DtRNA and DAAs are realized in terms of icosa-tetrahedral picture (the dark counterpart of information molecules X will be denoted by DX). This is because the amino acids are often neutral unlike DNA nucleotides which are negatively charged. The dark charge assignable to the icosa-tetrahedron can be controlled by pionic bonds with charges 0,+/- 1 so that it can be 3 units for DDNA and vanish for amino acids. A natural proposal is that the charge of icosa-tetrahedron compensates the charge of the amino acid and tRNA.

There are pairings of type DX-DY. The pairings DDNA-DRNA, DRNA- DtRNA and DtRNA-DAA induce the biochemical dynamics of transcription and translation. There are also pairings DX-X. DDNA-DNA and DRNA-RNA unique DtRNA-tRNA pairing is 1-to-many and relates to the wobble phenomenon. The pairings between dark nucleon variants of biomolecules and corresponding dark 3N-photons make possible biocommunications and control.

**Details of the bioharmony model**

Consider now a more detailed bioharmony model of the genetic code based on the geometries of icosahedron and tetrahedron.

- Icosahedron has 12 vertices and 20 faces, which are triangles. The idea is that the 12 vertices correspond to the notes of 12-note scale. Tetrahedron has 4 vertices and 4 faces and is self-dual whereas the dual of icosahedron is dodecahedron having 20 faces and 12 faces.
- 12-note scale can be represented as a Hamiltonian cycle at an icosahedron going once through all vertices. The frequencies at the neighboring points as edges of a face in cycle relate by a frequency scaling of 3/2: this gives rise to the Pythagorean variant of quint cycle.
Octave equivalence means the identification of frequencies differing by a multiple of octaves. Octave equivalence can be used to reduce all frequencies to a single octave. If the scaling is exactly 3/2 at all steps there is a slight-breaking of octave equivalence since (3/2)

^{12}does not quite correspond to an integer number (7) of octaves. Pythagoras was well aware of this.Given cycle assigns to faces 3-chords defining a harmony with 20 chords assignable to the faces of the icosahedron. For dodecahedron there is only single harmony with 12 chords and 20-note scale which could correspond to Eastern scales. For the tetrahedron the Hamiltonian cycle is unique.

- Icosahedral Hamiltonian cycles can be classified by symmetries. The group Z
_{6}, Z_{4}, or Z_{2}(rotation by π or reflection) as a group of symmetries

- The natural idea is that the faces of the icosa-tetrahedron correspond to both 3-chords and genetic DNA/RNA codons. If the orbits of faces could correspond to amino acids (AAs), the DNA codon would code for amino acid AA if the corresponding face is at the orbit corresponding to AA.
- One wants 64 DNAs: Z
_{6},Z_{4}ja Z_{2}cycle give rise to 20+20+20 =60 DNa codons. Tetrahedron gives the remaining 4 codons. - Does one obtain a correct number of AAs? Do the numbers of faces at the orbits correspond to numbers of DNAs coding for the corresponding AA?
- Z
_{6}decomposes to 3 6-orbits and 1 2-orbit (3× 6+2 =20). There are 3 AAs coded by 6 DNAs. 2-orbit corresponds to AA coded by two DNAs. - Z
_{4}decomposes to 5 4-orbits. There are 5 AAs coded by 4 codons. - Z
_{2}corresponds to 10 2-orbits predicting 10 AAs coded by 2 codons.There would be 11 2-orbits altogether. There are 9 AAs coded by 2 codons.Some kind of symmetry breaking is present as in the case of dark nucleon code. 2 AA doublets must split to singlets. If (ile,ile,ile,met) coded by UAX could correspond to (ile,ile) and (met,met) such that (met,met) is split to (ile,met). In absence of symmetry breaking one would have 11 doublets as predicted.

- Z
- There are also 4 tetrahedral codons.
There is (stop,stop) doublet (UAA, UAG) and (stop,trp) doublet (UGA,UGG). These doublets could correspond to the faces of the tetrahedron. Only one face would code for amino acid in the vertebrate code. Other faces would not have corresponding tRNA?

For bacterial codes, the situation can be different. Pyl and sec appear as exotic amino acids. Could (UAA,UAG) for code for (stop,pyl) and (UGA,UGG) for (sec,trp) instead of (stop,trp)? Orientation preserving rotations form a 12-element group having Z

_{2}and Z_{3}as subgroups. For Z_{2}the orbits consist of 2 vertices and for Z_{3}of 3 vertices (face) and 1 vertex. Z_{3}symmetry could correspond to trp as singlet and vertebrate stop codons as triplet. For bacterial pyl and sec Z_{2}with symmetry breaking is suggestive.

**Bioharmony, dark nucleon code, and icosa-tetrahedral code as a tessellation of H**

^{3}Bioharmony model involves icosahedron and tetrahedron. This looks ugly unless there is some really deep reason for their emergence. One can also ask why not also octahedron having triangular faces.

Hyperbolic 3-space H^{3} has interpretations as a mass shell of Minkowski space M^{4} at the level of M^{8} and as light-cone proper-time constant surface at the level of H. The 4-surface X^{4} in M^{8} contains mass shells of M^{4} corresponding to the roots of the polynomial P defining X^{4}. Hence one expects that H^{3} plays a key role in quantum TGD both discretized momentum as defining a cognitive representation with momenta, which are algebraic integers associated with extension of rationals defined by P. H^{3} has infinite discrete subgroups of the Lorentz group analogous to discrete groups of translations in E^{3} as isometries and H^{3} allows an infinite number of tessellations (lattices).

Perhaps the simplest tessellation is icosa-tetrahedral tessellation involving also octahedrons and thus all triangular Platonic solids. This tessellation could give rise to genetic code by induction of tesselation to 3-surfaces or lower-D objects such as linear biomolecules, and cell membranes (see this). I do not however understand the mathematical details well enough but the following discussion is general.

Consider first the model for DDNA and DRNA allowing us to understand the connection between dark nucleon and dark photon realization of the genetic code physically.

- The realization of DDNA/DRNA/DtRNA/DAA could correspond to a sequence of icosahedron-tetrahedron pairs at H
^{3}contained by the 4-surface X^{4}⊂ M^{8}and its H images which is also H^{3}. - Each icosa-tetrahedron would contain a dark codon realized both as a face and dark nucleon triplet associated with it. The dark photon chord associated with the face must be the same as the codon defined by dark nucleon triplet. The dark nucleon triplets correspond to cyclotron frequency triplets, which in turn correspond to dark photon 3-chords associated with the Hamiltonian cycles.
- The cyclotron frequencies are determined by magnetic fields at flux tubes so that Hamilton cycles must correspond to flux tube patterns. The simplest hypothesis is that the Hamilton cycle is a closed flux tube connecting all vertices of the icosahedron. Dark codon triplet corresponds to a face with 3 flux tube edges.
The simplest option is that the flux tubes defining the edges define the cyclotron frequencies defining the dark codon in bioharmony. The variation of flux tube thickness implies frequency modulation crucial for communications.

The realization of the Hamilton cycle requires that the magnetic field strength along the cycle is scaled by factor 3/2 to give a quint cycle.

- An interesting question relates to the relation of DDNA strand and its conjugate. The change of the orientation of the Hamiltonian cycle changes the chord of the harmony. For the ordinary 8-note scale one can roughly say that major and minor chords are transformed to each other. The orientation reversal could correspond to time reversal. The fact that the orientations of two DNA strands are opposite suggests that DNA and conjugate DNA are related by the orientation reversal of the Hamiltonian cycle inducing the map G→ C, U→ A a the level of DNA letters. The conjugation does not imply any obvious symmetry for the corresponding amino acids as the inspection of the code table demonstrates.

- DRNA codons pair with 32 DtRNA codons and DtRNA codons pair with trNA codons in 1-to-many manner. Therefore DRNA-DtRNA pairing could be universal and 2-1, although not in a codon-wise manner. This pairing should be the same for both bioharmony and dark nucleont triplets.
- The pairing by 3-resonances requires that DtRNA icosa-tetrahedron contains the DRNA codons, which pair with DtRNA codon. There would be 2 DRNA codons in DtRNA icosahedron for most DtrNA codons and 1 codon for DtRNA pairing with DAA corresponding to met and trp. The number 32 of DtRNA implies in the case of icosa-tetrahedral code that there are 10+10-10=30 icosahedral DtRNAs and only 2 tetrahedral DtRNAs so that two faces of tetrahedron cannot correspond to DtRNA codon so that corresponding DRNAs must serve as stop codons.
One of the DtRNAs could correspond to trp. The second one would correspond to a stop codon in the vertebrate code: either the DtRNA codon is not present at all or or it does not pair with tRNA. TAG and TGA can code for pyl and sec in some bacterial versions of the code and in this case the corresponding dark DRNA codon would be represented at the DtRNA tetrahedron.

- For bioharmony DDNA-DAA correspondence means that AAs correspond to orbits of the faces of icosahedron under the subgroup Z
_{6},Z_{4}, or Z_{2}which could correspond to reflection or to a rotation by π.Since DRNA-DtRNA correspondence is 2-1 although not codon-wise, the natural first guess is that Z

_{2}orbits of the faces define the DRNA codons at the DtRNA icosahedron so that it would contain 2 codons for most DtRNAs. At the DtRNA tetrahedron the only option is Z_{1}so there is a symmetry breaking.If Z

_{2}corresponds to a reflection, the orbit always contains 2 codons. If Z_{2}corresponds to a rotation by π, it might happen that the face invariant under π rotation and the orbit would consist of a single point. Could this explain why one has (ile,ile,ile,met) instead of (ile,ile) and (met,met)? The rotation axis should go through the invariant face and since the face is a triangle, π rotations lead out of the icosahedron. Therefore the answer is negative.

- The pairing of Z
_{2}related DDRNA faces with two different DtRNAs coding for ile and met rather than two mets means Z_{2}symmetry breaking at the level of bioharmony. Could the fact that AUG acts as a start codon relate to this? Could it be that both AUG and AUA cannot act as start codons? It is difficult to invent any raeson for this. - The symmetry breaking could occur in DtRNA-DAA pairing and replace Dmet with Dile. Is it possible that the 3-chords for coding for ile and second met are nearly identical so that the resonance mechanism selects ile instead of met? Could the situation be similar for the codons coding for (stop,stop) and (stop,trp) and cause the coding of pyl or sec in some situations? The scale for the quint cycle model with octave equivalence does not quite close. Could this have some role in the problem?
- Since similar ambivalence occurs for stop codons assigned to the tetrahedral Hamiltonian cycle, one can look at the tetrahedral Hamiltonian cycle. In this case a given edge of the cycle corresponds to a scaling by (3/2)
^{3}so that 4 steps gives (3/2)^{12}, which is slightly more than 7 octaves. For the quint scale in Pythagorean sense, one obtains 4 notes in the same octave.Exact octave equivalence corresponding to equally tempered scale in which half-note corresponds to frequency scaling 2

^{1/12}, implies that there is only one 3-chord CEG#: this would explain why there are 3 stop codons in the vertebrate code!If bacterial codes correspond to Pythagorean scale, there would be two different 3-chords since CEG# and EG#C are not quite the same. The reason is that the frequency ratios of chords are powers of 3/2)

^{12}. This situation is completely exceptional.In the quint scale there are small differences between the 4 chords. Could this explain why only one of these 3-chords codes for AA (trp) in vertebrate code and pyl or sec is coded instead of stop in bacterial codes? Amusingly, the chord CEG# ends many finnish tangos and therefore acts like a stop codon!

Could bacteria have a perfect pitch and live in a Pythagorean world? Could the transition to multicellulars mean the emergence of an algebraic extension of rationals containing 2

^{1/12}≂ 1.059(, which is considerably larger than to (3/2)^{12}/2^{7}≂ 1.0136)! Could people with perfect pitch have in their dark genome parts using Pythagorean scale or can they tune the magnetic flux tube radii to realize Pythagorean scale? - Could the ile-met problem have a similar solution? The chords associated with ile and met would differ by ascaling with (3/2)
^{3}or (3/2)^{6}using octave equivalence. These chords are not quite the same: could it happen that the 3-chord associated with the second met is nearer to that for ile? These 3-chords do not contain quint scaling and should correspond to the special chords for which no edge belongs to a Hamiltonian cycle.

- DAA icosahedron must contain the DtRNA codons pairing with DAA. This raises the question whether DDNAs could have a direct resonant coupling to DAAs. Could this pairing occur in DDNA-DAA occurring in transcription (see this) so that pieces of DDNA and DAA associated with an enzyme involved could pair with each other by 3N-resonance at DDA-DAA level? At the chemical level the base-amino acid interactions are extremely complex involving stereochemistry and formation of hydrogen bonds (see this) so that the reduction of these interactions to 3N-resonance would mean a huge simplification.
- Could this resonance pairing serve as a universal mechanism of bio-catalysis and take place for various enzymes and ribozymes? One example are promoters and enhancers involved with the transcription. Enhancers and promoters induce a highly non-local process generating a chromosome loop in which two portions of DNA become parallel and near to each other and dark 3N-photons could explain the non-locality as an outcome of quantum coherence in long scales.
- Why would DDNA-DAA pairing not occur? 3N-resonance relies on cyclotron frequencies and therefore on the magnetic field strength determined by the radii of the monopole flux tubes. One explanation would be that the frequency scales of DAA and DDNA are slightly different. Could the attachment of DRNA to translation machinery scale the magnetic field strengths of the flux tubes and their cyclotron frequencies so that only dRNA-DtRNA and DtRNA-DAA couplings are possible.

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