Rational Platonia
Pythagoras believed that rationals are all that is needed for a Universe and for him the discovery of \sqrt{2} represented geometrically by the diagonal of a unit square was probably a shock.
It is interesting that in the TGD framework the rationals appear naturally. In its simplest form, Galois confinement (see this and this) states that the total 4-momenta of physical states are Galois singlets invariant under Galois group permuting roots of a given polynomial (the notion generalizes if one considers functions in momentum space). This would allow only momenta with components, which are integers when a physical natural momentum unit is used. Platon would have been right in a certain sense!
However, Galois singlets would at fundamental level consist of quarks (in particular leptons and bosons would do so) having 4-momenta with components, which are algebraic integers in the extension of rationals defined by the polynomial defining the space-time region considered (see this and this). One could regard the algebraic integer valued momenta as virtual momenta characterizing the building bricks of physical states.
Special role of primes 2, 3 and 5
The number mysticism of Pythagoras involves the idea that the numbers 2 and 3 are very special. Using the language of modern number theory, one could say numbers 2 and 3 span a group with respect to multiplication consisting of numbers 2m3n, where m and n are integers. One could call this group B(2,3). If m and n are restricted to non-negative integers, the inverses do not exist and only a semigroup is obtained. This object could be called A(2,3).
If Pythagoras identified rational numbers as a kind of Platonia, this group might be said to define an important province of Platonia. A more general object would be ideal consisting of all integers proportional to, say, 6=2× 3 closed with respect to multiplication by any integer.
It should be noticed that any set (p1,...,pn) of primes and even integers defines a group with respect to multiplication as the group B(m1,m2,...,mn) of integers. Especially interesting example is the group B(2,3,5) containing B(2,3) and B(3/2).
p-Adic length scale hypothesis states that powers of small primes near to prime define important p-adic length scales. Powers of 2 are of special importance in p-adic mass calculations (see this and this) but there exists also evidence for powers of 3 (see this).
Decimal system is the decimal system used in everyday life and very often numerologists freely change the position of a decimal number and get results, which make sense only if the decimal system is in a special role. Could this be the case? If so, then the decimal system would not reflect only the fact that we have 10 fingers, and also the algebras B(2,5) and B(10) could be special.
There are some indications that this might be the case.
- The faces of icosahedron resp. dodecahedron are triangles resp. pentagons so that numbers of 3 and 5 are natural.
- DNA is a helical structure with a twist angle 2\pi/10 between to codons so that 10 codons make a 6\pi twist and define length scale 10 nm which is the p-adic length scale associated with Gaussian Mersenne prime MG,151= (1+i)151-1, one of the 4 Gaussian Mersennes defining p-adic length scales in the range 10 nm, 2.5 μm. These scales are a number theoretic miracle. Numbers 2,3, and 5 relate to the geometry of DNA.
Pythagorean scale
Pythagoras also studied music and introduced the notion of Pythagorean scale for which the frequencies of notes are in rational ratios. A standard manner to realized this scale is by quint cycle, which means that one forms the multiples (3/2)nf0 of fundamental frequency f0 and identifies them by octave equivalence with a frequency in the basic octave [f0,2f0]. The quint cycle appears very often in jazz.
For n=12 the frequency obtained is almost a full number of octaves but quite not. This imperfectness of Platonia troubled Pythagoras a lot. In an equal tempered scale one introduces powers 2m/2nf0 and avoids this problem. This means replacement of rationals by its algebraic extension generated by 21/12.
Obviously, the Pythagorean scale is very natural in the framework of group B(3/2). Pythagoras also had ideas about the relationship of music scale and Platonic solids.
Pythagorean scale and genetic code
In the TGD framework, the idea about a possible connection between music and Platonic solids inspired the proposal about realization of the 12-note scale as a Hamiltonian cycle at icosahedron. The Hamiltonian cycle is a closed curve connecting only neighboring points of the icosahedron and going through all its 12 vertices. There are quite a large number of icosahedral cycles and they assign to the 20 triangles of icosahedron 3-chords proposed to define icosahedral harmony with 20 chords. The non-chaotic icosahedral cycles have symmetry groups Z6, Z4, and Z2, which can act as a rotation or reflection.
The big surprise was that the model of icosahedral harmony leads to a model of genetic code. The code would involve a fusion of 3 different icosahedral harmonies with symmetry groups Z6, Z4, and Z2 giving 60 codons plus tetrahedral code giving 4 codons. The counterparts of amino-acids would correspond to the orbits of these symmetry groups: 3 orbits with 6 triangles and 1 with 2 triangles as orbits for Z6, 4 orbits with 4 triangles for Z4 and 10 orbits with 2 triangles for Z2. The number of triangles at the orbit is the number of DNA codons. Tetrahedron would give the missing 4 codons and stop codons and one missing amino acid.
For a given choice of the 3 Hamiltonian cycles, the realization would be in terms of 3-chords of light defining harmony for a music of light (and possibly also sound). Since music expresses and generates emotions, the proposal was that this realization of the genetic code expresses emotions already at the molecular level and that emotional intelligence corresponds to this realization whereas bit intelligence would correspond to the interpretation of codons as 6-bit sequences.
It should be mentioned that Hamiltonian cycles are solutions to the travelling salesman problem at the icosahedron: cities would correspond to the vertices. In the case of dodecahedron, which is dual of icosahedron, there is only one Hamiltonian cycle so that the harmony is now unique. If this corresponds to harmony, the first guess is that there would be a 20-note scale and 12 5-chords.
What about dodecahedral harmony and analog of genetic code?
Could also dodecahedron define a bioharmony and an analog of genetic code?
- The first guess is that dodecahedral harmony has 20 notes per octave and perhaps corresponds to the scale defined by micro-octaves used in Eastern music. There would be 12 5-chords and the harmony would be unique. There would exist only a single emotional mood, a kind of enlightened state.
- Since the harmony is unique, and there are no other Platonic solids with pentagons as faces. The analog of genetic code should correspond to dodecahedral harmony. The 5-chords would define 12 analogs of DNA codons.
The dodecahedral cycle Z3 acts as a symmetry group (\url{https://jrh794.wordpress.com/2021/04/01/the-original-hamiltonian-cycle-continued/}). This means that there are 4 orbits of Z3 with 3 codons at each and they would correspond to 4 different analogs of amino-acids.
- Could one consider instead of an icosahedral quint cycle with scaling 3/2 replaced with scaling 5/2? The tempered system would use powers of 21/20 to generate a 20-note scale. A single step along Hamilton's cycle connecting neighboring vertices of dodecahedron would correspond to a scaling f\rightarrow 5/2f plus octave equivalence.
By octave equivalence the scaling b y 5/2 would correspond to a transition from say C to a note between Eb and E. The microintervals between this note and either Eb or E appears in blues, jazz etc as a blue note. This interval is between minor and major.
- One can test this. The cold shower is that (5/2)20 is not near to a power of 2. However, one has (5/2)19/225= 1.084 (for the quint cycle in the icosahedral case the deviation that Platon was worried of, is about 1 percent). As if one had a 19-note scale. A completely analogous situation is encountered with bio-harmony. The scales assigned to Z6,Z4, and Z2 give rise to 19 amino-acids as orbits of these groups. One amino-acid is missing and the tetrahedral code gives this amino-acid plus 3 stop codons (see this).
The icosa-dodecahedral duality suggests that the scale should consist of 19 notes only. Note however that for an equal tempered variant of the scale one does not have this problem.
- Dodecahedral code predicts 4 analogs of amino acids. Could these "amino acids" correspond to the 4 DNA codons? 3 dodecahedral codons would be needed to code for a single genetic codon.
Could he dodecahedral codons, which correspond to 5-chords, be realized as dark 5-photons and sequences of dark 5-protons. One should check whether the states of 5 dark protons could give rise to 12 dark dodecahedral codons and whether something analogous to 12 dark RNA codons, dark tRNA codons, and 4 dark amino acids could emerge.
For the dodecahedral bioharmony, 5-chords would label the codons and they would serve as addresses based on communications relying on cyclotron resonance. Icosahedral harmony would control codons and dodecahedral harmony would code for their letters so that the codes would appear in different scales.
One can however consider alternatives. For instance, could the passive DNA strand correspond to a dodecahedral realization at the level of letters and the active strand to the icosahedral realization at the level of codons. Or could "junk" DNA and introns in promoter regions correspond to the dodecahedral realization with dark dodecahedral DNA controlling single letters.
See the article The realization of genetic code in terms of dark nucleon and dark photon triplets or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
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