### Sacred geometry and biology

Platonic solids and Golden ratio are in central role in what some people call sacred geometry. I know that the average colleagues cannot tolerate words like "sacred geometry". In my humble opinion names (nor even Names) do not matter much, and one should look for what is behind the names. These Platonic structures are extremely interesting since they relate to discrete and finite subgroups of 3-D rotation group characterizing the symmetries of also biomolecules. In particular, tey involve Golden ratio known to pop up again and again in biology. They also relate to the inclusions of hyperfinite factors and there exists a fascinating correspondence discovered by McKay between the discrete subgroups and laced Lie algebras. For instance, exceptional Lie groups correspond to tetrahedron, octahedron, and icosahedron in McKay correspondence.

The following considerations suggests that tetrahedral and icosahedral geometries might have deep connections to biology and genetic code in the sense that genetic code could have geometric representation in terms of tetrahedrons and icosahedrons. For more details see the article Could one find a geometric realization for genetic and memetic codes? and for background the article What are the counterparts of Einstein’s equations in TGD?. In the following I try to represent the ideas as "slides" (used originally as a synopsis of a Skype discussion) for an imagined audience (this is just a trial motivated by my laziness).

** A brief summary**

- The problem of sacred geometrician (icosahedron consists of 20 tetrahedrons which are quite not regular) has a solution which requires deformation of Euclidian space to 3-sphere locally.

This is possible in sub-manifold gravity. This is not however enough: this sub-manifold gravity must be strong. TGD predicts both sub-manifold gravity and a hierarchy of strong gravities.

- Tetrahedron has 4 faces (number of code letters A, T, C, G of genetic code) and icosahedron 20 faces (number of amino-acids: accident or connection of genetic code with sacred geometry?)

Could symmetry breaking for icosahedron assign to a given face of icosahedron a unique amino-acid? What could select this face so that one could say that one particular icosahedron corresponds to a given amino-acid? Does it involve gluing of amino-acid to particular face?

- TGD predicts besides genetic code also memetic code.

Combinatorial hierarchy defined recursively M

_{2}= 3 ,M_{3}=7, M_{7}=127, M_{1}27= 2^{127}-1,... The lowest Mersenne numbers are primes. The rest also if Hilbert was right.

One can assign codes to these Mersenne primes. Number of codons: 2

^{3-1}=4 for M_{3}, 2^{7-1}= 64 for M_{7}(genetic code). 2^{127-1}= 2^{126}for memetic code.

Genetic code corresponds to M

_{7}=127 and is followed by memetic code for which codewords correspond to 2^{126}sequences of 21 DNA codons.

!21=20+1. Is this an accident?

Could icosahedral geometry provide a geometric realization of memetic code?

- TGD suggests a dark realization of genetic code realized in terms of dark proton sequences.

Genetic code realized at the level of dark nuclear physics!

!Lowest magic numbers for nuclei correspond to the numbers of faces for tetrahedron (4), octahedron (8), and icosahedron (20). Could nuclear physics have a connection with dark geometry In nuclear string model the folding of nuclear strings completely analogous to protein folding to these shapes could indeed give rise to magic nuclei.

!?Could memetic codons be realized as sequences of 21 dark protons folded to icosahedron with 20 faces +1 dark proton attached together with amino-acid to one face! Amino-acid sequences accompanied by sequences of dark memetic codons. Do proteins carry a huge amount of memetic information which we know nothing about!!??

!?Could some parts of intronic portion of ordinary DNA realize memetic code chemically? Could cultural evolution as opposed to purely biological relate to the intronic portion of DNA? Could dark protons at cell membrane be connected by magnetic flux tubes to DNA codons could make possible topological quantum computations and evolution of software making possible cultural evolution? If so, cultural evolution would correspond also to the evolution of magnetic body leading to a hierarchy of levels of collective consciousness.

** The problem of sacred geometrician and the first observation**

Icosahedron (12 vertices, 20 faces ) is Platonic solid as are also tetrahedron, cube, octahedron, dodecahedron.

Observation: Icosahedron consists of tetrahedrons, which are however**not** quite regular. The ratio of "surface" edges to radial edges is 1.05 rather than 1. Scaling does not help. Sacred geometrician does not feel happy;-).

- Is there any manner to obtain 20
**regular**tetrahedrons?

!There is! If Euclidian 3-space E

^{3}is replaced with 3-sphere S^{3}, one obtains 20**regular**tetrahedrons.

Could gravitation modify distances by making space-time non-flat. Could gravitation allow to deform piece of E

^{3}containing icosahedron to a piece of S^{3}so that one would have sacred geometry?

!In TGD universe sub-manifold gravity allows to take a piece of E

^{3}realized as 3-surface and deform it it CP_{2}directions to make it look locally like 3-sphere! One obtains 20 *regular* tetrahedrons!

- Is gravitational deformation physically possible?

Gravitation according to General Relativity cannot produce local S

^{3}geometry in scales of biology and and condensed matter physics. For a system with mass M the scale of piece of S^{3}would be Schwartschild radius r_{s}= 2GM. This length scale is extremely short scale if one works in the scales of condensed matter. Even for Sun r_{s}is only 3 km! Huge gravitational constant is require in order to have r_{s}∼ hbar/M, the Compton length of particle with mass M. G=∼ hbar/2GM^{2}. Gravitation would be extremely strong.

! This need not be a problem! TGD predicts hierarchy of strong gravities assignable to p-adic length scale hierarchy. Gravitational constant has a spectrum of values. For instance, inside hadrons gravitation is strong and gravitational constant is about 10

^{38}times larger than ordinary gravitational constant. Only in very long length scales gravitation gets extremely weak. Size of piece of S^{3}of order G_{strong}M and can be of order of the size of say atomic nucleus or of icosahedral water molecule cluster.

! TGD predicts a fractal hierarchy of scaled copies of hadron physics and thus also strong gravities. In biology these scaled copies would be especially important in 10 nm- 5 micrometer length scale range.

TGD inspired estimate for gravitational constant is given by

G= (L

_{p}^{2}/hbar) × exp(-S_{K}(CP_{2})) , L_{p}^{2}= p × R^{2}(CP_{2}) .

Here R(CP

_{2}) is CP_{2}size of order 10^{4}Planck lengths.

S

_{K}(CP_{2}) is Kähler action for a deformation of CP_{2}type vacuum extremal representing graviton line of a generalized Feynman diagram as a deformation of CP_{2}type vacuume extremal. In long length scales, the magnitude for the value of Kähler action becomes maximal, and one obtains ordinary gravitational constant for p=M_{127}and α_{K}equal to fine structure constant. M_{127}is the p-adic prime characterizing electron and largest Mersenne prime which does not correspond to completely super-astrophysical length scale.

In short scales (short graviton lines) the strong gravitation limit G∼ L

_{p}^{2}is approached since Kähler action becomes small. Hadron physics and its various fractal copies would correspond to strong gravities. Schwartschild radius r_{s}= 2GM = 2L_{p}^{2}M/hbar ≈ 2 L_{p}would be of order Compton length!

** Second observation: sub-manifold geometry allows to realize quasi-lattices consisting of pieces of S ^{3}**

!In sub-manifold gravity one can construct from pieces of S^{3} realized as 3-D surfaces in M^{4}× CP_{2} large quasi-lattice like structures by gluing them together along boundaries. Icosahedrons decomposed of *regular* tetrahedrons could serve as a basic supercell.

Could icosahedral water clusters in nanometer length scale correspond to this kind of quasi-lattice like structures?

Could it be that strong gravitation is essential for living matter where ordered water plays a key role? Penrose and Hameroff speculated also that quantum gravitation important but the problem is that ordinary gravitation is so weak.

** Third observation/question: sacred geometries ↔ genetic code**

Two intriguing observations:

- Tetrahedron has 4 faces. DNA has for different code letters: A, T, C, G.

- Icosahedron has 20 faces. There are 20 different amino-acids.

Two questions:

- Could A,T,C,G somehow correspond to faces of tetrahedron?

- Could amino-acids somehow correspond to faces of icosahedron?

One must be able to achieve two things.

- To distinguish between faces of tetrahedron/icosahedron in order to assign them code letter or amino-acid. Symmetry breaking is required.

!One can look these objects in two manners. In S

^{3}geometry of the space-time sheet. Or in E^{3}subset M^{4}× CP_{2}geometry of imbedding space! 4- and 8-dimension perspectives! The regular tetrahedrons of S^{3}geometry do not regular in E^{3}geometry! In the same manner the faces of icosahedron need not be identical in E^{3}geometry although they are so in S^{3}geometry. Could this be enough for the needed symmetry breaking or is more needed?! Need the breaking be purely geometric?

- To select one of the 4 faces of tetrahedron and one of the twenty faces of icosahedron in order to say that this tetrahedron represents particular code letter or this faces represents particular amino-acids. How to achieve this?

** Fourth observation: genetic code → hierarchy of codes. Memetic code**

TGD leads to a generalization of genetic code.

- Combinatorial Hierarchy defined recursively:

M(m+1)= M

_{M(n)}= 2^{M(n)}-1 .

Mersenne numbers

M

_{2}= 3, M_{3}=7, M_{7}=127, M_{1}27= 2^{127}-1,...

The lifted Mersenne numbers are primes. The rest also if Hilbert was right.

- One can assign to these Mersenne numbers codes.

M

_{n}=2^{n}-1: 2^{n}sequences of n bits. Boolean algebra. One half of Boolean statements can be simultaneously true and define "axioms". Codes are axioms systems with 2^{n-1}axioms in the case of M_{n}.

Number of codons is 2

^{3-1}=4 for M_{3}, 2^{7-1}= 64 for M_{7}(genetic code). 2^{127-1}= 2^{126}for memetic code.

The interpretation of the hierarchy is in terms of statements about statements about statements.... Reflective hierarchy of codes would give Combinatorial Hierarchy. Hierarchy of codes labelled by certain Mersenne primes.

- Genetic code corresponds to M
_{7}=127. 64=2^{6}code letters.

- The next code after genetic code - memetic code - corresponds to Mersenne prime 2
^{127}-1, which is the p-adic prime characterizing electron and ordinary gravitation in long length scales. This code has 2^{126}=2^{6× 21}= (2^{6})^{21}codewords, which can be represented as sequences of 21 DNA codons. Huge amount of information! Could this code be realized for introns and be assignable to topological quantum computation.

There are two further observations.

- First observation: number of DNA codons in memetic codon is 21 =20+1= number of aminoacids +1 (stopping sign as "aminoacid")? What could this mean!? Is there a connection between memetic codon, the number of amino-acids, and icosahedral geometry?

Could it be that one could take DNA strand and fold it to icosahedral shape?! This would be like protein folding. Could one have 20 DNA codons forming icosahedron and 21:st glued to one of the 20 faces of the icosahedron?

Could this select icosahedral face ↔ amino-acid? Could this involve also gluing of real amino-acid to the face? Could amino-acid sequences be accompanied by sequences of icosahedrons so that one would have realization of memetic code? Could amino-acid sequences much more that mere building bricks: software instead of mere hardware? Could huge amounts of hidden/dark information not realized chemically!? Could memetic code perhaps responsible for the cultural evolution be associated with aminoacid sequences?

- Second observation: DNA possesses pearls-in-necklace structure. Pearls identifiable as nucleosomes. Could they correspond to 21 DNA codons in icosahedral configuration?

This need not be the case. There are 47 codons in single nucleosome. 2× 21+5. 2 memetic codons plus 5 surplus DNAs, which corresponds to 5 nm length: p-adic length scale L(149) defining thickness of lipid layer of cell membrane? Note that 5 DNAs correspond to a twist of π for DNA double strand. Why this? Could there be some other realization?

** Further observations and further ideas**

Further observations lead to further ideas.

- Fifth observation:nuclear string model ↔ protein folding ↔ sacred geometries. In nuclear string model atomic nuclei correspond to folded strings. Protons are folded into string and neutrons too. The string structure orders the nucleons and is something new and in principle it should be possible to find tests for this.

Folding of nuclear strings completely analogous to the folding of proteins!

4, 8, 20 are nuclear magic numbers. These are numbers of faces for tetrahedron, octahedron, and icosahedron. Sacred geometries! Is this a mere accident? Note that usual explanation in terms of nuclear shell model with harmonic oscillator potential.

- Sixth observation: in atto-second time scale water obeys chemical formula H
_{1.5}O. 1/4:th or protons invisible in neutron diffraction and electron scattering (see Chaplin's homepage). Where these protons lurk?

!Could they be dark in TGD sense? Could they reside at dark space-time sheets and have non-standard value of Planck constant. This leads to a model for dark protons as protons lost by water molecules to dark space-time sheets.

- Seventh observation/idea: dark realization of DNA, mRNA, tRNA, amino-acids and vertebrate genetic code

!The model for dark proton predicts that its states are in 1-1 correspondence with DNA, RNA, tRNA, and amino-acids and also predicts with natural assumptions that the numbers of dark "DNA type" protons which correspond to dark "amino-acid type" protons are same as the number of DNA codons mapped to a given amino-acid in vertebrate genetic code! Sequences of dark protons define dark nuclei as analogs of DNA codon sequences. Dark realization of genetic and possibly also memetic code .

!This could make possible R keno& D department in which various genetic modifications could be tested and successful variants could be taken in use by transcribing dark genetic codons to ordinary ones. The very rapid evolution of genes related to immune system could involve active experimentation in the virtual world of dark genes and dark amino-acids. Evolution would not be a completely random process but an outcome of active experimentation just as the technological evolution is.

Non-reductionistic side remark: No one considers seriously of throwing copper and silicon to pool and waiting the computers evolve from this soup! But this is how standard biologist would do.

!?Could it be that dark DNA 21-plets folded to icosahedral configuration accompany amino-acid sequences! Each amino-acid would correspond to dark protonic string of 20 dark protons folded to icosahedron + 1 additional dark proton to which amino-acid is glued. Completion to memetic codon would select unique amino-acid!! Two flies killed with single blow!

! The folded dark proton string of 20 nucleons would correspond to especially stable magic dark nucleus having an icosahedral shape.

** Eighth observation: water molecule clusters and geometric realization of memetic and genetic codes **

Tetrahedral and icosahedral geometries assignable to water molecules to which one can assign dark protons.

!?Could water molecule clusters accompany dark proton sequences? One can imagine two options.

- Water molecule has tetrahedral structure. H
^{+}nuclei and two pairs of lonely electrons are in vertices of tetrahedron. Could water molecule be associated with a piece of S^{3}instead of E^{3}? Could water molecules combine to form icosahedrons? One dark proton for each water molecule.

?Could this structure correspond to single dark DNA codon? The size might be too small. Ordinary DNA codons corresponds to 1 nm. Water molecules is roughly 10 time smaller.

- 14 water molecules form a regular tetrahedron and these in turn combine to form icosahedral water clusters (see the homepage of Chaplin about anomalies of water). This structure has nanometer scale. Could this structure define one dark genetic codon containing one dark proton? In this case one would have 1 dark proton per 14 water molecules.

**Background: progress in understanding of TGD**

The consideration above were preceded by a development of ideas about preferred extremals of K&ayml;hler action. For preferred extremals of Kähler action energy momentum tensor of Kähler action must have vanishing divergence.This corresponds to local conservation of energy momentum. If Einstein-Maxwell equations with cosmological constant assumed, this is true. Λ and G emerge as predictions. This distinguishes TGD from general relativity and allows a hierarchy of strong gravitations assignable to scaled variants of strong interactions. These would be important even in condensed matter scales and living matter.

?Problem: Einstein-Maxwell equations with cosmological constant might allow too limited set of solutions. Does not make sense for vacuum extremals for which energy momentum tensor vanishes but Einstein tensor is non-vanishing. Would require infinite gravitational constant and vanishing cosmological constant.

The interpretation of vacuum extremals suggests something more general.

- Einstein-Maxwell equations cannot be true for them. The solution ansatz guaranteing the vanishing of divergence of energy momentum tensor must be generalized. This is possible: two or three cosmological "constants", which are actually not constants! This was discussed in previous posting.

- Interpretational problem: Vacuum extremals are vacua of Kähler action but their Einstein tensor non-vanishing. Einstein equations say that there must be energy momentum associated with these vacuum extremals. What does this mean?

!As I have suggested earlier, Einstein tensor characterizes the topologically condensed matter, smaller space-time sheets present and glued to the vacuum extremal space-time sheet. The sheet carries information about the energy momentum tensor of topologically condensed matter in its Einstein tensor.

## 10 Comments:

Matti,

I begin to see you understand aspects of my quasic ideas, and they have been applied to biology intuitively since 68 and formally to the gene code since 1974... others have come near, in serious sacred geometry or not. After all the golden ratio is the significant quadratic length in four space.

I would wish the integration of your number system closer to my ideas and some shared terminology for this is a minefield to explore such equations in number theory probably due to problems of oddness and evenness in our notations so we do not always find all cases or imagine them wrong.

Lately I have posted a lot of things that discuss this sort of thing with flat graphs that does such a counting- including the all important edges, 30 of such Platonic solids as part of the general picture.

But thank you for this post of a very high order.

L. Edgar Otto

Dear Matti,

About Field- particle duality, I put together three different approaches of TGD:

1-X^4(X^3) and X^3 : “Light-like 3-surfaces X3-generalized Feynman diagrams- correspond to the particle aspect of field-particle duality whereas the physics in the interior of space-time surface X4(X3) would correspond to the field aspect.”

2- Ho-H duality:

“It seems possible to interpret HO-H and HQ-coHQ dualities as analogs of wave-particle duality in the infinite-dimensional context. “

3- Flip flop nature of CD: localization of the lower tip and delocalization of the upper tip. They are corresponded to particle and field aspects of the duality. This phenomena is related to measurement and for example say why do we see only particles and not waves.

The two fundamental relations E=h*nu and p=h/lambda relate energy and momentum of particle aspect of wave-particle duality to frequency and wavelength of wave aspect of the duality. If we translate the two relations to first approach of TGD about wave-particle duality, energy and momentum of particle (=X^3) is related to frequency and wavelength of wave (=X^4(X^3))

Is there anything wrong in the above?

Dear Hamed,

how was the seminar that you mentioned few weeks ago?

These three dualities correspond to three geometric levels: space-time, imbedding space, and WCW ("world of classical worlds"). First and third dualities would correspond to the wave particle duality at space-time level and imbedding space level respectively.

The H-M^8 or H-OH duality would be wave particle duality at the level of WCW - in infinite-D context. I have considered the possiblity that surfaces in M^8=OH could be interpreted as analogs for tangents of curve: replace curve with 4-surface in H and tangent of curve with 4-surface in M^8.

In terms of spectroscopy, there has to be some correspondence to the geometrical implications of theoretical physics and genetic search for biological correlations?

Best,

One last link for consideration.

Quasicrystal: Prof. Dan Shechtman

Best,

Dear Matti,

Thanks, date of the seminar is whenever I determine it. From today I got more free time to learn. When I feel enough skill to present the seminar, I will do it.

OK. Tell me when you have it.

Dear Plato,

thank you for the links to your blog. I managed even to make a comment (it was not easy:-)) about quasilattices.

We have a lot of beliefs about chemistry and organic chemistry which need not hold true. The notion of chemical bond is assumed to follow from Schrodinger equation: this need not be the case (TGD predicts that it has a correlate at the level of space-time topology). The behaviour of biomolecules - consider DNA replication as example - is a mystery in the existing conceptual framework but we pretend that we understand it.

Here theoretical physics could both give and get but reductionistic paradigm prevents the progress.

Conjecture: The combinatorial hierarchy corresponds to the normed division algebras. Exploring this correspondence will shed light on TGD.

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