In the sequel I will discuss this finding from TGD viewpoint and propose a modified interpretation based on the geometry of icosahedron, one of the 5 platonic solids, which play a key role in TGD, and TGD inspired quantum biology and theory of consciousness.
The dimension 11 in this context looked to me a rather formal notion but one could give it a mathematical meaning.
- In 3-D one can take tetrahedra, 4-simplexes as building bricks of a discretized manifold. In dimension 11 one has 12-simplexes. These are glued together, which means that n-faces with n varying from 1 to 11 are glued together along n-1-D faces.
- In the case of the brain, one would have groups of neurons, with 12 neurons connected in such a way that one has a connectedness of a 12-simplex. There would be 11- edges meeting at each 12 vertices. Each neuron would be connected to all the other 11 neutrons and would have maximal connectedness, which is very natural if one wants a maximally coherent functional unit.
The notion of orientation is essential: axons are oriented by the direction of nerve signals which is always the same. The orientation of axons could induce orientations of n-faces. 2-face would correspond to a loop in which signals can rotate in a single direction.
Since axons must be present, each neuron must be connected with every other neuron. The geometric connectedness possible in the case of neurons since the axon from a given neuron can branch and have a synaptic contact with the dendrites of several neurons: for n=11-simplex with all other (11) neurons (see this). Note that also a synaptic contact with the neuron itself (autapse) is possible.
Could one consider also a generalization of this geometric view of a simplex. Could functional coherence of the neuron group serve as a criterion for whether neurons form an n-face?
- The interpretation in terms of 11 real dimensions might assume too much and I am reluctant to believe that it has anything to do with M-theory. However, one could realize n-simplexes in this way in 3-space and the orientation of the axon, determined by the preferred directions of signals, would define orientations of higher level simplexes. The idea that these structures could have something to do with geometric cognition allowing us to imagine higher dimensional geometric structures is attractive.
- The 12 vertices of the argued 11-simplex could be also identified as vertices of icosahedron, one particular Platonic solid appearing repeatedly in molecular biology. For an icosahedron, the Hamilton cycle, going through all vertices just once, has 12 vertices. It would connect each vertex to all other vertices by a unique path having a varying number of edges: 1,2,... The selection of this Hamilton cycle could raise one particular edge path among all possible closed edge paths possible in the maximally connected 12-neutron network in a special position.
- This icosahedron need not correspond to ordinary Platonic solid in the Euclidian 3-space. The definition of nearness can be defined also in terms of functional nearness. Indeed, hyperbolic 3-space has been suggested to play a role in neuroscience for neutrons: neurons resembling each other functionally would be near to each other in the hyperbolic metric and in TGD framework this metric is assigned with hyperbolic 3-space H3 as Lorentz invariant light-cone proper time = constant surface to which the magnetic body (MB) of the brain is assigned as 3-D surface. The signals from neurons, which are near each other in functional sense, would be sent to nearby points of the MB so that functional nearness would be geometric nearness at the level of MB.
- Also tetrahedron with 4 vertices and faces and octahedron with 6 vertices and and 8 faces are Platonic solids which have triangular faces representing 2-simplex and could correspond to dimensions d=3 and d=5. Cube with 6 square faces and d=8 vertices is the dual of octahedron and dodecahedron with d=20 vertices and 12 pentagonal faces is the dual of icosahedron. It might be also possible to assign to them dimension as the number of vertices by using maximal axonal connectedness of vertex neurons as a criterion.
Platonic solids and Hamiltonian cycles as path going once through each vertex of the Platonic solid and identified as nuclear strings play a key role in the "Platonization" of nuclear and atomic physics leading to quite precise quantitative vision about basic numbers of nuclear and atomic physics and even hadron physics. The key observation is that the states of j=l+/-1/2-blocks of atoms and nuclei correspond to Platonic solids for l<6 (a highly non-trivial fact), which therefore provide geometric representation for the j-block (see this).
- Icosahedron is unique among Platonic solids in the sense that it allows a large number of Hamiltonian cycles. Icosahedron, tetrahedron and their Hamiltonian cycles play a fundamental role in the TGD inspired model of genetic code (see for instance this) involving the notion of icosa-tetrahedral tessellation of hyperbolic 3-space involving all Platonic solids with triangular faces.
Each combination of 3 icosahedral Hamiltonian cycles with symmetries Zn, n=6,4,2 defines a particular realization of the genetic code predicting correctly the number of DNA codons coding for a given amino acid.
- The model of the code emerged originally as a model of musical harmony. The faces of icosahedron are triangles and would define 3-chords realized as cyclotron frequencies assignable to the vertices of the triangle. Each Hamiltonian cycle would define 20 chords defining a particular harmony whereas the 12 vertices along Hamiltonian cycles would define a 12-note scale, with neighboring vertices representing frequencies related by scaling by 3/2 (quint) modulo octave equivalence.
One could speak of music of light and since music creates and expresses emotions, the proposal is that different bio-harmonies correspond to different emotional states realized already at DNA and RNA level. Could these 12 neuron units and possible tessellations (hyperbolic crystals) associated with them relate to the realization of emotions at the level of the brain?
Physically, the Hamiltonian cycle as a representation of 12-note scale is an analog of a closed string made of flux tubes representing the edges (pipes of organ!)
- What is fascinating is that hyperbolic 3-space H3 (mass shell in particle physics), playing a key role in TGD, has a unique tessellation/lattice involving all Platonic solids, whose faces are triangles (icosahedron, octahedron, tetrahedron) and also provides a model of DNA making quantitatively correct predictions. I have proposed that this tessellation defines a universal realization of the genetic code realized in all scales at the level of the MB of the system. Could the 12-neuron unit interpreted as 11-simplex relate to one particular realization of this tessellation.
- Also cubic, icosahedral, and dodecahedral regular tessellations are possible in H3 (Euclidean 3-space E3 allows only cubic regular tessellation) and they would define the analog of a homology of dimension n= 7, 11, or 19 space at neuronal level.
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.