In the TGD framework, one can imagine two starting points concerning the explanation of the findings.

- Could there be a connection with the redshift quantization along some lines ("God's fingers"") proposed by Halton Arp (see this) and Fang-Sato. I have considered several explanations for the quantization. In TGD cosmic=time constant surface corresponds to hyperbolic 3-space H
^{3}of Minkowski space in TGD. H^{3}allows an infinite number of tessellations (lattice-like structures).I have proposed an explanation for the redshift quantization in terms of tessellations of H

^{3}. The magnetic bodies (MBs) of astrophysical objects and even objects themselves could tend to locate at the unit cells of the tessellation. - Icosa-tetrahedral tessellation (lattice-like structure in hyperbolic space H
^{3}) plays a key role in the TGD model of genetic code (see this) suggested to be universal. Lattice-like structures make possible diffraction if the incoming light has a wavelength, which is of the same order as the size of the unit cell.

- In X ray diffraction, the diffraction pattern reflects the structure of the dual lattice: the same should be true now. Only the symmetries of the unit cell are reflected in diffraction. If CMB is diffracted in the tessellation, the diffraction pattern reflects the symmetries of the dual of the tessellation and does not depend on the value of the effective Planck constant h
_{eff}. Large values of Planck constant make possible large crystal-like structures realized as part of the magnetic body having large enough size, now realized at the magnetic body (MB). - Icosatetrahedral tessellation plays a key role in the TGD inspired model of the genetic code. Dodecahedron is the dual of icosahedron and tetrahedron is self-dual! [Note however that also the octahedron is involved with the unit cell although "icosa-tetrahedral" does not reflect its presence. Cube is the dual of the octahedron.]
So: could the gravitational diffraction of CMB on a local crystal having the structure of icosa-tetrahedral tessellation create the illusion that the Universe is a dodecahedron?

- In diffraction, the wavelength of diffracted radiation must correspond to the size of the unit cell of the lattice-like structure involved. The maximum wavelength of CMB intensity as function of wavelength corresponds to a wavelength of about .5 cm. Can one imagine a tessellation with the unit cell of size about .5 cm?
- The gravitational Planck constant ℏ
_{gr}=GMm/β_{0}, where M is large mass and m a small mass, say proton mass (see this, this, this, this and this). Both masses are assignable to the monopole flux tubes mediating gravitational interaction. β_{0}=v_{0}/c is velocity parameter and near to unity in the case of Earth. - The size scale of the unit cell of the dark gravitational crystal would be naturally given by Λ
_{gr}= ℏ_{gr}/m= GM/β_{0}and would be depend on M only and would be rather large and depend on the local large mass M, say that of Earth. Λ_{gr}does not depend on m (Equivalence Principle). - For Earth, the size scale of the unit cell would be of the order of Λ
_{gr}= GM_{E}/β_{0}≈ .45 cm, where β_{0}= 0=v_{0}/c ≈ 1 is near unity from the experimental inputs emerging from quantum hydrodynamics (see this) and quantum model of EEG (see this) and quantum gravitational model for metabolism (see this and this). Λ_{gr}could define the size of the unit cell of the icosa-tetrahedral tessellation. Note that Earth's Schwartschild radius r_{S}=2GM≈ .9 cm.Encouragingly, the wavelength of CMB intensity as a function of wavelength around .5 cm to be compared with Λ

_{gr}≈ .45 cm! Quantum gravitational diffraction might take place for dark CMB and give rise to the diffraction peaks! - Diffraction pattern would reflect astroscopic quantum coherence, and the findings of Luminet et al could have an explanation in terms of the geometry of local gravitational MB rather than the geometry of the Universe! Diffraction could also explain the strange deviations of CMB correlation functions from predictions for large values of the angular distance. It might be also possible to understand the finding that CMB seems to depend on the features of the local environment of Earth, which is in a sharp conflict with the cosmological principle. According to Wikipedia article (see this), even in the COBE map, it was observed that the quadrupole (l=2, spherical harmonic) has a low amplitude compared to the predictions of the Big Bang. In particular, the quadrupole and octupole (l=3) modes appear to have an unexplained alignment with each other and with both the ecliptic plane and equinoxes.
- Could the CMB photon transform to a gravitationally dark photon in the diffraction? This would be a reversal for the transformation of dark photons to ordinary photons interpreted as biophotons. Also in quantum biology the transformation of ordinary photons to dark ones takes place. If so the wave length for a given CMB photon would be scaled up by the factor ℏ
_{gr}/ℏ =(GM_{E}m/β_{0})/ℏ ≈ 3.5× 10^{12}for proton. This gives Λ=1.75 × 10^{7}km, to be compared with the radius of Earth about 6.4× 10^{6}km.

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

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