The coxeter helix is built by accumulating tetrahedra “glued” face to face in a way that maximizes the distance between the first and last elements of the chain. Between them an axis emerges as the chain gets longer. There is also a helicity, and globally the figure is a helix which is not periodic: a new tetrahedron will never become parallel to an older one.

Klee Irwin and Fang Fang had written a paper some years ago (referenced below) describing a new version of this object where each time we connect a new tetrahedron to the helix, we don’t make the 3-vertex correspond, but rotate it by 15.52 degrees to the left or to the right (but to the same direction every time).

The table explains that if both helicities are the same, the 6th tetrahedron becomes parallel to the first, which proves a periodicity of 5; and if the helicities are opposite, the periodicity is 3.

So both helices are built only with 5 different orientations of the tetrahedron instead of an infinity; which is highly efficient in term of reduction.

Moreover these two versions of the tetrahedral helix, now encoding a parity, could be made only with tetrahedra belonging to the QSN.

After observing this, Fang found that they are effectively present in the QSN, and form some helicoidal filaments, carrying flow between more stable structures like the 20-Groups.

More stunning, we have found and documented in the second below referenced paper that this ‘magic’ angle P=arccos((3 phi-1)/4) which restores the periodicity in two ways, and displays the 3 and 5 of the icosahedral symmetry, is also related to the Cabibbo angle C=arctan(phi^-3) related to P by phi^-3=tan(C) =tan(P/2)/ sin(arccos(1/3)/2), one of the most important fundamental parameters of the standard model, related to neutrino mixing matrices (how and with which probability the neutrinos emitted by the sun are transmuting between different family electron-muon-tauon during their travel to earth…)

Reference sources:

As Marni says (2:39) the actual measurement of the Cabibbo angle show 13.04 degrees, but the value of 13.28 degrees is still acceptable, and backed by a model supported by several peer reviewed papers

Y. Kajiyama, M. Raidal and A. Strumia, “The Golden ratio prediction for the solar neutrino mixing”. Phys. Rev. D 76, 117301 (2007) [arXiv:0705.4559 [hep-ph]].

L. L. Everett and A. J. Stuart, “Icosahedral (A(5)) Family Symmetry and the Golden Ratio Prediction for Solar Neutrino Mixing”. Phys. Rev. D 79, 085005 (2009) [arXiv:0812.1057 [hep-ph]].

I. de Medeiros Varzielas and L. Lavoura, “Golden ratio lepton mixing and nonzero reactor angle with A5”. J. Phys. G 41, 055005 (2014) [arXiv:1312.0215 [hep-ph]].

F. Feruglio and A. Paris, “The Golden Ratio Prediction for the Solar Angle from a Natural Model with A5 Flavour Symmetry”. JHEP 1103, 101 (2011). [arXiv:1101.0393 [hep-ph]].

Quantum Walk on a Spin Network and the Golden Ratio as the Fundamental Constant of Nature