So, let's recapitulate. The dihedral angle of the icosahedron is supposed to be the sum of three angles of physical significance, plus 60 degrees. The exponential of the surface area of the icosahedron, divided by the mass of all leptons expressed in units of electron mass, allegedly equals ... another whimsical expression relying on the Weinberg angle. And, counting all the fermions and gauge bosons, we get 24, and this has something to do with SU(5) and the icosahedron.
Also, there are a bunch of alleged identities involving the Weinberg angle, which I have not yet bothered to check numerically, nor have I checked whether a simultaneous solution to these equations is even possible. Also, I suspect that the closeness of the "DFQ angle" mentioned in the previous post, and the experimentally measured value of the Weinberg angle, may subliminally be at work here. (Let me note in passing that if you embedded an icosahedron in a slightly non-Euclidean geometry, it should be possible to make the DFQ angle exactly equal to the measured Weinberg angle - at one's preferred energy scale, that is, since the value of the Weinberg angle flows.)
The attentive reader may recall that the premise of this blog was, that abstracts from the snarxiv could - if used in moderation - serve as genuine inspiration for physics. The current experiment is to see whether an authentic bit of numerological physics, found in the wild (the physics blogosphere), can similarly provide inspiration. So rather than proceed with the numerological analysis, for a moment I want to switch tracks and ask, in what sort of physical theory could the relationships listed at the start of this post actually exist and actually play a role in physical causation and explanation?
Since Weinberg angle, Cabibbo angle, Higgs VEV, etc., all acquire physical significance on account of the roles they play in a particular quantum field theory (the standard model), we are presumably looking for a beyond-standard-model theory which reduces to the standard model in some limit, and in which icosahedra matter. Perhaps there are compact dimensions shaped like icosahedra; perhaps there are icosahedral branes. Perhaps there are interaction vertices dual to icosahedra, or perhaps we calculate certain amplitudes by integrating over icosahedra.
See, this is the fun part: take the work of mr nothing, and try to hybridize it with the conventional apparatus of physical theory. But I must say that the "sum of angles" worries me. We are supposed to be using the icosahedron to explain the standard model, not vice versa. And while the dihedral angle is certainly a natural property of the icosahedron, the peculiar decomposition into a sum of four angles does not appear to be natural. Even supposing that an icosahedral structure appears at some level of our theory, why would the theory be sensitive to the existence of that decomposition of the dihedral angle? Unfortunately, the similarity of the DFQ angle and the Weinberg angle doesn't seem to help, because the DFQ angle isn't a natural part of the dihedral angle, so far as I can see. I shall need to meditate on the geometry of the icosahedron for a little while...