What are we dealing with here? An attempt to derive various basic parameters of physical theory from quantities associated with the icosahedron. Which quantities? The dihedral angle (angle between two faces), and what I'll call the "inscribed radius" and the "circumscribed radius" (really, these are distances from the 3d center of the icosahedron to the middle of any face and to any vertex, respectively).
I'm a bit puzzled by "angle w". Although it's close to the "DFQ" angle in the link above, I can't see that it corresponds to any natural property of the icosahedron. Yet we have these two alleged identities
5 x ( 1 + sin^2(W) +sin(W) ) = cos ( 2pi/10 ) / [ cos( 2pi/5 ) ]^2
(3 + sqr(5) ) x [ sqr(3)/12] = cos(w)/[ (1 + sin(w) )^2 -1]
If these are both genuinely, exactly true for some w, then that's interesting... OK, duh, I bet "angle w" is supposed to be the Weinberg angle. In fact, mr nothing says so explicitly: "mW/mZ = cos(W)"
Because I'm lazy, I will not try, right away, to see if a w does exist that exactly satisfies the two equations above, or to otherwise guess what the geometric inspiration for those formulae might be. Instead, let's continue identifying the physical interpretation of the three icosahedral quantities proposed. For example, we are told that
The dihedral angle of the icosahedron = Weinberg angle + Cabibbo angle + "GUT angle" + pi/3
Despite its name, the "GUT angle" appears to be mr nothing's discovery, rather than, say, the value of the Weinberg angle in some GUT - it is defined as the dihedral angle divided by the square root of "one plus (half the Higgs VEV in units of electron mass)".
The other crucial element of the physical interpretation of the icosahedron turns out to involve its surface area A:
exp( A ) / (sum mass all leptons/ electron mass ) = [ ( sin(w) + cos(w) ) x sin(w) ]^-1
The "inscribed radius" and "circumscribed radius" don't seem to be playing much of a role...
Conceptually crucial, I believe, will also be the statement that
6 leptons + 6 quarks + 8 gluons + 1 fotón + 3 B ( w+, w- , z ) = 24 = 4! ===> SU(5)
followed by the introduction of the icosahedron. This is where mr nothing is trying to turn his identities into physics.