Might we deduce the masses of standard model fermions from symmetry principles alone?

There are 24 fermions in the standard model (if we consider left/right and particle/anti-particle as the same type).

Thus there are 24 masses to find. (Yes, it is generally considered quarks of different colors (red, green, blue) have the same mass but we’ll count it as 3 different masses.)

Now we assume that the theory of physics obeys some symmetry principles (after all that is what most theories say).

So without getting into the physics we should expect that the 24 masses are related by some kind of symmetry. (And hopefully it would have a nice geometric explanation).

For example in 4 dimensions a regular polytope (the 24-cell) exists. If we consider a projection of some orientation of the 24-cell into 1D this seems unlikely to give us the masses we are after. (Although it might depend on the kind of projection – such as a stereographic projection). As the masses in the standard model seem to differ by orders of magnitude from each other.

One might think that the masses might be the eigenvalues of some matrix connected with the 24-cell. Although I wouldn’t know quite the connection between polytopes and matrices.

Has any investigations been done (that you know of) in trying to match the masses of the fermions in the Standard model with projections of some geometrical shapes? It seems an obvious thing to try in looking for some kind of order to the masses. (Just like we try and look for order in other areas of physics).

If not polytopes, what other symmetrical objects might be related to the masses of the fundamental fermions?

I think, some might dismiss this as pseudoscience or numerology. But I don’t think this is any different to how Kepler tried to fit data to ellipses which are another geometric shape. Newton used this empirical result to help him with his theory of gravity.