It is now known that neutrons and protons are not fundamental particles but are made up of the strongly interacting quarks. The forces among the neutrons and the protons inside an atomic nucleus are therefore a complicated reside left over after the quarks are bound up. Thus the binding energies of the nuclei and their excitations are very difficult to compute. However the strong nuclear force itself, acting between quarks, is tractable, especially at high energies. In addition to the strong nuclear force, there is a weak nuclear force, about a million times weaker. It is responsible for the decay of an isolated neutron. It is also a crucial step in the fusion reaction cycle inside stars. because it is so weak and slow, the stars live billions of years. Were it for the strong force alone, stars would disappear over time scales of thousands of years!
However, crucial empirical facts needed to be discovered before Gauge invariance could be applied in these cases. During the 1960's, in separate proposals, Glashow, Weinberg and Salam applied the Yang-Mills theory to the weak nuclear force. The remarkable discovery from mid-1960's onwards is that both the strong and the weak nuclear forces are gauge forces. The symmetries associated with them are a generalisation of the case for Electromagnetism. In particular, one needs twelve f-functions, eight for the Strong, and four other such functions for the Weak force and Electromagnetism combined. It turns out that the Electromagnetic force and the Weak nuclear force are inter-related, that is, their f-functions mix.
If we now wish to make a geometrical picture, we have to modify geometry the way we did in going from fig.s 4, 5 of sec. 3.1 to fig. 6 in sec. 3.3. There we proposed attaching a circle, the space of the f-function, to each point of the ordinary space-time. Now we must attach to each point an agglomerate of three spaces: one for the eight f-functions of the Strong force, one for the three others that describe the symmetries of the Weak force, and one for an f-function, which after "mixing" with one of the f's of the Weak force will produce Electromagnetism. Each of these spaces will be either a circle, a solid sphere or an eight dimensional generalisation of a sphere.
There now seem to be too many new members to the act, but the power of the gauge principle is that one does not need twelve different coupling constants, one for each f-function. We need only three coupling constants, one for each of the spaces described above.