Dear James,
I enjoy teaching. The only thing better than teaching is teaching ideas that I developed. I apologize for not explaining it better the first time.
I would say that the trial and error arose in two places, first the series: 1, SQRT(3), SQRT(6), SQRT(10), SQRT(15), etc. These ratios fall out naturally in multi-dimensional spherical packing (i.e. simplices). Secondly, I played with Lisi's triangular/ hexagonal Color Theory lattice and my own Hyperflavor tetragonal lattice (in my book), and started seeing certain 'conserved' quantities popping out naturally. I started with 1/2's in certain places - like weak isospin (in my book), but quickly found that the requirements of simplices forced these lattice-like regularities. Then I took it to the extreme and tried to build a lattice that might have represented the TOE.
The simplest example is a 2-dimensional triangular/ hexagonal lattice. This is relevant as it fits Color Theory and Lawrence's G2 Theory. Table 6 and Figure 2 in my essay demonstrate this simple example.
The next simplest example is a 3-dimensional tetrahedral/ Face-Centered-Cubic lattice. I demonstrated this in Section 7.2 (and Table 6 and Figure 3 - Tables 7 and 8 are also relevant, but not as key as Table 6) of my book, and rephrased it in Table 1 and Figure 1 of my essay.
An assumption that is built into these Simplices is that all charges sum to zero in a particle multiplet.
Can we ever measure these fractional charges? Probably not. Color confinement keeps us from measuring quark properties directly, although we have tons of indirect information. And these scalar fermions/ tachyons are expected to be unstable. It is interesting that the interactions and stable particles work together in such a manner that only integral electric charges are stable, though 1/6, 1/3, 1/2, and 2/3 may exist in underlying structures. It seems that of all of these fractionally charged particles, the only stable ones are the up (2e/3) and down (-e/3) quarks. These comprise the basis for an SU(2) Isospin symmetry that leads to a triplet of pions (+/-,neutral) of similar mass and a doublet of nucleons (proton,neutron) of similar mass.
We can measure the value of 'e' directly (and indirectly via stellar spectra), and know that it has been 'constant' over the past billion years. Depending on the interpretation of data, it could have varied more in the early history of the Universe (5-13 Billion years ago). I talked about this in Section 6.2 of my book.
Thank you for your interest!
Ray
