Cube-in-plane model for Standard Model fermion charges, masses, Fine Structure constant

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It's been a while since I last visited. I don't remember whether I posted here about my geometrical model to represent the relative charges of Standard Model fermions. You take a cube, draw a body diagonal through it (which passes through the cube's center and two diametrically opposed vertices) and then embed this diagonal in a reference plane. It can then be used as a rotational axis for the cube. I found some years ago that if you rotate the cube such that one vertex subtends an angle of arctan(sqrt27) plus or minus any multiple of 30 degrees, then all the distances from the vertices to the plane parallel the fermion charges. The direction TO the plane from above or below it represent the polarities of the charges (plus or minus). Arctan(sqrt27) is 79.10660535.... The fact that you can add or subtract increments of 30 degrees from this angle seems to relate to the fact that in a 360 degree circle there are exactly 12 such cuts, and there are exactly 12 fermions in each set of matter and antimatter fermions. For every increment of 30 degrees you have to switch between sine and cosine functions for the angles. And there are a number of sum and difference relations for all these sine and cosine values.

I also found some years after this that the same system can be used to model the rest masses of the Standard Model fermions under the Koide Formula (see Wiki page), but with a different mathematical function. And I have some evidence you can do the same with the gluon colors.

A couple of days ago, on a whim, I started using arctan(sqrt27) to see if any other relations could be derived from it. I tried the ratio of the Fine Structure Constant's reciprocal, 137.035999 and change to arctan(sqrt27), and found that the result was almost exactly sqrt3 (1.732...), varying from it by well under 1%. And when I tried a smaller value for the FSC that would have been relevant in the early moments of our universe (so closer to 127), I found that using 128 divided by arctan(sqrt27) gave almost exactly the Golden Ratio, 1.618... and change, again with a variation from the actual value of well under 1%. 

Anyone want to discuss??

 

Jess Tauber