I was about to write a post about solving the equations for asymmetric refraction. But as I got deeper into the equations, I realized that I had gotten the primary angle of refraction (φr) kind of wrong. But the wrongness is a very odd sort of wrong, because I had it correct in my first Theory of Everything -manuscript. I just seemed to have forgotten it.
So, what is this wrongness about? It’s easier to show with images. When refraction is viewed in the X-Z projection, you can only see the secondary angle of refraction (φR).
However, I you look at the X-Y projection, you only see the primary angle of refraction (φR).
And if you look at the Y-Z projection, you only see traces of primary refraction, because the refractions are only along the XY and YZ planes to begin with, and you can just about see the ring of primary refraction turning from a line for dots d1 and d2 to a thin ellipse for dots d3 and d4.
You might be confused about why the circle of the primary angle of refraction is on the ‘wrong side’ of dot d1 and not centered between the four dots d1 to d4. This is because there is a symmetrical pair of dots somewhere in the orbit, but most of the time does not touch this pair. Only at two locations in the whole Higgs helix, do the pairs actually touch:
I have figured out most of the equations for the system, but now that I realize what the asymmetricity really looks like, I have to double-check everything to make sure there are no errors. Perhaps in my next post I show what this look like with actual vectors.
Update from 13.1.2024
While I made no changes to the actual refracted shape, I added the right sized primary circle of refraction and the yellow center points that follow along this primary circle.
If I had tried to come up with the equations with a circle of primaryrefraction that was half this size, I would have very soon encountered problems. With this image, I should be able to get the correct vector equations.
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