In my post Three Hinges I presented for the first time the simplest possible model of the reflection of elementary particles of energy (kaus/dots). In the post I stated that the model can be truly precise only if I can find sensible equations for all its geometrical nodes. Then in my last post, Known Unknowns I presented the logic behind most of these nodes. But in the post, I also realized that I didn’t know the logic behind all of them.
Continuing from Known Unknowns: As the location of the (green) centers of three neighboring unit spheres of reflection (A, K & L) are known, as well as the location of the central kaus of the (yellow) ripped grazing double helix, it should be possible to accurately determine the two other kaus (M and N) grazing kau H, as the length of the green vectors and yellow vectors is known. This relies on the fact that the length of the vectors L-N and K-M is equal to the vector A-H. For a long time, I was convinced that the length of this vector wasn’t r (the radius of a kau) when the hinge angle θ was higher than zero, but looking at the reflection model, it seems that I was wrong. At least as far as I understood now, the lengths of vectors A-H, K-M and L-N are always r.
And subsequently, as we know that the vectors K-M and L-N are halves of a vector twice as long, we are able to determine the locations of the next two kaus of the grazing double helix (O and P). Thus, we have logically found the relation between each kau, center points and the hinge point in the model.
This sounds wonderful! So, where are the promised equations? They’re on their way. But I thought I’d let you know what the logic is even before I’ve presented the equations. If the logic is sound and the equations make sense, I should be able to draw unit sphere with any hinge angle that I desire. Perhaps this is my goal: only post something If I’m able to present an accurate unit sphere with a hinge angle of ten degrees. And before I can, I’ll refrain from posting.
But because I’m also a realist, I’m still expecting surprises that show my logic to be flawed. And if there are no errors, I’m pleasantly surprised.
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