In today’s post I’ll talk about reflection vectors. I have a feeling that I won’t be able to explain continuous reflection that well with today’s pictures, but I’ll still give it a shot.
The main concept I wish to convey here is that reflection of elementary particles of energy (dots) in the hydrogen atom (and probably elsewhere as well) is always a sum of three vectors. These are the tangents of three arced paths of dots, each with a specific curvature. In the left image you have the y-z projection of the saint Hannes knot. I chose that the parallel movement of the dots to be along the y axis (I could’ve chosen the x and z axes just as well: there’s no magic in the choice of axes). At the point where a yellow dot along the first orbital reflects from a green dot along the second orbital (where the locations of the dots are shifted by 90 degrees along the axis marked by the red arc in the right picture), the parallel motion of the yellow dot with its pair is right, whereas the parallel motion of the green dot with its pair is right.
After creating the above image, I realize that it might be wrong. You see, the curvature (black circular arcs) you see on the left cannot come from the parallel motion. The source of curvature comes from reflection, which is shown in the image on the left. For each of the yellow dots, at locations φ = 0, π/2, π & 3π/2, reflection must be the sum of two vectors of reflection: the blue vector for reflection with radius of √(2+√2) and the red vector for reflection with radius of 1 (because I’ve normalized the equations). The neighboring red vectors, depicting curvature with a radius of one, are always opposite and cannot explain the curvature in the left image. However, the neighboring blue vectors could explain the curvature in the right image, but the problem is that I’ve drawn the vectors in the wrong direction.
So, it would make more physical sense to draw the blue reflection vectors downwards, like this:
The more astute reader might notice that the reflection vectors for the yellow dots on the upper right-hand corner on the (right) x-z projection are pointing upwards. However, these dots are not being reflected by the intersection of the two quadruplets of dots, so they don’t need to be reflected downwards.
I’m still not 100 % sure of the red vectors. At least for now I wouldn’t switch their direction. However, for visual clarity, I think I need to shorten them. You see, the vectors depict a physical quantity of reflection. As there is doubly more reflection along the blue axis, the blue vectors should be twice as long as the red vectors:
So, is this everything I (or you) need to know about continuous reflection? To be frank, it cannot be. You see, if the above image was all there was to it, the opposite red vectors for the pairs of yellow dots would cause the dots to be separated.
Rather, what probably happens is that the dots behind and in front of the yellow dots along the y axis also cause reflection towards the center of the yellow dot pair. However, to be able to describe this reflection, one would need to expand the (left) y-z projection image to include these (green) dots:
In the above (cluttered) image, I’m trying to convey this concept with just the red reflection vector. With only one plane of reflection, you would have just a single vector of reflection. However, this vector of reflection is again reflected from the plane of reflection in front of it, splitting it into a parallel and reflected component. This time the reflection is that of an already reflected vector.
If this sounds confusing to you, don’t worry. I think this is completely new mathematics. It’s bound to be hard in the beginning. None of this has been solved with mathematical proofs. It’s all just intuition and logic, with plenty of opportunities for major errors.
I already see several possible points of error in the last image. So, don’t take it as final truth, but as a step along the way to something more concrete.
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