I am well and truly stuck! There seems to be no point in trying to come up with proper vector representation of reflection if I can’t be 100 % sure that this kind of reflection makes sense.
My fancy equations that are supposed to show the two paths (below, in blue and yellow) of elementary particles of energy (dots) along a knotting orbital are not based on keeping the dots on track.
Rather, the equations are a topological solution to the idea of how to make dots follow a path that makes them form a knot. However, the equation assumes that each infinitesimally small section of the orbital along which the dots move, should be of exactly the same length. That is, the dots must move at the same pace (i.e. the speed of light) so that they truly are reflected by their neighbors. If they don’t, this whole concept loses its foundation.
The frustrating thing is that I have the equations and I should be able to draw a picture and draw the path and come up with equations that define the length of the path. But I can’t!
The problem is that the dots move along an arced path, whose radius at least in principle varies from R -√(2+√2) -1 to R +√(2+√2)+1. The value of R can be ignored for now but is a very big number in comparison to the other numbers. However, the dots also move along a second arced path, that is always perpendicular to the first path and whose radius, at least in principle varies from √(2+√2)-1 to √(2+√2)+1.
As I mentioned before, there should be a general equation that should define the length of the path along a φ = 0 to φ = 1/∞. Or we should probably replace ∞ with ⌘, where ⌘ denotes a very large number, which separates two neighboring dots along the same orbital. But I don’t seem to be good enough at mathematics to see what this equation is. If I were able, then I’d be able to check whether my logic is valid or not.
Simplifying a lot, the problem can be visualized like this:
In the above image you have two circles with a radius of √(2+√2) moving along a radius of 10 by 8 degrees. At the beginning there are two spheres with a radius of one on either side of the ‘highest’ point of the circle, at a 22.5-degree angle. With the circle curving by 8 degrees, the point along which the sphere revolve around is shifted to the opposite direction by 4 degrees. And the two spheres that rotate this point have shifted by 2 degrees to the opposite direction to the bigger circle.
And for the above image to make sense, the path of both yellow spheres must be of equal length. I call this, half-jokingly, the toughest mathematical riddle. It probably isn’t objectively the toughest riddle, but effectively it sort of is. Had someone realized that you can describe the movement of spherical particles to follow this logic, they would probably have been able to apply it to physics before.
If I were able to formulate a proper sentence around this phenomenon, I would call it the dot reflection conjecture, or even the Lintinen conjecture. The funny thing is that this sounds just a bit like the Poincaré Conjecture, one of the biggest mathematical proofs of this century.
Perhaps, I could just let go of my perfectionism and just start writing the manuscript on my “Theory of Everything” -hypothesis and emphasize that the manuscript does not yet include a proof. This way, I might finally be able to get someone else to review the hypothesis.
Comments