Almost two weeks back I made a promise not to post anything before I managed to produce accurate equations describing the reflection of elementary particles of energy (kaus/dots). I didn’t exactly follow this promise, but published three follow-up posts with some new insights on the reflection model. However, after my post “Pétanque of Reflection”, I’ve kept my promise and have paused for ten whole days, probably my longest pause in ages. One of the reasons for this is that I’ve been busy with Christmas, so I haven’t had as much time to work on the problem.
But today’s the day: I’ve finally figured out all of the equations describing the unit sphere of reflection. Perhaps I won’t go too deep into what the equations mean just yet, but here are the cryptic versions of the equations:
The important things to note are that r is the radius of a kau (which I normalize to 1 for simplicity) and γ is what I call a flyby angle. All the other units are derived from r and γ. I’ll probably change some names for future posts and the Theory of Everything -manuscript, so don’t be surprised if future posts will look a bit different. So, without further ado, this is the shape that the above equations produce:
It’s a sphere with a radius of RUS and with two pairs of circular planes of reflection. On either side, these planes are connected with flyby paths (FL) that have steeper angle (γ) further away from the hinge point and a shallower angle (φ) closer to the hinge point. The flyby paths are spacetime vectors, where the points of reflection at the frozen present is described with a connecting vector with a length of exactly 2r.
Here we need to backtrack a bit. You see, the flyby vectors don’t mark a point in time and space, but rather a specific path in spacetime. This means that the yellow connecting vector exists only briefly when the two neighboring kaus collide. From the image we see that one of the kaus has propagated for the (yellow) flyby path within the unit sphere before the reflection, whereas one of the kaus is only entering to its flyby path at the point of reflection.
If you look at the above rotating model carefully, you see that all of the vectors are connecting to very specific points along the toruses, each of which follows closely along the surface of the transparent green unit sphere of reflection.
I’d like to say already that the model is ready, but there’s one more thing I need to check before I can say this with perfect confidence. I need to connect four of these in a row and make sure that the flyby paths align. When I do this, I know the model is ready.
I’d be surprised if I wasn’t able to do this, but I’ve been wrong before, so I still need to be careful…
Comments