Something surprising has happened since my last post. In the post I laid out a conjecture that roughly said that the path lengths of all curved trajectories must be the same, when the same angle of movement. If this sounds complicated, it is because it is a tricky riddle in the first place. I half-jokingly called it the toughest mathematical riddle, because no one has even figured out to ask it, despite its ultimate significance.
In the post I said:
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.
So, this is what I’ve done. I started rewriting my Theory of Everything -manuscript, but this time replacing the still vague concept of refraction of elementary particles of energy (dots) and replaced it with reflection. As soon as I started writing, I realized that I could scrap most (or even all) of the rather vague language regarding refraction.
So, what is reflection? According to Wikipedia:
Also:
If the reflecting surface is very smooth, the reflection of light that occurs is called specular or regular reflection. The laws of reflection are as follows:
1. The incident ray, the reflected ray and the normal to the reflection surface at the point of the incidence lie in the same plane.
2. The angle which the incident ray makes with the normal is equal to the angle which the reflected ray makes to the same normal.
3. The reflected ray and the incident ray are on the opposite sides of the normal.
These three laws can all be derived from the Fresnel equations.
However, I don’t want to derive these laws from Fresnel equations. Rather, I’m confident that these laws are better derived from simple logic applied to the movement of dots. We can first consider the reflection of one dot from another dot, as seen in the picture below:
When there are just two dots, reflection can always be considered to take place on a plane. This plane is defined by two vectors: the first being the vector of the direction of movement of the dot by the other dot. The second vector is drawn through the centers of the two dots being reflected. In the upper picture, one observes that if one draws an orthogonal vector to the one connecting the two dots, one can define an angle of refraction, θ. The speed of the dot along this vector is c cos θ and remains constant in reflection. The motion that is being reflected is the component along the vector that connects the two dots, with a speed of c sin θ. This speed is retained in reflection, but its direction is reversed. The three laws of reflection are derived from this picture and no reference to Fresnel equations are required.
However, dots don’t really exist as pair, but as strings. Helical in light, and jagged helical toroids in matter.
This means that in jagged helices dots aren’t reflected from a single dot, but from two or more dots (I think three is the minimum, but I might be wrong). In this case, using planes that are defined by the full motion of the dots is no longer mathematically the best choice. Rather, the motion of the dots can be split into components in coordinates suitable for nice linear algebra. This way, one I can ignore the rotational component of a pair of dots moving along an orbital and just focus on their shared (parallel) motion, marked by v in the picture below:
In the picture, there are two dots that are placed vertically for our convenience. The parallel motion is horizontal, meaning that this component is not reflected by the lower dot. If the upper dot is moving in a toroidal helical trajectory, the motion v isn’t actually completely horizontal, but consists of a small vertical component, which we will ignore for simplicity.
I was about to go deeper into the mathematics of this concept but found myself falling down a rabbit’s hole and never got round to finishing this post. Suffice it to say, the mathematics involved is quite tricky and still a work in progress.
However, the mathematics relies on two additional rules of reflection. They are:
The additional laws of reflection are:
Only the non-parallel components of motion of elementary particles of energy are reflected.
When an elementary particle of energy is reflected from more than one particle, the reflected component of motion is reflected again by all points of reflection.
I’m not 100 % happy with the fifth law. It’s not exactly wrong, but it’s too esoteric to mean anything without further clarification. This is why I’ve sat on this post for about a week.
So, don’t take these two extra laws of reflection as written in stone, but rather as placemarks for something more elegant.
Rest assured, when I manage to come up with something more elegant, I’ll be sure to tell about it in a post.
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