Subtitled: “Of Vectors, Circles and Cross-Products”
In my last post I said that I had sorted out the folding of elementary particles, or dots, that move at the speed of light into double-helical orbitals of triangles folded into an origami. At the end of the post, I lamented that I had solved the equations for a linearly folded origami, with no curvature. However, I knew I had more work to be done to figure out the equations for the case where these dots were in actual orbit, or with a non-zero curvature.
It has taken a lot of effort and I still don’t have the equations sorted, but at leas now I have the exact shape of the curved orbital, from which I can derive these equations.
One might ask, how is it even possible that I have figured out the shape of matter without realizing how to get there? Well, the short answer is mathematics, but let me try to answer with a bit more depth.
I will not bore you with the dozens of ways that I tried to explain the helical orbitals of matter and just present the explanation that was mathematically sound. And it just might be what I’m about to explain to you will blow your mind.
You see, the dots behave very much like a double stars, in that they are in orbit around each other under thein own gravity. The center-to-center distance in this pair of dots is the diameter of a single dot. Almost all of the speed of movement of these dots is linear, or moving at the speed of light in a straight path. However, a small fraction of this movement is rotational, with the dots moving in around the ‘double-star’ orbit with a diameter of a single dot. And what is the reason for this rotation? The reason is refraction, roughly the same phenomenon why a prism causes a beam of light to split into a rainbow. But what causes this refraction, or bending of the path of the dots? Well, refraction is caused by other pair of dots surrounding them. In fact, each pair of dots is connected to a pair of dots on either side of it. But what I only realized a few days ago was that the refraction is caused by interaction between a single dot of the pair with a single dot of the neighboring pair. This means that an orbit of dots is a zig-zagging line of dots, where each dot is touching only two other dots.
This may be difficult to visualize, so I’ll try to illustrate it with a Blender image I drew first with vectors and rings and only added the spheres once had sorted out the shape. The small green and blue circles, mostly occluded by the spheres shows two ‘double-dot orbitals’. The yellow spheres shows two touching dots of neighboring orbital. The yellow circular arc indicates the path along which the ‘double-dot orbitals’ move. The larger green and blue circles are the most complicated ones to explain. In short, they show the allowable locations of the neighboring touching dot, with a specified angle to the double-dot. With zero curvature this angle is 60°. This means that the touching dot touches both dots of the double-dot, and you have a rod of double-dots. While in matter, this angle is just over 60°, we have to exaggerate this angle to observe this curvature. In the image below, the angle of touching is 70°, if I remember correctly.
So, how do we know what this curvature is? Apparently, you can derive equations if you align vector with circles attached to them along a torus with an inner diameter that is the length of the vector. The small circles have to encircle the ‘tube’ of the torus, whereas both large rings must connect with one of the dots of the neighboring double-dot. And as it happens, each angle of touching is only associated with one possible curvature.
And what is this correlation between the angle of touching and the curvature? I don’t know yet, but I already know of all of the geometrical elements required to figure this out. Besides the vectors defining the touching dots and the circles defining orbitals and the circles of touching, we also need to define the cross-products of neighboring vectors (red) and also define the cross-product of this vector with the vector defining the double dot (pink).
And before I figure out the equations, all of this is highly complicated and confusing. However, as always, once I figure the deeper interactions behind this curving, I’m confident things become clearer.
Before I’ll wrap up for today, I’ll offer you a thought experiment. If one considers that everything is made of understandable and physical entities, what would be the alternative to what I suggest matter to comprise of? If we consider that E = mc², any other answer becomes untenable.
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