Sometimes curious things happen. In my quest to explain the Theory of Everything with the reflection of elementary particles of energy (dots), I’ve been building a model of reflection with increasing level of complexity. In my last post I talked about the realization that reflection can be understood as comprising of reflection cymbals, or hi-hats. To understand what I mean, you have to read my last post.
But this isn’t the curious thing: just the prelude to it. The curious thing was that I finally had the basic building blocks of reflection, but they were buried under a ton of mathematical scaffolding. When I tried to take these building blocks and stack them up, I almost immediately realized that most the geometry was more or less redundant. There were extremely few elements that couldn’t just be taken out without any negative effects. Below is the basic shape, which I call the unit cell of reflection. As I’ve told in my previous posts, there are only six dots in the basic shape (shown in red). And there are only two connections between these dots (blue and yellow cylinders). There are two known paths between the dots (purple and turquoise cylinders). And two dots are located on a yellow circular plane and two on a blue circular plane. And the two remaining dots are along a circular plane around the center of the purple path (red ring). And this plane has to be perpendicular to the purple flight path. The distance between dots along this plane is a bit higher than two radii of dots (>2r, distance marked with a red cylinder). The length of this cylinder must be the same both in the center, the bottom and the top.
And after this is the only thing left to determine is to calculate the angle between the two central dots based on the angle between the yellow and blue circular planes. I have figured out most of the equations required to determine this angle, but there are a few unsolved questions remaining.
This is what the unit cell of reflection looks like in all three projections:
Next, you might ask: can I stack these cells already? I sort of can, but not 100 % correctly. You see, when the angle between the circular planes is this big, the model becomes descriptive, rather than fully accurate. In 100 % accurate stacking, there should be four dots of neighboring unit cells perfectly lined up when the shapes are stacked. Based on my attempts with Blender, this four-dot-stacking isn’t possible with this large an angle. However, a qualitative approximation can be made with two-dot-stacking. In this stacking, the middle unit cell of a three unit cell stack is omitted and only the top and bottom cells are stacked. Below you see the principle. The bottom unit cell is duplicated, lifted almost by the length of the purple cylinder and rotated around the vertical (z) axis by roughly 180 degrees. And as this stacking is only qualitative, the tweaks to get the shapes to match were done manually, so I won’t bore you with the details.
After stacking, the purple vector transitions into a turquoise vector and vice versa. When observed from more or less the y-z projection (as the purple and turquoise cylinders are parallel to the y-z plane), the turquoise and purple cylinders of the stacked until cell seem to be continuing in a straight line. This is because the reflection of the movement for both vectors is perpendicular to the y-z plane. When rotated by 90 degrees to reveal the x-z plane (upper right image), one can see that the path is reflected inwards. However, because this inward reflection is so small, the neighboring dots never touch, even though they graze with a zero distance between them.
And this is what the above model looks like rotating:
This looks very convincing. But I can’t proclaim that this is the whole truth until I’ve checked the mathematics. I wouldn’t be too surprised if this is more or less the complete solution. Then again, on the other hand I wouldn’t be too surprised if the mathematical check would through one more curveball at me.
Sorry if my updates seem to be revolving around minute details. This is because I seem to be so close to the final solution…
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