As a natural continuum of my previous posts, it is once again time for some equations. If you’ve read my posts in the past, you probably know that I don’t share too many equations, as I’m well aware that there is an understanding that laypeople are put off by equations. However, every once in a while, I show the equations that are the basis of the figures that I show in my posts.
When I’m in the process of exploration, I don’t really write equations. Rather, I test the ideas with Blender and if I can produce reasonable figures, I trust that there must be equations behind them. Only when I’m getting confident that my images are correct enough, do I try to reverse engineer the equations from the shape. This might not be the ‘correct’ way of doing mathematics, but I don’t really care. After all, I’m I chemist, not a mathematician.
Anyhow, these are the two sets of equations for the two helices of dots.
Blue helix:
Yellow helix:
Perhaps I should explain the terms a bit. R is the average radius of the saint Hannes knot, φ is a variable angle that ranges from zero to 2π.
is half the distance from the center of one pair of dots to the second pair, which it refracts with. π/8 = 22.5 degrees, or the angle at which the pair of dots begins.
The difference between the two sets of parametric equations are shown in the color of the orbitals. Identical terms are shown in black.
These are the full projections of the equations:
With the equations being so alike, it is better to zoom in on the only dots that experience refraction from the opposite side of the orbital. In the zooms below, I show dots at φ = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2 and 7π/4. In the left image, the blue dot that touched the orange dot is at φ = 0 and the orange dot is φ =3π/2. In the right image the blue dot is φ =3π/4 and orange dot φ =π/4.
I should note that I’ve ‘cheated’ in the above image. You see, with the size of the dots being so large in comparison to the size of the orbital, the dots shown above aren’t the closest dots. Using the above equations with R = 40, produced the image above. However, when R is so small the smallest distance between the opposite sides of the orbital is much smaller than
It’s hard to explain simply, but this means that R must be a much, much bigger number than 40. What this number must be is yet unknown to me. I’ll have to see whether I can finish the revision of the Theory of Everything -manuscript without finding this out.
At the moment it seems wisest to no longer worry about being perfectly correct. While there might be some corrections required for the refraction equations, they finally explain what takes place at the point of contact.
I can't imagine what else could a peer-reviewer ask...
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