Note: this is already edited from the original posted earlier today.
In yesterday’s post I made a Möbius donut with twisting and bending a ribbon bent into half a cylinder. I commented:
So is this an electron? It seems like it, but I'll have to see how it fits to everything else I've introduced to make a definite statement.
Well, today I went back to the donut and looked at the equations more carefully and realized that I had still made one assumption of a miracle. That is, I twisted the half-cylinder for 180 degrees, bent for 360 degrees and then made a miracle-mirror™. That is, I assumed that if I continued with the curve from 360 degrees to 720 degrees, it would make a mirror image of the first 0 to 360 degrees.
The problem is that the second half of the curve isn’t a mirror of the second half. Here we have the colored with two different colors.
Then what? I tried to solve it by mathing it out and found an interesting phenomenon. When the sum of the values of x with a phase difference of 360 degrees was plotted against the sum of the values of y, also with a phase difference of 360 degrees I could plot a circle, with a radius that is twice the radius of the center point around which the helix twists. Like this:
Also, when I compared the sum values of x with a phase difference of 360 degrees against the sum values of z with a phase difference of 360 degrees, as well as compared the sum values of y with a phase difference of 360 degrees against the sum values of z with a phase difference of 360 degrees, the graphs looked identical to what I had previously observed for the refraction of light.
But this case was different to light. These ‘refracted’ loops we bound to other refracted loops, making a closed loop orbital.
So I went back to blender and got rid of the final bit of miracle and scrapped the mirror image, which wasn’t a bad idea, but just didn’t stand further scrutiny.
This time I again made a half a cylinder, but instead of twisting it 180 degrees, I twisted it for 360 degrees. Next, instead of bending it for 360 degrees, I bent it for 720 degrees and voilá: a much prettier donut, that doesn’t require the miracle-mirror™:
In the original post, the image didn't look much like anything, as the surface was an even yellow. I then learned how to add images as the surface texture. This helps better to illuminate the twisted structure of the donut.
However, as with most simple, but pretty mathematical ideas, someone had already done something similar. I just googled Möbius donut and this popped up:
To be frank, the animation (in the link) looks cool and clearly displays the trajectory of the dots, but it necessarily isn’t identical to my model. You see, you need to really twist the half-cylinder for 360 degrees, whereas the cool image might not be twisted enough (even though my new textured Möbius donut really looks quite the same as the above black and white one).
Probably the easiest way to illustrate improper twisting is to show what happens if you still twist the half-cylinder for 360 degrees, but bend it just by 360 degrees instead of 720 degrees. You get this really funky looking ribbon that is at the same time a closed loop, but where it yet doesn’t form a uniform donut (toroidal) surface.
Probably the animated Möbius donut was never formed with half a cylinder, but by just twisting a cylinder. While this is perfectly fine for the animation, this kind of Möbius donut doesn’t represent the nature of an electron.
Another way to illustrate improper geometry is to have the right amount of twisting (360 degrees) and bending (720 degrees), but making the cylinder be less than half. Here we see from the lack of a part of the Möbius donut how the loops knot into a rather complicated shape.
Again, one fewer miracle in the structure. So, is a uniform twisted donut the structure of the electron? While it seems that I’m getting closer and closer, I still won’t make a definite claim. I still need to see how it fits with everything else before I can be certain.
I think I’ll still be surprised in the days to come.
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