Inspired by the pretty pearl necklace of the last post, I decided to try to render the Möbius double donut with pearls as well. It took a bit of trial and error. One of the first attempts looked like this:
Pretty in its own way, but obviously wrong, in more ways than one. One of the clearest signs that things were wrong was that with the whole in the middle being so small, there was plenty of space between the dots on the outer rim. And of course the twisting didn’t seem to work and the bending by 720 degrees doubled the number of the pearls in the structure.
So, I decided to go just to the basics. Now that I new how easy it is to make an array of spheres and just make them follow a curve, I made a helix with 200 turns (or ring-like segments), a radius of 0.495 (this was determined iteratively so that the alignment of the pearls look ‘right’), and the height of a turn being 0.18, or slightly smaller than the diameter of the pearl (0.2). Then I used this curve as a modifier for the array of pearls and determined that I needed 3117 pearls to make a uniform band (minus a gap smaller than a single pearl that probably arises because the size were relatively randomly selected. This way the pearls on the inside were hexagonally packed. Unfortunately, the outside didn’t yet look like the square packing that I envisaged. Rather, there is still a bit of a gap from ‘ring to ring’. At the moment, I’m unsure whether all of this just reflects me not fitting the rings properly, or whether there’s something a more fundamental problem with the concept of hexagonal packing in the inner rim and square packing on the outer rim. But aesthetically, I would give the ring of pearls a B+, or even an A- (using American grading).
Looking at the ring of pearls from a distance, it’s hard to see any finer detail. However, when zoomed in, one can see the individual rings that make it up. Once zoomed in, something become blindingly obvious: the rings are bent! Well, more specifically they are stretched to a helix, where the height of a single turn is the height of a single pearl. Really, the shape of the alignment of the pearls is vertical sine curve, which is also bent. However, for simplicity I drew a schematic, where the visible yellow pearls are depicted as a line of reddish circles bent counterclockwise and the pearls you can’t see in the backside are depicted as a line of black circles bent clockwise.
The schematic immediately shows that if the pearls are moving (at the speed of light) past the neighboring pearls, the movement of the reddish pearls is bent towards left for the reddish circles and towards the right for the black circles. This means that the twisting of the ring of pearls is their inevitable feature! This proves the general reason why an electron has to be twisted into a Möbius donut!
The only question is, why the twist has to be exactly 360 degrees over the circumference of the donut? The simple answer is that this amount of twisting is determined already at the point when light is looped into the original double-looped shape. The harder question is how we get from the below simple form to the more complex one?
This shape being the original one, I realized that the pearls in the pearl-donut would need to be aligned into half a wind into donut. That is having revolved for 360 degrees around the donut, the string of pearls isn’t yet at the original position, but at the other end of the circular cutout of the donut. The loop needs to revolve another 360 degrees to make a closed loop.
I illustrated this by altering the diameter of the torus just the right amount (using trial-and-error) so that the spheres formed a twisting square array of spheres. What I got was a Möbius corn-cob, which looked pretty cool.
Here we see that the constraint of the spheres in the ring forces a perpendicular path against the line formed by the moving pearl and its neighbors.
If I wasn’t so pedantic, I could be happy with what I already got, but in my heart of hearts I know this approach involves an unacceptable short-cut. I realized that I hadn’t bent the helix by 720 degrees, but rather by 360 degrees. Why was this? Because I had thought that there would be a point where a 2-D array of pearls is bent 180 degrees into a half-cylinder, then twisted 360 degrees and then bent 720 degrees. Converting the half-cylinder into something doable with a helix proved to be impossible.
Then it hit me: the half-cylinder isn’t cut half length wise, but rathe the half-cylinder is quite unintuitively realized by stretching. So, this time I made a similar helix as before, but the height of a turn was two pearls instead of one. And hey presto, it worked! There was just enough space between the turns to fit another helix to twist around it. When I bent the structure 720 degrees, rather comically, the shape looked identical to the original shape.
This wasn’t a problem: far from it. This shows that the principle works. However, the problem is that this is bad for illustration. When one does not see that there are interlaced helices, it’s hard to see the wonder of the structure. I could overcome the problem of illustration by making two versions of the donut in different colors and adding just half of the required pearls for each array. However, as the array always starts at the same point, I had a slight problem of alignment for the illustration, but when acknowledging that the illustration isn’t perfect, I could clearly show that the interlacing of the two helices of the first 0 to 360 bend and the second 360 to 720 bend. I rand into problems with insufficient memory on my computer, so here is just a screen capture from blender instead of a proper rendering of the shape.
So, here we get to the million-dollar question: is this now the shape of an electron? Now I’m the most confident that I’ve ever been. Now I just need to make an array of dots (or pearls) and make a helix of them an bend the helix for 720 degrees. That’s it! The half-cylinder -idea that I had was transformed into something quite a bit different from what I originally thought, but this time I can’t find any miracles in the logic.
In my last post I said:
Now I have an urge to try to figure out whether I could figure out the mathematics for this compactification. I really don’t have enough trust in my skills in handling equations. I can only hope that there is a straightforward geometrical solution to the process. That’s something that I could handle. Also, geometrical proofs can always be visualized. And visualized proofs are always much more convincing.
This approach is exactly the kind of a straightforward geometrical solution that I can handle. And the proof was easily visualized. I still need to write proper equations for the structure, because in mathematical proofs you need the equation. But I already have the equations I found from Wolfram Alpha for the structure from my previous posts, so I think even with my not-so-stellar mathematical skills, I’ll be to write the equations.
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