In my last post I presented both the way to draw a sphere with a string. I also presented a separate equation for the deformation of the amplitude, but it still looked a bit ugly. After looking at it a bit more closely, I realized that I just needed to alter the point where I started plotting my spherical curve. While before I started at θ = 0, I just started it from θmin = 2π(P+1)/-2, or θmin = -π(P+1) and ended it with θmax = 2π(P-1)/2, or θmax = π(P-1), and I could write the equations like this:
where Amin is the minimum amplitude of the helix, G the growth rate of the amplitude, and P the number of turns in the helix.
This meant that I could just write the equations into excel and choose the number of data points to be plotted. I started with smaller number of points but figured out that I really needed 5000 data points to show any sufficiently large structures.
Then a very obvious thing for anyone working with Excel, is that I put a running number next to the data points from 0 to 5000. This meant that I could see where I was at. As mentioned, the starting point was θ = -π(P+1), with each data point increased with 2πP/5000, or πP/2500.
So, I had three variables that I could play around with. And the best of all was that I could both checked the values in Excel in the three 2D-projections available (x-y, x-z and y-z), as well as generating Blender objects with them.
At first I started with a very ‘choked’ helix, with a relatively limited number of turns, with 99 turns, a minimum radius of 5.0 and a growth rate of 0.2. By showing the choked helix before it has been bent 720 degrees, it is convenient to see what really takes place. This structure looks like the juggling prop diabolo.
The only way to know whether this shape is ‘any good’, is then to turn it 720 degrees. By doing this, I got this:
A very lop-sided thingamajig that doesn’t really look like sphere.
Next, I kept the turns at 99 and the growth rate at 0.2 but increased the minimum radius to 20.0. The shape started to look a bit better after turning around for 720 degrees:
Next, I wanted to see whether the number of turns would make a large difference, so I increased the number of turns to 399, kept the minimum radius at 20, but to compensate for the quadrupling of the turns, I reduced the growth rate to 0.05 to keep the overall shape of the choked helix.
I wouldn’t say that the sphericity was improved, but the structure started looking much more ‘smooth’.
So, as the reduction of the growth rate was responsible for the improved sphericity of the curve-sphere, I kept the number of turns at 399, and the minimum radius at 20.0, but decreased the growth rate to 0.05 and I got this:
Way more spherical and still very smooth. This seems to indicate that the cylinder from which the electron is bent has to be very narrow for the electron to appear spherical. Of course, this poses the question whether the electron really is very spherical? To not rock the boat too much, let’s assume this is the case.
The next question is, what would an electron look like, if instead of 399 loop, the number of loops is ‘astronomical’, as in at least billions? One could no longer see individual loops, but the structure would still look hazy, due to the distance of one loop from another. And here we need to start thinking about what the growth rate actually means? If the elementary particles that make up the loops are the diameter of Planck length, then the growth rate is on the scale of one Planck length per turn. With each elementary particle in orbit around the spherical surface, it would appear to the observer as a ‘probability cloud’, especially if the observer has no way of knowing what the spherical surface is made of.
But once one has the idea that the surface is made of a single string, looped a myriad of times, there really is no turning back into considering as a probability cloud. Once you can plot a sphere comprised of a string moving at the speed of light, you can't unsee it. After this realization, there's no turning back.
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