The Electron is a Knot

In my last post I realized that the electron appears to be a Möbius double donut, where after having turned 360 degrees, the elementary particles, or dots, comprising the electron are ‘upside down’ and have to turn a second round of 360 degrees to reach their starting point.

So, what is this turning, then? The short answer is the photons making up the electron. Imagine a starting point, where you have a single photon in vacuum. Or more precisely a supraphoton, the closed ring of light that is the true quantum of light. However, for subatomic quanta of light, the whole concept of photons and supraphotons gets me confused.

Actually, to be frank, if with very large supraphotons, the supraphotons are comprised of several photons and the number of photons in the supraphoton decreases, there has to be a point where the size of the supraphoton is equal to photon. But this probably means that beyond this point, the photons are comprised of more than one supraphoton. Because how else could the energy of the photon increase as the wavelength of the photon decreases?

What this appears to mean is that the simple donut shape originally considered for an electron is actually true for a gamma photon. But this gamma photon isn’t just a single circular loop of dots, but rather a bundle of them. Apparently the photon being looped into tinier supraphotonic loops (I might have to change the name: supraphoton sounds wrong in this context) relates to its frequency. If the frequency of the photon is high, you can have more than one loop in a photon. If the frequency is low, you need more than one photon to complete a loop.

I might be wrong but it seems that when a photon/supraphoton is generated from matter (such as in black body radiation), the photon/supraphoton isn’t a smooth ring after all, but rather is more like a vibrating circular string. What this means is that the photons that are absorbed by Van der Waals molecules aren’t generated upon impact with them but are truly present in the supraphoton. This idea almost blew my mind! I had been scratching my head with the apparent discrepancy between the wavelength and frequency of light.

However, I’m still confused as to what creates the correlation between the diameter of the supraphoton and the frequency of light I know mathematically that it’s all to do with the Planck relation and Planck’s law, but to really understand what’s really going on there, I have to think about it more carefully.

But, for the moment, let's forget why we get the frequency we do in gamma photons and focus on what would a photon whose frequency is so high that it needs to fold in two to ‘fit’ its wavelength. You can think of a helix that twists

Admitting I’m only ok in mathematics, at least compared to real mathematicians, I went online to see what the equation for a helical torus is. Again, I went to Wolfram Alpha.

The parametric equation of the ith helix are

where R is the radius of the ring, r is the radius of the helix, m is the number of helices, n is the number of windings per helix and λ is the winding direction (+1 for right and -1 for left).

Here is a toroidal helix with tube radius 0.2, number of helices is 1 and number of winds per helix is 3 and the winding direction is right.

The more there are winds in the toroidal helix, the easier it is to see that we are dealing with helices. Already with two winds the shape looks off:

Then with a single wind, the shape looks more like a ring than a helix.

So what does an electron look like? There the number of winds (or n) is not a whole number, but rather it is ½.

I went to Wolfram alpha an wrote:

plot x = cos t (1 -.2*cos(2pi+t/2)), y = sin t (1 -.2*sin(2pi+t/2)), z = 0.2*sin(2pi+t/2)

It cleaned the text to this:

And this is what I got:

My double donut!

But to be fully honest, the above isn’t exactly the accurate plot for the electron, because I arbitrarily chose a tube radius of 0.2.

Actually figuring out the tube radius was another clue into the more precise structure of an electron. I started playing around with the thickness of the tube and as Wolfram Alpha only shows a very specific angle in the plot, I really couldn’t make out what was going on. That’s why I wrote the equations into excel and started looking at what happens. At first I worked with very large tube radiuses (or radii) and got really wonky shapes. Then I thought “what would happen if the tube radius was really small?”

Excel doesn’t really plot 3 D data (without plugins), but I could plot the projections in x-y, x-z and y-z axes. With really small tube radii I got something that at first looke just like a simple circle, but which started showing interesting features when zoomed in (see below). From the x-z projection it looked like a sideways parabola. And from the y-z projection it looked like an infinity symbol.

Then I realized that this shape seemed to be contradicting with the concept of double-donut with thick tubes. But what I didn’t do was scrap the idea. I started thinkin about what would happen if this structure formed in the Breit–Wheeler process? If the shape was really descriptive of gamma rays forming electrons (and positrons)? This would mean that the shape above would describe a photon knotted into a structure that would need to be compactified into an electron. Thus, the above shape would be an intermediate state prior to the formation of an electron (and a positron).

Actually, to be frank, I didn’t realize it at that point, but the shape isn’t even a prior state of an electron, but of an electron-positron pair, prior to splitting.

Ok, now you’re probably completely lost. What on earth am I talking about? The double donut structure above looks like two mirror image photons that are slightly refracted (i.e. not circles anymore) and fused from the ends. So what does this mean? Well, the dots in one photon try to turn in a helical path that is the mirror image of the other photon. This means that the only way for the dots to move is that the shape flattens to a two-dimensional ring, so that all of them move parallel to each other (i.e. in the same direction).

Where have we seen the conversion of gamma rays into positron-electron pairs? Only in the presence of a laser field. What if laser light can be considered an inpenetrable wall, where each photon (and supraphoton) are connected to one another? This way, if this flat double donut were to hit this wall, it could push through, but would need to stop. We no really stop, but to transfer the linear motion of the double-donut into rotation.

But I had already mentioned that you can’t push a string. What this means is that each of the dots ‘tries’ to move past another dot and not to push it. This looks exactly like the movement of light in a helical path, but with one exception. While the path of light can be considered linear, the movement in the double donut has to be a quasi (as in almost)circular path. This way none of the dots need to push the other donuts. Rather, the donuts will spread as much as the double donut structure allows.

Here we are introduced to two ‘competing’ paths. At some point the helical path within the donut cannot stretch anymore, because this stretching will shrink the outer diameter of the double donut, as the number of dots isn’t changing in this process. Also at some point an equilibrium of the two processes is reached. Probably at a point where the dots are packed in the way that I mentioned before.

So now we have the 2D double donut. Only formed in an impact with a laser field.

Next, we should consider that although we started with a case, where the dots had looped to rings that correspond to the cutout of the tube, once we’re at the magical point, the dots are no longer relatively narrow rings, but rather the length of the whole toroidal helix

But then I think we make a really funky turn to an unexpected direction. I have a very strong intuition that the formation of the double donut isn’t the end after all. Rather, if the conditions are just right, this double donut splits into an electron and a positron.

And what happens when an electron and positron collide? They immediately transform into two gamma photons in a process called electron positron annihilation. This means that the double donut above is an unstable state. That is, the double donut will only split into an electron and positron in very specific conditions. Otherwise it will uncoil back into two gamma photons.

Then what? I went back to Excel and looked at the toroidal helix with a thick tube. The XY-projection of the double-donut looked really wonky:

So I decided to look at just a single donut, or half of the double-donut:

The important thing here is to remember, the shape wraps around a donut, like this:

So what would happen, if we had a toroidal helix that was the mirror image of the one above and placed it over it? We have a fused, or knotted double donut:

Like this (unfortunately the images above and below don’t have the same tube thickness):

So is this an electron? The answer is… who knows? If it is, then the strings of dot need to interlace in rather wonky ways. This has a slight resemblance to a Klein bottle. Well sort of, at least if you don’t look at the definition of a Klein bottle too closely.

At least you can unravel this knotted double donut structure into light too easily. And I guess a positron would then be the mirror image of the electron knotted double donut.

But honestly, my head hasn’t hurt this much from thinking in ages. I have a strong feeling that I’m not nearly ready yet. I’ve taken mostly logical steps, but I’ve also taken a few leaps of faith here and there.

I’m always getting back to this Sidney Harris cartoon:

Not until I’ve reduced these leaps of faith to zero, can I be absolutely certain. But the problem is that you can’t have absolute certainty. I feel like a painter with this theory. When is a painting ready?

No one sees the imperfections like the artist does and you must learn to accept them. (If you see no imperfections in your art, look closer – they're there.) With imperfections accepted, you then have to decide if you are happy with the result. If the answer is “yes”, then you are finished

Is this true even for the theory of everything? Well of course. The simplest version of the theory is that there are just tiny dots moving at the speed of light. Everything is just further proof of the theory. But for it to be finished, I would have to explain every single detail of the world. And I know my limits. I have to stop somewhere.

But perhaps a bit further still. The ‘problem’ with physics and chemistry is that any theoretical work has to have a rigorous mathematical basis. Logic isn’t good enough. I fear I’ll still be butchered because of the mathematics, if I won’t dig still a bit deeper.

P.S. After finishing this post, I realized there were errors in the beginning, that only became apparent once I'd followed the logic to its conclusion. I decided to leave the errors to the post and only try to correct them in future posts. The errors just show what following ones intuition does. You make a lot of errors, but if you know what you're doing, you can spot the errors and correct them.