I always enter these challenges by saying they are horrifically difficult, in case I make no progress. But it seems as though I always make progress on these challenges.
Charge as a physics characteristic or attribute, is a rather difficult concept. Unlike speed where we know it is distance/time, with charge we struggle to relate to it.
The best we can do is examine and study magnetism. But even here, it seems that the true meaning of charge is foreign to our grasp. Because it looks like charge is force, or poetically, frozen force. Charge and magnetism are unsettling to our minds and we have to accept them, knowing they are stranger than most concepts that we can grasp and quickly understand.
But I have made some progress tonight.
I said in my prior posts, that to understand the basics of charge would have to be able to explain the exponent of 10^-19 C for the elementary charge of 1.6 x 10^-19 C.
If one looks at radius or diameter of the smallest particles in physics such as the proton or electron. We know the Bohr radius is on the order of 5.2 x 10^-11 m. So we are getting close to 10^-19. We know the proton diameter is about 1.6 x 10^-15 m. So here with the proton we are only 10^-4 away from 10^-19.
Now we know that the mass of the proton is about 2,000 times larger than the electron. Can we thence say that the diameter of the electron in collapsed wavefunction is approx 10^-19 meters?
Now notice that the definition of the Ampere is 6.241 x 10^18 electrons passing a given point per second of time.
Here again, I contend a more accurate number for One Ampere is 6.18 x 10^18 electrons given my speculation that the elementary charge "e" is actually the digits of the golden ratio or the phi number of 1.618 and so, One Ampere should be the digits 6.18.. and not 6.24.
Notice also that in the proton diameter, it is reported as 1.6 x 10^-15. So the "phi number" is blossoming out all over the place in physics.
So now, how do I derive the elementary charge "e" purely out of math without ever doing any physics experimentation? Well, it is not going to be as challenging or difficult as was deriving the speed of light out of pure math, and I still have some lingering issues with that derivation as to why 1/8 of the distance of the log-spiral is used. But the derivation of the speed of light out of pure math is a beautiful tour in physics and math, in that every sphere surface embodies the speed of light as a division of its meridians by a log spiral.
The adventure of deriving elementary charge from pure math is not as dazzling, but more straightforward. This is so, I suspect, because charge is more of a force than a parameter. With speed we can break it down into distance and time. With charge, well, it is a quasi force itself. That is why the definition of charge and ampere are really only a repeat of the Coulomb force law.
So, let me derive elementary charge out of pure math. We are going to say that in physics there is a smallest particle between the proton or electron, and that be the electron. The proton diameter is 10^-15 m. The electron diameter we are going to surmize as 10^-19 meters. Charge itself is geometry and we are going to say that the proton is Elliptic geometry and thus, the electron is Hyperbolic geometry such as the pseudsphere. This means the proton is positively charged and the electron is negatively charged.
Now what number is the elementary charge going to have? I have established that its exponent is going to be 10^-19 Coulombs because the smallest of the proton or electron is the electron. Now all I need to establish the elementary charge is what numeric value for the 10^-19 C. And the answer is of course the only number in mathematics that is self similar, no matter how big or small in size, the number is proportionally the same. Call it the logarithmic fractal number and it is phi or the golden ratio of (1 + sqrt5)/2 = 1.618.
If you take that number and multiply it by itself you end up with 2.618.. If you take that number 1.618.. and divide it into 1 you end up with 0.618..
So the number phi is like a charge in physics, it is conserved and quantized.
Now I maybe able to go further in a derivation by pointing out the significance of sqrt5 for the Atom Totality and the division by 2 to make the phi number. But I think I have derived it well enough to move on.