# Parametrization of the hyperbole where the multiplication is defined by i^2 = -1 + 3i.

With a new number tau (see previous post) there is always a log of an suitable imaginary unit found, but that does not mean you have a parametrization instantly. And with ‘suitable’ I mean the determinant of the imaginary unit must be one because I always want to make some complex exponential, in this case a complex exponential hyperbole.
The post is relatively short, I always try to write short posts and I always fail fail and fail in that. But now it’s only four pictures long so this comes close to a tiktok version of my ususal math postings. By the way it has nothing to do with this post, but do you like tiktok? I don’t like it very much, it is more for people that have a bit different mindset compared to the way my old brain works…
In this post I first neatly write down the parametrization using the sinh and cosh to express the two coordinate functions. And after that I more or less express it all as much as possible into the two eigenvalues of our beloved imaginary unit i that rules this plane via: i^2 = -1 + 3i.
May be you have never thought about imaginary units as having eigen values themselves. In that case I invite you to calculate the eigenvalues of the ordinary numbers z from the complex plane. You know that plane that is ruled by an imaginary unit i via the rule: i^2 = -1. You will find a very interesting answer and of course after that you wonder WTF are the eigen vectors?

But let us not digress and enjoy the beauty of a complex exponential that is a hyperbole in this case. Here we go:

I am sorry for those bad looking accolades, it is some small fault in the Latex math package or some other old computer feature. You see it in much more texts written in Latex; the { and the } are just not presented properly.
That was it for this post, as always thanks for your attention.

# The number tau for the hyperbolic multiplication i^2 = -1 + 3i.

Some posts ago I showed you how you can calculate the number tau (always the logarithm of a suitable imaginary unit) using integrals for an elliptic multiplication. To be precise you can integrate the inverse of numbers along a path and that gives you the log. Just like on the real line if you start integrating in 1 and integrate 1/x you will get log(x). If you have read that post you know or remember those integrals look rather scary. And the method of using integrals is in it’s simplest on the 2D plane, in 3D real space those integrals are a lot harder to crack. And if the dimension is beyond 3 it gets worse and worse.
That is why many years ago I developed a method that would always work in all dimensions and that is using matrix diagonalization. If you want the log of an imaginary unit, you can diagonalize it’s matrix representation. And ok ok that too becomes a bit more cumbersome when the dimensions rise. I once calculated the number tau for seven dimensional circular numbers or if you want for 7D split complex numbers. As you might have observed for yourself: For a human such calculations are a pain in the ass because just the tiniest of mistakes lead to the wrong answer. It is just like multiplying two large numbers by hand with paper and pencil, one digit wrong and the whole thing is wrong.
Now we are going to calculate a log in a 2D space so wouldn’t it be handy if at least beforehand we know in what direction this log will go? After all a 2D real space is also known as a plane and in a plane we have vectors and stuff.

So for the very first tme after 12 years of not using it, I decided to include a very simple idea of a guy named Sophus Lie. When back in the year 2012 I decided to pick up my idea’s around higher complex numbers again of course I looked up if I could use anything from the past. And without doubt the math related to Sophus Lie was the most promising one because all other stuff was contaminated by those evil algebra people that at best use the square of an imaginary unit.
But I decided not to do it because yes indeed those Lie groups were smooth so it was related to differentiation but it also had weird stuff like the Lie bracket that I had no use for. Beside that in Lie groups and Lie algebra’s there are no Cauchy-Riemann equations. As such I just could not use it and I decided to go my own way.
Yet in this post I use a simple idea of Sophus Lie: If you differentiate the group at 1, that vector will point into the direction of the logarithm of the imaginary unit. It’s not a very deep math result but it is very helpful. Compare it to a screwdriver, a screwdriver is not a complicated machinery but it can be very useful in case you need to screw some screws…

Anyway for the mulitiplication in the complex plane ruled by
i^2 = -1 + 3i I used the method of matrix diagonalization to get the log of the imaginary unit i. So all in all it is very simple but I needed 8 pictures to pen it all down and also one extra picture know as Figure 1.

That was it for this post, we now have a number tau that is the logarithm of the imaginary unit i that rules the multiplication on this complex plane. The next post is about finding the parametrization for the hyperbole that has a determinant of 1 using this number tau.
As always thanks for your attention and see you in the next post.

# Nice experiment: Magnetic field in the direction of an electron beam.

Now I’ve seen a lot of relatively boring videos the last years with electron beams and magnetic fields. And the only thing they often show is just the Lorentz force that is perpendicular to both the magnetic field and the direction of the electrons. Never ever do they jump to the conclusion you can do your own ‘Stern-Gerlach experiment’ by trying to separate the electron beam into two.
As such those guys, it’s almost always guys, often do nothing more as holding the magnetic field perpendicular to the electron beam. And no matter how hard I shout and curse at youtube on my television, they never listen… But serious, today I came across a video of a teacher who tried to make the magnetic field as parallel to the electron beam as possible.
In the past I have done a similar thing and I still have photo’s from that. But the way we had set up these experiments is rather dual to each other.

The way Francis-Jones does it in the video: His magnetic field is wide, he uses those Helmholtz coils and one steady electron beam.

Back in the time I could still buy an old black and white television that still works to this present day. Because it’s a black and white television it only has one electron beam that constantly covers the entire television glass tube. So my electrons were spread out and my magnets was more a point like thing because it was a stack of neodymium magnets.

If you look at such experiments as ‘wide’ against ‘narrow’ there are two other possibilities this way:
1) A Helmholtz coil against a television screen, I don’t think you will get interesting results but you never know.
2) A stack of magnets against one steady electron beam, I expect a central point on the screen for the middle of the electron beam and a vague ring around it from the electrons that get repelled.

Anyway the reason that still today I think electrons are in fact magnetic monopoles was simple: My own simple and cheap experiment could absolutely not disprove that electrons are not tiny magnets but monopoles. All that stuff from quantum theory that for some mumbo jumbo reason the dipole magnetic field of the electron will anti-align with external magnetic fields, it is just fucking bullshit.
It is so fucking stupid in say the electron pair we know from chemical bondings and also from super conductivity, why the hell should those tiny magnets anti align? A few months back I made a picture for what the official version of an electron pair is, of course this madness should also have an south pole to south pole variant, but here is that nonsense once more:

Let me stop ranting and lets turn to the video. At one point in time Francis turns the electron beam a little bit and there is where the next screen shot comes from. It is at 7.50 minutes into the video:

Well you can judge for yourself but the problem with looking at such video’s is that they just never ever try to split the electron beam in two… So it is hard to say if here are two electron streams or that we are looking at some light reflection. So I cannot use this video for making my point it is stupid to view electrons as tiny magnets since their magnetism is just like their electric field properties: Monopole and permanent.

After having said that, let me show you once more a photo of the old black and white television. And a miracle happened: Not only did my experimental setup succeed into two classes of electrons with regard to their monopole magnetic charge. It also turns the old black and white television into a color television!

Yeah yeah, that small circular region behind the magnet is what gave me a bit of confidence years ago. These electrons are magnetic monopoles and not tiny magnets or whatever what. But the professional physics people much more like to talk about stuff like “Spin orbit coupling” or other mysterious sounding stuff.

At the end I want to remark my total costs were 12€ for the old black and white television and about 50€ for the stack of neodymium magnets. But this Francis guy says the tube is about 500 pounds, so likely Francis is from the UK. So shall I buy me one of those things for myself?
No of course not, I am not interested in writing a publication that could be read by professional physics people. Why should I? In case electrons are the long sought magnetic monopoles, it is obvious you won’t get much published into such lines of thinking.

Lets leave it with that while noting it was fun for me to write a new post on magnetics.

Updated two days later: Today, that was 06 March so actually yesterday, I realized that if you have access to one of those beautiful cathode ray tubes, you can also use two stacks of those strong magnets.

Since the goal is to make the beam split in two, you must use the north pole of the one stack and the south pole of the other stack. If you have never worked with these kinds of magnets, practise first before you hold them near the glass.
If the magnetic fields are strong enough and the electron beam splits in two, what does that mean for if electrons are magnetic monopoles or bipolar tiny magnets? Well if you view the electrons as magnetic monopoles, it is logical from the energy point of view that the beam splits:
Both kinds of magnetic charges only try to lower their potential energy.

And suppose that electrons are tiny magnets, in that case the electrons that align themselves with the applied magnetic field will lower their potential energy. And if you believe that electrons anti-align where does the energy come from that makes them do this?
All that anti-align stuff of electrons is rather mysterious and I think that is important for the physics people. If you are interested in quantum mechanics you likely have heard the next phrase of saying a few times:

If you think you understand quantum mechancis,
you do not understand quantum mechanics.

Well that is an interesting point of view but you can also think: If I get crazy results with thinking that electrons can anti-align, may be there is something wrong with my theory? But you never see physics professors talking that way, after all talking out of your neck is a shared habit amongst them.

Now the idea of using two stacks of magnets must be executed carefully as you see in the next picture:

End of this update. Thanks for your attention.