But for the two 2D multiplications we have been looking at in the last couple of months, the method of taking the upper row as a conjugate did not work. I had to do a bit of rethinking and it was not that hard to find a better way of defining the conjugate that worked on all spaces under study since the year 1991. And that method is replace all imaginary units by their inverse.

As such we found the conjugate on 2D spaces like the elliptical and hyperbolic complex planes. And the product of a 2D complex number

Now when I was writing the last math post (that is two posts back because the previous post was about magnetism) I wondered what the matrix representation of the conjugate was on these two complex planes. It could not be the transpose because the conjugates were not the upper rows. And I was curious what it was, it it’s not the transpose what is it? It had to be something that if you do it twice, you do the identity operation…

All in all in this post the math is not very deep or complicated but you must know how te make the conjugate on say the elliptic complex plane. On this plane the imaginary unit

It turns out that the matrices of the conjugate are not the transpose where you flip all entries of the matrix into the main diagonal. No, these matrix representation have all their entries mirrored in the center of the matrix or equivalently they have all their entries rotated by 180 degrees. That is the main result of this post.

So that’s why I named it the “Cousin of the transponent” although I have to admit that this is a lousy name just like the physics people have with naming the magnetic properties of the electron as “spin”. That’s just a stupid thing to do and that’s why we still don’t have quantum computers.

Enough intro talk done, the post is five pictures long and each picture is 550×1200 pixels. Have fun reading it.

That was it for this post, one more picture is left to see and that is how I showed it on the other website. Here it is:

Ok, this is really the end of this post. Thanks for your attention and may be see you in another post of this website upon complex numbers.

]]>Since all those energy problems that I have with viewing electrons as bipolar tiny magnets are always skipped, I decided to use this video as a short post on magnetism. In videos like this the pattern is always the same: at the surface it all looks logical like in this video the spin configuration in that stuff known as spin ice. But video after video I have seen over the last years, always when we need to look at crazyland they always skip that. When the energy stuff gets crazy, they just skip it. Now this is absolutely not some form of a conspiracy, these people like Erica simply believe the bipolar magnetic electron is true and as such they have a blind spot into the problems: They just don’t see the problems because of their blind spot.

In the year 2015 I started to doubt that electrons were tiny magnets with two magnetic poles. I started doubting that after I tried for myself to explain the results of the so called Stern-Gerlach experiment. In my view the results were only explainable if we use magnetic monopole electrons. A few days later reading all those official explanations I understood I had to be cautious. And at the begining back in 2015 I knew nothing about electron spin, all I knew was that people from physics thought they were tiny (bipolar) magnets. It’s been a long journey from there back in 2015 and it will also be a long long journey going from our present year 2024. After all the belief that electrons are tiny macroscopic magnets is deeply rooted in 100% of the physics community.

In this post, for the first time since 2015, I included a simple expression about how the professional physics professors view the potential energy of electrons related to magnetism. It is somewhere below and it is the same as we have for macroscopic magnets like say two bar magnets.

If you hold two bar magnets south to north pole, that is the minimum potential energy because it costs energy to separate them. And if you hold two bar magnets say north pole to north pole, that is the situation of high potential energy.

The post itself is four pictures and two additional figures and of course the perfect video from Dr. Erica Carlson. Say for yourself, this video is a perfect 10 with all kinds of animations I can only dream of. Ok ok, there is just one tiny tiny error in it: electrons are not tiny magnets.

But for the rest it’s a “PERFECT 10” kind of video.

That was more or less the end of this post but I made one more picture depicting another big energy problem that the official version of electron spin has: The behavior of a single electron in an applied magnetic field.

After all if it were true that electrons are tiny magnets, if you apply a magnetic field to electrons shouldn’t they all perfectly align with that magnetic field and as such fall into their lowest potential energy state?

Yes in an ideal world they should, but we live in a world where we not only have a lot of professional physics professors but also television physics professors. And they never talk about the energy problems there are with the electron as being tiny magnets.

So this is a strange strange world where physics just ingores simple problems like the last picture of this post:

Oh yes the stability problems we have if it were true that electrons are tiny magnets. As you see in the video it is always skipped and their brains never go down that route… It is what it is and here is the video:

Lets leave it with that, the next post will be about matrix representations of conjugtes of 2D complex numbers. They are weird and also lovely now I have my new method of understanding the process of conjugation.

And as always thanks for your attention and not falling asleep before you read these last words of this post.

In the case of 2D and 3D numbers I always want to know the logarithm of imaginary units. A bit more advanced as what all started a long time ago: e^

An important feature of those numbers tau that are the sought logs is that taking the conjugate always the negative returns. Just like the in the complex plane the conjugate of

The idea is easy to understand: The proces of taking a conjugate of some number is also a linear transformation. These transformations have very simple matrices and there all you do is try to find the eigenvector that comes with eigenvalue -1.

The idea basically is that tau must like in the direction of that eigenvector.

That is what we are going to do in this post, I will give six examples of the matrices that represent the conjugation of a number. And we’ll look at the eigenvectors associated with eigenvalue -1.

At the end I give two examples for 4D numbers and on the one hand you see it is getten a bit more difficult over there. You can get multiple eigenvectors each having the eigenvalue -1. Here this is the case with the complex 4D numbers while their ‘split complex’ version or the circular 4D numbers have not.

Now all in all there are six examples in this post and each is a number set on it’s own. So you must understand them a little bit.

The 2D numbers we look at will be the standard complex plane we all know and love, the elliptic and hyperbolic variants from lately. After that the two main systems for 3D numbers, the complex and circular versions. At last the two 4D multiplications and how to take the conjugate on those spaces.

The post itself is seven pictures long and there are two additional pictures that proudly carry the names “Figure 1” and “Figure 2”. What more do you want? Ok, lets hang in the pictures:

Years ago it dawned on me that the numbers tau in higher dimensional spaces always come in linear combinations of pairs of imaginary units. That clearly emerged from all those calculations I made as say the 7D circular numbers. At the time I never had a simple thing to explain why it always had to be this pair stuff.

So that is one of the reasons to post this simple eigen vector problem: Now I have a very simple so called eigen value problem and if the dimensions grow the solution always come in pairs…

That was it for this post, likely the next post is upon so called ‘frustrated’ magnetism because the lady in the video explains the importance of understand energy when it comes to magnetism. After that may be a new math post on matrix representations of the actual conjugates, so that’s very different from this post that is about the matrices from the process of taking a conjugate…

As always thanks for your attention.

Well yes that is what we do, at present day there is that cute package or applet (in the past that was a so called computer program, why is all that kind of stuff an “applet” nowadays?) called DESMOS. With DESMOS the two lines that make up the set of non-invertibles is easy to graph.

These two lines are interesting because they are the asymptotes of all those hyperboles in this space, anyway those hyperboles that can be written as det(

I took the opportunity to introduce a more comprehensive notation to denote such spaces of 2D complex and split complex numbers. I wrote it much more as a set like in set theory and it includes the ‘rule’ for the imaginary component.

For readers who are new to this website and don’t have a clue what ‘hyperbolic’ or ‘elliptical’ 2D complex numbers are, it is all basically rather simple:

These numbers are complex because

All in all this post has five pictures of size 550×1200 pixels and an additional two figures from graphs from the DESMOS applet.

For me it was funny to write some ‘high school math’ with just a few parabole kind of stuff in it. During the writing of this post I came across the idea of making a matrix representation of all that conjugating stuff, as such I found a beautiful but still extremely simple way to find the direction of the number tau in a particular space. The great thing is that this time it works in all dimensions so not only the 2D complex numbers but likely much much more.

Lets try to upload my post to the internet and may I thank you for your attention.

I also wanted you to show how you can write the product of such a complex number

At last I wanted to pen down the formula for finding the inverse on this particular hyperbolic complex number space. It looks just like the way this is done on the ordinary complex plane with the exception that if you calculate it the conjugate is a bit different.

For me it is funny that we have exactly the same looking formula for the calculation of inverses. All in all it shows that the fixation the professional math professors have on all that

It has to be remarked however that the study of normed spaces is important in itself and also in practice: If you can find a good norm for some difficult problem like the successive aproximations in say differential equations and you can prove using that norm the stuff converges, that is BINGO of course. Yet a norm is only a tool and not all there is inside that strange space known as human math.

The post itself is 6 images long, in it I have two (pairs of) integrals going from 1 to

I included a so called Figure 1 that show the evaluation of these integrals by the Wolfram package for definite integrals, it’s a handy online tool in case you don’t want to evaluate your integrals with pencil and paper.

The last image is from the other website where once more I want to insult the math professors just a tiny tiny bit by using the standard formula of finding the inverso on the standard complex plane.

So all in all this post is 8 pictures long.

That was it more or less for this post, I hope you are a bit more confident by now that you can actually integrate spaces like this more or less just like you do in the standard complex plane. The last image is from the other website, it contains a female robot. Why there are female robots is unknow to me, after all with other tools you never have females like have you ever heard of a female screw driver or a female mobile phone?

Ok, that was it for this post. Thanks for your attention and just like the female robot look up into the light and start thinking about the wisdom behind **“The norm of the product is the product of the norms”**.

But physics professors behave just like math professors who after 33 years of doing just nothing will keep on doing that and never ever talk about the three dimensional complex numbers. What explains that kind of behavior, after all it’s all tax payer money so they should be a bit more humble don’t you think? The way I see it is that university people like math and physics professors are some elite. And I don’t mean an elite in the sense they are the very best at their science, no it’s just a collection of overpaid snobs. You must not think I am emotional or so by using the word snobs, no it’s a cold hearted classification of their behavior.

It is now 102 years since the original Stern Gerlach experiment and there is boatloads and boatloads of theory of how electrons should behave in case such an experiment is repeated (that is a squence of those magnetic fields) and it is easy to understand the very first experimental physics human that would do such a sequential SG experiment would likely be rewarded a Nobel prize. And in the physics community the Noble prize is what they all dream of. So in a century of time without doubt on many occasions such an attempt must have been undertaken.

But there is no trace of any such experiment in the literature, the only experiment that was done was by Frisch and Segrè where they tried to flip the electron spin and that all failed big time. But when building their experimental setup Frisch and Segre got advice from Albert Einstein and likely because of that they got their (non) results published and as such we can find it back in the present day literature.

What I found strange in the last 10 years is that a lot of scientists actually believe such experiments have been done. That goes for physics but also chemistry, a lot of them talk like such experiments have actually been done. Here is a link that abundantly shows that the author thinks such experiments have actually been done:

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_(Peverati)/22%3A_Spin/22.02%3A_Sequential_Stern-Gerlach_Experiments

Now why should a succesful sequential Stern Gerlach experiment lead to an almost 100% probability of getting a Nobel prize? That is easy to explain: It would validate in a deep manner that quantum states like spin states are probabilistic in nature and as such would be a fundamental thing in say all the present day attempts there are in building quantum computers.

Another way of understanding there are just no successful sequential Stern Gerlach experiments done in the last 100 years is simple to do: Go to Youtube and search for it, all you find is animations that explain how it “should work”. But none of those videos give a hint of an experiment actually done…

Is it true there are no Nobel prizes rewarded in the last 100 years related to a repeated or sequential SG experiment? Well in this year 2024 the Nobel prize committee has a website and guess what? They have a search applet for their very website. If you search for “Stern Gerlach” you get something like 12 results and if you serach for “Stern Gerlach experiment” you only get 6 results. None of those results says anything about experimental validation of all that spinor crap or anything that shows you can actually flip the magnetic spin of an electron. I made a picture for the other website as you can see below:

If you want you can go to the website of the Nobel prize committee and look for yourself if you can find such a prize rewarded. Here is the link: https://www.nobelprize.org/.

It’s all a big bunch of crap: Electrons are not tiny magnets, they carry magnetic charge just like they carry electric charge.

I am very well aware that logic does not work, but say to yourself about the crap of the electron pair they have over there in the physics community: The Pauli exclusion principle says that those electrons must have opposite spins so what does that mean if it is true that electrons are tiny magnets?

Well if they have anti-parallel or opposite spins, doesn’t it look like this:

But again logic does not work so I do not expect that in this year 2024 the physics people will stop talking their usual bullshit. No way, after all as a social community they are just another bunch of overpaid snobs…

After having said that, after about only one century of time there is only recently an English translation made of the publication of the original Stern Gerlach experiment. The translation is done by Martin Bauer and here is a link to the pdf as you can find it on the preprint archive

The Stern-Gerlach Experiment

Translation of: “Der experimentelle Nachweis der

Richtungsquantelung im Magnetfeld”

Link used: https://arxiv.org/pdf/2301.11343.pdf

That was it for this post, as always thanks for your attention.

]]>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

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

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.

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

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.

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.

But a long long time ago when I tried to find the conjugate for 3D complex numbers, this simple flip does not work. You only get advanced gibberish so I took a good deep look at it. And I found that the matrix representation of some complex

I never gave it much thought anymore because it looked like problem solved and this works forever. But a couple of months ago when I discovered those elliptic and hyperbolic versions of 2D numbers, my solution of taking the upper row does not work. It does not work in the sense it produces gibberish so once more I had to find out why I was so utterly stupid one more time. At first I wanted to explain it via exponential curves or as we have them for 2D and 3D complex numbers: a circle that is the complex exponential. And of course what you want in you have some parametrization of that circle, taking the conjugate makes stuff run back in time. Take for example

But after that it dawned on me there is a more simple explanation that at the same time covers the explanation with complex exponentials (or exponential circles as I name them in low dimensions

And finding the inverse of imaginary units in low dimensions like 2D or 3D complex numbers is very simple. An important reason as why I look into those elliptic complex 2D numbers lately is the cute fact that if you use the multiplication rule

If you use the idea of the conjugate is the inverse of imaginary units on those elliptic and hyperbolic version of the complex plane, if you multiply z against it’s conjugate you always get the determinant of the matrix representation.

For me this is a small but significant win over the professional math professors who like a broken vinyl record keep on barking out: “The norm of the product is the product of the norms”. Well no no overpaid weirdo’s, it’s always determinants. And because the determinant on the oridinary complex plane is given as x^2 + y^2, that is why the math professors bark their product norm song out for so long.

Anyway because I found this easy way of explaining I was able to cram in five different spaces in just seven images. Now for me it is very easy to jump in my mind from one space to the other but if you are a victim of the evil math professors you only know about the complex plane and may be some quaternion stuff but for the rest you mind is empty. That could cause you having a bit of trouble of jumping between spaces yourself because say 3D circular numbers are not something on the forefront of your brain tissue, in that case only look at what you understand and build upon that.

All that’s left for me to do is to hang in the seven images that make up the math kernel of this post. I made them a tiny bit higher this time, the sizes are 550×1250. A graph of the hyperbolic version of the complex exponential can be found at the seventh image. Have fun reading it and let me hope that you, just like me, have learned a bit from this conjugate stuff.

The picture text already starts wrong: It’s five spaces, not four…

At last I want to remark that the 2D hyperbolic complex numbers are beautiful to see. But why should that be a complex exponential while the split complex numbers from the overpaid math professors does not have a complex exponential?

Well that is because the determinant of the imaginary unit must be +1 and not -1 like we have for those split complex numbers from the overpaid math professors. Lets leave it with that and may I thank you for your attention if you are still awake by now.