I decided to take a deeper look into this because of a video on the Eisenstein primes, those Eisenstein primes and integers look a lot like my own elliptic complex integers. But the Eisenstein numbers are defined on the standard complex plane while I modified the way the multiplication is done via a tiny change to

Lets start with the video that I considered to be remarkable good:

For me it was all brand new because to be honest I never ever studied the Gaussian integers and how that gives rise to a unique way of factorizing them. But now I had to solve the problem for my own 2D elliptic numbers I have to say that yes it is beautiful once you grasp what’s going on.

A well known fact from the Gaussian integers is that the number 5 is now no longer a prime number because it can be factored into 2 + *i* and 2 – *i*. On the complex elliptic numbers it is 7 that is the smallest integer to have that same fate as 5 on the Gaussian integers. This post is mostly about the two factors that factorize 7, I named them *p* and *q* and the interesting thing is they are each others conjugate.

But the number 3 is also no longer a prime so all in all for a person like me who dislikes algebra so often for so many years, it was all in all a nice patch of math to understand.

This post is 8 pictures long, each 550×1250 pixels and on top of that two extra figures are addes. So what more do you want? I kept the stuff as less technical as possible while hoping that if you want to dig a bit deeper you can do that for yourself now. So lets start the picture thing:

Here is a link to the free Wolfram webpage where you can check that indeed 1729 has 48 integer solutions because you can make 8 different factors with the prime numbers involved: https://www.wolframalpha.com/input?i=x%5E2+%2B+xy+%2B+y%5E2+%3D+172

Here’s the wiki: https://en.wikipedia.org/wiki/Unique_factorization_domain.

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

It soon dawned on me that the Fermilab experiment was a bit strange. They use the Lorentz force to let the muons go round while the spin stays horizontal. Now muons are cousins of the electrons and the official theory is that they are tiny magnets just like electrons. And as so often observed, the professional physics people only say things that sound or look logical. All weird stuff that comes from what I name Crazyland is just not mentioned. Things from Crazyland are of course the electron pair and how is that configuration even possible?

An old experiment done in 1922 was the Stern-Gerlach experiment and there too do the experimetalists use a vertical magnetic field. (It could be that in the original experiment the field was horizontal but that’s not important for our discussion here.) What’s interesting is that if you read or see one hundred explanations for the Stern-Gerlach experiment it is always the official version that the spins align vertical or anti-vertical.

The anti-vertical stuff is also a thing from Crazyland; why would an electron turn against the magnetic field and as such gaining potential energy? But we skip that because the relevant obervation is that if you see a 100 explanations, the electrons always align in a vertical manner.

Here you see a screenshot from the first video:

In the above picture it is nicely shown what the professionals have made of it; the Hamiltonian clearly says that if electrons anti-align they gain potential energy but they never talk about that. And the expression for how an electron is accelerated in an inhomogeneous magnetic field is basically the same as say in gravity. The potential energy in a gravity field is *mgh* and if you differentiate into the vertical direction, that is in the direction of *h*, you are left with *mg* and that’s the force due to gravity.

I think this is BS because I think electrons (and muons) are magnetic monopoles. As such they should be accelerated by all kinds of magnetic fields and I myself don’t have experimental evidence for that. But the professional physics people don’t have evidence for their claim that in a homogeneous field electrons don’t get accelerated. Since 2014 I never stumbled upon any experimental result in that direction. It’s about time to go to the first video. It is from a channel named Abide By Reason and that’s a very good name only he doesn’t do it. There’s not much reason found but it’s the official explanation for the SG experiment.

Now for the Fermilab muon g-2 experiment: Despite the vertical magnetic field for some strange reason non of those muons change their magnetic orientation. Even stronger, the folks from Fermilab are so über-ultra-mega smart that they know that after one rotation in the ring, the muon spin has furned about 12 degrees more…

Of course nobody explains why that spin stays horizontal even though the vertical magnetic field has a strength of about 1.5 Tesla. But in this experiment they need that spin is horizontal stuff so like all physics people at some time they have to talk out of their neck. Physics is the science of talking out of your neck while maintaining that you are a five sigma kind of science.

The above screenshot is from a lady that has a video channel named “Think Like A Physicist” and sometimes that’s a good idea but when it comes to electron spin you better try to think as a logical person.

Video title: Measuring Muon g-2.

Link used: https://www.youtube.com/watch?v=IHgaapwwLN0

Now the lady that thinks like a physicist claims the magnetic field is vertical but in the last 9 years I have seen all kinds of weirdo’s making all kinds of claims when it comes to this or that. So again avoiding difficult to read pdf’s from the preprint archive there was indeed a video from Fermilab herself validating the magnetic field is vertical.

Please remark that from the outside when you look at that ring the Fermilab got from Brookhaven, it is hard to see what kind of magnetic field is inside. The video is about 3 minutes long.

Video title: Muon g-2 Experiment Shimming.

Link used: https://www.youtube.com/watch?v=4HlKN0rfdKA

That was it for this post, in this post we had zero people explaining that quantum states like electron spin are just so fragile. But we had some people just ignoring muon spin doesn’t flip even when it’s going round and round in some Fermilab experimental setting.

Likely the next post is about prime numbers in the plane of elliptic complex numbers. So it’s just some two dimensional stuff with numbers and integers. A lot of prime numbers like 7 are not elliptic primes. They can be factored inside the elliptic plane by two smaller primes. So that’s all very interesting but also time consuming but all in all in a week or two it should be finished.

In the picture below you can see what natural primes survive the elliptic onslought. They are the ones with ellipses that don’t have integer solutions.

As always thanks for your attention .

Well take the sine function, lets write it as

The question is: Can you say something about the maximum value that this sum can take? And no, I had no idea about how to approach this problem.

The interesting detail is of course that this sum of sines does not seem to converge or diverge in any significant way. You can check that for yourself in for example the DESMOS package, just type the word sum and you get the sigma symbol for a summation. I like the package and in case you have never seen it, here is a link: https://www.desmos.com/calculator?lang=en.

As far as I know this problem has no or little math meaning, it is just some recreational stuff. But if you in your life had the honor of calculating a bunch of Fourier coefficients again and again, you know that the summation of sines and or cosines and or complex exponentials can have very tricky convergence questions. Now with my little sum of sine time lags we don’t have any convergence at all, the funny thing is it also does not diverge.

This post was meant to be short but as so often it grew to five pictures long and on top of that there are three extra figures added to the mix. It’s all pretty simple and not deep complicated math that as so often is very hard for human brains to digest. Have fun reading it.

At the end of this post I want to remark that I framed the question for some finite sum of sines. That is because I wanted to avoid all things related to taking a supremum and stuff like that. Look at Example 1 above, here of course the maximum value of the two sines is not 2 because there is no real solution to this, but of course the supremum is 2 because you can come arbitrarily close to 2.

Of course in Example 3 I wanted to know the sup and the inf of the amplitudes, but I framed the question in a finite sum of sines anyway.

Ok that was it for this unclassified post. If you want you can think a bit about sums of sines and if you get bored of that you can try to figure out what an electron pair is if the Pauli exclusion principle says it must have opposite spins… Thanks for your attention and see you in the next post.

Of course there is nothing spinning, back in the time Wolfgang Pauli himself calculated that even if you concentrate all the electric charge of an electron on it’s ‘equator’, it must spin so fast that this is a huge multiple of the speed of light. A long time ago I did such a calculation myself, it is not very hard to do but I skipped it in this because that calculation has nothing to do with the content of the post. So you can easily do that yourself, after all it is just some advanced high school physics and if you do that the answer will of course depend strongly on how large you think the electron is if you view it as a tiny billiard ball.

The word ‘spin’ is a terrible wrongly chosen word to describe the magnetic properties of the electron. I have wondered so often as why the physics people think year in year out that the electron is a tiny magnet while you really do not need much brain power to see that this is nonsense. Beside all those fundamental energy problems there are also problems with the above mentioned conservation laws. The fact we have today so many people from the physics community talking about ‘the spinning electron’ is caused in part by that original stupid choice to name it ‘spin’. After all this word strongly suggests that we are dealing with tiny magnets, every electron must be a tiny bipolar magnet while if you view them as magnetic monopoles you don’t have all these weird energy problems.

In case you are new to this website: I think that electrons are magnetic monopoles, just like their electric charge, and furthermore this magnetic charge is permanent and as such it is impossible to flip the spin of an electron.

And if you are from the physics community yourself, may be you need to vomit from the idea that electrons are not tiny magnets. Or may be you pity me because I am a middle aged man and you think I want to save physics or the wider community known as humankind from wrong doing when it comes to electron spin. Well I have to disappoint you: I don’t give a shit about such stuff, ok in the beginning I did but after a few years I realized that likely physics will be trapped a few centuries longer before they start using logical thinking when it comes to electron spin.

In the two pictures below I also experiment a bit with using other backgrounds, here you see something like a big hand made with some generative AI video thing. May be it is time to replace my old background made with my old Windows XP computer by some fresh stuff.

This intro is getting far to long because I wanted this post to be short. So let me hang in the pictures and here we go:

In Figure 1 below all you see are two images I downloaded from the internet while using the search phrases as written above. You just never see those spinning arrows if you search for electron-positron pair creation. It is as so often: As soon as we get into crazyland, the physics people just don’t talk about it.

Well yes, this is indeed the end of this post.

]]>But if you look into the details it all makes sense and this representation is the basis for things like residu calculus that you can sometime use to crack an integral if all more easy approaches fail.

In most texts on the standard complex numbers (with standard I mean that the imaginary unit

I decided to skip all that although if you want, you can do that of course for yourself. I also skipped all standard proofs out there because I wanted to craft my own proof and therefore in this post we only integrate over the ellipses and nothing else.

Another thing to remark is that this is just a sketch of a proof, a more rigor approach would make the post only longer and longer and I think that people who are interested in math like this are perfectly capable of checking any details they think that are missing or swept under the carpet. For example I show in this post the important concept of ‘radial independence’ but I show that only for a very simple function

It also contains two extra figures and may be I will write a small appendix related to figure 1. But I haven’t done that yet so below is the stuff and I hope you like it.

Ok, all that is left is an appendix where I give a third parametrization of the elliptic complex exponential. It is just some leftover from some time ago when I wondered if the two coordinate functions might some some time lags of each other. And yes, they are. In the case of these elliptic complex numbers the time lag is one third of the period.

Before I end this post, why not place a link to all that official knowledge there is around the Cauchy integral representation there is. Here is a link:

https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula

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

Lets dive into it, this post is 3 images long and here we go:

That’s more or less all I had to say today, if you view the magnetic properties of electrons as monopole magnets just like their electric properties the standard electron pair becomes the lowest (potential) energy state. Before I close this post let me quote from a wiki an interesting detail about the non-bonding electron pairs: they have a tendency to be outside the so called ‘bonding region’ between atoms in molecules. Once more this only makes sense when electron carry a monopole magnetic charge.

In theoretical chemistry, an antibonding orbital is a type of molecular orbital that weakens the chemical bond between two atoms and helps to raise the energy of the molecule relative to the separated atoms. Such an orbital has one or more nodes in the bonding region between the nuclei. The

density of the electrons in the orbital is concentrated outside the bonding regionand acts to pull one nucleus away from the other and tends to cause mutual repulsion between the two atoms. This is in contrast to a bonding molecular orbital, which has a lower energy than that of the separate atoms, and is responsible for chemical bonds.

Here is the link to the wiki I quoted from:

https://en.wikipedia.org/wiki/Antibonding_molecular_orbital

That’s it, see you in some other future post or enjoy some old posts on say the 4D complex numbers because they are beautiful and that is something we cannot say about the behaviour of the average physics professor with their weird fixation on electrons as tiny magnets…

]]>Here the elliptic plane is the same as we always studied the last couple of months, it is ruled by the imaginary unit

To be honest I did not explain in detail why it works, I hope it is rather obvious. Take for example two commuting (square) matrices

The post itself is only two pictures long and I included a third picture that I used on the other website. Beside a female robot the third picture contains another factorization of the equation of the ellipse that is the determinant of the matrix representations.

So this equation for the ellipse can be factored on the standard complex plane and also on the elliptic complex plane. The interesting detail is of course that on the elliptic complex plane you have integer coefficients while on the standard complex plane this is impossible.

Basically the eigenvalue funtions are both a map from the elliptic complex plane to the ordinary complex plane. If it was made by a professional math professor he or she would likely call it an isomorphism but I name them eigenvalue functions. In the past I also made them for the 3D complex and circular numbers and of course for the 4D complex numbers that were under study years ago.

Enough of the introdutionary talk, lets go:

Now I left a lot of stuff out otherwise it would not be a Tik Tok short kind of math post. But you can also use the elliptic complex plane as your primary source of eigenvalues. For example at the other side of our galaxy there lives an alien race known as the Orcs. And for some kind of religious reason these Orcs just don’t want to use circles because as they all know circles are evil. But they found the elliptic complex plane and they use that for solving eigenvalue problems like eigenvalues from square matrices or even stuff that we humans know as the Hamiltonian energy operator. That should work just as good as we humans do in using the complex plane we have over here where the complex exponential is a circle.

So let us now look at the third picture that has both factorizations in it:

Let me leave it with that and as always thanks for your attention.

]]>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.