All posts by ReinkoVenema

The RI has a new video on magnetic monopoles.

Yesterday all of a sudden there was a new video upon magnetic monopoles; naive & dumb as I was I only thought ‘Great may be I can learn something new!’ and I started watching.

The video from the Royal Institution is entertaining and as such not boring to watch. But for me there was nothing new to learn, so I started thinking about why this guy Felex Flicker behaves the way he does. After all he is a scientist and given the fact that physics is a so called ‘hard science’ all claims made should be backed up by experiments. Yet this Felix guy when he claims that magnetic domains in metals and electrons are magnetic dipoles, there is once more zero mentioning of any experimental evidence.

Compare that for example to how at CERN they study anti matter. From positrons and anti-protons they managed to make a bit of anti hydrogen. And they do as much experiments with it as possible and try to find out it ther properties of anti hydrogen are such as expected. And that is the way it should be, that is what I view as standard behaviour for a hard science. But for electrons they never ever even tried it. Over the years I have made a long list of troubles with the electron as a magnetic dipole. I can’t name them all here of course so let me pick up just one detail:

If electrons are magnetic dipoles, why do we only observe electron pairs (and unpaired electrons) but never larger structures?

Here you see the new Brexit style in UK clothing, it looks great:

Take for example atomic and molecular hydrogen, there is only stuff with an unpaired electron (atomic hydrogen) and stuff with an electron pair (the molecular version of hydrogen) and nothing else. That kind of behavior is not what one should expect if the electron was a magnetic dipole… Electrons never behave like the bar magnets in the next picture:

May be I should have formulated this a bit less rude. It is not personel or so.

My dear RI folks, it is in so many ways not logical that electrons are magnetic dipoles. So I more or less only wonder that psychological stuff: why do the professors behave like they do? Ok, most of the time it is bad for your carreer to go against the insights as shared in the group, but this electron stuff you tell is just not logical. And, in my view, more logic is found when you think of electrons as having a magnetic charge.

Enough of my preaching, here is the video:

This guy hangs together from electron pair bindings,
why only electron pairs?

Let me leave it with that. Likely in the next post I will show a new way of taking a norm in the 3D complex and circular numbers. It is all based on eigen values, for the 3D numbers you can make a norm out of the eigen values while for general matrices you can’t.

Hurray! Nuclear electric resonance found.

Always when physics people explain stuff like nuclear magnetic resonance and it’s cousin electron resonance, it is always explained in terms of alignment of the particle spin with the applied external magnetic field. In my view that is a bizarre explanation because that would cause hardly any acceleration of the nuclei and electrons, so how can that give some measureable em radiation?

Yet in medical applications like MRI there is plenty of em radiation to make an image from. Where does that come from? In my view where particles like electrons and protons carry magnetic charge and as such are all magnetic monopoles, the resonance works because there is actually something resonating… It must look a lot like harmonic resonance or like a mass on a spring if you want. Basically it should not make much of a difference if you use oscillating magnetic fields or an oscillating electric field. Ok, in practice like medical MRI scanning I don’t think you can use electric fields because most atoms and molecules in your body are not ions, that is they are neutral under electric fields oscillating or not.

To my surprise in a video about a so called ‘Breakthrough in quantum computing’ all of a sudden the concept of nuclear electric resonance came along. Ok, it was on the Youtube channel named Seeker, so often it is not carefully thought through, but anyway. it might be Seeker but the concept of nuclear electric resonance should have large similarities with nuclear magnetic resonance if my idea’s upon magnetic charge are correct…

Let us take the time and look at a few screen shots I made from that Seeker video:

Wow man, NER instead of NMR?

At some points in time the video will get highly confusing, after all it is the Seeker channel combined with the insights of that Australian team trying to make quantum computer with qbits made from magnetic spins. Of course that is not going to work because if permanent magnetism is a charge you just cannot make a super position of it. So if I am right, all those kind of quantum computer will never work. Let’s go to the next screen shot:

This is the confusing part: Electricity makes the magnetic moment wiggle.

Of course this fantastic part of the video is inspired by how the university people explain magnetic resonance. If you view the video below, please remark there likely is no arrow of a magnetic dipole anyway.

It has to be remarked however that atomic nuclei can have many protons and neutrons and as such all kinds of magnetic configurations should be possible. Next screen shot:

These people are experts in understanding the electron pair.

The guy on the left, I don’t know his name, explains the electron pair as next: These two electrons are in a superposition of spin up and spin down. It is just like man and wife, there are two persons but you do not know if it is the man or the wife. Only when you make a measurement on one of the electrons, you instantly know the spin state of the other electron…

Don’t forget those people from blah blah land have zero experimental evidence for the electron being a magnetic dipole. After having said that, why not go to the next screen shot?

I never ever heard of this guy, but he was Dutch so shame on me.

You should not feel much pity for Mr. Bloembergen. After all he got a Nobel prize so he died while still having plenty of money. You are looking only at an old photograph of just one more perfumed prince. Also, Nobel prize or not, it’s just another perfumed human being not understanding it is impossible for the electrons to have two magnetic poles.

After so many screenshots, enjoy the deep thinking as in the next Youtube video:

Every year we have quantum breakthroughs but never a real computer.

Before we split I want to link to a few experiments that I posted on the other website on 11 May. One of those experiments is completely undoable, the second requires a lot of work because there a beam of electrons should get split in half in a cyclotron. The third experiment is showing that magnetic domains always have surplusses of either north pole or south pole electrons. That is stuff I cannot do myself in my kitchen, garden or living room. The likelihood that someone else will pick that up in the next 10 years is relatively low, it is a wild guess but at best it will be something like 1% to at most 4 or 5%.
As you see my expectations are not very high. Say for yourself: how likely is it that an article about an experiment that validates the magnetic monopole character of electrons passes the peer review process?
That is not very high… Ok, end of this post; live well and think well.

Calculation of the circular exponential circle via ‘first principles’.

Oh oh, this is one of those posts where I only calculate in the 3D circular numbers while I classify it as 3D complex numbers. In the past when I made those categories on this website I did not want to have too many categories so that is why I only have 3D complex numbers as a category.

All in all this post (number 146 already) is not extremely important because over the years I have given many proofs that the parametrization for the exponential circle indeed fulfills all those equations like the sphere-cone equation of the fact the determinant is always one. On the other hand, if you have an important mathematical object like the exponential circles, it is always good to have as many proofs as possible. Just like there are many proofs for the theorem of Pythagoras, it would be strange if we only had one proof and nobody cares about more proofs to that theorem that more or less the central to a giant mountain of math.

What do I mean with ‘first principles’? Very simple: that is the summation formula for the exponent of a linear operator or the matrix exponential if you want. In this post I use a somehow slightly different number tau; I use a number tau that gives a period of 2 pi for the exponential circle. The reason is simple: that makes the long calculation much more readable.

Another thing I want to mention is that the long calculation is nine lines long. For myself when I read the works of other people I do not like it if calculations go on and on and on. I always try to avoid too long calculations or I just don’t write posts about them. Almost nobody reads the stuff it it’s too long and gets too complicated so most of the time I simply skip that. Beside that there is always 0% feedback from the mathematical community, so although I always year in year out try to keep it so simple that even math professors can understand it, nothing happens. Just nothing, so after all those years it is not much of a miracle I don’t want to engage with these overpaid weirdo’s at all. Likely if you are born stupid you will die stupid & I have nothing to do with that. Mathematics is not a science that is capable of cleaning itself up, the weirdo’s keep on hanging to their fantastic quaternions and their retarded ideas of what numbers & complex numbers are. Too much money and too much academic titles have not lead to a situation where the science of math is capable of cleaning itself when needed.

Enough of the blah blah blah, after all the physics professors have the same with their electron spin: where is your experimental proof that the electron is a magnetic dipole? For over five years nothing happens except a lot of weird stuff like quantum computers based on electron spin…

This post is five pictures long, for me it was cute to see how those three cosine functions slowly rise from the start of the long calculation. Also of importance is to notice that I had to use the simple formula for cos(a + b) = cos(a)cos(b) – sin(a)sin(b) that comes from the exponential circle in the complex plane. Just once more showing that 3D complex & circular numbers are indeed emerging from the 2D complex plane. Not that the math professional will react, but anyway…

Let’s go to the five pictures:

I think you must calculate them for yourself, grab a pencil and some paper and use the
fact that the circular multiplication uses j to the third power is 1.

Again, this is not a ´very important´ post. Given all those results and proofs from the past it is logical such a long calculation has to exist. It´s relevance lies in the fact you simply cannot have enough proofs for the calculation of parametrizations of the 3D exponential circle.

Let me leave it with that. See you in the next post.

Three video’s for killing the time if needed.

This time a somewhat different post, just 3 video’s I thought are interesting to share for their own reasons. In the first video the American television physics professor Brian Greene goes beserk on the beauty of the exponential circle in the complex plane… Brian, like so many others, do not know what they are missing. So many spaces have exponential circles and curves and indeed they are beautiful.

The second video is about a question that is often asked: Is math invented or is it a discovery? I think this is a false way of looking at math, if you replace the word ‘math’ by ‘food’ you already understand this is a weird question: Is food invented or is it discovered? In my view that often goes hand in hand but opinions vary wildly on this subject. The video is an interview with the UK math professor Roger Penrose. I included this video because back in the 80-ties of the previous century Roger had written some books on the things known as spinors. A lot of so called scientists think that spinors have something to do with electron spin, there are even weirdo’s that think after the electron has encircled the nucleus once it’s spin state is altered so that after two rounds the electron has it’s original spin back… Oh oh for people like Roger and those others it will be a long way in understanding the electron cannot be a magnetic dipole. In all ways possible that is not logical. For example the unpaired electron is not magnetically neutral while the electron pair is. And there are a whole lot more examples to be given showing electrons simply can’t be magnetic dipoles. And you only have to use the thing called logic for that; no weird quantum mechanical stuff but just a magnetic charge on the electron gives much better results if you use the thing called logic.

The third video is about a weird line of reasoning that I have observed in many video’s. It is about explaining how those jets form that emerge from black holes and their accredion disks. The reasoning is that the plasma in the accretion disk goes around the black hole and if a charge goes round it produces a magnetic field & that is all explanation given always. That is nonsense of course, even spinning metals like when you are drilling a hole with your drill machine never produces a magnetic field because for every electron that goes round on average also a proton goes round and all in all there is no overall magnetic field created. But if the electrons are magnetic monopoles, they will have much more acceleration compared to the far more heavy protons and as such an accretion disk around a black hole should be positively charged all of the time and that explains why the magnetic fields are so strong over there.

Ok, I crafted 8 pictures from the stuff. For example I made a 4D generalization of the 3D outer product while explaining such math is an invention and not a discovery. After the 8 pictures I will post the three video’s that aroused my attention for one reason or another. Have fun reading it.

The link to Reason 82 as why electrons cannot be magnetic dipoles is
08 Feb 2020: Reason 82: More on solar flares.

And here are the three Youtubers to kill the time.

Ok, let´s try to upload this bunch of stuff and see what happens.

Integration on the complex and circular 3D number spaces.

A lot of math professionals rather likely still think that 3D complex numbers do not exist, may be for reasons like there are non-invertible numbers or whatever what other reason they have. This post more or less proved such views are nonsense; for example a lot of math on the 2D complex plane does not rely on the fact it is a field (and as such only division by zero is forbidden).

But on the 3D complex and circular number spaces indeed it brings some complications if you have non-invertible numbers in the function you want to integrate over a particular curve. And I have to say that problem could be solved by using the special properties that those numbers have. In this post I only show some examples with the non-invertible number alpha (alpha is the midpoint of the 3D exponential circles and all multiples of alpha are also non-invertible so the line through 0 and alpha are all not invertible).

For me writing this was a good distraction away from all that negative news we have day in day, all those countries reporting daily death toll can make you a bit depressed… So when I am through with the daily news I always do some other stuff like calculating a few of such integrals. That is a very good antidote against all that bad news. After all there is not much gained if you constantly think about things you cannot change at all.

This post is relatively long; at first I crafted 12 pictures but it soon turned out that was not enough. So while filling the 12 pictures with the math and the text I expanded some of the pictures so they could contain more math & text. That was not enough and in the end I had to craft two more background pictures. All in all it is 14 pictures long, that is a record length for this website.

If in your own mathematical life you have performed contour integration in the complex plane, you must be able to understand how this works in the 3D spaces. And for those who have done the thing known as u-substitution on the real line: it is just like that but now this u thing is the parametrization of a path. All that stuff below with gamma in it is either the path or the parametrization of that path. Please remark that you must use the complex or the circular multiplication on 3D, just like integrating over a contour in the complex plane uses the 2D complex multiplication.

In case if you are not familiar with the number alpha that is found at the center of the exponential circle, use the search function of this website and for example look up ‘seven properties of the number alpha’.

I hope I have removed all faults, typo’s etc so that later I do not have to repair the math because that is always cumbersome. Here we go: 14 pictures long so this is hard to grasp in detail in just a few hours. But it is beautiful math & that is why I do this. For me math is a lovely hobby.

Enough of the blah blah blah, here we go:

Ok, let´s first hit the button ´Publish´ and see what will happen…
It looks all right but a day after first publication I realized there was some missing text. It turned out I had to rename picture number 2 and now every thing was like it was planned.

Later I will flea through the rest of the text, if needed I will post more addenda. For the time being that was it so till addendums or till the next post.

Integration on the circular and complex 3D number spaces.

Ok, the math text is finally written. It took a long time but all in all I am very satisfied with the result. It will be a long post, I estimate about 12 pictures long and that is more or less a record length on this website. I have finished only two pictures and I will take my time to make the other ones because my mouse does not work properly. When I click with the computer mouse, very often that acts as a double click and that makes making pictures a laborsome task because of all the errors that double clicking gives. And when I have to repeat a series of clicks three or four times before it is ok, it will take some time. May be I should buy a new mouse,

Anyway to make a long story short: For years I stayed more or less away from crafting math about integration because it is hard to find a definition that would work always. My favorite way of using Riemann sums could not work always because of the existence of non-invertible numbers in the 3D spaces. And that gave some mathematical fear in my small human mind because path independence came with that way of Riemann summation. All in all it is beautiful math to think about: For example if in 3D you use a primitive to integrate over a closed loop, is it always zero?

So only the first two pictures are posted and I have no idea when all other pictures are finished. Here we go:

Oh oh, only later I observed a double click problem in this picture…

That was it, till updates.

Just a teaser picture for integration on 3D complex and circular numbers.

It is about time for a small update! Despite all that COVID-19 stuff going round, for myself after all those years I finally tried to put integration on higher dimensional number systems on a more solid footing.
All those years I just refrained from it because you cannot use my favorite Riemann sum approach because of the non-invertible numbers we have in 3D or even higher dimensions.
But now I am trying to finally make some progress and stop avoiding this subject, I find it is utterly beautiful. It has an amazing array of subtle details involved when you have some non-invertible numbers in your integration stuff.
I have no idea when I have finished this rather important detail in my cute theory of higher dimensional complex & circular numbers, so let time be time & in the meantime only post a teaser picture about that lovely integration stuff. In the first lines you see a very familiar integral, likely you have done such calculations in the complex plane. In the case of 3D circular and complex numbers you must (of course) use the multiplication on 3D space to make it all work. Basically you are evaluating (or calculating) three integrals at the same time, just like on the complex plane where you are evaluating two integrals at the same time in your calculation. If you work with a pencil & paper, make sure you have enough paper because all those 3 integrals also have 3 terms in it so your calculations can become quite long…
Here we go:

Please remark this only works for invertible X.

Ok, let me end this update now. Till updates and for some strange reason you must wash your hands while the proper authorities never point to 3D complex numbers… Till updates.

Hilarious video: Don Lincoln explaining the Stern Gerlach experiment.

I am always baffled by those folks explaining this important experiment; why do they not see that the explanation offered is just 100% bs? It could be that in physics there are all kinds of ‘patches’ that explain particular parts of magnetism. Let me write two of those patches down:

Patch 1: Since in the Stern Gerlach experiment a beam of silver atoms was split in two under the application of a magnetic field, the ‘logical explanation’ offered is always that when electrons enter a vertically applied magnetic field, 50% will have spin up and 50% is spin down. If the applied magnetic field is turned 90 degrees, say horizontal, again both beams will split again in 50% left spin and 50% right spin.

Patch 2: Making permanent magnets. A magnetic field is applied to some metal and now all unpaired electrons always align with the applied magnetic field. Sometimes the explanation is a bit more advanced; at first it is explained that in the magnetic domains of say iron all spins are aligned and when making it into a permanent magnets all the magnetic domains align according to the applied (strong) magnetic field.

On their own such ‘explanations’ might sound logical, but if you combine them you get total rubish. It cannot be that one the one hand if you apply a magnetic field 50% of the unpaired electrons anti align and the other 50% align with the magnetic field while on the other hand always 100% of unpaired electrons align nicely when you make a permanent magnet. Such ‘explanations’ or patches of knowledge should enforce each other, but here it gives total bs. Either it is always 50/50 or it is always 100% alignment, why do those professional physics folks never observe that tiny part of physical reality? In my view they cannot go outside the patches, the reasoning always stays local inside that particular patch (explaining the SG experiment versus making permanent magnets).

The 50/50 patch that should explain the Stern Gerlach experiment is always very strange if you just keep an iron nail next to a magnet; wow man it gets attracted! But if 50% of the unpaired electrons in that nail would anti-align and the other half would align, what would explain the attraction? In my view people like that a weird beyond comprehension.

At Fermilab the honorable Don Lincoln often explains all kind of physical things, his style in doing so is often a bit too arrogant in my view. If you want to study physics you must be humble and always operate from the fact you only have a human brain. So being an arrogant overpaid jerk is a quality you must loose; that human stuff will ensure you will never understand physical things because it prevents you to think a bit deeper on it when for example you try to check if you could be wrong…

The video is on more items, not only the SG experiment but also the Einstein-Rosen-Podolski paradox, the creation of an electron-positron pair from a spin 0 particle & more of that stuff. I made two pictures from two screen shots. By all standards it is hilarious because what spin 0 particle are we actually talking about? Of course that is not mentioned, with just a tiny bit of arrogant behavior it is simply stated and you as an onlooker of that video are supposed to bow for the wisdom of Don Lincoln…

Cooment: In my view this shows conservation of magnetic charge.

Please remark I have no experimental evidence that if electrons are magnetic monopoles, there is conservation of total magnetic charge just like with electric charge. I think it is the case but you also have constantly those physics people explaining that you can flip electron spin with micro waves. But all those patches they try to explain, for example spin flip inside a qbit for quantum computing, can also be the result of electron change. There are always more electrons in the surrounding and if you apply some micro wave radiation it could very well be that you ram out the anti aligning electron that simply gets replaced by an electron of the opposite magnetic charge. After all I have never ever seen an experiment where there is only one electron trapped in isolation and after a short pulse of em radiation it has changed it’s spin.

Ok, let us go on with the hilarious stuff:

Comment: Never forget that Stern was the first assistant professor to Einstein. So Einstein never ever had a clue about electron spin in the first place…

Ok, let’s go to the video itself. The Lincoln guy is a bit irritating because of his arrogant attitude, but it is soon funny & hilarious when he props up his 50/50 spin alignment nonsense. For me it is funny because if electrons carry magnetic charge, a more or less conservative estimate as when the professional physics professors will find that out is about 5000 to 5 million years into the future.
Just like the speed math professors understand a bit more upon 3D complex numbers.
Title of the video: Quantum Entanglement: Spooky Action at a Distance.

Ok that was it for this post, think well and work well.
Updated on 18 March 2020: Lately I found a cute video from Veritaisum and the MinutePhysics guy where indeed they use both patches of ‘explanation’ in just one video. The Stern-Gerlach experiment is explained via electrons doing the 50/50 thing while for permanent magnets all electrons align & we can safely conclude these guys are lunatics.

But if you look at other video’s of Veritasium & the MinutePhysics guy, they often look so smart and it all looks like they have more or less healthy brains… These guys are not idiots and that leaves we with a big question I still have: Why do the people of physics never understand that separate patches of human knowledge should enforce each other?
Why do these two guys not see that giving two explanations is highly contradictionary? If you have a permanent magnet in your hands and you approach a piece of iron, if 50% of the electrons align and the other half anti-align, iron would not be magnetic. But Iron is very magnetic as any body knows, so why do weirdo’s like Veritasium & the MinutePhysics guy not see that? Here is that cute video from two idiots not capable of seeing their wisdom is not perfectly optimesed:

Ok, let’s close this post for the second time.

A simple thought experiment on electron spin Sean Carroll style.

The television physics professor Sean Carroll is very good at explaining electron spin: either the electron spins clockwise or anti clockwise. In the past I often got annoyed by such ‘explanations’ because it shows shallow thinking and a complete rufusal to even try to understand electron spin.

At present day I can only laugh about it: If overpaid people like that want electron spin to be such stupid stuff, may be it is better to say it is your cake so why not eat it? If you leave all those shallow puddles of thinking a more or less normal person would like to know what experiments are there that actually prove or strongly suggest that electrons are indeed magnetic dipoles? If you try to find out about such heroic and historical experiments, it is once more a bit hollow and disappointing: Never ever as far as I know was there a physics experiment trying to actually prove the electron is a magnetic dipole…

So let us do a little thought experiment where the electron spin is ‘Sean Carroll style’ caused by the classical electron spinning around some axis, Here we go:

Suppose a pair of particles is created, say an electron and a positron.
Suppose total spin must be zero, say the electron is spin up and the positron is spin down.
If the electron spins clockwise, the positron should also spin clockwise otherwise total spin ‘Sean Carroll style’ would not be zero. After all the positron has a positive electric charge so it has to spin around some mysterious axis in the same way as the electron otherwise total spin ‘Sean Carroll style’ would not be zero.
But ha ha ha: That would violate the classical law of conservation of angular momentum because both the electron & the positron must rotate around some weird axis in the same way.
Conclusion once more: It is not possible for the electron to be a magnetic dipole. End of the simple thought experiment.

A rather recent video from our deep thinker Sean Carroll was out via the UK Royal Institution, it is not science but it has a high entertainment quality in it. Therefore if you like good but shallow entertainment, go to people like the television physics professor Sean Carroll:

The video from the Royal Society is highly hilarious, Sean is complaining about the fact that most physics professors have stopped to try understand quantum physics. That is funny because of course if you use words like ‘spin’ to describe the magnetic properties of something like an electron, how can you be not confused? And I have the same problem in my little unworthy life: the math professors have given up a long time ago about finding the 3D complex numbers. That must have been about a century ago when they were not capable of understanding it is all about prime numbers and 3 is a different prime number from 2.
I stole a sceen shot from the video, Sean used the grapes as ‘understanding quatum mechancis’ where I use it as ‘3D complex numbers’.

Let me end this post with the same thought experiment as above: the creation of an electron positron pair. Only now it is ‘my style’ and not some stupid ‘turning around some axis style’.

Suppose an electron positron pair is created, total electric charge must be zero. Hence the positron has positive electric charge.
The magnetic charges should also be zero, hence one of the created particles will have a north magnetic charge while the other will have a south magnetic charge.

In my view that is all there is. End of this post.

The scalar replacement theorem.

Ok ok I was a bit lazy but it is finished now so let’s finally post this scalar replacement theorem. Never in this post I formulate or proof this scalar replacement theorem, but basically this theorem says that if you replace the real numbers (scalars) in the way you describe say 2D split complex numbers by numbers from the complex plane, the result is a space who’s numbers also commute and it even has viable Cauchy Riemann equations. In this post I will write z = x + yj for the 2D circular numbers (also known as the split complex numbers) and write z = x + yi for numbers from the complex plane. If you combine such spaces it must have imaginary units that are different in notation, so j is the imaginary unit that does j^2 = 1 while the good ol i from the complex plane is known for it’s important property that i^2 = -1.

If we replace the x and y in z = x + yj by complex numbers we get a new 4D space where both j and i place there role. All in all those 4D numbers will be written as Z = a + bi + cj + dij. Of course the a, b, c and d are real numbers and as such this new space is 4D.

A long time ago I once used this to calculate the logatithm of j, it worked perfectly and that is why I more or less gave idea’s like that the name of ‘scalar replacement’. Later I found that way of using diagonalization of the matrix representations in order to calculate the logarithm, that is a far more general useable way of calculating logarithms but anyway the original calculation for log j was so cute, I could not abondon it and say to that calculation: From now on you are a poor orphan and no one will help you survive from day to day… How could I abandon such a calculation, better loose the UK a 100 times on a row than abandoning such nice calculations… 😉

But let’s go back to being a serious and responsible adult; the post is relatively long with 10 pictures. As usual I had to leave a lot out and I hope it is more or less easily readable. After all a lot of math out there looks like it is written by people who eat a plate of coal for breakfast. And if you eat coal for breakfast, likely this has an influence on the math you will produce on a particular day… Ok, here we go:

Ok, the goal of this post is of course to make you think a little bit about this 4D space and compare it to the quaternions and stuff. But last year on 2 March I posted the diagonalization method for finding the logarithm of an arbitrary split complex number. Below is a link.

Let me end this post with a funny mathematical joke about how to NOT WRITE MATH. Using a fucking lot of indices is not a way to make your work readable, here is a picture of what I view as some kind of mathematical joke.

In case you desire a serious headache, go read that file.

Ok, end of this post.