Integrals for the number tau for the 2D multiplication defined by i^2 = -1 + 3i.

Yes yes I know we already calculated the number tau for this space equipped with a hyperbolic multiplication. (That was a few posts back using matrix diagonalization.) But I had a few reasons to write this anyway, one reason was just curiosity. I wanted to know how those integrals looked and since we had calculated the number tau anyway we did not need to solve these integrals with pencil and paper.
I also wanted you to show how you can write the product of such a complex number z against it’s conjugate. On the standard complex plane this defines a circle and on our hyperbolic space it is of course a hyperbole.
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 “The norm of the product is the product of the norms” kind of stuff is only true because on the standard complex plane the determinant equals the square of the norm of a complex number
z. In our present case of hyperbolic complex numbers we devide the conjugate by the determinant and those determinant define hyperboles and not circles. So nothing of that “The norm of the product is the product of the norms” kind of stuff. The deeper underlying mechanism is just always the determinant of the matrix representation.

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. The integrals are of course the inverse of a complex number because the derivative of the logarithm is the inverse and we want to know the log of i because by definition that is the number tau.

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”.

Nobel prize for a sequential Stern Gerlach experiment? Nope, nada, njet, nein & NEE!

This is now year nine or may be the tenth year that I started doubting that electrons were tiny bipolar magnets because it makes much more sense that they are magnetic monopoles. Over the years I have found out that logic just does not work and given the fact that physics people get a salery from tax payer money, that is weird behavior.
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:

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:
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:

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

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:

Figure1: Don’t mind the ‘female robot’ because all female robots are fake.

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:

Really true: Maxwell’s little demon holds all electrons in place…

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:

It could be some light reflection but is it still one electron beam?

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!

Please note the small white region, it’s circular but you can’t see it on the photo.

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.

I have no idea what that teddy bear is doing there.

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.

Comparison of the conjugate on five different spaces.

To be a bit precise: I think two spaces are different if they have a different form of multiplication defined on them. Now everybody knows the conjugate, you have some complex number z = x + iy and the conjugate is given by z = x – iy. As such it is very simple to understand; real numbers stay the same under conjugation and if a complex numbers has an imaginary component, that gets flipped in the real axis.

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 z = x + iy number has an upper row that you can view as the conjugate. So I tried the upper row of my matrices for the 3D complex and circular numbers and voila instead of gibberish for the very first time I found what I at present day name the “Sphere-cone equation”.

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 e^it in the standard complex plane where the multiplication is ruled by i^2 = -1. Of course you want the conjugate of
e^it to be e^-it or time running backwards.

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 n = 2, 3). And that more simple thing is that taking the conjugate of any imaginary unit always gives you the inverse of that imaginary unit.

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 i^2 = -1 + i, in that case the third power is minus one: i^3 = -1. And you do not have to be a genius to find out that the inverse of this imaginary unit i is given by -i^2 .
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.

A de Moivre identiy for the i^2 = -1 + i multiplication.

We already have found some parametrizations for the complex exponential (that ellipse, see previous posts below) we do not really need such an identity. But they are always fun to make such identities in a new number system under study like lately those elliptical and hyperbolic multiplications in the plane.
Lets recap what we have done all these posts:

1) We looked at the matrix representations for 2D numbers ruled by i^2 = -1 + i. The determinant of such matrices was x^2 + xy + y^2. Therefore we wanted to know more about the ellipse that gives a determinant of 1.

2) We found a way to take the logarithm of the imaginary unit i by taking the integral of the inverse from 1 to i. That is the number tau for this kind of multiplication. As such we had a complex exponential only now it covered that ellipse.

3) After that we had to find parametrizations of the complex exponential, actually we found two of them via very different idea’s. It was left totally unproven that the two were the same although in for example the Desmos graphing package they covered the same ellipse. (But that is not a real math proof of course).

4) That is this post: Penning down the de Moivre identity or formula for this particular kind of 2D multiplication. So we can end the recap here.

The post is seven pictures long and I made them a bit larger this time: 550×1200 pixels. There’s also two additional figures so all in all very much pictures for so little math. I like the end result a lot, ok ok it is not very deep math but it looks damn cute in my opinion.
So I hope you will have fun reading it and thinking about the math involved in this post on 2D multiplications.

Often when found for the first time, the math was formulated very differently.

Here is a link to a more general wiki upon Abraham de Moivre:

In the next figure you can see that three of the equations give rise to the same graph in the Desmos package while of course again remarking this is not what a math proof should be…
Please ignore the typo (6) in the equation for the blue graph while remarking that Desmos still spits out the correct answer.

Once more: This is not a proof but you can use it for a bit of relief that you’re on the right road.

That was it for this post. May be the next post is on the conjugate or may be a post on magnetic stuff. I don’t know yet but I do know I want to thank you for your attention.

That mysterious electron pair and so called VESPR theory.

Some time ago I stopped writing posts about magnetism because the number of such posts would exceede the number of posts on the 3D complex numbers. And that was of couse not the long term strategic goal of this website, so I stopped posting it here.
But on the other website I kept on writing small sniplets and what I consider the best two sniplets is now reposted here in a new post on magnetism.

For readers who are new: For the last 9 years I have been trying to figure out if electrons are truly tiny magnets yes or no. About 9 years ago I started to doubt that electrons or electron spin is indeed a bipolar magnet. At the time I tried to explain the results from the Stern-Gerlach experiment and I arrived at the conclusion that very likely electrons were magnetic monopoles. My main argument has been all those years: If electrons are magnetic dipoles, because they are so small they must be neutral under application of (large) magnetic fields.

Since the SG experiment it is know that lone electrons are not magnetically neutral but all and everything observed was always explained by electrons as tiny magnets. Why at the time (1922 and later) they never observed that there are all kinds of problems with electron spin as tiny magnets, is unknown to me. For example the scientists at the time had correspondence between each other and some of those letters literally started with the Gauss law for magnetism and stating that a solution must be found inside the framework of the Gauss law for magnetism…

It never dawned on them that doing science is that you must prove the Gauss law for magnetism does apply for lone or unpaired electrons. But they never did that, no one doubted that magnetism was without magnetic charges and as such even a very small particle like the electron had to be a tiny magnet.

Since last year I often phrase my view on electron magnetism as follows:

The magnetic properties of the electron are just like it’s electric properties: Permanent and Monopole.

It is a bit strange that after 9 years I still have to try and find nice sounding slogans like the above as if I were some marketing bureau.

Anyway one of the big mysteries of the official version of electron spin is that in an electron pair the spins must be opposite. Nobody remarks this is totally crazy because if we allow for that we also give up the observation that opposite charges atract while same charges repel. I made an extra picture for this weird official version of electron spin:

Well take your time to think about it, this is the official version of the electron pair if electrons were tiny magnets. The physics professors never ever mention such details, no you often get a boatload of complicated math but they never ever talk about what anti alignment for tiny magnets actually means.

I also want to remark that journalists never ever ask such questions when they interview physics professors on magnetic related stuff. It’s fucking taxpayer money and we must believe this kind of crap?
Well yes, according to Cornell university we must. The next picture is one I actually used on the other website:

You don’t make this nonsense up: Like two bar magnets with opposite poles together.

VESPR theory. VESPR stands for Valence Electron Shell Pair Repulsion. This theory comes a bit more from the chemical sciences where they try to explain the shapes of the electron clouds of atoms and molecules.

The important detail is that electron pairs are neutral to magnetism and that as such electron pairs around an atomic nucleus repel each other.

If you use the idea that electrons are magnetic monopoles this all is very logical: Coulomb forces pull electrons in and the electrons form pairs because they have opposite monopole magnetic charges.

If you use the idea that electrons are tiny magnets this all is very crazy: Coulomb forces pull electrons in and they only form pairs? Why not form other configurations that are possible with tiny magnets? Why only electron pairs my dear physics professors?

My dear reader you have a brain for yourself so look in the picture below as why this particular atomic nucleus has two electron pairs that repel each other. And don’t mind the female robot or ponder the question as why there are female robots at all…
Just think a bit around the nonsense that comes along with electrons being tiny bipolar magnets. Here is the picture as used on the other website:

It’s time to publish this post, thanks for your attention and see you in a next post.

Two parametrizations for the ‘unit’ ellipse in the i^2 = -1 + i kind of multiplication.

Basically this post is just two parametrizations of an ellipse, so all in all it should be a total cakewalk… So I don’t know why it took me so long to write it, ok ok there are more hobbies as math competing for my time. But all in all for the level of difficulty it took more time as estimated before.
In the last post we looked at the number tau that is the logarithm for the imaginary unit i and as such I felt obliged to at least base one of the parametrizations on that. So that will be the first parametrization shown in this post.
The second one is a projection of the 3D complex exponential on the xy-plane. So I just left the z-coordinate out and see what kind of ellipse you get when you project the 3D exponential circle on the 2D plane. Acually I did it with the 3D circular multiplication but that makes no difference only the cosines are now more easy to work with. Anyway the surprise was that I got the same ellipse back, so there is clearly a more deeper lying connection between these two spaces (the 3D circular numbers and these 2D complex multiplication defined by i^2 = -1 + i).
A part of the story as why there is a connection between these spaces is of course found into looking at their eigenvalues. And they are the same although 3D complex numbers have of course 3 eigenvalues while the 2D numbers have two eigen values. A lot of people have never done the calculation but the complex plane has all kinds of complex numbers z that each have eigenvalues too…
Anyway I felt that out of this post otherwise it would just become too long to read because all in all it’s now already 10 images. Seven images with math made with LaTex and three additional figures with sceenshots from the DESMOS graphical package.
By the way it has nothing to do with this post but lately I did see a video where a guy claimed he calculated a lot of the Riemann zeta function zero’s with DESMOS. I was like WTF but it is indeed possible, you can only make a finite approximation and the guy used the first 200 terms of the Riemann zeta thing.
At this point in time I have no idea what the next post will be about, may be it’s time for a new magnetism post or whatever what. We’ll wait and see, there will always pop something up because otherwise this would not be post number 254 or so.
Well here is the stuff, I hope you like it or enjoy it.

Figure 1: This parametrization is based on the number tau.
Figure 2: The projection in red, stuff without 1/3 and 2/3 in blue.
Figure 3: The end should read (t – 1.5) but I was to lazy to repair it.

That was it for this post, of course one of the reasons to write is that I could now file it under the two categories “3D complex numbers” and “2D multiplications” because we now have some connection going on here.
And I also need some more posts related to 3D complex numbers because some time ago I found out that the total number of posts on magnetism would exceed those of the 3D complex numbers.

And we can’t have that of course, the goal of starting this website was to promote 3D complex numbers via offering all kinds of insights of how to look at them. The math professors had a big failure on that because about 150 years since Hamilton they shout that they can’t find the 3D complex numbers. Ok ok, they also want it as a field where any non-zero number is invertible and that shows they just don’t know what they are talking about.
The 3D complex numbers are interesting simply because they have all those non-invertible numbers in them.

It is time to split my dear reader so we can both go our own way so I want to thank you for your attention.

An inverse and a number tau for the i^2 = -1 + i multiplication.

This way of doing the complex multiplication keeps on drawing my attention because of the funny property that i^3 = -1. As such it has interesting parallels to the 3D complex number. For example the eigenvalues of this defining imaginary unit is the third root out of -1 (and it’s conjugate). That is in line with the results from the 3D numbers although over there 3D numbers have 3 eigenvalues and not 2.

In this post I want to show you a way to find the logarithm of this imaginary unit i via integrating the inverse from 1 to i. Just like on the real line if you integrate 1/x from 1 to say some positive a, you get log a. It is important to remark there are more methods to find such logarithms. For example you can diagonalize the multiplication and take the log of the eigenvalues and as such you can find the log of the imaginary unit.

Anyway back in the time I did craft my first complex exponential for the 3D complex numbers this way (using the integral of the inverse) so for me it is a bit of a walk down memory lane. You always get integrals that are hard to crack but if you use the WolframAlpha website it’s easy to find. Remarkably enough the two values for the integrals we will find below are also found in 3D and even the 6D complex numbers. So for me that was something new.

For myself speaking I loved the way the inverse of a complex number based on i^2 = -1 + i looks. You have to divide by the determinant once more proving that norms do not have very much to do with it. (In standard lessons on complex numbers it is always told that the norm of the product is the product of the norms, but that’s only so for the complex plane and the quaternions. So if you keep on trying such idea’s you won’t come very far…)

This post is five pictures long, lets go:

May be this parametrization is the next post.

Ok, that was it more or less for this post. Since we are now getting more and more posts on two dimensional complex (and split complex) numbers may be I will open a new category for those posts. On the other hand you must not open a new category every time you things that are a bit different from what you usually do…