# Comparison of the ‘Speed = the Square’ equation on 7 different spaces.

This post is very simlilar to a few back when we calculated the results on 4 different spaces. This time I hardly pen down any calculation but only give the results so we can compare them a little bit.
The way most professional math professors tell the story of complex numbers it goes a bit like this: We have the real number line, the complex plane and on top of that a genius named Hamilton found the quaternions. On top of that there are a bunch of so called Clifford algebra’s and oh we math professors are just so good. There is no comparison to us, we are the smartest professionals in the world!

Well that is very interesting because it is well known these so called ‘professionals’ could not find the 3D complex numbers for about 150 years. So how come they all say we have this and that (complex plane and quaternions) and that’s enough, we are just perfect! Why they keep on saying rubbish like that is the so called Dunning-Kruger effect. That’s something from psychology and it says that people who lack understanding of some complicated stuff also lack the insight that they are stupid to the bone when it comes to that particular complicated stuff. So the views of professional math professors is very interesting but can be neglected one 100 percent, it’s just Dunning-Kruger effect…

If you look at the seven results of the ‘Speed = the Square’ equations, the solutions form a strickt pattern that only depends of the number of dimensions and if it is the complex or the circular multiplication. So every time a math professor goes from the complex plane to the wonderful world of quaternions you now know you are listening to a weirdo.

I said I only give results but since I have never ever introduced the 4D circular numbers I just extrapolated the other six spaces to the solution that lives in that beautiful space. So the last example is a bit longer.

Anyway although the math depth of this post is not that very deep (solving a differential equation that wants the derivative to be the square of what you differentiate), it clearly demonstrates solutions of all 7 different spaces look strikingly similar.
But because of the Dunning-Kruger effect likely the math professors will keep on telling total crap when it comes to complex numbers. Why am I wasting my time on explaining math professor behaviour? Better go to the five pictures of our post. Here we go & bye bye math professors.

May be I should write some posts about general complex number theory on spaces of arbitrary dimension. On the other hand I found the 3D complex numbers back in the year 1990. So if after all those years I will once more try to write some general theory one thing will be clear: Math professors will keep on trying to convince you of the beauty of quaternions or that garbage from the Clifford algebra’s.

Why, as a society, do we keep on wating tax payer money on math professors? Ok, they do not everything wrong but all in all it is not a great science or so where the participants are capable of weeding the faults out and grow more of the good stuff.
Let me end this post and thank you for your attention.

# Solving the ‘Speed = the Square’ equation on the space of 4D complex numbers.

Unavoidable I had to write some post after the video on the quaternion from Hamilton. Now my 4D complex numbers commute so they are very different from the standard version of quaternions. Just like in the complex plane the multiplication is ruled by the imaginary unit i that has the defining property of i^2 = -1. On the space of four dimensional complex number I mostly write l for the first imaginary component, the defining property is of course that now the fourth power equals minus one: l^4 = -1.
In 2018 I wrote about 20 introdutionary posts about the 4D complex numbers. That is much more as you would need for the quaternions of Hamilton but on the quaternions you can’t do complex analysis and that explains almost all of the difference.
You can view the quaternions as three complex planes fused together by the common use of the real line. My 4D complex numbers can be viewed as a merge of two complex planes in the sense that there are two planes clearly ‘the same’ as a complex plane.
This post is once more one of the ‘Speed = the Square’ equations and just as on the other spaces we looked at we choose the initial condition such that it is the first imaginary unit l. As such our solution is easily found to be f(t) = l / (1 – lt) because if you differentiate that you get the square. So from the mathematical point of view this is all rather shallow math because all we have to do is find the four coordinate functions of our solution f(t). For that you need to calculate the inverse of 1 – lt and to be honest after so much years I think almost all math professors are just to fucking stupid to find the inverse of any non real 4D compex number Z let alone if you have something with a variable t in it like in 1/(1 – lt).

I did my best to write this as transparant as possible while also keeping it as short as possible. For an indepth look at how to find the inverse of a 4D complex number, look for Part 17 in the intro series to the 4D complex numbers. (Just use the search function for this website for that.)

This post is just three pictures long so lets hope that is inside your avarage attention span. And it’s math so without doubt a lot of people will digest this with a speed of one picture a week! No I am not being sarcastic or so, I just like as how I evolved to the math place I am now. Often that also goes very slow but it has to be remarked the math professors are much more slow slow slow because they could not find the 3D complex numbers in all of human history.
Let’s dive into the picture stuff:

One of the funny things of the math of this post is that on the one hand it is very simple: You only need high school math like the quotient rule for checking my claims are true and differentiation mimics the multiplication on the 4D complex numbers. On the other hand you have those math professors likely not capable of finding these easy coordinate functions for themselves.
But this post is not meant as an anti math professor rant but more upon the beauty of simple math you can do on say the space of 4D complex numbers.
See you in the next post.

# Why could Hamilton not find the three dimensional complex numbers?

This very short post was written because of a video from the video channel Kathy loves physics. It is one of those “Quaternions are fantastic” video’s. And Kathy just like a lot of other physics people think indeed that quaternions are fantastic. But you cannot differentiate or integrate on the quaternions so I guess this stronly limits it’s use in physics.
But quaternions are very handy in describing rotations in 3D space, I never studied the details but it was said that on the space shuttle it was used for nagvigation. And because of these rotation properties at present day they are used in the games industry.

Anyway in the video Kathy explains that Hamilton did try for a long time to find the three dimensional complex numbers. And he never succeeded in that. Of course I know this for decades right now but in the past I never looked into a tiny bit more detail in what Hamilton was actually doing.
And he was looking at complex numbers of the form X = x + yi + zj where the imaginary components both equal to minus one: i^2 = j^2 = -1.

If you check the easy calculations in this post for yourself, it is amazing how much it already looks like the stuff as found on the quaternions. As such it is all of a sudden much less a surprise that Hamilton found the quaternions. As a matter of fact it was only waiting until he would stumble across them. But at the time the concept of a four dimensional space was something that made you look like a crazy lunatic, there were even vector wars and lots of crazy emotional stuff.

At present day it is accepted that 3D complex numbers do not exist, in my experience the professional math community is still emotionally laden but now into the direction of total neglect. Stupid shallow thought like “If Hamilton could not find them, they likely don’t exist”.

Back in the 19-th century they were always looking for an extenstion of the complex plane to three dimensional space. Of course they failed in that attempt because it is a fact of math life that you cannot solve the equation X^2 = -1 on the space of 3D complex (and also circular) numbers.

The content of this post is just two pictures, after that two more pictures as I used them on the other website and after that you can finally dive into the Video from Kathy. If you are interested in physics and also the history of physics, Kathy her channel is a thing you should take a look at if you’ve never seen it. Here we go:

YES, that is what he should have done. Hamilton tried for about one decade to find the numbers that form the title of this very website, so may be he tried this kind of approach. I don’t know, but the 3D complex numbers are not some extension of the complex plane because 2 is not a divisor of 3. You know that prime number stuff is going on here.
But the math professors are not interested in that kind of stuff.

Here is how I used it on the other website:

As you see in the above picture I was already working on the previous post because if you differentiate the three functions that mimics the 3D circular multiplication. You can also mimic the multiplication on the complex plane, that is in the next picture:

At last you can view the famous video of Kathy! It’s only 30 minutes or so but if you see too many so called TIKTOK videos that is infinitely long: Wow 30 minutes long looking at just one video?

End of this post, likely the next post is about 4D complex numbers.

# Five highlights of the year 2021.

Despite my slowly detoriating health the last year was a remarkable fruitfull year when it comes to new stuff. So I selected five highlights and of course that is always a difficult thing. Two of the highlights are about magnetism and the other three are just math. Once more: The fact that I include two magnetic highlights does not mean I am trying to reach out to the physics community in any meaningful way. If these idiots and imbeciles keep on thinking that electrons have two magnetic poles, be my guest. There is plenty of space under the sun for completely conflicting insights: Idiots and imbciles thinking that electrons have two magnetic poles and more moderate down to earth people that simply remark: for such a bold claim you need some kind of experimental evidence that is convincing.
But 2021 was a very good year when it came to math; I found plenty of counter examples to the so called last theorem of Pierre de Fermat. I was able to make a small improvement on the so called little theorem of Fermat. A very important detail is that I was able to make those counter examples to the last theorem so simple that a lot of non math people can also understand it. That is important because if you craft your writings to stuff only math professors can understand, you will find yourself back in a world of silence. Whatever you do there is never any kind of response. These math professors were not capable of finding three or four dimensional complex numbers, they stay silent year in year out so I have nothing to do with them. In the year 2021 I classified the physics professors to be the same: Avoid these shitholes at all costs!

After having said that, this post has eight pictures of math text and it has the strange feature that I am constantly placing links of posts I wrote in the last year. So lets go:

I think that if you show the above animated gif to a physics professor and ask for an explanation, likely this person will say: “Oh you see the electrons aligning with the applied external magnetic field, this all is well understood and there is nothing new under the sun here”.
Of course that kind of ‘explanation’ is another bag of bs, after all the same people explain the results of the Stern-Gerlach experiment via the detail that every electron has a 50% probability that it will align with the applied external magnetic field (and of course 50% that it will anti-align). In my view that is not what we see here. As always in the last five+ years an explanation that electrons are magnetic monopoles with only one of the two possible magnetic charges is far more logical.

This year in the summer I wrote an oversight of all counter examples to the last theorem of Pierre de Fermat I had found until then. It became so long that in the end I had three posts on that oversight alone. I wrote it in such a way that is starts as easy as possible and going on it gets more and more complicated with the counter example from the space of four dimensional complex numbers as the last example. So I finished it and then I realized that I had forgotten the space of so called split complex numbers. In the language of this website the split complex numbers are two dimensional circular numbers. It is just like the complex plane with two dimensional numbers of the form z = x + iy, only now the square of the imaginary unit is +1 instead of i^2 = -1 as on the complex plane. So I made an appendix of that detail, I consider this detail important because it more or less demonstrates what I am doing in the 3D and 4D complex number spaces. So let me put in one more picture that is the appendix of the long post regarding the oversight of all counter examples found.

All that is left is place a link to that very long oversight:

Ok, so far for what I consider the most significant highlights of the previous year. And oops, since I am a very chaotic person before I forget it: Have a happy 2022! It is time to say goodbye so think well and work well my dear reader.

# Oversight of all counter examples to the last theorem of Pierre de Fermat, Part 2.

Post number 191 already so it will be relatively easy to make it to post number 200 this year. If you think about it, the last 190 posts together form a nice bunch of mathematics.
In this post we will pick on where we left it in the last post; we start with the three dimensional complex and circular numbers. In the introduction I explain how the stuff with a pair of divisors of zero works and from there it is plain sailing so to say. When back in Jan of this year I constructed the first counter example to the last theorem of Pierre de Fermat I considered it a bit ‘non math’ because it was so easy. And when one or two days later I made the first counter example using modular arithmetic I was really hesitant to post it because it was all so utterly simple…
But now half a year later it has dawned on me that all those professional math professors live up to their reputation of being overpaid under performers because in a half year of time I could find not one counter example on our beloved internet. And when these people write down some calculations that could serve as a counter example, they never say so and use it only for other purposes like proving the little theorem of Fermat. It has to be remarked however that in the past three centuries of time, when people tried to find counter examples, they likely started with the usual integers from the real line and as such tried to find counter examples. Of course that failed and this is not because they are stupid or so. It is the lack of number spaces they understand or know about that prevented them in finding counter examples to the last theorem of Pierre de Fermat.
If you do not know anything about 3D complex or circular numbers, you are not a stupid person if you cannot find counter examples to the last theorem. But you are definitely very very stupid if you do not want to study 3D complex numbers, if you refuse that it proves you have limited mathematical insights and as such likely all your other math works will be limited in long term value too.
While writing this post all of a sudden I realized I skipped at least one space where counter examples are to be found: It is on the space of so called split complex numbers. I did not invent that space, that was done by the math professors. The split complex numbers are a 2D structure just like the complex plane but instead of i^2 = -1, on the split complex plane the multiplication is ruled by i^2 = 1. Likely I will write a small post about the split complex number space. (Of course in terms of the language of this website, the 2D split complex numbers are the 2D circular numbers.)

This post is 8 pictures long, I kept on to number them according to the previous post so we start at picture number 11. They are all in the size of 825×550 pixels. I hope it is worth of your time. Here we go:

In this post I used only ‘my own spaces’ like 3D complex and circular numbers and the 4D complex numbers. As such it will be 100% sure the math professionals will 100% not react on it. Even after 30 years these incompetents are not able to judge if there is any mathematical value in spaces like that. Why do we fork out so much tax payer money to those weirdo’s? After all it is a whole lot of tax payer money for a return of almost nothing. Ok ok a lot of math professors also give lectures in math to other studies like physics so not all tax payer money is 100% wasted but all in all the math professors are a bunch of non-performers.

I think I will write a small post about the 2D split complex numbers because that is a space discovered by the math pro’s. So for them we will have as counter examples to the last theorem of Pierre de Fermat all that modulo calculus together with the future post on the split complex numbers. Not that this will give a reaction from the math pro’s but it will make clear you just cannot blame me for the non reactive nature of the incompetents; the blame should go to those who deserve it… Or not?

May be the next post is about magnetism and only after that I will post the split complex number details. We’ll see, anyway if you made it untill here thanks for your attention and I hope you learned a bit from the counter examples to the last theorem of Pierre de Fermat.

# Two things and a proof that the 4D complex rationales form a field.

I finished the proof that was originally meant to be an appendix to the previous post. And I have two more or less small things I want to share with you so lets get started with the first thing:

Thing 1: Tibees comes up with a very cute program of graphing 3D surfaces. It’s name is surfer, the software is very simple to use and it has the giant benefit of making graphs from implicit equations like
f(x, y, z) = 0. For example if you want the unit sphere in 3D space you must do x^2 + y^2 + z^2 – 1 = 0. Now for this website I always used an internet applet that uses ray tracing and by doing so over the years such graphs always look the same. But this surfer program has cute output too and it has the benefit you don’t need to be online. Here is how such a graph looks, it is the determinant in the space of 3D complex numbers, to be precise it shows the numbers with a determinant of 1:

By the way, the surface of this graph is a multiplicative group on it’s own in 3D space. I never do much group stuff but if you want it, here you have it. And for no reason at all I used GIMP to make one of those cubes from the above graph. It serves no reason beside looking cute:

The Tibees female had a video out last week where she discusses a lot of such surfaces in three dimensional space using that surfer software. And she is a pleasant thing to look at, it is not you are looking at all those extravert males drowning in self-importance only lamentating shallow thoughts. The problem posed in the video is an iteresting one, I don’t have a clue how to solve it. Title of the video: The Shape No One Thought was Possible. It is a funny title because if you start thinking about all the things that math professors thought were not possible you can wonder if there is enough paper in the entire universe to write that all out..
https://www.imaginary.org/program/surfer.

Thing 2: The last weeks it is more and more dawning on me that all those centuries those idiots (the math professors) did not find counter examples to the last theorem of Fermat. Nor was there any improvement on the little theorem of Fermat. Only Euler did some stuff on the little theorem with his totient function, but for the rest it is not much…
Well since Jan of this year I found many counter example to the last theorem of Fermat and in my view I made a serious improvement on the little theorem of Fermat.
So is the improvement serious or not?
Here is a picture that shows the change:

On a wiki with a lot of proofs for the little theorem of Fermat they start with a so called ‘simplification’. The simplification says that you must pick the number a between 0 and p. So if you have an odd prime, say a = 113, does the little theorem only make sense for exponents above 113?
And can’t we say anything about let’s say the square 113^2?

With the new version of the little theorem we don’t have such problems any longer. Here is a screen shot from the start of that wiki, the upper part shows you the improvement:

If you follow that link you can also scroll down to the bottom of the wiki where you can find the notes they used. It is an impressive list of names like Dirichlet, André Weil, Hardy & Wright and so on and so on. All I want to remark is that non of them found counter examples to the last theorem nor did they improve on the little theorem of Fermat. Now I don’t want to be negative on Dirichlet because without his kernel I could never have crafted my modified Dirichlet kernel that is more or less the biggest math result I ever found. But the rest of these people it is just another batch of overpaid non performers. It’s just an opinion so you don’t have to agree with it, but why do so many people get boatloads of money while they contribute not that much?

End of thing 2.

Now we are finally ready to post the main dish in this post: the proof that the subset of four dimensional rational numbers form a field. Math professors always think it is ‘very important’ if something is a field while in my life I was never impressed that much by it. And now I am thinking about it a few weeks more, the less impressed I get by this new field of four dimensional complex numbers.
Inside the theory of higher dimensional complex numbers the concept of ‘imitators of i‘ is important: these are higher dimensional numbers that if you square them they have at least some of the properties of the number i from the complex plane. They rotate everything by 90 degrees or even better they actually square to minus one.
Well one of the imitators of i in the space of 4D complex number is dependent of the square root of 2. As such it is not a 4D rational complex number. That detail alone severely downsizes my enthousiasm.
But anyway, the next pictures are also a repeat of old important knowledge like the eigenvalue functions. Instead of always trying to get the eigenvalues from some 4×4 matrix, with the eigenvalue functions with two fingers in your nose you can pump out the eigenvalues you need fast. This post is six pictures long each size 550×825 pixels.
Here we go:

Yes that is the end of this post that like always grew longer than expected. If you haven’t fallen asleep by now, thanks for your attention and don’t forget to hunt the math professors until they are all dead! Well may be that is not a good idea, but never forget they are too stupid to improve on the little Fermat theorem and of course we will hear nothing from that line of the profession…

# Inverses for the field of 4D complex rationales.

This year starting in January I found more and more counter examples to the last theorem of Fermat. As a by product when we looked at the stuff on the 4D complex numbers, we found that if we restrict ourselves to the 4D rationales, they were always invertible. And as such they form a field, this is a surprising result because the official knowledge is that the only possible 4D number system are the quaternions from Hamilton. So how this relates to those stupid theorems of Hurwitz and Fröbenius about higher dimensional complex numbers is something I haven’t studied yet. But that Hurwitz thing is based on some quadratic form so likely he missed this new field of 4D complex rationales because the 4D complex numbers are ruled by a 4 dimensional thing namely the fourth power of the first imaginary unit equals minus one: l^4 = -1.
Compare that to the complex plane that has all of it’s properties related to that defining equation i^2 = -1.

And because we now have a 4D field I thought like let’s repeat how you find the inverse of a 4D complex rational number. And also prove that we have a field as basic a proof can be. But while writing this post I had to abandon the second thing otherwise this post would grow too long. Of course in the past I have crafted a post for finding the non-invertible 4D complex numbers but in that post I never remarked that rational 4D complex numbers are always invertible. To be honest in the past it has never dawned on me that it was a field, for me this is not extremely important but for the professional math people it is.

When back in Jan of this year I found the first counter example to the last theorem of Fermat I was a bit hesitant to post it because it was so easy to find for me. But now four months further down the time line I only found two examples where other people use some form of my idea’s around those counter examples and both persons have no clue whatsoever that they are looking at a counter example to the last theorem of Fermat. But in a pdf from Gerhard Frey (that is the Frey from the Frey elliptic curve that plays an important role in the proof to the last theorem of Fermat by Andrew Wiles) it was stated as:
(X + Y)^p = X^p + Y^p modulo p.
That’s all those professionals have, it is of a devastating minimal content but at least it is something that you could classify as a rudimentary counter example to the last theorem of Fermat. It only works when your exponent if precisely that prime number p and it lacks the mathematical beauty that for example we have in expressions like:
12^n = 5^n + 7^n modulo 35.

Anyway this post contains nothing new but there is some value in repeating how to find inverses of higher dimensional complex numbers. All you need is a ton of linear algebra and for that let me finish this intro on a positive note: Without the professional math professors crafting linear algebra in the past, at present day for me it would be much harder to make progress in higher dimensional complex numbers. And it is amazing: Why is linear algebra relatively good while in higher dimensional number systems we only look at a rather weird collection of idea’s?
This post is made up of seven pictures each of size 550×800 pixels.

Ok ok this post is not loaded with all kinds of deep math results. But if you have a properly functioning brain you will have plenty of paths to explore. And the professional math professors? Well those overpaid weirdo’s will keep on neglecting the good side of math and that is important too: That behavior validates they are overpaid weirdo’s…

For example the new and improved little theorem of Fermat: The overpaid weirdo’s will neglect it year in year out.
That’s the way it is, here is once more a manifestation of the new and improved little theorem of Fermat:

Let’s leave it with that. Thanks for your attention.

# Another counter example to Fermat’s last theorem using 4D complex numbers.

All in all I am not super satisfied with this post because the math result is not that deep. Ok ok the 4D complex numbers also contain non-invertible numbers, say P and Q, and these are divisors of zero. That means PQ = 0 while both P and Q are non-zero. And just like we did in the case of 3D circular and complex numbers because of the simple property PQ = 0 all mixed terms in (P + Q)^n become 0 and as such: (P + Q)^n = P^n + Q^n.

In the space of 4D complex numbers an important feature of the determinant det(Z) of a 4D complex number Z is that it is non-negative. As such there is not a clear defined layer between the part of the number space where the determinant is positive versus the negative part. During the writing of this post it dawned on me that Gaussian integers in the 4D complex space always have a non-zero determinant. As such the inverse of such a Gaussian exists although often this is not a Gaussian integer just like the inverse of say the number 5 is not an integer. A completely unexpected finding is that the 4D complex fractions form a field…

That made me laugh because the professional math professors always rejected higher dimensional complex numbers because they are not a field. For some strange reason math professors always accept or embrace stuff that forms a field while they go bonkers & beserk when some set or group or ring is not a field. This is a strange behavior because the counter examples that I found against Fermat his last theorem are only there because 3D and 4D numbers are not a field: there are always non zero numbers that you cannot invert.
As such a lot of math professors are often busy to make so called field extensions of the rational numbers. And oh oh oh that is just soo important and our perfumed princes ride high on that kind of stuff. And now those nasty 4D complex numbers from those unemployed plebs form a field too
I had to smile softly because 150 years have gone since the last 4D field was discovered, that is known as the quaternions, and now there is that 4D field of rationals that are embedded into something the cheap plebs name ‘4D complex numbers’? How shall the professional math professors react on this because it is at the root of their own behavior over decades & centuries of time?

Do not worry my dear reader: They will stay the overpaid perfumed princes as they are. Field or no field, perfumed princes are not known to act as adult people.

After having said that, this post is only five pictures long all of the ususal size of 550×775 pixels. For myself speaking I like the situation on the 3D numbers more because there you can easily craft an infinite amount of counter examples against the last theorem of Fermat.
Ok, here we go:

Yes I have to smile softly: all this hysteria from overpaid math professors about stuff being a field or not. And now we are likely into a situation where the 4D complex numbers are not a field but the space of 4D complex rationals is a field…

Will the math professors act as adults? Of course not.
Ok, let’s end this post because you just like me will always have other things to do in the short time that we have on this pale blue dot known as planet earth. Till updates.

# The total differential for the complex plane & the 3D and 4D complex numbers.

I am rather satisfied with the approach of doing the same stuff on the diverse complex spaces. In this case the 2D complex plane and the 3D & 4D complex number systems. By doing it this way it is right in your face: a lot of stuff from the complex plane can easily be copied to higher dimensional complex numbers. Without doubt if you would ask a professional math professor about 3D or higher dimensional complex numbers likely you get a giant batagalization process to swallow; 3D complex numbers are so far fetched and/or exotic that it falls outside the realm of standard mathematics. “Otherwise we would have used them since centuries and we don’t”. Or words of similar phrasing that dimishes any possible importance.

But I have done the directional derivative, the factorization of the Laplacian with Wirtinger derivatives and now we are going to do the total differential precisely as you should expect from an expansion of the century old complex plane. There is nothing exotic or ‘weird’ about it, the only thing that is weird are the professional math professors. But I have given up upon those people years ago, so why talk about them?

In the day to day practice it is a common convention to use so called straight d‘s to denote differentiation if you have only one variable. Like in a real valued function f(x) on the real line, you can write df/dx for the derivative of such a function. If there are more then one variable the convention is to use those curly d’s to denote it is partial differentiation with respect to a particular variable. So for example on the complex plane the complex variable z = x + iy and as such df/dz is the accepted notation while for differentiation with respect to x and y you are supposed to write it with the curly d notation. This practice is only there when it comes to differentiation, the opposite thing is integration and there only straight d‘s are used. If in the complex plane you are integrating with respect to the real component x you are supposed to use the dx notation and not the curly stuff.
Well I thought I had all of the notation stuff perfectly figured out, oh oh how ultrasmart I was… Am I writing down the stuff for the 4D complex numbers and I came across the odd expression of dd. I hope it does not confuse you, in the 4D complex number system I always write the four dimensional numbers as Z = a + bl + cl^2 + dl^3 (the fourth power of the imaginary unit l must be -1, that is l^4 = -1, because that defines the behavior of the 4D complex numbers) so inside Z there is a real variable denoted as d. I hope this lifts the possible confusion when you read dd

More on the common convention: In the post on the factorization of the Laplacian with Wirtinger derivatives I said nothing about it. But in case you never heard about the Wirtinger stuff and looked it up in some wiki’s or whatever what, Wirtinger derivatives are often denoted with the curly d‘s so why is that? That is because Wirtinger derivatives are often used in the study of multi-variable complex analysis. And once more that is just standard common convention: only if there is one variable you can use a straight d. If there are more variable you are supposed to write it with the curly version…

At last I want to remark that the post on the factorization of the Laplacian got a bit long: in the end I needed 15 pictures to publish the text and I worried a bit that it was just too long for the attention span of the average human. In the present years there is just so much stuff to follow, for most people it is a strange thing to concentrate on a piece of math for let’s say three hours. But learning new math is not an easy thing: in your brain all kind of new connections need to be formed and beside a few hours of time that also needs sleep to consolidate those new formed connections. Learning math is not a thing of just spending half an hour, often you need days or weeks or even longer.

This post is seven pictures long, have fun reading it and if you get to tired and need a bit of sleep please notice that is only natural: the newly formed connetions in your brain need a good night sleep.

Here we go with the seven pictures:

Yes, that’s it for this post. Sleep well and think well & see you in the next post. (And oh oh oh a professional math professor for the first time in his or her life they calculate the square Z^2 of a four dimensional complex number; how many hours of sleep they need to recover from that expericence?)
See ya in the next post.

# Factorization of the Laplacian (for 2D, 3D and 4D complex numbers).

Originally I wanted to make an oversight of all ways the so called Dirac quantization condition is represented. That is why in the beginning of this post below you can find some stuff on the Dirac equation and the four solutions that come with that equation. Anyway, Paul Dirac once managed to factorize the Laplacian operator, that was needed because the Laplacian is part of the Schrödinger equation that gives the desired wave functions in quantum mechanics. Well I had done that too once upon a time in a long long past and I remembered that the outcome was highly surprising. As a matter of fact I consider this one of the deeper secrets of the higher dimensional complex numbers. Now I use a so called Wirtinger derivative; for example on the space of 3D complex numbers you take the partial derivatives into the x, y and z direction and from those three partial derivatives you make the derivative. And once you have that, if you feed it a function you simply get the derivative of such a function.

Now such a Wirtinger derivative also has a conjugate and the surprising result is that if you multiply such a Wirtinger derivative against it’s conjugate you always get either the Laplacian or in the case of the 3D complex numbers you get the Laplacian multiplied by the famous number alpha.

That is a surprising result because if you multiply an ordinary 3D number X against it’s conjugate you get the equation of a sphere and a cone like thing. But if you do it with parital differential operators you can always rewrite it into pure Laplacians so there the cones and spheres are the same things…

In the past I only had it done on the space of 3D numbers so I checked it for the 4D complex numbers and in about 10 minutes of time I found out it also works on the space of 4D complex numbers. So I started writing this post and since I wanted to build it slowly up from 2D to 4D complex numbers it grew longer than expected. All in all this post is 15 pictures long and given the fact that people at present day do not have those long timespan of attention anymore, may be it is too long. I too have this fault, if you hang out on the preprint archive there is just so much material that often after only five minutes of reading you already go to another article. If the article is lucky, at best it gets saved to my hard disk and if the article has more luck in some future date I will read it again. For example in the year 2015 I saved an article that gave an oversight about the Dirac quantization condition and only now in 2020 I looked at it again…

The structure of this post is utterly simple: On every complex space (2D, 3D and 4D) I just give three examples. The examples are named example 1, 2 and not surprising I hope, example 3. These example are the same, only the underlying space of complex numbers varies. In each example number 1 I define the Wirtinger derivative, in example 2 I take the conjugate while in the third example on each space I multiply these two operators and rewrite the stuff into Laplacians. The reason this post is 15 pictures long lies in the fact that the more dimensions you have in your complex numbers the longer the calculations get. So it goes from rather short in the complex plane (the 2D complex numbers) to rather lengthy in the space of 4D complex numbers.

At last I would like to remark that those four simultanious solutions to the Dirac equation it once more shouts at your face: electrons carry magnetic charge and they are ot magnetic dipoles! All that stuff like the Pauli matrices where Dirac did build his stuff upon is sheer difficult nonsense: the interaction of electron spin with a magnetic field does not go that way. The only reason people in the 21-th century think it has some merits is because it is so complicated and people just loose oversight and do not see that it is bogus shit from the beginning till the end. Just like the math professors that neatly keep themselves stupid by not willing to talk about 3D complex numbers. Well we live in a free world and there are no laws against being stupid I just guess.

Enough of the blah blah blah, below are the 15 pictures. And in case you have never ever heard about a thing known as the Wirtinger derivative, try to understand it and may be come back in five or ten years so you can learn a bit more…
As usual all pictures are 550×775 pixels in size.

Oh oh the human mind and learning new things. If a human brain learns new things like Cauchy-Riemann equations or the above factoriztion of the Laplacian, a lot of chages happen in the brain tissue. And it makes you tired and you need to sleep…
And when you wake up, a lot of people look at their phone and may be it says: Wanna see those new pictures of Miley Cyrus showing her titties? And all your new learned things turn into insignificance because in the morning what is more important compared to Miley her titties?

Ok my dear reader, you are at the end of this post. See you in the next post.