# Factorization of the determinant inside the space of 3D circular numbers. Aka: The conjugate ‘determinant style’.

A few weeks back I was thinking in writing finally some post about general theory for spaces with arbitrary dimension. It soon dawned on me that the first post should be about the impossibility of solving X^2 = -1 on spaces of odd dimension for both the complex and the circular method of mulitplication on those spaces. So post number one should be about the fact the famous number i does not exist in spaces with dimensions 1, 3, 5 etc.
And what about the second post? Well you can always factorize the determinant inside such spaces, that is a very interesting observation because the determinant is also the product of all eigenvalues. These eigenvalues live traditionally in the complex plane and as such a naive math professor could easily think that the determinant can only be factorized inside the complex plane. So that would be a reasonable post number two.
Since all these years I only did such a factorization once I decided to do it again and that is this post. The basic idea is very simple: If you want to find an expression for the inverse of a general 3D circular number, you need the determinant of that number. From that you can easily find a factorization of the determinant. It’s as simple as efficient.

But now I have repeated it in the space of 3D circular numbers I discovered that part of this factorization behaves very interesting when you restrict yourself to the subset of all 3D circular number that are not invertible. That is that taking the conjugate ‘determinant style’. The weird result is that taking this kind of a conjugate increases the number of eigenvalues that are zero. So this form of conjugation transports circular numbers with only one eigenvalue zero to the sub-space of numbers with two eigenvalues zero.

For years I have been avoiding writing general theory because I considered it better to take one space at a time and look at the details on just that one space. May be that still is the best way to go because now I have this new transporting detail for only what would be the second post of a general theory, it looks like it is very hard to prove such a thing in a general setting.

Luckily the math content of this post is not deep in the sense if you know how to find the inverse of a square matrix, you understand fast what is going on at the surface. But what happens at the level of non-invertibles is mind blowing: What the hell is going on there and is it possible to catch that into some form of general theory?

I tried to keep it short but all in all it grew to a nice patch of math that is 8 pictures long. Here is the stuff:

At the end of this post I want to remark that the quadratic behaviour of our conjugate ‘determinant style’ is caused by the fact it was done on a 3D space. If for example you are looking at 17 dimensional number, complex or circular, this method of taking a conjugate is a 16 degree beast in 17 variables. how to prove all non-invertible numbers get transported to more and more eigenvalues zero?

May be it is better to skip the whole idea of crafting a general theory once more and only look at the beautiful specifics of the individual spaces under consideration.

End of this post and thanks for your attention.

# 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.

# Correction: My spinning plasma model for sun spots is likely not correct.

Oops I likely made a many year mistake when it comes to the magnetic stuff. Many years ago I had the idea that the magnetism as found in sun spots could possibly explained by spinning plasma underneath the solar surface.
After all if electrons are magnetic monopoles, a spinning cylinder shaped plasma should eject lots of electrons along it’s magnetic field lines. That makes the spinning plasma a terrible good magnet because is a lot of positive charged plasma is spinning that creates strong magnetic fields.

After the original idea it took me about one year there could be a possible mechanism on the sun that creates such spinning plasma structures: The sun rotates faster at the equator as it does at the poles.

Now sun spots often come in pairs with opposite magnetic polarity and in my view I thought the leading sun spot was the one created by a bunch of rotating plasma under it.

It is easy to understand that if the root couse of sun spots was a rotating column of plasma underneath them, on the opposite hemispheres of the sun the leading sun spot should have opposite magnetic polarity. That one always checked true, but for the rotational hypothesis to be true over the solar cycles the polar magneticity of the leading sun spot should always be the same.

And that is where likely my old idea is crashing right now. In the next picture you see what is more or less observed in the last change of the solar cycle, for me this is not funny.

SC24 and SC25 stand for Solar Cycle 24 & 25 and again: for me this is not funny:

Yes it is what it is. But at least as soon as I discover I have made a serious mistake I tell that as soon as possible.
All in all this mistake does not have any impact on the tiny fact that it is impossible for electrons to be tiny magnets, electrons are magnetic monopoles and as such we have two variants of them. So the Gauss law for magnetism is just not true for an individual electron, it is nonsense to say magnetic field lines always loop in on themselves.

But after seven years of explaining this kind of mistakes, that stuff known as the science of physics is not capable of cleaning herself of stupid ideas.

Let’s leave it with that, this correction is a set back but the weirdo’s classified as the physics professors still have to give some experimental proof that electrons are indeed ‘tiny magnets’.

# 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.