Part 14: The Cauchy integral representation for the 4D complex numbers.

4D complex numbers, Exponential circle, Exponential curves, Matrix representation

It took me longer than expected to craft this update. That is also the nature of the subject; you can view and do math with Cauchy integral representation in many ways. In the end I settled on doing it just for polynomials of finite degree and even more simple: these polynomials are real valued on the real line. (So they have only real coefficients and after that are extended to the space of 4D complex numbers).

In another development, last week we had the yearly circus of Noble prizes and definitely the most cute thing ever was those evolving protein molecules. Because if you can use stuff like the e-coli bacteria you can indeed try if you can (forcefully) evolve the proteins they make… That was like WOW. Later I observed an interview with that chemistry Nobel prize winner and she stated that when she began she was told ‘gentlemen don’t do this kind of thing’.

So she neglected that ‘gentlemen stuff’ and just went on with it. That is a wise thing because if you only do what all those middle age men tell you to do you will find yourself in the very same hole as they are in…
The physics prize was also interesting, for myself speaking I was glad we did not observe those physics men totally not understanding electron spin but with the usual flair of total arrogance keep on talking about spin up and spin down.

You can also turn that spin nonsense upside down: If elementary particles only carry monopole electrical charge than why should electrons be bipolar when it comes to magnetism? That Gauss law of magnetism is only a thing for macroscopic things, there is no experimental proof it holds for quantum particles…

But let’s talk math because this update is not about what I think of electron spin. This is the second Cauchy integral representation I crafted in my life. Now the last years I produced a whole lot of math, my main file is now about 600 pages long. But only that very first Cauchy integral representation is something that I printed out on a beautiful glossy paper of size A0. That first Cauchy integral representation was on the space of 3D numbers and there life is hard: The number tau has determinant zero and as such it is not invertible. But I was able to complexify the 3D circular numbers and it was stunning to understand the number tau in that complexification of the 3D circular numbers. Just stunning…

Therefore I took so much time in trying to find an easy class of functions on the space of 4D complex numbers. I settled for easy to understand polynomials, after all any polynomial gives the same value everywhere if you write them as a Taylor series.
Since this property of polynomials is widely spread I can safely say this in this part 14 of the basics to the 4D complex numbers we have the next Theorem:
THEOREM: The math will do the talking.
PROOF: Just read the next 12 pictures. QED.

As usual all pictures are 550 x 775 pixels in size. I also use a thing I name ‘the heart of the Cauchy integral’, that is not a widely known thing so take your time so that the mathematical parts of your brain can digest it…

I truly hope the math in this update was shallow enough so you can use it in your own path of the math that you like to explore.

End of this post, may be in Part 15 we will finally do a bit more about the diagonalization of 4D complex numbers because that is also a universal way of finding those numbers tau in the different dimensions like the 17D circular numbers & all those other spaces.

Have a nice life or try to get one.

Part 13 of the basics of 4D complex numbers: Factorization of the determinant.

4D complex numbers, Matrix representation

It is about high time for a small update. Originally I wanted to include a little rant against all those math professors that have stated that the so called Euler formulae for the exponential circle in the complex plane is the most beautiful piece of math ever.

How can you say that and after all those other exponential circles and curves I found stay silent year in year out? We now have a fresh academic year and likely the new year nothing will happen again.

The same goes for magnetism, if it is true electrons carry magnetic charge one way or the other this will have huge economical impacts in the long run. Not only can you better understand how spintronic devices can work but as a negative also understand how nuclear fusion with those Tokomak things will never work because of the electron acceleration.

In my view this is important given the speed of climate change and how slow we react on it while at the same time we are always promised golden mountains of almost free energy if only we had nuclear fusion…

But likely university people are university people so we will just observe one more academic year of just nothing.

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After having said that, in this post we calculate the determinant via multiplying the four eigenvalues every 4D complex number Z has. Of course that is always when such a Z is viewed as it’s matrix representation M(Z).

You can do much more with that kind of stuff, in the previous part number 12 we unearthed the eigenvalues and eigenvectors so we can also do the nice thing of matrix diagonalization. I have not planned anything about the content in the next few parts of these small series on 4D complex numbers. So we’ll see.

This post is five pictures long, at the end I also show you the so called cylinder equation from 3D space (circular version).

Have fun reading it and thinking about it a little bit. See you around.

Before the actual post I will show you the teaser picture as published on the other website, it contains the matrix who’s determinant we are going to factor. When I read all these posts on 4D complex numbers backwards it was only in the basics number 3 where I mentioned this explicit matrix representation M(Z).

In the teaser picture you also see the main result. I never worked out the determinant of the matrix M(Z) via a method like expansion via the minors or so. Just going straightforward for the eigenvalue functions and multiply these in order to get the determinant for any 4D complex number Z.

So it is the determinant of the matrix representation M(Z) we are going to factorize.
Here we go:

 

 

 

 

 

Ok, let´s not rant because why waste all that emotional energy on university people?
But it is now almost your years back since I crafted a pdf about the first 10 exponential circles and curves I found:

An overview of exponential circles and curves in …
http://kinkytshirts.nl/pdfs/10_exponential_circles_and_curves.pdf

That was from 22 December 2014. We can safely conclude that at most universities a lot is happening, but it is mostly weird stuff. Weird stuff like ‘The sum of all integers equals minus 1/12…’ Oh oh, if you have people like that inside your ranks how can that bring any good?

So is it science or comedy in the next video?

It is more like comedy I just guess…

But what would life be without comedy? That would also be a strange place to live, a life without observable comedians likely is a less funny place to live in.

Hey let’s pop open one more pint of beer. By the way in the quantum world every thing is different and at first I could not believe that, why would that be? It was years and years later I found out that quantum particles like electrons never drink beer. Just never. And at that point in time I finally understood just how different the quantum world is compared to our human world.

Till updates.

Part 12 of the 4D complex numbers: The four eigenvalue functions for a arbitrary 4D complex number.

4D complex numbers, Matrix representation

In Part 11 we found the eigenvalues and eigenvectors of the first imaginary unit l (with of course the property l^4 = -1). But if we have those eigenvalues for l,it is easy to find the eigenvectors of powers of l.

But every 4D complex number is a sum of a real part and the three imaginary parts with the units  l,  l ^2 and l ^3. So from a linear algebra point of view it is also easy to find the eigenvalues of such combinations of  l. Needless to say that in this post you should read  l and it powers mostly as it’s matrix representation M(l).

I have to admit that I very often avoid the constant need for the matrix representation, for example I mostly write det(l) for the determinant of M(l). But it makes texts just so unreadable if you constant write det(M(l)), I do not like that.

When you understand how to calculate the eigenvalue functions as shown below, please remark it looks a like the so called discrete Fourier transform.

This post is five pictures long (each 550×775 pixels) although the last picture is rather empty… And why no an empty picture? After all when you had a paper book in the good old days, there were always empty pages in it. Sometimes it was even written that ‘This page is left empty intentionally’. And in those long lost years that was the crime of the century because if you write on a page that it is empty, that is never true…

In the next post we will take a look at the missing equation we still have when it comes to the calculation of the 4D exponential curve when it comes to the  sphere/cone equation. we missed one equation to arrive at a 1D solution for our exponential curve.

If you also include the demand that the determinant of the exponential curve is 1 all of the time, you can squeeze out more equations inside the 4D complex numbers.

Before we say goodbye, here is a link to the matrix representation of the discrete Fourier transform. But take you time when thinking about matrices like in the link or the Omega matrix as above.

DFT matrix
https://en.wikipedia.org/wiki/DFT_matrix

Ok, that was it. Till updates.

Part 11 of the 4D complex numbers: The eigenvalues and eigenvectors from the first imaginary unit.

4D complex numbers, Magnetism, Matrix representation

First a short magnetic update:

In reason 66 as why electrons cannot be magnetic dipoles I tried to find a lower bound for the sideway acceleration the electrons have in the simple television experiment.

To put it simple: How much sideway acceleration must the electrons have to explain the dark spots on the screen where no electrons land?

The answer is amazing at first sight: about 2.5 times 10^15 m/sec^2.
This acceleration lasts only at most two nano seconds and in the end the minimum sideway speed is about 5000 km/sec so while the acceleration is such a giant number it does not break relativity rules or so…
Here is the link:

16 Aug 2018: Reason 66: Side-way electron acceleration as in the television experiment.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff03.htm#16Aug2018

You know I took all kinds of assurances that it is only a lower bound on the actual acceleration that takes place. For example I took the maximal sideway distance as only 0.5 cm. Here is a photo that shows a far bigger black spot where no electrons land, so the actual sideway distance if definitely more than 0.5 cm.


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The math part of this post is not extremely thick in the sense you can find the results for yourself with the applet as shown below. Or by pencil & paper find some 4D eigenvalues and the corresponding eigenvectors for yourself.

But we need them in order to craft the so called eigenvalue functions and also for the diagonal matrices that come along with all of the matrix representations of the 4D complex numbers Z.

I hope I wrote it down pretty straightforward, this post is five pictures long. And if you like these kind of mathematical little puzzles: Try, given one of the eigenvalues like omega or omega^3, find such an eigenvector for yourself. It is really cute to write them down, multiply them by the eigenvalue and observe with your own eyes that indeed we have all that rotation over the dimensions included that omega^4 = -1 behavior.

This post is five pictures long, it is all rather basic I hope.

The applet used is from the WIMS server (https://wims.sesamath.net/), look for the Matrix calculator in the section on Online calculators and plotters.

For the time being I think that in Part 12 we will craft the eigenvalue functions for any 4D complex number Z. Ok, that was it for this update.

Part 10 of the 4D basics: The 4D imitators of i rotate everything by 90 degrees.

4D complex numbers

This is a short and simple update that nicely fits into the series of the basics of 4D complex numbers. It is only 3 pictures long and all we do is multiply a 4D complex number A by one of the imitators of i and after that calculate the inner product between A and A times the imitator of i.

Compare it to the complex plane: Take a 2D complex number z from the complex plane, write it as a vector and also write iz as a vector. Take the inner product and conclude that it is zero so that z and iz are perpendicular to each other.

That’s all we do in the space of 4D complex numbers:

I hope this was basic enough…  See you in the next post.

A few numerical results related to the 4D sphere-cone equations using the four coordinate functions of the 4D exponential curve.

4D complex numbers, Exponential circle, Exponential curves

This is Part 9 in the basics to the complex 4D numbers. In this post we will check numerically that the 4D exponential curve has it’s values on the 4D unit sphere intersected with a 4D cone that includes all coordinate axes. In 3D space the sphere-cone equations ensure the solution is 1 dimensional like a curve should be. In 4D space the sphere-cone equations are not enough, there is at least one missing equation and those missing equations can be found in the determinant of a matrix representation M(Z) for a 4D complex number Z.

But we haven’t done any determinant stuff yet (because you also need a factorization of the 4D determinant in four variables and that is not a trivial task). So this post does not contain numerical evidence that the determinant is always one on the entire exponential curve.

If you want to compare this post to the same stuff in the complex plane:
In the complex plane the sphere-cone equation is given by x^2 + y^1 = 1 (that is the unit circle) and if you read this you probably know that f(t) = e^{it} = cos t + i sin t.

You can numerically check this by adding the squares of the sine and cosine for all t in one period and that is all we do in this post. Only it is in 4D space and not in the two dimensional complex plane…

This post is seven pictures long (all of the usual size 550 x 775).

All graphs in this post are made with the applets as found on:

WIMS https://wims.sesamath.net/wims.cgi?lang=en

For the two graphs from above look for ‘animated drawing’ choose the 2D explicit curves option. There you must use the variable x instead of time t.

Here is the stuff you can place in for the sphere equation:

(cos(pi*x/2)*cos(pi*x/4))^2 +
(0.707107*(cos(pi*x/4)*sin(pi*x/2) + sin(pi*x/4)*sin(pi*x/2)))^2 +
(-cos(pi*x/2)*sin(pi*x/4))^2 +
(0.707107*(cos(pi*x/4)*sin(pi*x/2) – sin(pi*x/4)*sin(pi*x/2)))^2

If you just ‘cut & paste’ it should work fine…
That should save you some typo’s along the way

Ok, that is what I had to say on this numerical detail.

On a possible model for solar loops: rotating plasma.

4D complex numbers, Magnetism, Matrix representation

Yes that is all there is: spinning plasma… At the end of last year’s summer I had figured out that if indeed electrons have far more acceleration compared to the protons, if on the sun the solar plasma starts rotating this caused a lot of electrons flying out and as such the spinning plasma would always be electrically positive.

But at the time I had no clue whatsoever about why there would be spinning plasma at the surface of the sun but lately I found the perfect culprit: The sun spins much faster at the equator compared to the polar regions.

This spinning plasma is visible at the surface of the sun as the famous sun spots and it is known these sun spots are places of strong magnetic fields.

There is a bit of a weak spot in my simple model that says all spinning plasma creates a strong magnetic field because if the solar spots are at there minimum none of them are observed for a relatively long time. The weak spot is: Why would there be no tornado like structures be made during this minimum of solar spots? After all the speed difference is still there between the equator regions and the polar regions.

Anyway the good thing is that my simple model is very falsifiable: If you can find only one spinning tube-shaped or tornado-shaped plasma structure that not makes magnetic fields, the simple model can be thrown into the garbage bin.

The simple model is found in Reason number 65 as why electrons cannot be magnetic dipoles on the other website:

Reason 65: A possible model for solar loops going between two solar spots.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff03.htm#22July2018

The main feature of the solar loops is that before your very own eyes you see all that solar plasma accelerating while according to the standard model of physics this is not possible.

Now there are plenty of physics professors stating that electrons can be accelerated by a magnetic field but if you hear them saying that you know they have never done the calculations that make it at least plausible that non constant magnetic fields are the main driver of electron acceleration.

Here are two nice pictures of what I am trying to explain with my simple model.

The above picture is in the UV part of the spectrum.

After having said that, the next post is like planned about numerical evaluations related to the four coordinate functions of the new found exponential curve f(t) for the 4D complex numbers. I hope to finish it later this week.

Now we are talking about cute numerical results anyway, in the next picture you can see numerical validation that the number tau in the 4D complex space is invertible because the determinant of it’s matrix representation is clearly non-zero.

You might say ‘so what?’.  But if the number tau is invertible on the 4D complex numbers (just like the complex plane i has an inverse) in that case you can also craft a new Cauchy integral representation for that!

Again you might say ‘so what?’. But Cauchy integral representation is highly magical inside complex analysis related to the complex plane. There is a wiki upon it but the main result is a bit hard to swallow if you see it for the first time, furthermore the proof given is completely horrible let alone the bullshit after that. Anyway here it is, proud 21-century math wiki style:

Cauchy’s integral formula
https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula

Ok, let’s leave it with that. Till updates my dear reader.

Part 8 of the 4D basics: Wirtinger derivatives.

4D complex numbers, CR equations, Laplacian

This is the 91-th post on this website so surely but slowly this website is growing on. This post was more or less written just for myself; I don’t know if the concept of Wirtinger derivates is used a lot in standard complex analysis but I sure like it so that’s why we take a look at it.

The idea of a Wirtinger derivative is very simple to understand: You have some function f(Z) and by differentiating it in the direction of all four basis vectors you craft the derivative f'(Z) from that.

At the basis for all the calculations we do in this post are the Cauchy-Riemann equations that allow you to rewrite the partial derivatives we put into the Wirtinger derivative.

The main result in this post is as follows:
We take our Wirtinger operator W and we multiply it with the 4D complex conjugate of W and we show that this is a real multiple of the Laplacian.

The 4D case is more or less the same as on the complex plane, that is not a miracle because in previous 4D basics we already observed two planes inside the 4D complex numbers that are isomorphic to the complex plane. So it is not much of a surprise the entire 4D space of complex numbers behaves in that way too; all functions are harmonic that is the Laplacian of such a function is zero.

This post is ten pictures long, most of them are size 550 x 775 but a few of them are a bit broader like 600 x 775 because the calculations are rather wide.

On the scale of things this post is not ultra important or so, it is more like I wrote it for myself and I wanted to look in how much this all was different from the three dimensional case.

Here are the pics:

Further reading from a wiki (of course that is only about 2D complex numbers from the complex plane):
Wirtinger derivatives
https://en.wikipedia.org/wiki/Wirtinger_derivatives

Ok, that was it. Till updates.

Planning of posts + small magnetic update.

4D complex numbers, Magnetism

The next post is about so called Wirtinger derivatives for functions defined on the space of 4D complex numbers. That would also be part 8 in the basics of complex numbers but you can ask yourself if it looks like this is it basic?

It looks like you can rewrite these horrible looking operators always as the Laplacian.
It’s amazing. So that will be the next post.

After that, for the time being it is in the planning, a few numerical results from the sphere-cone equation for 4D complex numbers. That could serve as another post.

In another development I decided to skip all possible preperations for an ‘official publication’ when it comes to electron spin. The bridge between what I think of electron spin (a magnetic charge) and the official version (magnetic monopoles do not exist) is just too large. As such the acceptance in a peer reviewed scientific journal are not that high given the ‘peer review hurdle’.

Beside this hurdle there are much more reasons that I throw this project into the garbage bin. I just don’t feel good about it.

May be in five or ten years I will change my view on this, but I think it is better for every body that the official standpoint on electrons just stays as it is:

Electrons are magnetic dipoles.

No, why should I try to get into some physics scientific journal saying it ain’t so?

Until now all experimental evidence I have is this lousy picture that I made with an old television set, it is from April 2016:

Once more it is very hard to explain this away with the Lorenz force only. By all mathematical standards the Lorentz force is continuous when it comes to electron velocity and the applied magnetic field.

What we observe with this old 12 € television is that the electrons behave not that way; they behave discontinuous…

Let’s leave it with that. Till updates.

Calculation of the four coordinate functions for the 4D exponential curve (complex version).

4D complex numbers, Exponential circle, Exponential curves

Like promised in this post I will show you in the greatest detail possible how to find those rather difficult looking four coordinate functions.

I had thought about crafting these four coordinate functions before but the method I had in mind was rather labor some so I balked a bit at that. Not that I am lazy but I also had to work on the basics for the 4D complex numbers like in the last posts…

So one day I decided to look into the specific details of what I name ‘imitators of the number i’ and I was very surprised by their behavior. As a matter of fact these imitators imitate i soo good that you can make exponential circles of them.
And I wrote down the two exponential circles, I looked at them and realized you can factorize the 4D exponential curve with it and as such you will get the four coordinate functions…

That was all, at some point in time on some day I just decided to look at the imitators of the number i from the complex plane and within 5 at most 10 minutes I found a perfect way of calculating these four coordinate functions.

It always amazes me that often a particular calculation takes a short amount of time, like 10 or 20 minutes, and after that you always need hours and hours until you have a nice set of pictures explaining the calculation…

Anyway, this post is five pictures long and as such it contains also Part 6 and 7 of the Basics to the 4D complex numbers.

I hope that in the long run it will be the result in this post that will make 4D complex numbers acceptable to the main stream mathematical community.
But may be once more I am only fooling myself with that, after all back in the year 1991 I was only thinking stuff like ‘If you show them the 3D Cauchy-Riemann equations, they will jump in the air from joy’.
They (the math professors) never jumped from joy, no significant change in brain activity was ever observed by me. So when I write ‘in the long run’ as above, may be I should more think like a geological timescale…
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But let’s not complain because once you understand the factorization, it is so beautiful that it is hard to feel angry or whatever what.
Here are the five pictures:

 

Ok, that is how you calculate the four coordinate functions.

Till updates.