In my own pdf I did a similar calculation in three dimensional space but that was with a pyramid or a simplex if you want. You can view that as three projections too although at the time I just calculated the areas and direved that 3D version of Pythagoras.

Within the hour I had a proof that was so amazingly simple that at first I laid it away to wait for another day or for the box for old paper to be recycled. But later I realized you can do this simple proof in all dimensions so although utterly simple it has absolutely some value.

The biggest disadvantage of proving more general versions of the theorem of Pythagoras and say use things like a simplex is that it soon becomes rather technical. And that makes it hard to read, those math formula’s become long can complex and it becomes harder to write it out in a transparant manner. After all you need the technicalities of your math object (say a simplex or a parallelogram) in order to show something is true for that mathematical object or shape.

The very simple proof just skips that all: It works for all shapes as long as they are flat. So it does not matter if in three dimensional real space you do these projections for a triangle, a square, a circle, a circle with an elleptical hole in it and so on and so on. So to focus the mind you can think of 3D space with some plane in it and on that plane is some kind of shape with a finite two dimensional area. If you project that on the three coordinate planes, that is the

I only wrote down the 3D version but you can do this in all dimensions. The only thing you must take in account is that you make your projections along just one coordinate axis. So in the seven dimensional real space you will have 7 of these projections that each are 6 dimensional…

This post is four pictures long, I did not include a picture explaining what those angles alpha and theta are inside one rectangular triangle. Shame on me for being lazy. Have fun reading it.

So all in all we can conclude the next: You can have any shape with a finite area and as long as it is flat it fits in a plane. And if that plane gets projeted on the three coordinate planes, the projected shapes will always obey the three dimensional theorem of Pythagoras.

Ok, thanks for your attention and although this inner product kind of proof is utterly simple, it still has some cute value to it.

The second pdf is from myself, likely I wrote it in 2012 because a proof of a more general version of the theorem of Pythagoras was the first math text I wrote again after many years. After that at the end of 2012 I began my investigations into the three dimensional complex numbers again and as such this website was needed in 2015.

Anyway I selected 3 details from these two pdf’s that I consider beautiful math ideas where of course I skip a definition of what ‘beautiful’ is. After all the property ‘mathematically beautiful’ is not a mathematical object but more a feeling in your brain.

Let me start with four pictures where I look into those 3 selected details, after that I will hang the two pdf texts into this post.

The first pdf is from Charles Frohman. May be you must download it first before you can read it, I should gain more experience with this because the pdf format is such a hyper modern development…;)

The first text is from 2010:

At last my old text from 2012:

(Later I saw there were some old notes at the end of my old pdf, you can neglect that, it has nothing to do with the Pythagoras stuff.)

There is little use in comparing these texts, I only wanted to make a proof that uses natural induction so I could prove the theorem in all dimensions given the fact we have a proof (many proofs infact) for the theorem of Pythagoras with a rectangular triangle. Charles his text is more broader and the main piece is the proof for that determinant version of the theorem of Pythagoras.

At last a remark about the second detail of mathematical beauty: Charles gave the example of a parallellogram where the square of the area equals the sum of squares of the three projections on the three coordinate planes. I think you can take any shape, a square of a circle it does not matter. It only matters it is a flat thing in 3D space. After I found that within the hour I had a proof for the general setting of this problem in higher dimensional real space, may be this is for some future post.

For the time being let us split and go our own ways & thanks for the attention of reading this.

This is based on the assumption that at low energies two colliding electrons will not touch each other. It seems that this kind of behaviour keeps on going on at high collision energies.

Another detail that is interesting are questions about the size of an electron. In this video a number like smaller then 10 to the minus 18 cm is mentioned. Since I think that if it is true that electrons are ‘tiny magnets’ so they are bipolar in the magnetic sense, they cannot be accelerated by magnetic fields in a significant manner.

The physics professors think that a non constant magnetic field can accelerate an electron. Non constant can mean it varies over time, varies over space or both. If you apply a magnetic field to such a bipolar electrons, say if the north pole of the electron is repelled by that, the other side of the electron must feel an attractive force. The difference should account for the acceleration of the electron.

Lets do an easy calculation: Using the radius being this 10^-18 cm or 10^-20m, the density of an electron is about 2.2 times 10^29 kg per cubic meter of ‘electron stuff’. Suppose we have a ball shaped electron with a volume of one cubic meter thus it has a mass of 2.2 times 10^29 kg, it’s radius is about 60 cm.

So the diameter of our superlarge electron is about 120 cm and it has this rediculous huge mass. Do you think you can accelerate this thing with a magnetic field that has some nonzero gradient?

There are so many problems with the model of the electron being a magnetic dipole. Why should electrons ‘anti align’ themselves with an applied magnetic field? That is strange because they gain potential energy with that. That is just as strange and crazy as the next example:

You have a bunch of stones, one by one you grab them and hold them still in place. You let them loose. Some fall to the ground, the others fly up.

This never happens because nature has this tendency to lower the potential energy.

Another problem is that it is known that the electron pair is magnetically neutral. The ‘explanation’ is that the two electrons have opposite spin and ‘therefore’ cancel each other out. That is a stupid explanation because if it is true that the electron is a bipolar magnetic thing it should be magnetically neutral to begin with.

**The second video** is from Brian Keating, Brian is an experimental physics guy. This is one of those ‘Where are the magnetic monopoles’ videos that people who like to demenstrate they are dumb post on Youtube and the likes. It makes me wonder: **What the hell are they doing with our taxpayer money?** The concept of a magnetic monopole is just plain fucking stupid; it is a particle with no electric charge but only one of the two possible magnetic charges.

Why is this fucking stupid? Just look at the electron: if their fairy tales are true, the electron is an electric monopole and a magnetic dipole. If I would look to some dual version of an electron and have drunk lots of beer I would propose a **particle that is an electric dipole but also a magnetic monopole**.

You never hear those physics people talk about that, it is always that stupid talk of where are the magnetic monopoles or if there is just one magnetic monopole in every galaxy it is ok. What I consider the weirdest thing that if you advertise the electron as a magnetic dipole, should you not give a tiny bit of experimental validation for this? But no, Brian has no time for such considerations.

So where are all the magnetic monopoles? If my view on magnetism is correct they are in every electron pair that holds your body together.

Ok, lets leave this nonsense behind. Don’t forget people like this might be infuential but they are too stupid to understand only the smallest part of say three dimensional complex numbers.

End of this post.

And once more: The math professors are doing it wrong when it comes to finding the conjugate for

The setup of this post is as next:

1) Explaining (once more) how to find the conjugate.

2) Calculating the two sphere-cone equations.

3) The solution of these S-C equations is the exponential circle that is,

4) parametrisized by three so called coordinate functions that we

5) substitute into both S-C equations in order to get

6) just one equation.

Basically this says that the complex and circular multiplication on our beloved three dimensional space are ‘very similar’. Just like that old problem of solving

Anyway this post is six pictures long.

That was it for this post on my beloved three dimensional complex numbers.

]]>You can do a lot with exponential circles and curves. A very basic thing is making new de Moivre identities. From a historical point of view these are important because the original de Moivre identity predates the first exponential circle from Euler by about 50 years. In that sense new de Moivre identities are very seldom so you might expect some interest of the professional math community…

Come on, give me a break, professional math professors do a lot of stuff but paying attention to new de Moivre identities is not among what they do. But that is well known so lets move on to the four pictures of our update. After that I will show you the two video’s.

Let us proceed with the two video’s. Below you see a picture from the first video that is about 3D circular numbers and of course the conjugate is done wrong because math folks can only do that detail wrong:

By all standards the above video is very good. Ok the conjugate is not correct and may be the logarithm is handled very sloppy because a good log is also a way to craft exponential circles. But hey: after 30 years I have learned not to complain that much…

The next video is from Michael Penn. He has lots of videos out and if you watch them you might think there is nothing wrong with that guy. And yes most of the time there is nothing wrong with him until he starts doing all kinds of algebra’s and of course doing the conjugate thing wrong. Michael is doing only two dimensional albebra’s in the next video but if you deviate from the complex plane very soon you must use the conjugate as it is supposed to be: **The upper row of the matrix representatio**n.

Here a screen shot with the content of the crimes commited:

Here is his vid:

Ok, that was it for this appendix to the previous post.

]]>On the other hand one of the most famous expressions in math is and stays the exponential circle in the complex plane.

That stuff like

Anyway a couple of days back at the end of a long day I typed in a search phrase in a website with the cute name duckduckgo.com. Sometimes I check if websites like that track this very website and I just searched for “3D complex numbers”. The first picture that emerged was indeed from this website and it was from the year 2017. I looked at it and yes deep in my brain it said I had seen it before but what was it about? Well it was the product of two coordinate functions of the exponential circle in 3D. It is a very cute graph, you can compare it to say the product of the sine and cosine function in the complex plane.

So I want to avoid repeating all that has been written in the past of this website but why not one more post about the 3D exponential circles?

In the end I decided to show you how likely one of those deeply incompetent “professional” math professors would handle the concept of conjugation. Of course one hundred % of these idiots and imbeciles would do it as “This is just a flip in the real axis or in the

But let’s post the six pictures, may I will add an addendum in a few days, may be not. Here we go:

Ok, may be in will write one more appendix about how these kind of coordinate functions of exponential circles give rise to also new de Moivre identities. That is of interest because the original de Moivre identity predates the Euler exponential circle by about 50 years.

Yet once more: Likely there is just nothing that will wake up the branch of overpaid weirdo’s known as the math professors…

So for today & late at night that was it.

Thanks for your attention.

So in a permanent magnet the electrons must align in order to be attractive to each other while in chemistry the opposite must happen. My dear reader

What I self consider a strange thing is that people from the physics community never ever themselves say that all their views on magnetism are often not logical. Are they really that stupid or do they self censor in order not to look stupid?

Anyway five years back in the year 2017 I was studying a new way of making computer memory by IBM: so called racetrack memory in nano wires. I was highly puzzled by that because one of the main researchers said that you cannot move the domain walls of magnetic domains with magnetic fields. You could move the domains themselves but not the walls and I was as puzzled as can be. Yet that same day I found a possible answer: the magnetic domains of say iron can be moved by magnetic fields because they have a surplus of a particular kind of electrons. So two magnetic domains separated by a domain wall must have opposite magnetic charges. In the next picture you get the idea of what IBM tried to do:

It was a cute idea but IBM had to give up on it because they did not use insights that are logical but kept on hanging to the tiny magnet model.

So in the long pdf that is squarely based on the official version of electron spin (the tiny magnet model) has all kinds of flaws in it. For example in the next picture that all does not pan out because those small arrows are not there in reality if electrons carry magnetic charge just like they carry electric charge:

Ok for me it is an experiment to try include a pdf file, if it fails I will hang this pdf in the pdf directory of the other website and link to that file.

Lets give it a try:

I leave it this way and do not try to make the pdf visible. After all if you are interested in stuff like this you must download it anyway because it is a few hundred pages long. And it is a funny read so now and then, for example yesterday I came across a section where they took the outer product of two (vector) electron spins and I just wonder **WHY**?

Ok, let me push the button named **Publish** and say salut to my readers.

There are many interesting aspects to this problem. For example take a piece of paper and a pencil and draw the graph of the exponential function and the identity function. It is now very easy for every point on your graph of the exponential function to find the graph of the double composition

It is very well possible that this problem is solved in the theory of dynamical systems. If memory serves we once had a lesson in when a family of functions could be interpolated but that was 30 years back and what I want is explicit expressions and formula’s and not only a vague existencial proof without a way to find an explicit answer.

Back in time before the logarithm was invented, the people of those long lost centuries had a similar problem understanding what exponential behaviour was. And you can go a long way in understanding exponential behaviour but say for yourself; without knowledge of the logarithm that kind of knowledge is far from being optimized.

In this post I only talk about the composition square root but of course any

This post is only two pictures long so here we go:

And it is also the end of this post. Give it a thought and if you are able to make some inroads on this that would be great. But all in all I think we do not have the math tools to crack this old old problem.

See you around in the next post.

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.

But it was the magnetism video that made me look up the average size of the so called magnetic domains and that was when I found that PERFECT gif. So I cannot say it was all a waste of time, that perfect gif is made with something that is named a Kerr microscope and with such a device you can make magnetic domains visible.

Years ago, if memory serves it was Feb 2017, I was studying so called ‘racetrack memory’ that was under development by IBM. That IBM project failed because they kept on hanging to electrons being ‘tiny magnets’ with two magnetic poles, because that is likely not true all their work failed. Anyway they came up with the fact that you cannot move magnetic domains with magnetic fields and I totally freaked out. Late at night I realized that within my broader development of understanding magnetism at the electron level, the IBM findings were logical if magnetic domains in say Iron or so, always have a surplus of either north pole monopole electrons or south pole monopole electrons. Domain walls separate the two kinds of magnetic domains. Itis a pity that about five years back I never heard of those Kerr microscopes.

Again I want to highlight that I do not want to convince anybody that electrons are the long sought magnetic monopoles. I have done that for six or seven years and it was only in this year 2021 that I arrived at the conclusion that physics professors are just as stupid as the average math professor. It is a pile of garbage so it is not much of a miracle that six or seven years of trying to apply logic did not work at all. So from this year on going into the future the physics professors have the same status as the math professors: A pile of rotten garbage that you must avoid at all times at all costs needed.

After having said that, this post is five pictures long where I comment on the two horrible videos. Below that I will post the two videos so you can see for yourself (or may be you want to see them first). And at the end you can see that perfect gif where magnetic domains change in size due to the application of an external magnetic field. Also back in 2017 I more or less figured out how magnetic domains will change if you approach a piece of iron with a permanent magnet. What you see in the gif is more or less precisely that: Some domains grow while domains next to that shrink.

Ok, here we go:

Now we can go to the first video, the math one:

I found the magnetics video by doing an internet seach on ‘The Stern-Gerlach experiment for iron’. It is disappointing that almost no significant results are there. Some of stuff out of the 2030-ties of the last century but that was all behind pay walls. Very high in the rankings came the next video that uses iron filings to mimic or imitate the Stern-Gerlach experiment. The video guy should have used magnets on only one side, if that resulted into attraction & repulsion of the iron filings he would have gotten a standing ovation from me. Without any insult; the way he executed this experiment is a true disaster only showing he does not understand why the SG experiment is so important.

And by the way: If my idea of electrons being magnetic monopoles is in fact correct, you do not have to use inhomogeneous magnetic fields. Everything will do; even the most constant magnetic field in space and in time will do. But again after so many years of talking to deaf ears from stupid physics people, I have lost my desire to convince anybody any longer..

Ok, I have never hung any animated gif into this WordPress website so let’s check it out if it works properly:

I found this animated gif in a wiki: Magnetic domain.

That was it for this post. Thanks for your attention.