The video is very interestig because it compares the earth versus some of the other planets in our solar system. It is not much of a secret that I think electrons carry magnetic charge and that is what makes them ‘move along magnetic field lines’. If electrons carry magnetic charge means they are magnetic monopoles and not the magnetic dipoles that is more or less included in the standard model of particle physics.

As usual we only apply the thing called ‘logic’ and we do not get emotional because the academic field does not respond year in year out. Ok ok, I am human too so let me allow a tiny amount of emotion: All those physics professors that think electrons are magnetic dipoles are just like math professors: incompetent to the bone because of groupthink. In the case of understanding magnetism the groupthink is easy to explain: it is the Gauss law for magnetism (magnetic monopoles do not exist) while there is zero experimental evidence for that Gauss law.

Why do particles with non zero spin move along magnetic field lines? I think that is because they carry net magnetic charge. The weirdo’s from the universities think that it is done because of the gradient of planetary magnetic fields. Of course it is never backed up by some calculations because: 1) Planetary magnetic fields are rather weak in the first place and because of that: 2) The gradient of such fields is completely neglectible. You see once more: All you need is a bit of the thing known as ‘logic’. Why the university people do not want to apply the thing known as logic is unknown to me. In my view it is far better to use logical reasoning if you want to make a bit of progress in understanding the stuff out there in the universe; but after talking like that for the last six years or so it has become clear university people just don’t want to think ‘logical’.

Let’s move on, why waste time on people that are mentally handicapped anyway? In the next picture you see a perfect accumulation of how not understanding electrons in a magnetic field leads to all kinds of weird representations of what actually is going on. Yes the earth magnetic fields acts as a ‘shield’ for the solar wind, but it is not that the particles that make up the solar wind ‘bounce off’ that shield. The next representation is rather retarded but that is what you get when humans just hold on the the Gauss law and hold on and hold on & just want to be retarded idiots.

Moving on, the video mentions computer simulations. But if you craft computer simulations where the electron is a magnetic dipole while the thing known as ‘logic’ say they cannot be magnetic dipoles, how can these computer models be a realistic representation of what is actually going on? Of course those computer models can’t do that, so these computer models must have some feature inside them that makes particles with non zero spin accelerate in magnetic fields.

Moving on, those computer models suggest leakage from the Mars atmosphere in the past if it had a rather weak planetary magnetic field. The reason I write this post is that they arrived at the conclusion that a weak planetary magnetic field leads to a situation where the magnetic field lines are not closed. They originate at the planet but never return to it.

Talking about idiots: That detail alone violates the Gauss law for magnetism (all magnetic fields always close in upon themselves).

But the insight of how a weak magnetic field could lead to more planetary atmosphere loss is brilliant.

All of my life I was too stupid to make it up:

At last here is the video that aroused my attention:

Let me close this post with two more ‘things’.**Thing 1**: Almost by definition if the electron is a magnetic dipole it is neutral when it comes to magnetism. Just atomic hydrogen has one proton and one electron and as such it is neutral under the influence of electric fields. Let’s do a thought experiment: Suppose a planet as a whole has a strong electric charge either positive or negative of say a few million volts. Furthermore this planet has an atmosphere of atomic hydrogen (ok that is not very realistic but anyway). Now does the electric potential cause a dramatic atmospheric loss of the atomic hydrogen that is neutral in electric fields?

No of course not, because the atomic hydrogen is electrically neutral it has no net force acting on it. Hence a planetary size eletric potential should not lead to a loss of non-ionic atoms.**Thing 2**: They once tried to figure out if the neutron was an electric dipole (or may be an electric tripole because after all the neutron seems to be composed of 3 quarks). They failed hard. But if we compare electron size to neutron size, likely the electron is orders of magnitude smaller than the neutron so why should the eletron not be neutral when it comes to magnetism?

Ok ok, the goodie old Stern Gerlach experiment says that electron is not neutral under magnetic fields hence **elementary logic** says the electron cannot be a magnetic dipole. As such all electrons must be magnetic monopoles…

As you see, when doing ‘scientific stuff’ it is always better to use logic and not silly emotions. Of course I get irritated nothing changes but why get overly emotional? And don’t forget: suppose somebody has done the perfect experiment that indeed validates electrons cannot be magnetic dipoles. Well such a person will be at the end of his or her career because no ‘respectable scientific journal’ will post such a result. That’s the way it is, so I don’t care about those journals.

Let me leave it with that. See you in the next post.

]]>In this post I would like to share a simple experiment that every body can do, it does not blow off one of your arms it is totally safe, and shows that those Pauli matrices are a very weird pipe dream. Here we go:

The official explanation of the Stren Gerlach experiment always contains the next: If electron spin is measured into a particular direction, say the vertical direction, if later you measure it again in a direction perpendicular on the vertical once more it has 50/50 probability. So if it is measured vertically and say it was spin up, if you after that measure it in say a horzontal manner once more the beam should split according to the 50/50 rule.

Ok, the above sound like highly IQ level based on lots of repeated laboratorium experiments. Or not? And what is a measurement? A measurement is simply the application of a magnetic field and look what the electron does; does it go this way or that way?

Electron pairs are always made up of electrons having opposite spins, in chemistry a pair of equal spins is named a non-bondig or an anti-bonding pair. Chemical bonds based on electron pairs cannot form if the electrons have the same spin.

Now grab a strong magnet, say one of those strong neodymium magnets and place it next to your arm. Quickly turn the magnet 90 degrees or turn your arm 90 degrees, what does happen? Of course ‘nothing happens’ but if electron spin would follow that 50/50 rule, in that case 50% of your electron pairs would become an anti bonding pair. As such your flesh and bones whould fly apart…

Now does that happen? Nope njet & nada. As far as I know it has never been observed that only one electron pair became an anti-bonding pair by a simply change of some applied external magnetic field…

As far as I know the above is the most easy day to day experiment that you can do in order to show that electrons simply do not change spin when a different magnetic field is applied…

I have been saying this for over five years but as usual when it comes to university people there is not much of a response. In that regard physics is just like the science of math: It has lost the self cleaning mechanisms that worked in the past but now in 2020 and further those self cleaning mechanisms do not work anymore. It is just nothing. It is just a bunch of people from blah blah land. So let’s wait & see if one of those ‘electron idiots’ will get the Nobel prize tomorrow.

Luckily I have a brain for myself. I am not claiming I am very smart, ok may be compared to other humans I do well but on the scale of things like understanding the universe I am rather humble. I know 24/7 that a human brain is a low IQ thing, but just like all other monkeys it is the only thing we have.

Very seldom the human brain flares up with a more or less bright idea that simplifies a lot of stuff. A long time ago I wanted to understand the general theorem of Pythagoras, I knew of some kind of proof but I did not understand that proof. It used matrices and indeed the proof worked towards an end conclusion but it was not written down in a transparent way and I just could not grasp what the fundamental idea’s were.

So I made a proof for myself, after all inside math the general theorem of Pythagoras is more or less the most imporatant theorem there is. I found a way to use natural induction. When using natural induction you must first prove that ‘something’ is true for some value for *n*, say *n = 2* for the two dimensional theorem of Pythagoras. You must also prove that if it holds for a particular value of *n*, it is also true for* n + 1*. That is a rather powerful way to prove some kind of statement, like the general theorem of Pythagoras, holds for all *n* that is holds in all dimensions.

I crafted a few pictures about my old work, here they are.

It is from March 2018 when I wrote down the ‘inverse’ theorem of Pythagoras:

And from March 2017 when I wrote the last piece into the general theorem of Pythagoras:

Ok, let me leave it with that and in about 10 hours of time we can observe if another ‘electron idiot’ will win the 2020 Nobel prize in the science of physics. Till a future post my dear reader. Live well and think well.

]]>Video 1) From the Youtube channel Scishow a video with the title

‘Satellite Squad Goals: The Cluster Mission to the Magnetic Field’.

For me that video contains relatively much completely new stuff, the fact that there are 4 satellites out there constantly monitoring the earth magnetic field was unknown to me.

And the presenter of the video claims that after the so called ‘magnetic reconnection’ the charged particles from the solar wind slam into the north & south pole of the earth with a staggering 10 thousand km/sec. I did not know it was that fast…

The official explanation for the acceleration of for example single electrons is that you must have an inhomogeneous magnetic field. After all these folks think that electrons have two magnetic poles and if the electron goes through a magnetic field that varies in space the two forces on the north and south pole of the electron do not cancel out and there is a net force responsible for the acceleration. There is only one problem: they simply multiply the electron magnetic moment against the gradient of the magnetic field and voila: that’s it. But if the acceleration is explained as a difference in opposing forces, should you not take into consideration the size of the electron? Yes of course, but since physics professors are so terribly smart why don’t they do this? Well if you take the size of the electron into your calculations, there is no acceleration or better it is basically zero.

Now years ago I tried to estimate how stong a magnetic field had to be to accelerate one of those dipole electrons with a acceleration of only 1 meter per second squared. If memory serves I used an ‘electron size’ of 10 to the power -15 meter (in reality it is even much smaller) and again if memory serves you needed magnetic fields with a gradient of over 100 thousand Tesla per meter.

And if you think about that estimation it makes a lot of sense: electrons are very small and as such have an extreme density given their size and mass. Say it is in the order of the density of a neutron star. And if you try something with the density of a neutron star to accelerate with the difference of a magnetic field, likely you won’t go far…

Ok, suppose for the moment that the electrons are the long sought magnetic monopoles. So they are not magnetic dipoles but the electrons themselves are magnetic monopoles just like they are electric monopoles.

Now look at the picture below: it is about when the magnetic reconnetion just closed. Just before the closing along the magnetic field lines emergin from the earth north & south pole, the particles were expelled because they carry the wrong magnetic charge. But when reconnection takes place, the particles that were expelled by say the earth south pole find themselves back on a trajectory going to the earth north pole. And as such they will get accelerated into that direction.

Yet a couple of years ago when I published those estimations that show you need crazy gradients for all that shit to be true, of course nobody reacted. All those university professors in physics, when you tell them that extra ordinary claims like the electron being a magnetic dipole also needs extra ordinary proof, all of a sudden they are deaf deaf deaf.

These people they don’t have any experimental proof that the electron is a magnetic dipole. And worst of all: They don’t even think about it…

Finally, here is the SciShow video:

Video 2) From the Standupmath guy a video about the perimeter of an ellipse. Weirdly enough it is not possible to find a more or less simple expression for the perimeter of an ellipse. Of course a long long time ago I tried to find an expression myself but using the standard stuff like arc length brings very fast a lot of headache. With the present day of math tools it is completely not possible to derive a good expression for the perimeter of an ellipse.

What I did not know is that there is a world of approximation stuff out there for estimation such ellipse perimeters. And of course in itself this has it’s own logic: after all an ellipse is more or less completely defined by saying what it’s two half axes *a* and *b* are. You can always fix one of those axis to 1 say *b* = 1 and study the perimeter problem as a function of the variable *a*. You do some curve estimation, you drink a few pints of beer and later when you are sober again you drink some green tea.

And you conclude some curve estimation is relatively good but that all in all the ellipse perimeter problem is just too large for our human brains that in general are not good at doing math.

There is only one exeception; Ramanujan.

In the next picture you see one of those Ramanujan approximations and once more you see how the human mind should work if we were living in a better world:

The video is here, 21 minutes long but worth the time:

Ok, that was it for this post. Think well, live healty and try to make some bio fuel from the basic ingredient known as ‘math professor’.

In that case we will find ourselves back in a better world, or not?

I always make my pictures with an old graphics program named Picture Publisher 10. It is so old that on most windows 7 and windows 10 it does not run but it has all kinds of features that even the modern expensive graphics programs simply still don’t have. Silently I was hoping that I could use GIMP for my math texts and yes that could be done but in that case I have to use old background pictures forever. Or I have to craft a ‘new style’ for making the background in the math pictures that can last at least one decade.

But let’s not nag at what GIMP cannot do, if you install just one large addon you have about 500 filters extra and my old program PP10 comes from an era when the word ‘addon’ was not a word used ever. Before we jump to the math, let me show you a nice picture you can make with the tiling filter inside GIMP. It is about my total bicycle distance since I bought this bicycle computer, it says 77 thousand km so the Tour the France racers can suck a tip on that:

Ok, let us look at the math of this post. This is part 22 in the introduction to the 4D complex numbers. The 4D complex numbers have three imaginary units, *l, l^2* and* l^3*. And the stuff that makes it ‘complex’ is the fact that* l^4* = -1, you can compare that to the complex plane where the square of the imaginary unit equals -1.

On the complex plane, if you know what the logarithm of *i* is, you can use that to find the exponential circle also known as the complex exponential. This is what the number tau always is in all kinds of spaces: It is always the logarithm of the first imaginary unit that has a determinant of +1. In this post we will calculate the eigenvalues of this important number tau. That will be done with two methods. In the first method we simply use the eigenvalue functions, plug in the number tau and voila: out come the four eigenvalues. In the second method we first calculate the four eigenvalues of the imaginary unit* l *and ‘simply’ take the logarithm of those four eigenvalues.

It is not much of a secret that my style of work is rather sloppy, I never order my work in theorems, lemma’s or corrolaries. It is not only that such an approach if too much a straight jacket for me, it also frees me from a lot of planning. I simply take some subject, like in this case the eigenvalues of the number tau and start working on explaining that. While writing that out there always comes more stuff around that I could include yes or no. In this post what came around was that only after writing down the four eigenvalues I realized that you can use them to prove that the exponential curve (the 4D complex exponential) has a determinant of 1 for all points on that curve. That was an important result or an important idea so I included it because that makes proving that the determinant is 1 much more easy.

Now a few posts back with that video from that German physics guy Alexander Unzicker I said that he (and of course all other physics professionals) could always use the 4D complex exponential curve for the ‘phase shifts’ that those physics people always do. But for doing such 4D ‘phase shifts’ or unitary transformations in general, you need of course some kind of proof that determinant values are always +1. Well Alexander, likely you will never read this post but below you can find that very proof.

The previous post was from the end of August and now I think about it: Have I done **so little math** during the **last four weeks**? Yes there were no results simply left out, it was only penning down these eigenvalues of tau and the idea you can use these eigenvalues for proving the 4D exponential curve always has a determinant of 1. It is amazing that GIMP can hinder the creation of fresh math…

The math pictures are seven in number, all in the usual size of 550×775 pixels. I hope you like it and see you in the next post.

So these are the four eigenvalues of the number tau and based on that the four eigenvalues of the 4D complex exponential for a values of time.

That’s it for this post. See you in a future post.

Added on 27 Sept 2020: This proceeds the two pictures made with GIMP that started this post. I just made the whole stuff on a cube (actually it is a beam because the starting picture is not a square). It is amazing how good such filters in GIMP are:

Ok, you are now at the real ending of this post.

]]>When I was reading back a few of my own old writings like part 20 from 02 Feb it struck me that those non-invertible numbers are all just linear combinations of the so called imitators of

Another interesting detail is that in the 4D complex number system the set on non-invertible numbers consists of just two lines that are perpendicular to each other. That is very different from the 3D situation where the set of non-invertibles is always a plane combined with a perpendicular line through zero. The reason that in 4D complex space the set is so small lies of course in the matrix representation and the determinant. On the 4D complex space the determinant is non-negative, just like the determinant is non-negative on the complex plane. Every 4D complex number has four eigenvalues and they come in conjugate pairs, so the product of these four eigenvalues gives the determinant hence the determinant cannot be a negative real number.

And say for yourself: aren’t the eigenvalue functions a very handy thing? If you want to find the eigenvalues of let’s say the 4D complex number

This post is seven pictures long, as usual in the 550×775 pixel size.

Ok, that was it for this post. Till updates my dear reader.

]]>Another example: why is glass transparent? Well the photons in the visible range have energies that do not interfere with the electrons in the glass. That is why these photons simply pass through and we can use stuff like glass for the windows in our homes.

So an element, say atomic hydrogen, is capable of emitting light at particular frequencies and at the same time that atomic hydrogen can absorb the same frequencies.

Now we go to the famous 21 cm wavelength: the standard explanation for the source of this em radiation is that in atomic hydrogen you find that both the proton and the electron have the same spin. The spins are aligned so to say and that state has a tiny bit more energy compared to the situation where the electron spins are not aligned. If the spins are aligned (that can be both up or both a down spin) there is a tiny probability that the electron spin flips. That releases a photon of 21 cm wavelength. It is never explained as why it is the electron spin that should flip, after all if the proton spin would flip this should give rise to the emission of a 21 cm photon also…

It is not much of a secret that I think that electrons are not magnetic dipoles but magnetic monopoles. Electrons get accelerated into the direction of the applied magnetic field, but if electrons were magnetic dipoles they would be neutral to external magnetic fields. Ok ok, professional physics professors come up with non homeogenous magnetic fields that should do the acceleration but if I do an easy estimate I find crazy gradients are needed. Something like 100 thousand Tesla per meter or so. It is important to remark that all those people doing the blah blah thing about inhomeogenous magnetic fields only do the blah blah thing: they never show a calculation that supports the blah blah. And yes, they also have a Hamiltonian kind of thing, but in the Hamiltonian the size of the electron is not incorperated. But the smaller in size a magnetic dipole is, the less it will get accelerated by such magnetic fields.

Another example that is hard to believe is the deflection of the solar wind by the earth magnetic field. Not only is the earth magnetic field very weak out there in space, it is hard to believe it has a serious gradient there in outer space. It must be very constant. Yet the solar wind gets deflected by the weak magnetic field of the earth. In my view this can only be done if electrons and protons are magnetic monopoles.

Here is an old ‘picture of the day’ from December 18, 1996 ‘A sky full of hydrogen’.

The spin flip that ’causes’ the 21 cm radiation seems to be a seldom thing; about once in 10 million years. And it is always mentioned that it is spontaneous. In the next picture from a wiki you see how this supposedly works. Link: Hydrogen line https://en.wikipedia.org/wiki/Hydrogen_line

Ok, what is my version of events? Very simple: Suppose there is an hydrogen atom in outer space that has it’s proton and electron carry the same magnetic charge. So both proton and electron have a north or a south magnetic charge. Suppose it is a north hydrogen atom. This atom simply bumps into a single electron having the south charge. Under the right conditions (bump not to slow and not to fast) this leads to a hydrogen atom with a north proton and a south electron. The north electron gets ejected because like magnetic charges repel. And the radiation emmited simply has wavelength of about 21 cm.

The 21 cm em radiation is also used in atomic clocks like those in the EU Galileo global satellite positioning system. In an apparatus named maser the hydrogen atoms get separated due to their magnetic charge and released in a resonance chamber. The point I want to make is that such a resonance chamber must have a very special coating on the inside otherwise the hydrogen atoms all ‘spin flip’ much to fast. So the coating must be a material without any unpaired electrons in it’s outer shell and of course it can’t be a metal because metals often have an electron sea that just sloshes around. From the ESA, here is a scetch of a hydrogen maser:

This was more or less what I had to say on this magnetic subject. If you start thinking about if it is possible that electrons are magnetic dipoles always you will find it is not logical. For example if it is true that the electron goes round the proton in atomic hydrogen, why the hell would the stuff get anti-aligned? If magnetism is just a vector pointing into some direction, if the electron goes round and round the up & down state would lead to precisely the same hydrogen atom…

All you need is a bit of logic.

Ok, we are at the end of this post. All I want to share with you is a teaser picture for the next post. The math of the next post is more or less finished but I still have to turn the stuff into the standard jpg pictures. The next post is more or less Part 21 into the basic introduction to the 4D complex numbers. I stopped those intro files back in Feb 2019, at this point in time I do not know if I will proceed but at least in a couple of days I will add post number 21 into that 4D number system. It is mainly about the so called ‘imitators of *i*‘, these imitators mimic the number i from the complex plane. Here is the teaser picture:

Here is an internal link to Part 20 intro to the basics of the 4D complex numbers:

That was it, thanks for your attention & in a few days the new post will be ready to publish. So see you around my dear reader.

]]>Now before I started brewing beer I often made wine. That was a hobby that started when I was a student. In the past it was much more easy to buy fruit juice that was more or less unprocessed, like 100% grape juice for 50 cents a liter. And with some extra sugar and of course yeast in a relatively short time you have your fresh batch of 20 liters wine. And somewhere from the back of my mind it came floating above that I had seen such irregularities arising from wine bottles if you stack them horizontal. But I never knew it had a solution like shown in the video.

Video title: The secret of the 7th row – visually explained

The next video is from Alexander Unzicker, the vid is only five minutes long. First I want to remark that I like Alexander a lot because he more or less tries to attack the entire standard model of physics. That not only is a giant task but you also must have some alternative that is better. For example when I talk or write about electrons not being magnetic dipoles, I never end in some shouting match but I just apply logic.

Let me apply some logic: In the Stern Gerlach experiment a beam of silver atoms is split in two by an inhomeogenous magnetic field. The magnetic field is stronger at one side and weaker at the other. One of the beams goes to the stronger side while the other goes to the weaker side of the applied magnetic field. But the logical consequence of this is that the stream silver atoms going to the weaker side **gains potential energy**. This is not logical. If you go outside and you throw a few stones horizontal, they always will fall to the earth and there is the lowest potential energy. The stones never fly up and accelerate until they are in space. In order to gain the logical point it is enough to assume that electrons are magnetic monopoles and that is what makes one half of the beam of silver atoms go to the weaker side of the applied magnetic field. If the electrons come in two varieties, either monopole north or monopole south, both streams do what the rest of nature does: striving for the lowest energy state.

Talking about energy states: Did you know that the brain of math professors is just **always** in the **lowest energy state possible**?

But back to the video: Alexander is always stating that often when progress is made in physics, all in all things become more easy to understand. That also goes for electrons, all that stuff about electrons being magnetic dipoles is just very hard to understand; why do they gain potential energy?

In his video Alexander gives a bad space as example where a so called three sphere is located. On the quaternions you cannot differentiate nor integrate, they are handy when it comes to rotations but that’s more or less all there is. So Alexander I don’t think you will make much progress in physics if you start to study the quaternions. And by the way don’t all physics people get exited when they can talk about ‘phase shifts’? They use it all the time and explain a wide variety of things with it. I lately observed Sabine Hossenfelder explaining the downbreak of quantum super positions into the pure ground states (the decoherence) as done by a bunch of phase shifts that make all probabilities of super positions go to zero. Well, the 4D complex numbers have a so called exponential curve and voila; with that thing you can phase shift your stuff anyway you want…

Video title: Simplicity in Physics and How I became a Mathematician

Yesterday I started working on the next post. It is all not extremely difficult but ha ha ha may be I over estimate my average reader. After all it is about the non-invertible numbers in the space of four dimensional complex numbers. The stuff that physics and math professors could not find for centuries… So you will never hear people like Alexander Unzicker talk about stuff like that, they only talk in easy to understand common places like the quaternions. And when I come along with my period of now about 18 years completely jobless, of course I understand the high lords of all the universities have more important things to do. All these professors are just soooooo important, they truly cannot react on social slime that is unemployed for decades. I understand that, but I also understand that if such high ranked people try to advance physics with the study of quaternions, the likelihood of success is infinetisimal small…

Anyway, here is a teaser picture for an easy to understand problem: if two squares are equal, say* A^2 = B^2*, does that always mean that either *A = B *or that *A = -B*?

In another development for decades I always avoided portraits and photo’s of myself on the internet at all costs. Of course after 911 that was the most wise strategy: you stay online but nobody know how you look. But over the years this strategy has completely eroded, if for example I just take a walk at some silly beach about 30 km away people clearly recognize me. So I more or less surrender, likely I will still try to prevent my head being on some glossy and contacts with journalists in general will also be avoided for decades to come.

But in the present times why not post a selfportrait with a mask?

The upper half of the picture below is modified in the ‘The Scream’ style and the lower half is modified with something known as ‘vertical lines’.

Ok, that was it for this post.

]]>Why do they not reconstruct properly? Well often you need to take the sum or the difference of a 4D complex number

Yet I was not crazy, as far as the reconstruction works it does it more or less as expected only you get only half of the signal reconstructed. That is not that worse but the garbage that enters the reconstruction is what makes this kind of making a 4D Fourier series something that will never have any practical benefit. But again does that mean the glass is half empty?

I remember that a long time ago in something like 1991 or 1992 I had found the product of a 3D complex number X and it’s conjugate X*. My naive idea was that this should only give the unit sphere in 3D space, but this product that also two imaginary components that I considered garbage at the time. Back in the time, it was just before the internet era, I could not know that this ‘garbage’ was actually the equation of a cone. And if you intersect this cone with the unit sphere in the space of 3D complex numbers, you get the 3D exponential circle. So it wasn’t garbage, it was the main prize in 3D complex number theory.

Back in the time in 1991 it was stuped from me to expect the 3D complex numbers would behave ‘just like’ the two dimensional complex numbers from the complex plane. May be in this year 2020 I am making the same mistake again by expecting a 4D kind of Fourier series must behave ‘the same’ as those defined on the real line (the original Joseph Fourier proposal) and the more advanced version from the complex plane.

With the 4D Dirichlet kernel just like with the 2D Dirichlet kernel from the complex plane, you must take the difference of a number and it’s conjugate. In the complex plane this makes the real component zero and this difference is just an imaginary number. The 2D Dirichlet kernel is the quotient of two such imaginary numbers and as such it is always a real number. For the 4D Dirichlet kernel stuff is not that easy but for me it was surprising that you can show relatively easy the 4D Dirichlet kernel has to be a ‘self conjugate’ number. That means* Z = Z** (on the complex plane when you have a number *z* such that *z = z** it means it is a real number).

This post is 8 pictures long, 7 of them have the standard size of 55×775 pixels but I had to make one picture both a bit more broad and higher in order to get the math fitting in it. Ok, let’s upload the math pictures with the stuff around the 4D Dirichlet kernel.

Ok, credits have to go to where that is deserved. I remember that back in the time like in 1990 I found it relatively hard to calculate the 2D Dirichlet kernel. It took me over 15 or 20 minutes but again: that was before the internet era. Yet at present day I was all so simple and why was that? That is because there is a nice Youtube video doing the easy stuff, it is from ‘Flameable math’:

May be at the end I can say the glass is half full because now this reconstruction stuff does not work properly, luckily I do not have to construct the Fejer kernel for 4D complex numbers…

Ok, let’s call it a day and let me end this post.

Yet in this post I define a set of possible signals that I name ‘pure tones’ and these clearly have a 4D Fourier expansion because by definition they have a finite number of non-zero Fourier coefficients. Of course when you only have a finite amount of non-zero coefficients, you don’t have any kind of convergence problem. So for the time the convergence problems are avoided.

In this post, number 154 already, I hope I demenstrated enough that the basis functions used in the definition are all perpendicular to each other. After all that was a nasty hurdle we met when it was tried with just the four coordinate functions of the 4D exponential curve as our basis vectors. So we do not meet that problem again using the exponential curve as a whole. If I denote the exponential curve as

For myself speaking I have no idea at all if crafting a Fourier series like this has any benefits of using just the definition as on the complex plane. After all I only have more or less basic knowledge about the use of Fourier series, so I just don’t know if it is ‘better’ in some regards and ‘worse’ in others.

At last without doubt under my readers there will be a significant fraction that wonder if those 4D complex number system is not some silly form of just the complex plane? After all if that 4D space is based on some imaginary unit *l* with the property that now the fourth power* l^4 = -1*, how can that be different from the complex plane? The answer to that lies in the logarithm of the first imaginary unit* l*. If this 4D space was just some silly extension of the complex plane, this log of *l* should be nicely bound to *i* pi/4 where *i* is of course the imaginary unit from the complex plane. But log*(l)* is the famous number tau because with that you can make the exponential curve *f(t) = e^(*tau *t)*. Basically the main insight is that *i* pi/4 makes the complex exponential go round with a period of four because* i^4 = 1* and the 4D number tau makes the exponential curve go round with a period of 8 because* l^8 = 1*.

This post is six pictures long, all 550×775 pixels in size.

As usual I skipped a lot of stuff. For example, how did Fourier do it? After all at the time all this stuff with inner products was poorly developed or understood. That alone would be a cute post to write about. Yet the line of reasoning offered by Joseph Fourier was truly brilliant.

In case you are lazy or you want to avoid Google tracking you, here is a link to that cute symbolab stuff: symbolab.com

Link used: https://www.symbolab.com/solver/fourier-series-calculator

Ok, that is what I had to say for this tiny math update.

Orginally I planned to define the 4D way of taking a Fourier series in this post, but while writing this post I realized to would become too long. So this post is seven pictures long (all 550×775 pixels) therefore I hope it does not cross the length of your attention span. For myself speaking always when I read some stuff that is ‘too long’ I start scrolling till the end and as such you often miss a lot of important details.

An important feature of exponential curves is that you have those de Moivre identities that come along with all those exponential curves. In this post I did not prove the 4D version of a de Moivre identity for the 4D complex numbers, but I give some numerical evdence. It has to be remarked that when I wrote the 20 posts around the basics of the 4D complex number system, I did not include the de Moivre identity. So that is more or less an ommision. On the other hand it is of course much more important to be able to find the exponential curve in the 4D space that is the basic material needed to write down such a de Moivre identity.

I categorize these posts about the 4D Fourier series also under ‘integration’. Not that I have many fantastic insights about integration but the reason for this category is much more down to earth: You have to perform an integration for every Fourier coefficient you calculate. In case you missed it: this year I finally wrote that post about how to define integration in the 3D complex numbers. Use the search function of this website in order to find that post in case you are interested.

Ok, that is what I had to say. Let us go to the beef of this post number 153 and that is of course the seven pictures that are all hungry for your attention. Here are the seven pictures:

There is an online Fourier series applet as it is named in these years. In the past we did not have applets but only computer programs. Anyway it is important those applets are there so I want to give a big thumbs up to the people that maintain that website! Here is a link to the Fourier thing:

Fourier series calculator. Link used: https://www.symbolab.com/solver/fourier-series-calculator

At last I want to point at the importance of such free websites. It is very good if a society has enough of this kind of ‘free stuff’. For example when in the year 2012 I picked up my study of the 3D complex numbers for me it was very important that there were free online applets for the logarithm of a matrix. Without such a free website I would have taken me many more years to find those exponential circles & curves. Or I would not have found them at all because after all my biggest breakthrough was when I did read that numerical evidence from log* j *on the 3D complex numbers: why are the two imaginary components equal I was just wondering? Later I found how crucial that was: only if the imaginary components are equal the eigenvalues are purely imaginary in the complex plane.

So I am breaking a lance for free websites where you can find good applets (read: computer programs) that help your understanding of math.

Ok, that’s it for this post. In the next post I will finally give that definition and after that I do not have a clue. I still do not have any good proof for convergence of these 4D Fourier series so we’ll see. Till updates.