Impending Nobel prize & recycled Pythagoras theorem & it’s ‘inverse’.

Tomorrow is the new Noble prize in physics out, actually it is already past midnight as I type these words so it is actually today. But anyway. I am very curious if this year 2020 the Nobel prize in physics will once more go to what I name those ‘electron idiots’. An electron idiot is a person that just keeps on telling that electrons are magnetic dipoles because of something retarded like the Pauli matrices. May be idiot is a too harsh word, I think that a lot of that kind of behavior or ideas that can’t be true simply stay inside science because people want to belong to a group. In this case if you tell the official wisdom of electron spin you simply show that you belong to the group of physics people. And because people want to belong to a particular group they often show conformistic behavior, when it comes to that there is very little difference between a science like physics or your run of the mill religion.

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.

Two video’s to kill the time.

Two very different subjects: the earth magnetic field and the standupmath guy has a great video about the perimeter of an ellips.

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?

On a simple yet curious integral identity.

In a pile of paper notes I found back this curious identiy, shall I throw it away or write a small post upon it? Most things I throw away, if I would write posts about everything that comes along this website would be 1340 posts long…

I found it in a video from Presh Talwalkar, Presh runs the video channel ‘Mind your decisions’ on Youtube. There is only a tiny problem: I can’t find back the original video. And since Presh has posted about 518 video’s it would take a long long time to find that video back. So no video included.

Anyway the video started more or less like next: Presh throws in three difficult looking integrals and asks his viewers to take five minutes and try out if they can find the answer. It looks like those integrals are for relatively fresh students and I was just like ‘you can’t ask such integrals for starting students!’ But likely those students had seen this identity and as such those nasty looking integrals could be solved with two fingers in the nose if they just recognized it to be this curious identity…

By the way, Presh his channel has about 1.4 million subscribers. My applause goes to Presh. One point four million is not a bad result, for example the university sponsored channel Numberphile has over three million subscribers so on his own Presh is doing just fine.

So this post is not about 3D numbers, complex or circular but upon this identity. It is only three pictures long so it won’t take much of your precious time. Let’s go:

Of course with symmetric I mean a function that is even with respect to the midpoint of the interval [a, b]. Let’s try if we can post a link to the Presh Youtube channel: Mind your decisions.

Ok, that was it for this post. No idea yet what the next post is about, after all most things I just throw away. So till updates my dear reader.

Teaser picture for the next post.

After a lot of rainy days it was perfect weather today for the time of the year. It has been 3 weeks already since the last post and it is not that I have been doing nothing but the next post still isn’t finished. I told you that we would be looking at a parametrization that solves all 5 equations from the last post. So let me give you the parametrization in the teaser picture below. I also included the parametrization based on the modified Dirichlet kernels, by all standards the discovery of those modified kernels was one of the biggest discoveries in my study of higher dimensional number systems. To be precise: I found the first modified Dirichlet kernel years ago when I studied the 5D complex space.

In the last post I may have sounded a bit emotional but that is not the case. I am more or less one 100% through with the behavior of the so called math professors. They are incompetent to the bone and although that is not an emotional thing, it is that coward behavior that I do not like in those people. No, if it is highly overpaid, utterly incompetent and on top of that day in day out a coward, better show them the middle finger.

After having said that (I wasn’t expecting an invitation anyway) let’s look at the teaser picture because it is amazing stuff. I remember when I wrote down the parametrization for the very first time. At the time I did not know if the cosine thing would work because say for yourself: if you have a periodic function and you make two time lags of it, how likely is it they will form a flat circle in 3D space? But the cosine together with the two time lags does the trick because it is not hard to prove the parametrization lies in the plane with x + y + z = 1.

Ok, here is the cute parametrization for the 3D exponential circle:

I think next week everything is ready so likely I can finally upload the next post. So thanks for your attention and till updates.

On Benford’s law.

Benford’s law is a statistical observation. If I remember the story more or less correct, Benford found at some point in time that those old logarithm tables had pages that were far more worn out compared to others. And it seems that people used those old (but at the time very important) tables much more for numbers starting with small digits like 1 or 2 and much less for high leading digits like 8 or 9. The observation was that the probability of a leading digit of d was given by log(1 + 1/d). I remember that during a train ride to the city of Utrecht about two decades ago I found a very simple distribution that gives the Benford law perfectly for numbers written in the usual base 10. Basically if you use a uniform distribution in the exponent, that more or less always gives rise to some approximation of Benford’s law.

A few weeks back for no reason at all, I did a search on the preprint archive on the subject of Benford’s law and a rather strange article popped up. It is written by Kazifumi Ozawa. Title: Continuous Distributions on $(0, \infty)$ Giving Benford’s Law Exactly.

Yet the way I did the stuff two decades back suggests that it is very hard to make a continuous distribution on such a large interval like 0 to infinity (the positive real line). So I decided to give a bunch of examples that all give Benford’s law perfectly. And I skipped studying the theorem at the end of the article of Kazifumi Ozawa. Yet that theorem of Suihara looks very interesting so may be that is something for a future post although this website is more or less dedicated to higher dimensional complex numbers of course.

The math in this post is more or less as basic as it gets: You need to know what the logarithm with base 10 is and it would be great you need the first course basics of probability theory. So it is handy you know about probability density functions p(x) and if you integrate them you get the probability for a particulat set. I guess all university introductionary courses in probability theory cover that for a lot of studies like math, physics, chemistry, biology and so on. Ok, it is handy if you can integrate such probability densities a bit but I have to admit I skipped all things related to that (so basic is this post). But I skipped a lot of things yet this post is still 10 pictures long. All pictures are of size 550 x 775.

A dog named Loïs, David Pakman, Sean Carrroll and Harry Potter.

I am sitting on my couch watching Youtube videos while the dog is lying next to me. Is the dog always shaking from fear that I will beat her up? Come on, she is a dog and not a math professor! Anyway to my amazement there is a video from the David Pakman show with Sean Carroll, Sean is one of those television physics professors that you see relatively often on television or other media outlets. I get a warm feeling in my stomach because I can listen to the intelligent words of Sean, he truly is a high shot smart ass. So I play the video and everything looks fine but all of a sudden Harry Potter materializes behind the show presentator David Pakman and why does Harry Potter look so angry? Something to do with Brexit or so?

All of a sudden Harry Potter pulls his magic wand, points to to Sean Carroll and he shouts ‘Imbicilus Totalus!’. A sudden flash of lightning leaves the magic wand and enters the head of Sean. Slowly David Pakman is turning around to see what is behind him but Harry is much faster; out of a bag he pulls a short broom with a big handle and I can read the inscription on the broom. It is a SmartAss3000, fast and with a routine Harry sticks it in his ass and he flies away in the darkness of the night. Wow man that is totally different from what I remember from those Harry Potter movies! How life changes over time… May be those old Harry Potter movies are now in some parallel universe far far away.

I need to calm the dog because the dog understands I am very agitated so I explain to her that not all humans do such weird stuff with a SmartAss3000. Loïs does not seem to understand what I am trying to say but since I am calm again she waggles her tail and soon she calms down again. Finally I started to watch the video while hoping that Harry Potter did not do too much damage with the Imbicilus Totales curse. My hopes were idle, after all Harry is a very good wizard, and after about 8.30 minutes Sean Carroll explains electron spin: “If you measure electron spin, the electron can only spin clockwise or counter clockwise” Sean explains… Oh oh, Harry Potter clearly succeeded with his curse because just a few posts ago we calculated that the electron needed to spin many times the speed of light in order to explain the magnetic properties the electron has. Say it needs to spin about 100 times the speed of light, in that case any electron spinning can account for at most 1% of it’s magnetic behaviour. No problem for Sean Carroll: it is spinning clockwise or counter clockwise and that is enough explanation for him…

Well not for me because I have a long list of problems with electrons being magnetic dipoles (the official version of an electron is that it is a magnetic dipole, of course we have zero experimental proof of that but who bothers?) The clockwise / anti clockwise spinning of an electron is nonsense because if it is a magnetic dipole, that vector can point into any direction. But as far as I see reality, all experiments point much more towards electrons having a magnetic charge. That is electrons are all magnetic monopoles and not magnetic dipoles.

But Sean is not the only to hang on the electron magnetic dipole thing, at CERN there are thousands of physics professors that actually think that electrons are pure point particles, that is they have no volume but are true points. If that were true, if the electron has no size, it would be completely impossible to accelerate electrons with a magnetic field like in the Stern Gerlach experiment. By the way, here is the video:

Ok, after having said that kind of stuff, the next post will be about Benford’s law. If I remember it correctly I worked a short time on that nice law back in the year 1999. One or two weeks back I made a search on the preprint archive and after a bit of thinking I decided to craft a post for this website out of it. It is very easy to find and craft all kinds of probability distributions that fit Benford’s law perfectly. Benford’s law is about numbers as we find them in nature and gives the probability distribution for the leading digit. Do an internet search if you never heard from it. Here is a teaser picture to get started for the next post:

That was it for this post.

Using the Cayley-Hamilton theorem to find ‘all’ multiplications in 3D space.

It is a bit vague what exactly a multiplication is, but I always use things that ‘rotate over the dimensions’. For example on the 3D complex space the imiginary unit is written as j and the powers of j simply rotate over the dimensions because:

j = (0, 1, 0)
j^2 = (0, 0, 1) and
j^3 = (-1, 0, 0). Etc, the period becomes 6 in this way because after the sixth power everything repeats.

In this post we will look at a more general formulation of what the third power of j is. The Cayley-Hamilton theorem says that you can write the third power of 3 by 3 matrices always as some linear combination of the lower powers.

That is what we do in this post; we take a look at j^3 = a + bj + cj^2. Here the a, b and c are real numbers. The allowed values that j^3 can take is what I call the ‘parameter space’. This parameter space is rather big, it is almost 3D real space but if you want the 3D Cauchy-Riemann equations to fly it has to be that a is always non zero. There is nothing mysterious about that demand of being non zero: if the constant a = 0, the imaginary unit is no longer invertible and that is the root cause of a whole lot of trouble and we want to avoid that.

It is well known that sir Hamilton tried to find the 3D complex numbers for about a full decade. Because he wanted this 3D complex number space as some extension of the complex plane, he failed in this detail and instead found the quaternions… But if the 3D numbers were some extension of the 2D complex plane, there should be at least one number X in 3D such that it squares to minus one. At the end I give a simple proof why the equation X^2 = -1 cannot be solved in 3D space for all allowed parameters. So although we have a 3D ocean of parameters and as such an infinite amount of different multiplications, none of them contains a number that squares to minus one…

I gave a small theorem covering the impossibility of solving X^2 = -1 a relative harsh name: Trashing the Hamilton approach for 3D complex numbers. This should not be viewed as some emotional statement about the Hamilton guy. It is just what it says: trashing that kind of approach…

This post is 7 pictures long, each of the usual size of 550×775 pixels.

Sorry for the test picture, but the seven jpg pictures refused to upload. And that is strange because they are just seven clean jpg’s. Now it is repaired although I do not understand this strange error.

Anyway have a cool summer. Till updates.

The Cayley-Hamilton theorem neglected for 25 years?

That is strange, if you don’t know the Cayley-Hamilton theorem; it is the finding that every square matrix A, if you calculate the characteristic polynomial for the matrix A it is always zero. At first this is a very surprising result, but it is easy to prove. It’s importance lies in the fact that in this way you can always break down higher powers of the matrix A in lower powers. In the study of higher dimensional complex and circular numbers we do this all the time. If in 3D space I say that the third power of the imaginary component is minus one, j^3 = -1, we only write the third power as a multiple of the zero’th power…

In this post I will give two simple proofs of the Cayley-Hamilton theorem and although in my brain this is just a one line proof, if you write it down it always gets longer than anticipated.

At the end I show you an old video from the year 1986 from the London Mathematical Society where it is claimed that the CH theorem was neglected for 25 years. Now Hamilton is also famous for having sought the 3D complex numbers for about a full decade before he gave up. And I still do not understand why Hamilton tried this for so long but likely he wanted to include the imaginary unit i from the complex plane in it and that is impossible. Or may be he wanted a 3D complex number system that is also a field (in a field all elements or numbers that are non-zero have an inverse, in algebra wordings; there are no divisors of zero). A 3D field is also impossible and in this post I included a small proof for that.

Furthermore in this post at some point may be you read the words ‘total incompetents’ and ‘local university’. You must not view that as some emotional wording, on the contrary it is a cold clinical description of how math goes over there. So you must not think I am some kind of frustrated person, for me it is enough that I know how for example to craft a 3D complex number system. If they don’t want to do that, be my guest. After all this is a free country and we also have this concept of ‘academic freedom’ where the high shot math professors can do what they want.

And what is this ‘academic freedom’ anyway? If for example unpaired electrons are never magnetically neutral but electron pairs always are magnetically neutral, can the physical reality be that electrons are magnetic dipoles? Of course not, that is a crazy idea to begin with. But 97 years of academic freedom since the Stern-Gerlach experiment have never ever brought any meaningful understanding of the magnetic properties of the electron. If it acts as a magnetic charge and you say it is not a charge it is easy to understand how you can fool yourself for about one century of time.

This post is seven pictures long although the last picture is empty.
The two proofs of the Cayley/Hamilton theorem is how I would prove such a thing but good theorems always have many proofs. All pictures are of the size 550×775 pixels.

Here is the old video from 1986 where it is claimed the Cayley-Hamilton theorem was neglected for about 25 years. Oh oh oh what a deep crime. But the human mind is not made to produce or understand math, so in my view 25 years is a short period of time if in the good old days math professors were equally smart as the present day math professors. The title of the video is The Rise and Fall of Matrices.

Ok let me leave it with that an not post a link to the top wiki on the Cayley-Hamilton theorem where all kinds of interesting proofs are given. Till updates my dear reader.

The logarithm of all 2D circular numbers (the split complex numbers).

Yesterday I was editing the six pictures for this update and all of a sudden I realized I had made a dumb dumb mistake: The pictures count down from number 7 to number 2…

I had processed them in the wrong order; I had made seven background pictures but I filled in the math text in the wrong order.

All in all I decided to leave it this way; it might be a stupid mistake but it is not a critical mistake like making a critical math error or having wrong ideas about what is actually happening on the math level. It is just an editing error and also funny. So I leave it this way.

In this post we look at the so called split complex numbers, they are the cousin of the numbers from the complex plane. The only difference is that where in the complex plane the square of the imaginary unit equals minus one, for the split complex numbers this equals plus one.

Although this is a minor change, split complex numbers are not a field because it contains non-invertible numbers outside the number 0. All I do in this post is finding the eigenvalues and eigenvectors of all split complex numbers and via taking the log of the eigenvalues we calculate what the log of an arbitrary split complex number is.

In the speak of this website the split complex numbers are just the 2D circular numbers. Remeber in all dimensions numbers are complex or circular depending if the first imaginary unit equals -1 or +1. You can find many more ways of crafting a multiplication but the best math results are always found in the complex and circular version of the numbers in that particular dimension…

Ok, in this post I left all things out that talks about the 4D hybrid space that is a mixture of the 2D circular and complex numbers. But as you see on inspection of the above six pictures, the eigenvalues might be always real but they can be negative. As such always pay attention when you apply that function named the log…

That was it for this post, at this point in time I have no idea what the next post will be about. After all we had this long rout of over 20 posts on the 4D complex numbers and I left a whole lot of other stuff out in that period. Stuff like 3D Gauss integers or a general definition for integration that works in all dimensions. Till updates my dear reader.

And life? Life will go on.

Originally I planned on showing you some numerical results from the circular 4D numbers while explaining there is also a number alpha in 4D. For me that would be a nice holiday away from all that 4D complex number stuff from the last months…

But the numerical applet did not work, it is still dead in the water:

Ok ok, I could have done those numerical showings also in rigid analysis but I guessed that calculating a 4D tau for circular numbers via analysis was too much. And I settled for a much more easy to understand thing:

The logarithmic function for every 2D circular number. In the field of professional math professors the 2D circular numbers are known as the split complex numbers.

So that is what the next post will be about: Finding log(z) for all invertible split complex numbers.

I only wrote one previous post on the 2D circular aka split complex numbers and that dates back to Nov 24 of the year 2016:

The second hybrid: a 4D mix of the complex and the circular plane.

Ha ha, now I can laugh about it but back in the time it was some hefty pain. Anyway to make a long story short: In that old post from 2016 I calculated the log for just one split complex number namely the first imaginary unit j.

Let me show you my favorite part of that old post from 2016:

So the next update will only contain 2×2 size matrices while I skip the detail that the log lives mainly in the hybrid number system from the old post.