It is about time for a small update! Despite all that COVID-19 stuff going round, for myself after all those years I finally tried to put integration on higher dimensional number systems on a more solid footing. All those years I just refrained from it because you cannot use my favorite Riemann sum approach because of the non-invertible numbers we have in 3D or even higher dimensions. But now I am trying to finally make some progress and stop avoiding this subject, I find it is utterly beautiful. It has an amazing array of subtle details involved when you have some non-invertible numbers in your integration stuff. I have no idea when I have finished this rather important detail in my cute theory of higher dimensional complex & circular numbers, so let time be time & in the meantime only post a teaser picture about that lovely integration stuff. In the first lines you see a very familiar integral, likely you have done such calculations in the complex plane. In the case of 3D circular and complex numbers you must (of course) use the multiplication on 3D space to make it all work. Basically you are evaluating (or calculating) three integrals at the same time, just like on the complex plane where you are evaluating two integrals at the same time in your calculation. If you work with a pencil & paper, make sure you have enough paper because all those 3 integrals also have 3 terms in it so your calculations can become quite long… Here we go:
Please remark this only works for invertible X.
Ok, let me end this update now. Till updates and for some strange reason you must wash your hands while the proper authorities never point to 3D complex numbers… Till updates.
I am always baffled by those folks explaining this important experiment; why do they not see that the explanation offered is just 100% bs? It could be that in physics there are all kinds of ‘patches’ that explain particular parts of magnetism. Let me write two of those patches down:
Patch 1: Since in the Stern Gerlach experiment a beam of silver atoms was split in two under the application of a magnetic field, the ‘logical explanation’ offered is always that when electrons enter a vertically applied magnetic field, 50% will have spin up and 50% is spin down. If the applied magnetic field is turned 90 degrees, say horizontal, again both beams will split again in 50% left spin and 50% right spin.
Patch 2: Making permanent magnets. A magnetic field is applied to some metal and now all unpaired electrons always align with the applied magnetic field. Sometimes the explanation is a bit more advanced; at first it is explained that in the magnetic domains of say iron all spins are aligned and when making it into a permanent magnets all the magnetic domains align according to the applied (strong) magnetic field.
On their own such ‘explanations’ might sound logical, but if you combine them you get total rubish. It cannot be that one the one hand if you apply a magnetic field 50% of the unpaired electrons anti align and the other 50% align with the magnetic field while on the other hand always 100% of unpaired electrons align nicely when you make a permanent magnet. Such ‘explanations’ or patches of knowledge should enforce each other, but here it gives total bs. Either it is always 50/50 or it is always 100% alignment, why do those professional physics folks never observe that tiny part of physical reality? In my view they cannot go outside the patches, the reasoning always stays local inside that particular patch (explaining the SG experiment versus making permanent magnets).
The 50/50 patch that should explain the Stern Gerlach experiment is always very strange if you just keep an iron nail next to a magnet; wow man it gets attracted! But if 50% of the unpaired electrons in that nail would anti-align and the other half would align, what would explain the attraction? In my view people like that a weird beyond comprehension.
At Fermilab the honorable Don Lincoln often explains all kind of physical things, his style in doing so is often a bit too arrogant in my view. If you want to study physics you must be humble and always operate from the fact you only have a human brain. So being an arrogant overpaid jerk is a quality you must loose; that human stuff will ensure you will never understand physical things because it prevents you to think a bit deeper on it when for example you try to check if you could be wrong…
The video is on more items, not only the SG experiment but also the Einstein-Rosen-Podolski paradox, the creation of an electron-positron pair from a spin 0 particle & more of that stuff. I made two pictures from two screen shots. By all standards it is hilarious because what spin 0 particle are we actually talking about? Of course that is not mentioned, with just a tiny bit of arrogant behavior it is simply stated and you as an onlooker of that video are supposed to bow for the wisdom of Don Lincoln…
Cooment: In my view this shows conservation of magnetic charge.
Please remark I have no experimental evidence that if electrons are magnetic monopoles, there is conservation of total magnetic charge just like with electric charge. I think it is the case but you also have constantly those physics people explaining that you can flip electron spin with micro waves. But all those patches they try to explain, for example spin flip inside a qbit for quantum computing, can also be the result of electron change. There are always more electrons in the surrounding and if you apply some micro wave radiation it could very well be that you ram out the anti aligning electron that simply gets replaced by an electron of the opposite magnetic charge. After all I have never ever seen an experiment where there is only one electron trapped in isolation and after a short pulse of em radiation it has changed it’s spin.
Ok, let us go on with the hilarious stuff:
Comment: Never forget that Stern was the first assistant professor to Einstein. So Einstein never ever had a clue about electron spin in the first place…
Ok, let’s go to the video itself. The Lincoln guy is a bit irritating because of his arrogant attitude, but it is soon funny & hilarious when he props up his 50/50 spin alignment nonsense. For me it is funny because if electrons carry magnetic charge, a more or less conservative estimate as when the professional physics professors will find that out is about 5000 to 5 million years into the future. Just like the speed math professors understand a bit more upon 3D complex numbers. Title of the video: Quantum Entanglement: Spooky Action at a Distance.
Ok that was it for this post, think well and work well. Updated on 18 March 2020: Lately I found a cute video from Veritaisum and the MinutePhysics guy where indeed they use both patches of ‘explanation’ in just one video. The Stern-Gerlach experiment is explained via electrons doing the 50/50 thing while for permanent magnets all electrons align & we can safely conclude these guys are lunatics.
But if you look at other video’s of Veritasium & the MinutePhysics guy, they often look so smart and it all looks like they have more or less healthy brains… These guys are not idiots and that leaves we with a big question I still have: Why do the people of physics never understand that separate patches of human knowledge should enforce each other? Why do these two guys not see that giving two explanations is highly contradictionary? If you have a permanent magnet in your hands and you approach a piece of iron, if 50% of the electrons align and the other half anti-align, iron would not be magnetic. But Iron is very magnetic as any body knows, so why do weirdo’s like Veritasium & the MinutePhysics guy not see that? Here is that cute video from two idiots not capable of seeing their wisdom is not perfectly optimesed:
The television physics professor Sean Carroll is very good at explaining electron spin: either the electron spins clockwise or anti clockwise. In the past I often got annoyed by such ‘explanations’ because it shows shallow thinking and a complete rufusal to even try to understand electron spin.
At present day I can only laugh about it: If overpaid people like that want electron spin to be such stupid stuff, may be it is better to say it is your cake so why not eat it? If you leave all those shallow puddles of thinking a more or less normal person would like to know what experiments are there that actually prove or strongly suggest that electrons are indeed magnetic dipoles? If you try to find out about such heroic and historical experiments, it is once more a bit hollow and disappointing: Never ever as far as I know was there a physics experiment trying to actually prove the electron is a magnetic dipole…
So let us do a little thought experiment where the electron spin is ‘Sean Carroll style’ caused by the classical electron spinning around some axis, Here we go:
Suppose a pair of particles is created, say an electron and a positron. Suppose total spin must be zero, say the electron is spin up and the positron is spin down. If the electron spins clockwise, the positron should also spin clockwise otherwise total spin ‘Sean Carroll style’ would not be zero. After all the positron has a positive electric charge so it has to spin around some mysterious axis in the same way as the electron otherwise total spin ‘Sean Carroll style’ would not be zero. But ha ha ha: That would violate the classical law of conservation of angular momentum because both the electron & the positron must rotate around some weird axis in the same way. Conclusion once more: It is not possible for the electron to be a magnetic dipole. End of the simple thought experiment.
A rather recent video from our deep thinker Sean Carroll was out via the UK Royal Institution, it is not science but it has a high entertainment quality in it. Therefore if you like good but shallow entertainment, go to people like the television physics professor Sean Carroll:
The video from the Royal Society is highly hilarious, Sean is complaining about the fact that most physics professors have stopped to try understand quantum physics. That is funny because of course if you use words like ‘spin’ to describe the magnetic properties of something like an electron, how can you be not confused? And I have the same problem in my little unworthy life: the math professors have given up a long time ago about finding the 3D complex numbers. That must have been about a century ago when they were not capable of understanding it is all about prime numbers and 3 is a different prime number from 2. I stole a sceen shot from the video, Sean used the grapes as ‘understanding quatum mechancis’ where I use it as ‘3D complex numbers’.
Let me end this post with the same thought experiment as above: the creation of an electron positron pair. Only now it is ‘my style’ and not some stupid ‘turning around some axis style’.
Suppose an electron positron pair is created, total electric charge must be zero. Hence the positron has positive electric charge. The magnetic charges should also be zero, hence one of the created particles will have a north magnetic charge while the other will have a south magnetic charge.
In my view that is all there is. End of this post.
Ok ok I was a bit lazy but it is finished now so let’s finally post this scalar replacement theorem. Never in this post I formulate or proof this scalar replacement theorem, but basically this theorem says that if you replace the real numbers (scalars) in the way you describe say 2D split complex numbers by numbers from the complex plane, the result is a space who’s numbers also commute and it even has viable Cauchy Riemann equations. In this post I will write z = x + yj for the 2D circular numbers (also known as the split complex numbers) and write z = x + yi for numbers from the complex plane. If you combine such spaces it must have imaginary units that are different in notation, so j is the imaginary unit that does j^2 = 1 while the good ol i from the complex plane is known for it’s important property that i^2 = -1.
If we replace the x and y in z = x + yj by complex numbers we get a new 4D space where both j and i place there role. All in all those 4D numbers will be written as Z = a + bi + cj + dij. Of course the a, b, c and d are real numbers and as such this new space is 4D.
A long time ago I once used this to calculate the logatithm of j, it worked perfectly and that is why I more or less gave idea’s like that the name of ‘scalar replacement’. Later I found that way of using diagonalization of the matrix representations in order to calculate the logarithm, that is a far more general useable way of calculating logarithms but anyway the original calculation for log j was so cute, I could not abondon it and say to that calculation: From now on you are a poor orphan and no one will help you survive from day to day… How could I abandon such a calculation, better loose the UK a 100 times on a row than abandoning such nice calculations… 😉
But let’s go back to being a serious and responsible adult; the post is relatively long with 10 pictures. As usual I had to leave a lot out and I hope it is more or less easily readable. After all a lot of math out there looks like it is written by people who eat a plate of coal for breakfast. And if you eat coal for breakfast, likely this has an influence on the math you will produce on a particular day… Ok, here we go:
Ok, the goal of this post is of course to make you think a little bit about this 4D space and compare it to the quaternions and stuff. But last year on 2 March I posted the diagonalization method for finding the logarithm of an arbitrary split complex number. Below is a link.
Let me end this post with a funny mathematical joke about how to NOT WRITE MATH. Using a fucking lot of indices is not a way to make your work readable, here is a picture of what I view as some kind of mathematical joke.
In case you desire a serious headache, go read that file.
It is about high time to make a new post around here. I have not done much lately but I have to admit sometimes I can be lazy as hell. At present I am working on the scalar replacement theorem so likely that is the next post on this website. This replacement theorem says that if you want you can replace the real numbers you use in for example the 3D circular or complex numbers by numbers from the complex plane. I only replace it by complex numbers from the complex plane, the more general version of the replacement theorem is much wider but I often dislike math that is ‘too general’.
On the other website I opened page five on magnetics. Page five means this is the fifth year of writing about electrons and why it is highly unlikely they are magnetic dipoles but magnetic monopoles. As such I stumbled upon the Bohr-van Leeuwen theorem, but that was nothing new only that I did not know it had that name. For me this theorem is just the description used for the forces on an electrically charged particle in and electric and magnetic field. And it says that in a constant magnetic field (that means both constant in time and constant in space), the magnetic field does not do any work. That is there is no acceleration or stuff like that, needless to say I disagree with that. Just take a look at the sun, with the Bohr-van Leeuwen theorem in your hand all that stuff that is going on is hard to explain. Why does the solar plasma accelerate along magnetic field lines? Why is the solar atmosphere, the corona, so hot compared to the surface of the sun? If indeed magnetic fields do not do any work, the sun is hard to understand… (Actually if electrons are magnetic monopoles, the sun is also hard to understand.)
I also found a cute video from about 5+ years back, so likely that was the time it started to dawn upon me that it was impossible that electrons are magnetic dipoles. At that time I tried and tried to understand the results of the Stern Gerlach experiment but how hard I tried it only worked when electrons carried magnetic charge. Until now in the last five years I could not disprove myself, I tried and tried but the longer you think about it the more nonsense it becomes that the electrons are dipole from the magnetic point of view. And to put it simple: Why is there no electric dipole particle? But those people, and I mean of course the people from physics, never talk about stuff like that. Anyway, here is the cute video:
From a screen shot from the video combined with the words from a wiki about the Bohr-van Leeuwen theorem I made the next picture:
I hope this is easy to understand…
As a funny side remark, the guy from the video is from Australia and over there in New South Wales or so they try to make quantum computers based on electron spin qbits. And they think electrons can be in a super position of spin up and spin down, that is in their view as why they can be used as qbits on a quantum computer. Until now (five years later) they still have nothing to show for. Of course when electrons carry magnetic charge, it is very hard to place one electron in a super position state of being a north and a south magnetic monopole at the same time. Just like in a hydrogen atom where there are two particles named electron and proton that both particles are in a super position of positive electric charge and negative charge. No, the proton always has a positive electric charge and the electron a negative one. So good luck with making a quantum computer based on electron spin… Ok, I have done enough of the writing words and stuff. May be it is high time to split and may the magnetic force be with you. Till updates.
All in all it was a nice day today. Brewing is completed and tomorrow the wort can go into the fermentation bottles and the wonderful process of fermenting can take place. For those of you that also like to brew: A couple of months back I found a cute video explaining that you can also brew beer without cooking it. And I was like seeing water burning or I was like a professional math professor understanding 3D complex numbers for the very first time in their life… Anyway if you are interested search for ‘Raw ale no boil brewing’ on Youtube. It is of interest because if you brew without boiling, only after that you understand what you usually cook away in things that might taste good (or bad).
But let’s go to this post: It is about probability amplitudes as they are used in quantum physics where all those kind of amplitudes are multiplied against their conjugate and that gives a real positive number known as the probability. If you write it in polar coordinates on the complex plane, it is easy to see that those probability amplitudes can have all kinds of phases (the argument of the complex plane number). So for that to work on 3D complex or circular numbers, it would be great if you can write it more or less like the polar coordinates as in the complex plane. And that is easy to do in 3D space: Once you have found and also understand the exponential circles, it is evident that all numbers on those exponential cones are some real multiple of a number from the exponential circle.
As such the numbers found on the exponential cone can be written just like the polar stuff from the complex plane, also now the r as used in polar coordinates can also be negative. That is a strange result because for millions of years we were always indoctrinated by a positive r … 😉
Another important difference with the complex plane lies in the fact that the complex plane is closed under addition. That is obvious, but it is also obvious that on a cone it is very different. Most of the time if you add up two numbers you are either inside or outside the cone. But probability amplitudes are always multiplied against their conjugate and added up only later, so we can still use the exponential cone for things like that. I don’t see that ship stranding, so let’s do it.
I also want to remark I am using the so called ‘pull back map’ once more. The professional professors also have a pull back map but that is a very different thing compared to what I use. So don’t be confused by that: the way I use it is to fix higher dimensional exponential circles (and curves) on the exponential circle in the complex plane. (This for fine tuning the period in time and stuff like that, or for understanding why the numbers are what they are: WTF that square root of 3 in it???
This post is 7 pictures long, most are the usual size of 550×775 pixels. At last I want to remark that for myself speaking I do not know if there is any benefit in trying this kind of use of 3D complex and circular numbers. It is funny to think about positive and negative values for r like for example in electron spin or a wave function for the electron pair. But I just do not know if this add any value or that you can use the complex plane only and miss nothing of all you could have learned.
Ok, here we go:
Ok, that was it for this post. Till updates my dear reader.
First a household message: In about two weeks time this website should go to new very fast servers. In order for that to work properly I have to do all kinds of things that I have never done before. Stuff like updating PHP. Ok, that does not sound too difficult but as always the work explodes because first I have to backup everything. And before I can backup everyting I need a new ftp account. The only luck is I still have a running ftp client on my own computer…
In case this website is gone in two weeks, somewhere I got lost in the woods. And there is no hurry: this math website is just a hobby of me. An important hobby because it is a bit of exercise for the brain… End of the household message.
What is the yeast of this post? Historically the de Moivre identity (or theorem) predates the very first exponential circle on the complex plane. If you use the exponential circle, a proof of the de Moivre identity becomes very very easy. In this short post we will use the 3D exponential circle for circular numbers. Two posts back I showed you a possible parametrization via those 3 cosine expressions, in this post we use those parametrizations to formulate a 3D de Moivre identiy. Because we already have an exponential circle, we do not need to give a rigid math proof for this identity. Once you have and exponential cricle, stuff like that comes for free along with it…
As usual I skipped a lot of things while writing this post. For example I skipped using those modified Dirichlet kernels. I skipped giving the 4D de moivre identity for the 4D complex numbers. All in all I was satisfied to cram this all in a very short post; only three pictures long! In case you are still reading this while having no clue whatsoever what a de Moivre identiy is, here is some stuff from brilliant.org: De Moivre’s theorem Http stuff in the link: https://brilliant.org/wiki/de-moivres-theorem/
Ok, only three pictures long. Here we go:
That was it for this post. If I don’t change plans, in the next post we will look at the 3D exponential cone because on that cone you can do all those quantum probability calculations just like in the 2D complex plane. But before that I have to go though that horrible PHP update…
So see you in the next post or let’s split indefinitely and end this stupid website for no reason at all… 😉
About a year ago I proposed as small contest with the Wendelstein fusion reactor folks from the Max Planck institute in Germany. The proposal was done on Oct 25 last year. Here is a link:
Yes they have done nothing yet so it looks like the contest can go on in the year 2020. A few days back there was a new Youtube video out with Hartmut Zohm where he gives a lecture for a ‘general public’. Since I have seen plenty enough videos like that, for me it was a very boring experience but I decided to suffer hefty for a nobel purpose… The video is also in the German language, for some this might be a problem. So it is a boring video but since I use it as a ‘source’ let’s post it:
At Hartmut his side, everything was exactly the same as one year ago: All physics is sound understood, this must be it. But this time he also mentions the turbulence, in a professional manner he sweeps that one under the rug by stating: We don’t even understand turbulence in water, so with plasma it is even a bit more difficult. I had to laugh hard, Hartmut is a great comedian…
In my view where I think it is more likely electrons carry magnetic charge, the main magnetic field for containing the plasma is the root cause for a ton of turbulence. And that is simply explained by the large acceleration the electrons have while the two different magnetic charges will travel in opposite directions. That should give tons and tons of turbulence. Anyway that is my take on it: It will never work because the electrons get accelerated to relativistic speeds…
At the universities nothing will change. No proof will be given that the electron is a magnetic dipole. (The most retarded explanation I ever observed was: The electron is a magnetic dipole! And how do we know that? Because of the Stern-Gerlach experiment! It goes in two directions and therefore it is a magnetic dipole!) And also no experimental proof (a better word is evidence) that electrons carry a magnetic charge beside the electric charge.
Let me end this post with a little joke: The ppp (professional physics professors) always say the electron pair is one spin up and one spin down electron. So they pair up north pole to north pole or south pole against south pole…
Ok, it is not a funny joke, but you can also cry about it if you want to. Anyway I hope that in Oct 2020 I will not forget to update on this very important contest by showing you next year once more nothing has happened… Till updates my dear reader.
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:
It is just a u-substitution…
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.
It is about high time I post the solution in parametrization form of those five equations from 03 Oct 2019. That is almost 2 months back and oh how ashamed am I for my laziness… But for me math is a hobby, an important hobby but a hobby anyway. So other hobby’s are allowed to interfere with my little math hobby.
This post is 10 pictures long and at the end there is a horrible bad video from the Youtube channel Seeker. Begin this week I crossed that video with an intriguing title; Could These Numbers Unravel New Dimensions in Space? I was just curious but it is that Cohl Furey stuff again. It is an attempt to explain particle physics via complex number, quaternions and octonions… What do they have in common? These number systems are always fields that means all non-zero numbers have an inverse. Why the professional math professors find that so important is unknown to me, it is more like they have nothing else in the toolbox. If you are interested you can find the Cohl Furey video’s on Youtube.
In this post I too write about things that are common in the complex plane, complex and circular 3D numbers and 4D complex numbers. You can use the modified Dirichlet kernels as the building blocks for all possible exponential circles or in the case with 4D complex numbers: the exponential curve (in 4D space the curve is in a 3D hyper plane).
But I also wanted to show you the original cosine solution that I found years ago. To this day it is still amazing that the cosine can pull it off; that the cosine can be a building block for a 3D exponential circle. Next year it will be three decades ago when I found the 3D complex numbers and got interested in them. At present day you can wonder why there is never a healthy response from the math communuty. It is all very logical: if there is no healthy response that means the math community in itself cannot be healthy. It is just a community of perfumed princes and that’s it.
After so much blah blah it is high time to go to the ten pictures:
So from the complex plane in two dimensions to 4D complex space; a binding element is how you can use the modified Dirichlet kernels and their time lags to construct these very interesting parametrizations. Of course there is much more that binds those spaces together; the matrix representations are all very similar, just like the eigen values and eigen vectors. But above 2D it is never a field. And again why the professional math professors have this weird fixation on fields is completely unknown to me. At last, here is that wonderful video that will make your toes curl
