Probability amplitudes on the 3D exponential cones (circular and complex version).

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 … ­čśë

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.

A new de Moivre identity.

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… ­čśë

A small update on the Wendelstein ‘contest’.

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:

Now Oct 2019 they have done nothing yet…

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.

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:

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.

Two parametrizations for 3D exponential circles.

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

End of this post and thanks for your attention!

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:

The cosine & the modified Dirichlet kernel parametrizations

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

The sphere-cone equation in a matrix notation.

It is about time for a new post on 3D numbers, circular and complex. In this post I write the sphere-cone equation in a matrix notation so see the previous post on conjugates if you feel confused. The sphere-cone equation gives us two equations, as the name suggests these are a sphere and a cone and on the intersection we find the famous exponential circle.

Beside the sphere-cone equation I also demand that the determinant equals 1, now we have three equations and every intersection of those 3 equations has as it’s solution the exponential circle. Can it become more crazy? Yes because it is possible to factorize the third degree determinant into a linear and a quadratic factor. Those factors must also be 1 and now we have five equations! And since you can pick 10 pairs out of five, we now have 10 ways of solving for the intersection where the exponential circle lives…

It is strange that after all these years it is still easy to find 10 video’s where so called ‘professional math professors’ sing their praise upon the exponential circle in the complex plane. They really go beserk over the fact that e to the power it gives the cosine and sine thing. And after all those years still silent, yeah yeah those hero’s really deserve the title of honorable shithole… It is honorable because they often have relatively large salaries and they are shitholes because of their brave behavior when it comes to 3D complex numbers. Bah, I am getting a bad taste in my mouth when I think about the behavior of professional math professors. Let me stop writing about that low form of life.

This post is 8 pictures long. May be, I have not decided yet, is the next post about parametrizations of the exponential circle. In these 8 pictures I work out the case for the circular multiplication, that is the case where the imaginary unit j behaves like j^3 = 1. At the end I only give the 3D complex version of the matrix form of the sphere-cone equation and the rest you are supposed to do yourself.

Ok, again do not confuse this with quadratic forms. A matrix equation as written above has a real and two imaginary components while quadratic forms are often just real valued.

Let┬┤s try to upload this stuff. See you in the next post.

A correction and a few new photo’s upon electron spin.

The correction is rather simple: In the past I always said that those old televisions run on something like 50 thousand Volt. That is of course the kathode electric potential and not your input voltage. That is not entirely correct: all photo’s I showed you in the past were made with a small television set and those seem to need a lower kathode voltage. May be something in the 25-30 kV range.

So that is a small correction but I have written posts where we tried to calculate the sideway acceleration and I based the speed of the electrons going from the electron cannons to the glass screen on the 50 thousand Volt. I memory served that gave a giant speed of almost one third of the speed of light and that gave giganormous numbers when it came to sideway acceleration. If indeed those small tv sets run on a lower electric potential, that was a bit over the top.

Well that does not impede the fact that electrons are likely magnetic monopoles and not magnetic dipoles as the standard model of physics says. So far for my correction on past statements.

I am still having a bigger television and I finally made a few photo┬┤s of the behavior of electrons with that oldie. It is best to make those pictures in a dark room so that your camera has a relatively long shutter speed. I tried it once at daylight outside but that gave lausy results because in between the rereshment of the screen it often is black because no electrons have landed there recently. If you take photo┬┤s in a dark room it gets better. When I looked with my human eyes to the television without any magnets around, I see a clear blue uniform is color and intensity everywhere. In the next photo you see how the camera ┬┤sees┬┤ it. Not very uniform…

Without any magnet.

In the next photo I come in from the right with a stack of magnets. It is amazing from how far away the screen already starts changing.

As memory serves, there are three electron cannons in it.

In the above photo you likely see already the separation of the ┬┤blue┬┤ electrons in those that are attracted by my stack of magnets. That should be the blue spot on the lower right. The other blue region should have at it┬┤s right lower boundary mostly repelled electrons while that large blue region could also contain a lot of electrons that are not disturbed enough.

Ok, the next photo is more important because even at 50 thousand Volt with the relatively sharp tip of my neodymium magnets you get that dark disk where no electrons land. So we have a clear separation of electrons that are attracted versus those that are repelled by the magnetic field from the stack of magnets.

Remark it is very hard to explain the dark region where clearly no electrons land with the Lorentz force. The standard model has only this Lorentz force in it and ok ok they also use that weird term for the potential energy for an electron in an inhomogeneous magnetic field but in my view that is not correct because it does not include the size of the electron. And by the way, it should be a cakewalk to separate the electrons according to their spin using magnetic fields that are as uniform as possible. There are still plenty of those in physics labs all over the world, if electrons are magnetic monopoles it should not make much difference if you use a uniform or non/uniform magnetic field.

If electrons can ┬┤only follow┬┤magnetic field lines, what explains the dark region

In the last photo I turned my stack of magnets around. On the other side I often have 2 or 4 ring shaped magnets that I removed from two magnetron ovens. They have a hole through the middle and I tried to photograph it such you can look through that very hole.

Wow man, a giant region where no electrons land…

Model model on the wall, where is the standard in this all…

I hope you see that tiny spot in the middle where the attracted electrons create a while light. Ok that was it for this update. In the meantime I am working on a post around the sphere/cone equation written in matrix form. But that is far from finished so see you somewhere next month!

End of this post.

Making a permanent magnet within 5 minutes.

Today was a good day because I scored 80 kg of malt and almost one kg of hops so I can brew for a long time… Upon arriving home there was an envellope on the doormat, is it what I hoped for? Yes it was, a new compass. I bought it two days ago and it costs only 8.50ÔéČ.

Yesterday I made a permanent magnet and the new compass says it actually is a real bipolar magnet (made from an iron nail). For me that was a nice historical moment because it was the very first time that a human made a permanent magnet that was solely based on the principle that electrons are magnetic monopoles…

Three days ago on the other website I updated the magnetic pages with reason number 77 as why electrons cannot be magnetic dipoles. Here is a link: 04 Sept 2019: Reason 77: More on the Curie temperature of iron. http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff04.htm#04Sept2019

The Curie temperature is that temperature where ferro magnetic materials like iron loose their magnetic properties. But what does that mean? That means two things: A permanent magnet gets destroyed, it is no longer a permanent magnet when it becomes too hot. And the second surprising thing is that a piece of hot iron above the Curie temperature is no longer attracted by a permanent magnet. Let me repost two photo’s on this website to make the point clear. In the first photo you see an iron nail attracted by a small stack of neodymium magnets, I heat the nail up with a simple burner and all of a sudden the nail is no longer attraced. That is what you see in the next two photo’s:

This is before the burn.
And when it gets too hot, it falls down

Ok, how to make a permanent magnet with only a small stack of neodymium magnets? Very simple: I hanged the nail a bit higher and placed my small stack of neodymium magnets under the nail. About 2 mm of distance between the tip of the nail and the magnets. And you burn the nail until it is hot enough. After that you just let it cool down and voila: you have made yourself a permanent magnet using only idea’s derived from electrons being magnetic monopoles.

That is one magnetic pole in the pocket.
And there is the other..

Ok, that was it for this post. See you in the next post.

Update from 13 Sept: My nail magnet is so weak you cannot lift other nails with it. It works fine because a compass reacts to it but it is not very strong. On the other side of the spectrum I found a cute video today where they claim to have achieved a magnetic field strength of 20 Tesla…

The video is from Tokamak Energy, that is one of those startup companies that try to craft workable fusion reactors for commercial electricity production. Like explained before: if indeed electrons are magnetic monopoles and because they react much stronger to the applied magnetic field compared to the plasma protons, this will cause a ton of turbulence. And stronger magnets do not solve that problem; on the contrary the turbulence will appear much sooner in a stronger magnetic field.

You may hope that the university people finally pop up some kind of proof that electrons are indeed magnetic dipoles. But it is now Sept 2019, the start of a new academic year. And to be honest I don’t expect such a proof this year. So in the meantime while the climate is changing, lot’s of people dream about nuclear fusion as an energy source and the university people will do nothing day in day out this new academic year.

From a video from 5 June this year where Tokamak Energy promotes itself by pointing at the climate change I made this small screen shot:

Tokamak Energy – A faster way to fusion..

Here is the video I found today: if only electrons were not magnetic monopoles it would be a great find. Ok, end of this update.

The two self-conjugate planes for 3D circular and complex numbers.

This is another lightweight easy going summer update. It is about matrix representations and how to find the conjugate of a 3D complex or circular number. I use the case of the complex plane of 2D conplex numbers to show that conjugation is not some silly reflection just always but rather simple will always be the upper row of a proper matrix representation. As a matter of fact it is so easy to understand that even the biggest idiots on this planet could understand it if they wanted. Of course math professors don’t want to understand 3D numbers so also this new school year nothing will happen on that front…

Did you know that math professors study the periodic system? Yes they do, anyway in my home country the Netherlands they do because every year they get a pay rise and that pay rise is called a periodic. And as such they study the periodic system deep and hard…

I classified this post only under the categories ‘3D complex numbers’ and ‘matrix representations’ and left all stuff related to exponential circles out. Yet the exponential circle stuff is interesting; after reading this post try to find out if the numbers alpha (the midpoint of the exponential circles) are symmetrix (yes). And the two numbers tau (the log of the first imaginary unit on the circular and complex 3D space) are anti-symmetrix (yes).

This post is just over 7 pictures long. As the background picture I used the one I crafted for the general theorem of Pythagoras. (Never read that one? Use the search funtion for this website please!) All pictures are of the usual size namely 550×775 pixels.

It is a cute background picture, I remember it was relatively much work but the result was fine.

Ok, that was it for this update. Although it is so very simple (for years I did not want to write of just two simple planes that contain all the self-conjugate numbers) but why make it always so difficult? Come on it is summer time and in the summer almost all things are more important than math. For example goalkeeper cat is far more important compared to those stupid 3D numbers. So finally I repost a video about a cat and that makes me very similar to about 3 billion other people.

Till updates & thanks for your attention.