# 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…

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.

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.

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.

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 8.50€.

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:

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.

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:

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.

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.

# Making a permanent magnet using 20 thousand amps of electric current.

A lovely video was found where a guy from the Nottingham university is showing his workplace around. And they have that heavy equipment for making permanent magnets in just one blast. In the first five years of looking at magnetism I only told you about that slow process of heating up the material till above the Curie temperature, applying an external magnetic field, cool everything down slowly and voila: there are your permanent magnets!
But you can do it in one blast too, all you need is a very strong applied magnetic field. For me that is nothing new because my father worked at the local electricity plant and decades back they too had the equipment to make a permanent magnet in one short blast. At an electricity plant they have plenty of electricity anyway so why waste that?
(My father worked at the electric meter department and in those times they used permanent magnets in the electricity meters you had in your home or your business.)

Anyway, if my version of electron spin is true and electrons are not magnetic dipoles but come in two varieties carrying magnetic charge, in that case the ‘permanent’ in a permanent magnet arises from the fact that the unpaired electrons are shielded in the inner atomic orbitals. That is what makes them permanent… All that blah blah of electron spins aligning themselves to the applied magnetic field is pure nonsense, that blah blah does not explain why the magnetism is permanent.
Of course professional physics professors will always point to the tiny detail that if you think you understand quantum mechancis, you don’t understand quantum mechanics…
Now I too have a lot of things that I do not understand in quantum mechancis, but I think the electron being a magnetic dipole is 100% bullshit. They carry magnetic charge because that makes more sense and is a far more simple explanation of what we observe…

After having said that, if a permanent magnet always has it’s unpaired electrons that give rise to the emergent magnetism always in it’s inner orbitals, in that case if you blast them with a giant external magnetic field they should always heat up. They heat up because the unpaired electrons feels a relatively giant force from the applied magnetic field and as such are ripped out of those inner orbitals. And all ripped out electrons are replaced by electrons of the opposite magnetic charge…
It’s as simple as it is.

Here is the video it is only 15 minutes long:

Ok, let me end this post with a picture made from two screen shots from the video. In the top screen shot you see at the left those strips of metal. Wow man, those strips of metal are the wires that transport the 10 to 20 thousand Ampere blast.
In the lower screen shot you see the blue machine where the magic seems to happen.

Ok that was it. If you make permanent magnets and they are not heated at the end, I am wrong about my electron idea’s… Only a professional physics professor will lamentate that applying a short energy burst of only 20 thousand amps will likely heat up everything.

End of this post.

# But are these quadratic forms?

This is a lazy easy going summer post, it does not have much mathematical depth. Let’s say the depth of a bird bath. But with most posts I write you also need a lot of knowledge about what was in previous posts and for the average person coming along that is often too time consuming… So we keep it simple today; quadratic forms on 3D space.

If you have had one or two courses of linear algebra you likely have encountered quadratic forms. They are often denoted as Q(X) where the X is a column matrix and the quadratic form is defined as Q(X) = XT A X. Here XT is the transponent of X so that would be a matrix row. As you might guess, the X column matrix contains the variables while the constant square matrix A is the source of coefficients in the quadratic form Q(X).  In most literature it is told the matrix A is symmetric, of course there is no reason at all for that; any square matrix will do. On the other hand it is easy to see or to show that if a square matrix is anti-symmetric the corresponding quadratic form will always be zero everywhere.

In this post we will take matrices that are always the matrix representation of 3D complex & circular numbers. Matrix representations are a complete category on this website so if you don’t know them you must look that up first. (Oh oh, here I go again: this was supposed to be easy but now the average reader must first try to understand matrix representations of higher dimensional multiplications…)

Compared to the previous update on the likely failure of all fusion reactors this post is far less dramatic. If in the future I am right and we will never have fusion power, that will be the difference between life and death of hundreds of millions of people in the long run… So in order to be a bit less depressing let’s lift the spirits by a lightweight new post on quadratic forms! Why not enjoy life as long as it lasts?

Ok, the actual post is seven pictures long, all in the usual size of 550×775 pixels.

I have to admit that for me the use of the number alpha was important because that is at the center of the exponential circles in the 3D complex and circular spaces. So I have a legitimate reason to post this also under the category ´exponential circle´. And from the non-bird bath deep math, that is the big math ocean that is very deep, I like to classify as much posts under that category ´exponential circles´.

Ok, let´s leave it with that and try to upload this post. Till updates my dear reader.

# On nuclear fusion reactors and why they will all fail.

To be precise: I am talking about all nuclear fusion reactor designs that use magnetic confinement for the fusion plasma. So these are the standard tokamak reactors that are build in a lot of places but also the stellarator fusion reactor from the Max Planck institute in Germany. Some years ago it came to my attention that the USA based company Lockheed Martin was also going into the fusion reactor thing and they were bragging about new technology and making mobile 100 Mega Watt nuclear fusion reactors. But their talk was a little bit strange, it was some CEO kind of guy that explained how their new technology would outbeat the tokamak design because with the new much stronger magnetic fields they could make, the magnetic field would be much stronger at the place of the fusion vessel wall. According to the Lockheed Martin CEO type of guy, plasma was diamagnetic and as such would stay away from the fusion reactor wall. Needless to say I had to laugh because in my view on physical reality electrons carry magnetic charge and will always make fusion plasma instable. In fusion reactions the protons (or the isotopes of hydrogen to be precise) need to fuse and that cannot be done if electrons constantly get accelerated to relativistic high speeds.

A few years back a lot of folks were bragging that by the year 2019 Lockheed Martin would have those mobile nuclear fusion reactors on large trailors, something like 100 Mega Watt per mobile unit. If they would have pulled that off, Lockheed Martin might be the first company to achieve a market capitalization of 10 trillion US$. That would be gigaenormous because after all 10 trillion = 10 thousand billion… It was supposed to look like this: As you see, the 2013 pipe dream is still not at the scene now in 2019. Why not? Well if plasma theorists keep on using the electron as a magnetic dipole, all of the advanced models they have for plasma behaviour will never depict an accurate picture of physical reality. If in practice electrons come in two varieties, monopole north and south ones, in all of those fusion reactor designs they will move in opposite directions. It is more or less ‘along the magnetic field lines’ because all acceleration caused by the magnetic field makes the electrons accelerate in that direction. And this acceleration in two directions that will grow in a turbulent fashion and make the fusion plasma uncontrolable. In this regard, I mean how turbulence arises, the density of the plasma is an important factor. In a low density plasma the electrons will have plenty of acceleration before they interact with other plasma particles. The denser the plasma is the shorter the length of those interaction free paths, that is obvious. In all present day plasma models, there is nothing that makes electrons accelerate along the magnetic field lines. That means all those models are wrong. Before we go on with wrong mathematical models of plasma behaviour, sometimes the news upon nuclear fusion can be very funny. I just made an intenet search upon ‘Lockheed Martin nuclear fusion’ and I stumbled upon the next hilarious title from Yahoo finance: Lockheed Martin doubles down on cold fusion. Link: https://finance.yahoo.com/news/lockheed-martin-doubles-down-cold-120300203.html As you see in this world there is just never a shortage of idiots; at Lockheed Martin they do not understand why the fusion plasma gets so instable but compared to the total idiots of Yahoo finance the Lockheed people look like pure Einsteinian human material… How stupid and 100% uninformed you must be to think that Lockheed is chasing cold fusion reactors… Ok, a bit more on those math models they use to simulate plasma behaviour. It was in 2016 I came across that weird news from MIT, they simulated very large scale with lots of computer time how plasma should behave. There was actually an electron going round the entire plama vessel. Now my dear reader I was dumbfounded, if the electron is a magnetic dipole, how could it go round? In my world where plasma electrons carry magnetic charge, the only thing the electrons want to do is going round and round… I never found how the model works that was used by the MIT smart asses, but here is a short video of the ‘result’ of the MIT plasma simulation: In the video description there is a link where it even get more hilarious. Let me quote it: A long-standing discrepancy between predictions and observed results in test reactors has been called “the great unsolved problem” in understanding the turbulence that leads to a loss of heat in fusion reactors. Solving this discrepancy is critical for predicting the performance of new fusion reactors such as the huge international collaborative project called ITER, under construction in France. Comment: Don’t worry, ITER will never work if electrons carry magnetic charge. Plus the famous standard model says electrons are magnetic dipoles, so why worry that ITER will fail & fall flat on it’s stupid face? Link where you can find the quote: New finding may explain heat loss in fusion reactors http://news.mit.edu/2016/heat-loss-fusion-reactors-0121 Come on, don’t worry & be happy. # 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.

That´s it for this post. Till updates my dear reader.

# 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.

# New roots of unity (the 3D complex ones) & The rain theorem.

I just finished brewing the 100-th batch of a beer under the names Dark Matter and Spin 1/2 beer. All in all that is an amazing amount of beer; in the past I brewed 35 liter per batch but now it is 40 liter per batch so all in all an amount of something between 3500 and 4000 liters… So ein prosit my dear reader.

The ‘new’ roots of unity aren’t that new, this post is a re-editing of something I posted on 05 Jan 2014 on the other website. Later that year I started this website. Actually these roots of unity are just over five years old. In mathematical terms that is still very young so in that sense they are still new.

Recall the roots of unity in the complex plane are solutions to z^n = 1 and as such these roots are found on the exponential circle (the complex exponential) in the complex plane. As you have found the exponential circle or exponential curve in some space, from that you can always make new roots of unity. That is hardly a mathematical achievement because it is so simple to do once you have found your exponential circle or curve. But in the diverse spaces these new roots of unity behave very different, for example in this post we will add them all up but unlike in the complex plane they do not add up to zero. That is caused by the fact that in the complex 3D space the number alpha is at the center of the exponential circle, as such if we add n roots of unity in 3D space the result is n times alpha. Last year we studied the space of 4D complex numbers and if you would craft new roots of unity in that space it will behave much more like those in the complex plane because in the 4D complex numbers we have 0 as the center of the exponential curve. (For dimensions above 3 the exponential curve always lies in a hyperplane so it can never be a circle.) It always amazes me that you have all those physics people who study string theory but as far as I know never use exponential curves…

Life is beautiful, because how can you do string theory without math like that? But in physics almost everything is beautiful, for example if they explain the outcome of the Stern-Gerlach experiment always 50% of unpaired electrons align with the applied magnetic field and the other 50% for some mysterious reason do the anti-align thing. And if one hour later the same physics professor explains how a permanent magnet can attract some piece of iron, all of a sudden 100% of the unpaired electrons align and all that talk of 50/50 suddenly is not observed… Life inside the science of physics is wonderful; all you have to do is a bit of blah blah blah and if people complain this is not logical at all you simply say: Quantum physics is such that if you think you understand it, you don’t understand it… How wonderful is the life of physics professors; talk some blah blah blah and if people complain you blame them for ‘not understanding quantum things’. For sure that is a beautiful form of life.

But enough of the talking, somewhere in the next seven pictures I did forget to insert a graph of the determinant. Yet I showed you the structure of the non-invertible numbers so often, I think I post it with that fault included. After all why should life be perfect? If life would be perfect you would have no way of improvement and likely that is the moment you die: no more possibility of improvement. As usual the pictures are 550×775 pixels but I had to make the first one a tiny bit longer. Good luck with digesting it & have a bit of fun in the process.

For myself speaking it was fun to read my own two proof for the rain theorem again after five years. Please do not forget that new roots of unity on other spaces can be very different in behavior, after all they are always part of the exponential circle or curve in that space so they will derive their math properties from that. Till updates.