If you change the way the multiplication in the complex plane works, instead of a unit circle as the complex exponential you get ellipses and hyperbola. In this post I give a few examples, where usually the complex plane is ruled by i^2 = -1 we replace that by i^2 = -1 + i and i^2 = -1 + 3i. In the complex plane the unit circle is often defined as the solution to the complex variable z multiplied against it’s conjugate and then solve where this product is one. There is nothing wrong with that, only it leads to what is often told in class or college and that is: The norm of a product of two complex numbers is the product of the norms. And ok ok, on the complex plane this is true but in all other spaces the I equipped with a multiplication it was never true. It is the determinant that does all the work because after all on the complex plane the determinant of a matrix representation of the complex variable z is x^2 + y^2. (Here as usual z = x + iy for real valued variables x and y.)
Therefore in this post we will solve for det(z) = 1 for the two modified multiplications we will look at. I did choose the two multiplications so that in both cases det(i) = 1. That has the property that if we multiply and z against i, the determinant stays the same; det(iz) = det(z).
I simply name complex z with integer x and y also integers, a more precise name would be Gaussian integers to distinguish them from the integers we use on the real line. Anyway I do not think it is confusing, it is rather logical to expect a point in the plane with integers coordinates to be an integer point or an integer 2D complex number z.
Beside the ellipses and hyperbola defined by det(z) = 1, or course there are many more as for example defined by det(z) = 3. Suppose we have some integer point or z on say det(z) = 3, if we multiply that z by i you stay on that curve. Furthermore such a point iz will always be an integer point to because after all the multiplication of integers is always an integer itself. That is more or less the main result of this post; by multiplication with the modified imaginary unit i you hop through all other integer points of such an ellipse or hyperbole. (By the way I use the word hyperbola to be the plural of hyperbole but I do not know if that is the ‘official’ plural for a hyperbole.)
What I found curious at first is the fact that expressions like z = -3 + 8i can have an integer inverse. But it has it’s own unavoidable logic: The 2×2 matrix representation contains only (real) integers and if the determinant is one, the inverse matrix will have no fractions whatsoever. The same goes for any square matrix with integer entries, if the determinant is one the inverse will also be a matrix with only integer entries.
This post is six pictures long, each size 550×1100 and three additional screen shots where I used the desmos graphics package for drawing ellipses and a hyperbole. At last I want to remark that I estimate these results as shown here are not new, the math community is investigating so called Diophantine equations (those are equations where you look for integer solutions) and as such a lot of people have likely found that there are simple linear relations between those integer solutions. Likely the only thing new here is that I modify the way the complex number i behaves as a square, as far as I know math folks never do that. So let me try to upload the pictures and I hope you have fun reading it.
Ok, that was it for this post. I hope you liked it and learned a bit of math from it. I do not have a good category for 2D numbers so I only file this under ‘matrix representations’ because those determinants do not fall from the sky. And file it under ‘uncategorized’. Thanks for your attention and see you in a new post.
Even if you change the multiplication in the plane away from the complex or split complex multiplication it is always easy to find the famous CR-equations. And if you have those in the pocket you can differentiate functions defined on the space you made ‘just like’ on the real line.
After all the CR-equations only need that the numbers commute, you also need to make sure that all basis vectors have an inverse but that are the only restrictions. As far as I know in the math world of the universities the CR-equations are only used in the complex plane, may be some stuff with multiple complex variables and that’s all there is in that part of the math universe.
Originally I wrote the post in just one go and one day later when I read it I was rewarded with a lot of stupid typo’s. Stuff like mindlessly typing x and y where it should have been a and b, also I added the inverse of the imaginary unit because after all I said it was important in the previous post where we looked at CR-equations in the case of a more general n-dimensional space.
An interesting feature of the two dimensional plane is that the two basis vectors 1 and i always commute no matter what you come up with for i^2. I hope the reader is familiar with the fact that on the complex numbers you have the square of i being minus one and plus one for the split complex numbers. In this post we will look at a 2D multiplication that is ruled by i^2 = 1 + i, so you can view this as a minor modification of the split complex numbers.
At first I named the multiplication a ‘strange multiplication’ but one day later I realized that the age old golden ratio has the same property as my imaginary unit i. If you square the golden ratio, that is also the same as the golden ratio plus one. So I renamed it to the golden ratio multiplication. I know it is a little click baity because the golden ratio itself is not used but only the polynomial equation you need to calculate the golden ratio. Universities have their multi-million marketing budgets, still can’t find 3D complex numbers by the way and I have my free tiny click baity golden ratio multiplication. I think it is an allowed sin.
Did you know that if there is something wrong with a product, it always needs massive marketing budgets. Just look at Coca Cola, without the advertisements the stuff should gradually sell less and less because it is not a healthy product. You can say it is an unhealthy product so there is something wrong with it and as such it needs all that marketing stuff in order to survive.
This post is only four pictures long, I hope it is a bit more easy to digest because it is only two dimensions. So lets go to the math in the four pictures:
Ok, that was it for this post. Thanks for your attention.
There are many ways to introduce CR-equations for higher dimensional complex and circular numbers. For example you could remark that if you have a function, say f(X), defined on a higher dimensional number space, it’s Jacobian matrix should nicely follow the matrix representation of that particular higher dimensional number space. I didn’t do that, I tried to formulate in what I name CR-equations chain rule style. A long time ago and I did not remember what text it was but it was an old text from Riemann and it occured he wrote the equations also chain rule style. That was very refreshing to me and it showed also that I am still not 100% crazy…;) Even if you know nothing or almost nothing about say 3D complex numbers and you only have a bit of math knowledge about the complex plane, the way Riemann wrote it is very easy to understand. Say you have a function f(z) defined on the complex plane and as usual we write z = x + iy for the complex number, likely you know that the derivative f'(z) is found by a partial differentiation to the real variable x. But what happens if you take the partial differential to the variable y? That is how Rieman formulated it in that old text: you get f'(z) times i. And that is of course just a simple application of the chain rule that you know from the real line. And that is also the way I mostly wrote it because if you express it only in the diverse partial differentials, that is a lot of work in my Latex math typing environment and for you as a reader it is hard to read and understand what is going on. In the case of 3D complex or circular numbers you already have 9 partial differentials that fall apart into three groups of three differentials each. In this post I tried much more to hang on to how differentiation was orginally formulated, of course I don’t do it in the ways Newton and Leibniz did it with infitesimals and so on but in a good old limit. And in order to formulate it in limits I constantly need to divide by vectors from higher dimensional real spaces like 3D, 4D or now in the general case n-dimensional numbers. That should serve as an antidote to what a lot of math professors think: You cannot divide by a vector. Well may be they can’t but I can and I am very satisfied with it. Apperently for the math professors it is too difficult to define multiplications on higher dimensional spaces that do the trick. (Don’t try to do that with say Clifford algebra’s, they are indeed higher dimensional but as always professional math professors turn the stuff into crap and indeed on Clifford algebra’s you can’t divide most of the time.)
May be I should have given more examples or work them out a bit more but the text was already rather long. It is six pictures and picture size is 550×1100 so that is relatively long but I used a somehow larger font so it should read a bit faster.
Of course the most important feature of the CR-equations is that in case a function defined on a higher dimensional space obeys them, you can differentiate just like you do on the real line. Just like we say that on the complex plane the derivative of f(z) = z^2 is given by f'(z) = 2z. Basically all functions that are analytic on the real line can be expanded into arbitrary dimension, for example the sine and cosine funtions live in every dimension. Not that math professors have only an infitesimal amount of interest into stuff like that, but I like it. Here are the six pictures that compose this post, I hope it is comprihensible enough and more or less typo free:
Ok that was it, thanks for your attention and I hope that in some point in your future life you have some value to this kind of math.
On this entire website when I talked about a matrix representation it was always meant as a representation that mimics the multiplipaction on a particular space as say the 3D complex numbers. And making such matrices has the benefit you can apply all kinds of linear algebra like matrix diagonalization, or finding eigenvalues (the eigenvalue functions) and so on and so on. So the matrix representation was always the representation of a higher dimensional number. Big E is very different, this matrix describes the multiplication itself. As such it contains all possible products of two basis vectors and since this is supposed general theory I wrote it in the form of an nxn matrix. For people who like writing computer code, if you can implement this thing properly you can make all kinds of changes to the multiplication. As a matter of fact you can choose whatever you want the product of two basis vectors to be. So in that sense it is much more general as just the complex or the circular multiplication. I do not like writing computer code myself that much but I can perfectly understand people who like to write that. After all every now and then even I use programs like PARI and without people that like to write code such free programs are just not there. The math in this post is highly descriptive, it is the kind of math that I do not like most of the time but now I finally wrote this matrix down it was fun to do. If you are just interested in some fixed number space as say the 3D or 4D complex numbers, this concept of big E is not very useful. It is handy when you want to compare a lot of different multiplication in the same dimension and as such it could be a tool that comes in handy.
The entries of this matrix big E are the products of two basis vectors so this is very different from your usual matrix that often only contain real numbers or in more advanced cases complex numbers from the complex plane. I think it could lead to some trouble if you try to write code where the matrix entries are vectors, an alternative would be to represent big E as n square nxn matrices but that makes it a bit less overseeable.
May be in linear algebra you have seen so called quadratic forms using a symmetric matrix. It is a way to represent all quadratic polymonials in n variables. Big E looks a lot like that only you now have vectors as entries.
I did choose the number 1 to be the very first basis vector, so that would give the real line in that particular space. Of course one of the interesting details is that all analytic functions that you know from the real line can easily be extended to all other spaces you want. For example the sine and cosine or the exponential function live in all kinds of spaces in all kinds of dimensions. As such it is much much broader as only a sine on the real line and the complex plane. This post is five pictures long each 550×1100 pixels. I made them a bit larger because I use a larger font compared to a lot old posts. There are hardly mathematical results in this post because it is so descriptive. Compare it to the defenition of what a group is without many examples, the definition is often boring to read and only comes alive when you see good examples of the math involved.
If you want to try yourself and do a bit of complex analysis on higher dimensional spaces, ensure your big E matrix is symmetric. In that case the multiplication commutes, that is AB = BA always. If you also ensure all basis vectors are invertible you can find the so called Cauchy-Riemann equations for your particular multiplication. Once you have your set of CR equations you can differentiate all you want and also define line integrals (line integral are actually along a curve but that does not matter). A simple counter example would the 4D quaternions, they do not commute and as such it is not possible to conduct any meaningful complex analysis on the space of quaternions. End of this post and thanks for your attention.
I had a bit of computer problems lately, my oldest computer did not allow me to enter the BIOS and as such I could not tell from what hard disk to boot and that was the end of the motherboard in that old computer. It was my last Windows XP computer and looking at the latest Windows 7, 10 or 11 stuff there is not that much progress. Windows OS systems become more and more a fashion thing because most of it is outdeveloped and hard to improve upon… But lets go to this small post about the long title above: total angular momentum.
To put it simple: In an atom or molecule with unpaired electrons in it physics people often think of the spin of the unpaired electrons as vectors that get projected against the vertical direction if a magnetic field is applied. Those projections are treated as scalars and as you might have guessed it is either +1/2 or -1/2. (Here I leave the constant of Planck out for simplicity.)
Ok my version of magnetism is that electrons are magnetic monopoles so they have either a north pole charge or a south pole magnetic charge. Adding these up works just like adding up the electrical charges in an atom or a molecule; because after the humble opinion of your servant electron magnetism looks just like electron electricity: permanent and monopole. So that is all to know, basically it is very simple once you have seen a few times what they do and mean with that “Total angular momentum in the x-direction” because that sounds so complicated. But all in all the calculations they do are just like adding up monopole magnetic charges so they were on the right track more or less a long time ago. This post is four pictures and for me that is a bit hard because I always made the backgrounds on the computer that broke down. And that old graphics program can’t run on Windows 7 or higher while modern graphics programs like GIMP just don’t have the features to make such a background in a few clicks. Of course a long time ago this doomsday day was foreseen and I saved a lot of backgrounds as png files. So I opened that folder and what? It’s fucking empty…:( Well lets go to the pictures:
The most hilaric picture is the fourth one where two electrons with opposite spins are in different energy levels in an atom or molecule. And if you want your electron spin as a vector, how the hell should you get opposite pointing vectors while the electrons are in different parts of the atom or molecule? That is very weird behavior and as such it always goes unexplained to the extend it is never mentioned as a problem. Just like all that other stuff that is wrong with electron magnetism as you view them as tiny magnets and not as magnetic monopoles.
Well thanks for your attention and see you in another post.
This is a 56 year old documentary about the machine depicted below and imitates the original Stern-Gerlach experiment in many ways. Now in the year 1967 (I became 4 years old that year) you could also have used any television they had in those long lost years and split the electron beam with a magnetic field like I once did. It is all done beautiful, there is even an oven to vaporise the metal, in this case cesium. All these decades the explanation of the result of this experiment has not changed: Everything is explained by tiny magnets that align or anti-align with the applied magnetic field. And, just like at present day, the anti-alignment is not explained at all. How can that happen in the first place and how can it be stable? From five screenshots I made four pictures and after that I will link to the video that has a length of about 25 minutes.
In the next picture you see the oven, I did not know you could vaporise metal so easily.
Professor Jerrold R. Zacharias is going to do some explanation of how this all is supposed to work and as such you see the well known iron filings and a magnetic field made by these two coils you see below.
As usual a lot of the explanation is swiped under the carpet (just like in 2023) and at the bottom of the picture you see that magnet being alligned or anti-aligned with the magnetic field. But why does that piece of cardboard hang from four lines or four thin ropes? What would happen if he just glued two of the lines at the north and south pole of the magnet? In that case it would be very very hard to get an anti-alignment of the bar magnet with the magnetic field. Just as impossible as balancing a pencil on it’s top for say a few minutes of time. But the physics professors just never talk about stuff like that, it is only “The standard model of particles physics is so amazingly correct” and more of that nonsense. Well I agree that if you swipe all this stuff under the carpet it all looks tidy..
The cesium used has atomic number 55 and a mass of say 133 atomic units. If here just like in the original SG experiment it is only one unpaired electron that causes the effects observed, in that case the electron moves a mass about 250 thousand it’s own electron mass. On top of that, the electron is very tiny and it is supposed that a magnetic field with a gradient can give a force on the electron. In my opinion that is where fantasyland begins. Viewing electrons as magnetic monopoles removes a lot of what so hard is to understand, like the anti-alignment thing.
As picture number four you see the results of different field strengths of the magnetic field, it is a lovely old fashioned machine.
Here is the video:
The video is only posted on Youtube about 9 months ago, that is why I missed it in the past. After all you don’t go out every month looking for the lastest video about the SG experiment! Let me end this post with a comical note. For a picture on the other website I combined an idea about electron spin with that old painting known as “The Scream”. Here it is:
That was it for this post. May be it is time to do some math again, now it looks as if I only discuss video’s. Thanks for your attention.
This stuff is supposed to work on 12 of those qubits also known as silicon qubits. Anyway these are loose electrons, so one electron per qubit. It is well known that if you have a computer, the thing must be possible to flip individual bits and on a quantum computer it must be possible to flip the spin of an electron if you want to use that as a qubit. It is no secret that for a couple of years I think electrons are magnetic monopoles and the last time I have arrived at the conclusion that the magnetic charge of the electron is just as permanent as it’s electric charge. In particular this means that it is just not possible to flip the spin of an individual electron and therefore this whole Intel 12 electron spin thing can never ever work properly. You cannot flip the spin of an individual electron and since their spin is permanent there is also not much possibility that you can bring any pair into entanglement or stuff like that. A few posts ago I showed you some kind of rule for electron transitions in atomic or molecular orbitals; the only transitions allowed are those without spin flip. That is very interesting and one of the details that validate spin is a permanent monopole magnetic charge and not some kind of tiny magnet or a vector for that matter. I made the next screenshot from an advertisement from Intel:
So that stuff will never work as I am correct in my view on electron spin (not tiny magnets but magnetic monopoles). The fact that likely the magnetic charge is permanent has all kinds of far reaching consequences, for example in chemistry you now must often have the right kind of electron at the right time for a chemical reaction to occure or proceed. So it is not true that at the last moment the electron will flip it’s spin if that is needed in the chemical reaction (this despite the tiny tiny energy difference if spin flip would be possible), so if a particular electron just isn’t there the chemical reaction will stop or alter or whatever what. Here is the advertisment video from Intel:
The whole Intel 12 spin qubit thing is explained a bit by some folks from NYU. Likely NYU will stand for New York University. For this kind of video they found in interesting format; you are looking at a group of people while one of them does the most of the talking. That is far less boing as looking at one person sitting in a room. Here is how it looks:
The interesting thing on the social side is that Intel will send one or more of these spin things to the Dutch unversity of Delft. Of course Delft is famous for discovering the Majorana particle, build a quantum computer on that thing that did never exist, collect 40 million or more US$ from Microsoft, had to withdraw the Majorana particle claim and so on and so on. If some weirdo’s (like in Delft) just do not want to listen to my long list of problems with electron spin, they are always allowed to make a bunch of fools of themselves one more time. A long long time ago Britney Spears sang “Baby hit me one more time” and that beautiful song they must sing once more over there at TU Delft. Here is the video:
That was it for this post, it is about time that I start writing an old fashioned math post instead of all these posts around a video. Well thanks for your attention and lets wait for the likely failure of the TU Delft on this small but important detail.
Molecular oxygen has the interesting property that it has a so called non-binding electron pair and it is know that in such a non-binding pair the electrons have the same spin. The way I view it for a couple of years is that the electrons are magnetic monopoles and that explains their behavior, so they don’t have a spin orientation so to say. There is a relative simple experiment that I can’t do but if you have access to liquid oxygen and have some heavy electric magnets, you can do a simple experiment to see if the individual oxygen molecules behave like magnetic monopoles yes or no. On Youtube there are plenty of videos showing the magnetic properties of oxygen by pouring the liquid stuff over a strong magnet or better: Pour it in between the two poles of two magnets. After some time the oxygen will have separated over the two magnetic poles, if you can flip the magnetic polarity with the electric magnet, all oxygen should go loose and try to get to the other pole. Once you have a setup like this or may be you have only strong permanent magnets, once the oxygen is separated you can try to put some of the liquid into a plastic bag. Let the oxygen become gas and see if the bag as a whole has magnetic monopole properties. That would be funny.
The only experiments I have done myself are those old ones with two televisions and try to that separation in the electron stream. It has to be remarked however that I once almost bought an old oscilloscope because you can let the electron beam go around so it becomes a circle on the glass tube. So that kind of experiment is still waiting…
Back to the oxygen molecule with it’s non-binding electron pair: Once I heard that I thought that likely oxygen must be in a lower energy state this way. May be that having an electron pair that pushes things apart give rise to a lower energy state. As far as I remember I never tried to look up the somehow more fine details. So the video from the Action Lab came around for free and has all the information I needed. It has even information I was not waiting for like in the next screen shot:
In the next picture I grouped it a bit from four screen shots. And may I thank the folks from the chemical sciences for measuring all that energy stuff? If the shapes below are correct, you see the molecule with the bonding pair becomes a linear shaped molecule that is clearly very different from triplet oxygen.
The video has the title Singlet Oxygen Is Scary! And yes there is something to say for a classification like that. Here is the video:
The reason as why I like this kind of stuff is that molecular orbitals make chopped meat of all that stuff that looks so sacred. Stuff like the Pauli exclusion principle, or the Aufbau principle and or the Hund rule for placing an extra electron. Stuff like that always assumes that all electrons are the same and it is just a matter of some magnetic vector pointing in this or that direction that solves the thing. But I think electrons have a monopole magnetic charge that is permanent. So there are two kinds of electrons and that has all kinds of far reaching consequences. Well that is all for some other day. In the meantime I am going to pop up a second beer, upload this post and after that I will ask the electrons in the beer the following question: Why do electrons never drink beer? May be it is time to split. Goodbye.
This is one of those videos based in large part on that old theorem of Frobenius that says there are no 3D fields containing the complex numbers from the complex plane. There is another similar proof out, also rather old, that says the same. Well there is nothing new about that, after all there is nothing in the 3D complex numbers that squares up to -1. (See a few posts back when I wrote the first so called general theory where I show that in all odd-dimensional number systems you cannot solve for X^2 = -1). It is also not much of a secret that beside 0 there are much more non-invertible numbers in 3D real space. So no, it is not a field but I knew that for over 30 years my dear reader. Over the years I have joked at many occasions that the only thing math professors are good at is at saying “We cannot find the 3D complex numbers”. And that is so deep ingrained for some strange reason that actually they can’t. Ok ok there is also the question of competence into doing new math research. As a matter of fact most math professors are relatively bad at doing researh on new math stuff. Math professors are rather good at reproducing the good math from the past, in such a comparison they are much more like a classical orchestra that plays old works of music from Bach or any other long dead composer. But most professional musicians do not compose very music for themselves, it is their job to repeat the old music from the past and that’s it. Math professors are just like that often, now that is not only negative or so. For example the proof of the prime number theorem (how much primes are there asymptotically under a given magnitude?) was hard to find and it is long. If you can repeat that proof in front of a class, that is for sure a highly cognitive thing to do. I am not saying all math professors are dumb.
But as far as I know it most of them have a strong tendency for shallow thought, better fast than accurate and much slower. For example in 30 years of time every now and then some folks discover the 3D complex numbers again and oh oh what exiting it is to find the conjugate! They do it all wrong, yes one 100% take the wrong kind of conjugate and after that all calculations fade into total nonsense and as such they too conclude there is not much going on with 3D complex numbers. I have never seen anyone finding the 3D complex exponential, that is what I mean with shallow thinking. People just project on the new stuff things they know from the old stuff and the new stuff ‘must’ for some strange reason obey the old stuff and when that does not happen they begin to cry like the crybabies they are. I did that too on a few occasions but after doing stupid twice I understood what I was doing wrong: I just was not open minded about how the new stuff actually worked.
Now when I viewed the video for the first time a few weeks back I was relatively annoyed, not only because of the dumb content but also because of the title (3D complex numbers do not exist). But the days after when this new post came to my mind I just felt so tired. Why is it always so fucking stupid and shallow? Why is that year in year out why are all those math professors never evolving, why is it always fucking stupid and no oversight at all? So I did not have a good feeling about it and because the video is so stupid I could easily write 20 different posts and that made me feel even more tired and exhausted because these people just don’t improve. Anyway I decided to keep it short and simple and only point out that the two most important new numbers in the 3D complex numbers are the number tau and alpha that are related to the 3D complex exponential. These two numbers if you multiply them that gives zero, the property that non-zero numbers can multiply to zero is called divisors of zero.
And guess what, the video from Michael Penn begins with a small list of the properties that 3D numbers have to obey in case you will have something meaningful or so as Michael says it. Well in his small list is indeed it must be a field and also it does not have divisors of zero. Furthermore Michael did not figure it out himself but uses the work of another person so we have everything combined: Shallow thinking combined with repeating the work of other people.
Here is the video:
Left is placing the two pictures that compose this post here below. And I have an extra pdf with that Frobenius stuff in it, it is hard to read and is precisely the kind of math I avoid to write down. You can find the Forbenius classification from page 26 of the pdf.
For myself speaking: I consider this pdf a disaster to read. It is all overly complicated formulated with not much math content to it. But it contains a bit of the old school Frobenius stuff so it has some value although it has weird notation that is hard to grasp…
Over time I have come to understand that the 3D complex numbers are an expansion from the real line just like the complex plane is. For example: multiplication in the 3D complex numbers is ruled by the imaginary unit j who’s third power equals minus one: j^3 = -1. But if you have seen only the complex plane your entire life, in that case you are likely tempted to try and think of it as e to some imaginary power like on the complex plane. But you cannot do that because the number i from the complex plane does not live inside the 3D complex numbers. So for me they have equal right to exist or to study, they are very different but at least you can do complex analysis on the 3D complex numbers and you can’t do that on the famous quaternions.
Now I don’t want to sound as a sour old man, after all this post was not very funny to write but on the scale of things it is just a luxery problem. Let me end this post with thanking you for your attention.
I have this pdf about a year now. Last year when I was doing that matrix form of the theorem of Pythagoras I came across this pdf and I saved it for another day. Last year when working on that matrix Pyth stuff you always get a large bunch of those determinants of those minor matrices. And I thought like “Hey you can put them in an array and in that case the length of that array is the volume of that parallelepipid”. Well in this pdf Peter is just doing that: He does not use a nxd matrix but only two columns because the goal is to make a product. Last year after say an hour of thinking I decided to skip that line of math investigation and instead concentrate more on if it would be possible to make that none square nxd matrix into a square nxn matrix.
Another reson to save this pdf for a later day was the detail it was also about differential operators. Not that I am very deep into that but the curl is a beautiful operator and as you likely know can be expressed with the help of the usual cross product in three dimensions. To be honest I considered this part of the pdf a bit underwhelming. I really tried to make some edible cake from that form of differential operators but I could not give a meaning to it. May be I am wrong so if you want you can try for yourself.
The pdf contains much more as the above, a lot of cute old identies and also some rather technical math that I skipped because sometimes I am good at being lazy…;)
My comments are four pictures long and after that I will try to hand the dpf from Peter into it.
What is it that makes those present day picture generators so bad at things having five fingers? I think those genrative ai machines don’t count on their fingers.
I posted this in the category Pythagoras stuff because it is vaguely related to that matrix version of the Pythagoras theorem. Ok that was it for the time being. May be the next post is from some nutty professor who claims in a video that 3D complex numbers don’t exist. Or something completely different. Who knows where our emotions bring us?