This channel is of course not meant for political statements but this fucking war is a fucking distraction from doing math. While writing this post in small pieces I was constantly dissatisfied with the level of math (too simple, done too often in the past etc). But when I was finished and read it all over, all in all it was not bad. It is a short oversight of how to find shere-cone equations and once more how to find a conjugate. And once more: The math professors are doing it wrong when it comes to finding the conjugate for 32 years now & the clock keeps on ticking. On the one hand this is remarkable because if you do internet searches a lot of people understand that the Jacobian matrix should be the matrix representation for the derivative of a complex valued function in say three dimensional space. So that goes good, but when it comes to taking the conjugate for some strage reason they all keep on doing it wrong wrong and wrong again so they will never find serious math when it comes to number systems outside the complex plane or for that matter the quaternions.
The setup of this post is as next: 1) Explaining (once more) how to find the conjugate. 2) Calculating the two sphere-cone equations. 3) The solution of these S-C equations is the exponential circle that is, 4) parametrisized by three so called coordinate functions that we 5) substitute into both S-C equations in order to get 6) just one equation.
Basically this says that the complex and circular multiplication on our beloved three dimensional space are ‘very similar’. Just like that old problem of solving X^2 = -1 is impossible in these spaces while the cubic problem of X^3 = -1 has only trivial solutions like basis vectors. That too is ‘very similar’ behaviour.
Anyway this post is six pictures long.
That was it for this post on my beloved three dimensional complex numbers.
I know I know I have published stuff like this before and over again. But that was also years ago and now I do it again it is still not boring to me. After all the professional math professors still are not capable of finding those beautiful exponential circles and curves simply because they all imitate each other. And they imitate each other with how to use and find a so called conjugate. And if you use the conjugate only as some form of ‘flipping a number into the real axis’ all your calculation will turn into garbage. Anyway by sheer coincidence I came across two videos of math folks doing it all wrong. One of the videos is even about the 3D circular numbers although that guy names them triplex numbers.
You can do a lot with exponential circles and curves. A very basic thing is making new de Moivre identities. From a historical point of view these are important because the original de Moivre identity predates the first exponential circle from Euler by about 50 years. In that sense new de Moivre identities are very seldom so you might expect some interest of the professional math community…
Come on, give me a break, professional math professors do a lot of stuff but paying attention to new de Moivre identities is not among what they do. But that is well known so lets move on to the four pictures of our update. After that I will show you the two video’s.
Let us proceed with the two video’s. Below you see a picture from the first video that is about 3D circular numbers and of course the conjugate is done wrong because math folks can only do that detail wrong:
Below you can see the video:
By all standards the above video is very good. Ok the conjugate is not correct and may be the logarithm is handled very sloppy because a good log is also a way to craft exponential circles. But hey: after 30 years I have learned not to complain that much…
The next video is from Michael Penn. He has lots of videos out and if you watch them you might think there is nothing wrong with that guy. And yes most of the time there is nothing wrong with him until he starts doing all kinds of algebra’s and of course doing the conjugate thing wrong. Michael is doing only two dimensional albebra’s in the next video but if you deviate from the complex plane very soon you must use the conjugate as it is supposed to be: The upper row of the matrix representation.
Here a screen shot with the content of the crimes commited:
Most of his other video’s are better, but his knowledge is just a reflection of what professional math professors think about conjugation. It is always just a flip in the real axis.
Here is his vid:
Ok, that was it for this appendix to the previous post.
It is past midnight, this evening I brewed hopefully a lovely beer. It is late so let me keep the intro short. The last time I often lack stuff for new posts because most of the theory of 3D complex and circular numbers has been posted in this collection of 200+ posts. And you cannot keep it repeating over and over again, if all those years in the past the math professionals did just nothing, why would they change their behaviour in the future? Beside that I do not want have anything to do with them any more, it is and stays a collection of overpaid weirdo’s and there is nothing that can change that. On the other hand one of the most famous expressions in math is and stays the exponential circle in the complex plane. That stuff like e^it = cos t + isin t is what makes many hearts beat a tiny bit faster. So when someone comes along stating that he found an exponential circle in spaces like 3D complex numbers, you might expect some kind of attention. But no, once more the math professionals prove they are not very professional. Whatever happens over there I do not know. May be they think because they could not find this in about 350 years no one can so it must all be faulty. For me it was a big disappointment to get discriminated so much, on the other hand it validates that math professors just are not scientists. Ok they have their salary, their social standing, their list of publications and so on and so on. But putting lickstick on a pig does not make it a shining beauty, it stays a pig. So a math professor can have his or her prized title of professor, that does not make such a person a scientist of course. At best they show some form of imitating how a scientist should behave but again does such behaviour make these people scientists? Anyway a couple of days back at the end of a long day I typed in a search phrase in a website with the cute name duckduckgo.com. Sometimes I check if websites like that track this very website and I just searched for “3D complex numbers”. The first picture that emerged was indeed from this website and it was from the year 2017. I looked at it and yes deep in my brain it said I had seen it before but what was it about? Well it was the product of two coordinate functions of the exponential circle in 3D. It is a very cute graph, you can compare it to say the product of the sine and cosine function in the complex plane. So I want to avoid repeating all that has been written in the past of this website but why not one more post about the 3D exponential circles?
In the end I decided to show you how likely one of those deeply incompetent “professional” math professors would handle the concept of conjugation. Of course one hundred % of these idiots and imbeciles would do it as “This is just a flip in the real axis or in the x-axis” and totally spoil the shere-cone equation and only find weird garbage that indeed better cannot be published. After all our overpaid idiots still haven’t found the 3D complex numbers, I am still living on my tax payer unemployment benefit and life, well life will go on. But it is not only math, with physics there are similar problems and they all boil down to that often an idiot does not realize he or she is an idiot.
But let’s post the six pictures, may I will add an addendum in a few days, may be not. Here we go:
Isn’t that a cute graph or not?
Ok, may be in will write one more appendix about how these kind of coordinate functions of exponential circles give rise to also new de Moivre identities. That is of interest because the original de Moivre identity predates the Euler exponential circle by about 50 years.
Yet once more: Likely there is just nothing that will wake up the branch of overpaid weirdo’s known as the math professors… So for today & late at night that was it. Thanks for your attention.
It is no secret that I think electrons are not “tiny magnets” having two magnetic poles but that electrons are magnetic monopoles just like they are electric monopoles. Viewing electrons as small tiny magnetis leads to all kinds of logical contradictions. For example a permanent magnet is always explained as a thing where all electron spins of unpaired electrons align and as such together they build that macroscopic magnetic field as you know from stuff like a bar magnet. But in chemistry an important binding element in molecules is the electron pair. Yet now there is something like the Pauli exclusion principle and the two electrons must have opposite spin. End example. So in a permanent magnet the electrons must align in order to be attractive to each other while in chemistry the opposite must happen. My dear reader this is not logical. Also, why do we find only electron pairs? Well if you look at it as there are two kinds of electrons with both a magnetic charge either ‘north pole’ or a ‘ south pole’ charge, that explains why we only observe electron pairs. If the ‘tiny magnet’ model was true, we should observe all kinds of electron configurations like 5 electrons in a circle or whatever you can make with tiny magnets. What I self consider a strange thing is that people from the physics community never ever themselves say that all their views on magnetism are often not logical. Are they really that stupid or do they self censor in order not to look stupid?
Anyway five years back in the year 2017 I was studying a new way of making computer memory by IBM: so called racetrack memory in nano wires. I was highly puzzled by that because one of the main researchers said that you cannot move the domain walls of magnetic domains with magnetic fields. You could move the domains themselves but not the walls and I was as puzzled as can be. Yet that same day I found a possible answer: the magnetic domains of say iron can be moved by magnetic fields because they have a surplus of a particular kind of electrons. So two magnetic domains separated by a domain wall must have opposite magnetic charges. In the next picture you get the idea of what IBM tried to do:
It was a cute idea but IBM had to give up on it because they did not use insights that are logical but kept on hanging to the tiny magnet model.
So in the long pdf that is squarely based on the official version of electron spin (the tiny magnet model) has all kinds of flaws in it. For example in the next picture that all does not pan out because those small arrows are not there in reality if electrons carry magnetic charge just like they carry electric charge:
And life, well life will go on…
Ok for me it is an experiment to try include a pdf file, if it fails I will hang this pdf in the pdf directory of the other website and link to that file.
I leave it this way and do not try to make the pdf visible. After all if you are interested in stuff like this you must download it anyway because it is a few hundred pages long. And it is a funny read so now and then, for example yesterday I came across a section where they took the outer product of two (vector) electron spins and I just wonder WHY?
Ok, let me push the button named Publish and say salut to my readers.
This is a problem I found about thirty years ago and I was never ever able to solve it. The problem as I formulate it is about finding a so called ‘composition root’ to the exponential function. Just keep it simple, say the composition ‘square root’. If we denote that as r(x) what I mean is that this function if composed with itself gives the good old exponential function: r(r(x) = f(x) = e^x. There are many interesting aspects to this problem. For example take a piece of paper and a pencil and draw the graph of the exponential function and the identity function. It is now very easy for every point on your graph of the exponential function to find the graph of the double composition f(f(x) = e^(e^x)). But, as far as I know, you cannot go back and given the function f(x) find it’s composition root r(x). It is very well possible that this problem is solved in the theory of dynamical systems. If memory serves we once had a lesson in when a family of functions could be interpolated but that was 30 years back and what I want is explicit expressions and formula’s and not only a vague existencial proof without a way to find an explicit answer.
Back in time before the logarithm was invented, the people of those long lost centuries had a similar problem understanding what exponential behaviour was. And you can go a long way in understanding exponential behaviour but say for yourself; without knowledge of the logarithm that kind of knowledge is far from being optimized.
In this post I only talk about the composition square root but of course any n-th root should be possible and as such giving rise to the idea that you can iterate or compose the exponential function also a real number amount of times. I have to admit I also have no proofs for the solution to this all being unique, but you should be able to differentiate all stuff found and it should still be coherent so my gut feeling says the solution is unique. My guess is there is only one ‘composition square root’ r(x) that is as smooth as f(x) itself…
This post is only two pictures long so here we go:
And it is also the end of this post. Give it a thought and if you are able to make some inroads on this that would be great. But all in all I think we do not have the math tools to crack this old old problem.
Despite my slowly detoriating health the last year was a remarkable fruitfull year when it comes to new stuff. So I selected five highlights and of course that is always a difficult thing. Two of the highlights are about magnetism and the other three are just math. Once more: The fact that I include two magnetic highlights does not mean I am trying to reach out to the physics community in any meaningful way. If these idiots and imbeciles keep on thinking that electrons have two magnetic poles, be my guest. There is plenty of space under the sun for completely conflicting insights: Idiots and imbciles thinking that electrons have two magnetic poles and more moderate down to earth people that simply remark: for such a bold claim you need some kind of experimental evidence that is convincing. But 2021 was a very good year when it came to math; I found plenty of counter examples to the so called last theorem of Pierre de Fermat. I was able to make a small improvement on the so called little theorem of Fermat. A very important detail is that I was able to make those counter examples to the last theorem so simple that a lot of non math people can also understand it. That is important because if you craft your writings to stuff only math professors can understand, you will find yourself back in a world of silence. Whatever you do there is never any kind of response. These math professors were not capable of finding three or four dimensional complex numbers, they stay silent year in year out so I have nothing to do with them. In the year 2021 I classified the physics professors to be the same: Avoid these shitholes at all costs!
After having said that, this post has eight pictures of math text and it has the strange feature that I am constantly placing links of posts I wrote in the last year. So lets go:
Once more: Zero reaction from the overpaid idiots & imbeciles.This is the tau for the three dimensional circular numbers! Not for 3D complex numbers.Next link contains the proof of the improved little theorem as posted on 20 April:
I think that if you show the above animated gif to a physics professor and ask for an explanation, likely this person will say: “Oh you see the electrons aligning with the applied external magnetic field, this all is well understood and there is nothing new under the sun here”. Of course that kind of ‘explanation’ is another bag of bs, after all the same people explain the results of the Stern-Gerlach experiment via the detail that every electron has a 50% probability that it will align with the applied external magnetic field (and of course 50% that it will anti-align). In my view that is not what we see here. As always in the last five+ years an explanation that electrons are magnetic monopoles with only one of the two possible magnetic charges is far more logical.
This year in the summer I wrote an oversight of all counter examples to the last theorem of Pierre de Fermat I had found until then. It became so long that in the end I had three posts on that oversight alone. I wrote it in such a way that is starts as easy as possible and going on it gets more and more complicated with the counter example from the space of four dimensional complex numbers as the last example. So I finished it and then I realized that I had forgotten the space of so called split complex numbers. In the language of this website the split complex numbers are two dimensional circular numbers. It is just like the complex plane with two dimensional numbers of the form z = x + iy, only now the square of the imaginary unit is +1 instead of i^2 = -1 as on the complex plane. So I made an appendix of that detail, I consider this detail important because it more or less demonstrates what I am doing in the 3D and 4D complex number spaces. So let me put in one more picture that is the appendix of the long post regarding the oversight of all counter examples found.
I hope this brings some clarity to the minds of math people.
All that is left is place a link to that very long oversight:
Ok, so far for what I consider the most significant highlights of the previous year. And oops, since I am a very chaotic person before I forget it: Have a happy 2022! It is time to say goodbye so think well and work well my dear reader.
If you start commenting on bad videos you will have a busy hobby for the rest of your life. But there are also reasons to take a look at these videos, for example the math video is horrible but the path of calculation shown is rather beautiful. The other video is about magnetism and when I viewed it for the first time it was really late at night and only after a good night sleep I realized how horribly bad that video was. But it was the magnetism video that made me look up the average size of the so called magnetic domains and that was when I found that PERFECT gif. So I cannot say it was all a waste of time, that perfect gif is made with something that is named a Kerr microscope and with such a device you can make magnetic domains visible. Years ago, if memory serves it was Feb 2017, I was studying so called ‘racetrack memory’ that was under development by IBM. That IBM project failed because they kept on hanging to electrons being ‘tiny magnets’ with two magnetic poles, because that is likely not true all their work failed. Anyway they came up with the fact that you cannot move magnetic domains with magnetic fields and I totally freaked out. Late at night I realized that within my broader development of understanding magnetism at the electron level, the IBM findings were logical if magnetic domains in say Iron or so, always have a surplus of either north pole monopole electrons or south pole monopole electrons. Domain walls separate the two kinds of magnetic domains. Itis a pity that about five years back I never heard of those Kerr microscopes. Again I want to highlight that I do not want to convince anybody that electrons are the long sought magnetic monopoles. I have done that for six or seven years and it was only in this year 2021 that I arrived at the conclusion that physics professors are just as stupid as the average math professor. It is a pile of garbage so it is not much of a miracle that six or seven years of trying to apply logic did not work at all. So from this year on going into the future the physics professors have the same status as the math professors: A pile of rotten garbage that you must avoid at all times at all costs needed.
After having said that, this post is five pictures long where I comment on the two horrible videos. Below that I will post the two videos so you can see for yourself (or may be you want to see them first). And at the end you can see that perfect gif where magnetic domains change in size due to the application of an external magnetic field. Also back in 2017 I more or less figured out how magnetic domains will change if you approach a piece of iron with a permanent magnet. What you see in the gif is more or less precisely that: Some domains grow while domains next to that shrink.
Ok, here we go:
Now we can go to the first video, the math one:
I found the magnetics video by doing an internet seach on ‘The Stern-Gerlach experiment for iron’. It is disappointing that almost no significant results are there. Some of stuff out of the 2030-ties of the last century but that was all behind pay walls. Very high in the rankings came the next video that uses iron filings to mimic or imitate the Stern-Gerlach experiment. The video guy should have used magnets on only one side, if that resulted into attraction & repulsion of the iron filings he would have gotten a standing ovation from me. Without any insult; the way he executed this experiment is a true disaster only showing he does not understand why the SG experiment is so important. And by the way: If my idea of electrons being magnetic monopoles is in fact correct, you do not have to use inhomogeneous magnetic fields. Everything will do; even the most constant magnetic field in space and in time will do. But again after so many years of talking to deaf ears from stupid physics people, I have lost my desire to convince anybody any longer..
With magnets on two sides; of course it will spread out! This is stupid!!!!!
Ok, I have never hung any animated gif into this WordPress website so let’s check it out if it works properly:
As you see: Some domains grow while adjecent ones shrink.
I found this animated gif in a wiki: Magnetic domain. That was it for this post. Thanks for your attention.
This year in January I started finding more and more counter examples to the last theorem of Pierre de Fermat. I started in the space of 3D complex numbers using the so called divisors of zero you can find there. Rather soon after that I found those very easy to understand counter examples using modular arithmetic modulo N where N was the product of two prime numbers. Along the way I found out there is also a cute improvement to make to the so called little theorem of Fermat, that was one of the many cute results of this year. Simple example: Take 107^2, the old little theorem of Fermat says you must take it modulo 2 and you get the number 1. With the improved little theorem you do not take it mod 2 but mod 2*107 or mod 214. And voila: 107^2 mod 214 = 107. So the problem of the little theorem of Fermat with small exponents is more or less solved in this way. It was a cute result for sure…:)
Yet since January I have been wondering if all counter examples must use some form of divisors of zero. If so there would be a small possibility to strongly improve on the result from Andrew Wiles. The video below shows a counter example using only integers modulo a prime number p. Since those number spaces modulo p do not have divisors of zero, actually these spaces are fields, I can now say that it is not needed to have divisors of zero. So that is why I decided to include this video in this post: it saves me a lot of time and funny enough this result goes back to at least the year 1917.
Lately there are much more math video’s on the usual video channels and a lot of them have these beautiful animations. A significant part of those new video’s can be traced back to the work of the guy that runs the 3Blue1Brown channel who more or less started to craft those beautiful animations. Somewhere this summer or may be spring he called for the public to make their own math video’s and even I thought about it a short time. But I have no video editing software, I have zero experience in using such software, I have no microphone and the only video camere I have is my photo camera that I never use to make video. So I decided not to try and make a video about 3D complex numbers, it was just too much work. Here is how the small picture from the video looks on Youtube, it looks cute but likely it takes a lot of time to make such animations:
That graph sure looks cute!
Ok, back in the year 1917 a guy named I. Schur managed to find a counter example using the integers modulo a prime number. He himself in that old article cites another person so it is a bit vague how this all panned out. Anyway I do not have to be ashamed as not to find this result modulo p; it is based on the colorings of graphs like you see above. I do not know much about that so don’t blame me for missing these results. Besidew that, another cute result of this year was that I found out that my own 4D complex numbers form a field if you restrict them to the rational numbers. That simply disproves stupid but accepted theorems like the Hurwitz theorem that says that the only higher dimensional complex numbers are the quaternions and total garbage like the octonions. That whole fucking stupid theorem from Hurwitz is just not true and of course the mathematical community says nothing. Anyway, the video is just 10 minutes of your time and my post is only two pictures long so have fun with it!
And now we go for the video:
The paper that was unpicked is written in German but don’t panic: It is not in that gothic font they used for a long time. You can find it here: https://eudml.org/doc/145475. So if you can read the German language you can try to unpick it for yourself… Ok, end of this post. Thanks for your attention.
It is just past midnight on a Saturday so why not start this new post? Two weeks ago all of a sudden I felt like doing a magnetics post once more and to be honest that is a tiny miscalculation because this is also post number 200. This should actually be about higher dimensional complex numbers or so but it is what it is so we are doing magnetics. This year I stopped working on the magnetic pages on the other website, after more than five years and zero response I decided to classify the physics professors the same as the math professors: Just another bunch of incompetents that you better avoid being around. That policy stays in tact so although this post is very long you should not view this as an attempt to change the views physics professors have on magnetism in particular the idea that electrons must be magnetic monopoles because all other interpretations are not logical. To be honest I feel a bit more free now, now I can say you are a total idiot or a complete moron or a natural born imbecile instead of every time trying to come up with more reasons as why electrons cannot be magnetic dipoles. This post contains five easy calculations around magnetism, to be a bit more precise: there are three forces that can act on an electron (Coulomb, Lorentz and say the Stern-Gerlach force) and the five easy calculations cover that. Ok, gravity also works on electrons but that is out of this story. We will look at the speed and the acceleration of electrons and we will try to estimate how much of a gradient a magnetic field must have to explain the observed experimental results. I found a gradient of about 450 million Tesla/meter and that all in my living room… I cannot find a fault in it and of course a gradient so large is not realistic so I reject the way electron potential energy related to magnetism is incorperated into the standard model of particles. Last week I found a perfect photo that can serve as a model of how physics professors are behaving in my view (a bunch of overpaid incompetents). The ‘blah blah blah’ can serve as my five easy calculations; for the idiots and imbeciles that the physics professors are, these calculations will be ‘blah blah blah’ material. Come on, next year 2022 it will be one 100 years since the Stern-Gerlach experiment was performed. If a collection of overpaid idiots have it wrong for one 100 years, the most logical thing is that they will keep it wrong for at least another 100 years.
Excluded the picture above, this post is 14 pictures long. And even then I skipped a lot of stuff like when talking about the solar wind I just assume all particles go with the same speed of say 750 km/sec. Of course that is an oversimpification; there are many different particles in the solar wind that all behave rather different. I hope you don’t loose oversight and therefore I would like to explain how I arrived at that crazy magnetic gradient of 450 million Tesla/meter. Here we go:
1) I estimate how fast electrons go in an old color television where I assume these electron have a kinetic energy of 40.000 eV. This is about 120 thousand km/sec or about one third the speed of light.
2) If you place a stack of those strong neodymium magnets at the television screen you see a black spot where no electrons land. I estimate the sideway acceleration the electrons make and that is a crazy huge number: about 4.5 times 10^15 m/sec. That might sound very huge but it is just simple Newtonian mechanics: If the electron has a sideway acceleration this big for about 2 nanoseconds, the sideway displacement is about 1 cm. That is in line with what the photo of the television screen says..
3) I calculate the force on the electron for this sideway acceleration and plug this into the expression the professional physics professors use for the force related to inhomogeneous magnetic fields and voila: there is your gradient of only 450 million Tesla/meter.
All in all the five calculations are say advanced high school level / first year university level. Given the fact that physics professors keep on thinking that electrons are tiny magnets with a north and a south pole we can safely conclude that even a science like physics is much more some kind of social construct and not a hard ball science in itself.
Enough of the introductory talk, here are the 14 pictures:
Ok, that was it for this post. Please remark that the polar aurora’s do exist for real while the blah blah blah of physics professors explaining this makes not much sense. For example they do not explain why the electrons go so fast that they ionize the air and most of all: These idiots are not aware they are missing something here.
At last I want to remark that for me it feels rather refreshing to talk about physics professors as idiots and imbeciles. That is much better as always being polite and respectful. Why should overpaid idiots and imbeciles also face a lot of completely misplaced respect? See you in the next post & thanks for your attention because this post was a long reading for you.
Already a few years back I wanted to write a post on the so called de Padé apprximations because they are so good at taking the logarithm of a matrix. For me the access to an internet application that calculated those logs from matrix representations was a very helpful thing to speed things up. It would have taken me much much longer to find the first exponential circle on the 3D complex numbers if I could not use such applets. But in the year 2018 pure evil struck the internet: the last applets or websites having them disappeared to never come back. Ok by that time I had perfected my method of simply using matrix diagonalization for finding such logs of matrices. You can still find it easily if you do an internet search on ‘Calculation of the 7D number tau‘. Yet in the beginning I only had such applets as found on the internet and I soon found out that using the so called de Padé approximation always gave much better results compared to say a Taylor approximation. It is not very hard to understand how to perform such a de Padé approximation. Much harder to understand is how de Padé found them, after all it looks like a strike of genius if this works. The genius part is of course found in the stuff you can simply neglect in such approximations, at first it baffles the mind and later you just accept it that you are more stupid as de Padé was…;) Anyway this week I stumbled upon a cute video and as such I decided to write a small post upon this de Padé stuff. (On the shelf are still a possible new way of making an antenna based on the 3D exponential circles and some updates on magnetism.) So let us first take a look at the video, here we go:
As you see the basic idea is pretty simple: you use those two polynomials to ‘approximate’ that Taylor series and as a bonus you have a much better approximation of the original function. All in all this is amazing and it makes you wonder if there are methods out there that are even better compared to this de Padé approximation.
Now you can choose beforehand what degree polynomial you use in the nominator and denominator. There are plenty of situations where this brings a big benefit like in the video they point out the divergence problems of say the sine function that is bounded between +1 and -1 on the real axis. The Taylor approximations always go completely beserk outside some interval where they fit quite well. With the appropiate choice of the degrees of the polynomials in the de Padé approximation you can avoid this kind of stuff. In my view the maker of the video should also have pointed out that a de Padé approximation can have it’s own troubles when you divide by zero. And when the original function never has such a pole at that point, the de Padé approximation also goes very bad. These de Padé approximations are indeed much better compared to the average Taylor approximation but they are not from heaven. You still have to use your own brain and may be that is a good thing.
De Padé still rules in the year 2021, likely it was a good idea to start with.
In this post I did not cover the matrices and why a de Padé approximation of the logarithm of a matrix is good or bad. If you want to find exponential circles and curves for yourself, use those applets mostly on imaginary units who’s matrix representation has a determinant of +1. In case you want to find your very first exponential circle, solve the next problem:
Ok, it is late at night so let me hit that button ‘publish website’ and see you around.
