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 late at night, my computer clock says it is 1.01 on a Sunday night. But I am all alone so why not post this update? This post does not have much mathematical depth, it is all very easy to understand if you know what split complex numbers are. In the language of this website, the split complex numbers are the 2D circular numbers, In the past I named a particular set of numbers complex or circular. I did choose for circular because the matrix representations of circular numbers are the so called circulant matrices. It is always better to give mathematical stuff some kind of functional name so people can make sense of what the stuff is about. For me no silly names like ‘3D Venema positive numbers’ or ‘3D Venema complex numbers’. In math the objects should have names that describe them, the name of a person should not be hanged on such an object. For example the Cayley-Hamilton theorem is a total stupid name, the names of the humans who wrote it out are not relevant at all. Further reading on circulant matrices: Circulant matrix. I also have a wiki on split complex numbers for you, but like all common sources they have the conjugate completely wrong. Professional math professors always think that taking a conjugate is just replacing a + by a – but that is just too simplistic. That’s one of the many reasons they never found 3D complex numbers for themselves, if you do that conjugate thing in the silly way all your 3D complex math does not amount to much… Link: Split-complex number.
This is the last part on this oversight of counter examples to the last theorem of Pierre de Fermat and it contains only the two dimensional split complex numbers. When I wrote the previous post I realized that I had completely forgetten about the 2D split numbers. And indeed the math results as found in this post are not very deep, it’s importance lies in the fact that the counter examples now are unbounded. All counter examples based on modular arithmetic are always bounded, periodic to be precise, so professional math professors could use that as a reason to declare that all a bunch of nonsense because the real integers are unbounded. And my other counter examples that are unbounded are only on 3D complex & circular number spaces and the 4D complex numbers so that will be neglected and talked into insignificance because ‘That is not serious math’ or whatever kind of nonsense those shitholes come up with.
All in all despite the lack of mathematical depth I am very satisfied with this very short update. The 2D split numbers have a history of say 170 years so all those smart math assholes can think a bit about why they never formulated such simple counter examples to the last theorem of Fermat… May be the simplicity of the math results posted is a good thing in the long run: compare it to just the natural numbers or the counting numbers. That is a set of numbers that is very simple too, but they contain prime numbers and all of a sudden you can ask thousands and thousands of complicated and difficult questions about natural numbers. So I am not ashamed at all by the lack of math depth in this post, I only point to the fact that over the course of 170 years all those professional math professors never found counter examples on that space.
This post is just 3 pictures long although I had to enlarge the lastest one a little bit. The first two pictures are 550×825 pixels and the last one is 550×975 pixels. Here we go:
That was it for this post, one of the details as why this post is significant is the use of those projector numbers. You will find that nowhere on the entire internet just like the use of 3D complex numbers is totally zero. Let’s leave it with that, likely the next post is about magnetism and guess what? The physics professors still think there is no need at all to give experimental proof to their idea of the electron having two magnetic poles. So it are not only the math professors that are the overpaid idiots in this little world of monkeys that think they are the masters of the planet.
I finished the proof that was originally meant to be an appendix to the previous post. And I have two more or less small things I want to share with you so lets get started with the first thing:
Thing 1: Tibees comes up with a very cute program of graphing 3D surfaces. It’s name is surfer, the software is very simple to use and it has the giant benefit of making graphs from implicit equations like f(x, y, z) = 0. For example if you want the unit sphere in 3D space you must do x^2 + y^2 + z^2 – 1 = 0. Now for this website I always used an internet applet that uses ray tracing and by doing so over the years such graphs always look the same. But this surfer program has cute output too and it has the benefit you don’t need to be online. Here is how such a graph looks, it is the determinant in the space of 3D complex numbers, to be precise it shows the numbers with a determinant of 1:
By the way, the surface of this graph is a multiplicative group on it’s own in 3D space. I never do much group stuff but if you want it, here you have it. And for no reason at all I used GIMP to make one of those cubes from the above graph. It serves no reason beside looking cute:
The Tibees female had a video out last week where she discusses a lot of such surfaces in three dimensional space using that surfer software. And she is a pleasant thing to look at, it is not you are looking at all those extravert males drowning in self-importance only lamentating shallow thoughts. The problem posed in the video is an iteresting one, I don’t have a clue how to solve it. Title of the video: The Shape No One Thought was Possible. It is a funny title because if you start thinking about all the things that math professors thought were not possible you can wonder if there is enough paper in the entire universe to write that all out.. Link to the Surfer program in case you want to download & install it: https://www.imaginary.org/program/surfer.
So far for thing 1.
Thing 2: The last weeks it is more and more dawning on me that all those centuries those idiots (the math professors) did not find counter examples to the last theorem of Fermat. Nor was there any improvement on the little theorem of Fermat. Only Euler did some stuff on the little theorem with his totient function, but for the rest it is not much… Well since Jan of this year I found many counter example to the last theorem of Fermat and in my view I made a serious improvement on the little theorem of Fermat. So is the improvement serious or not? Here is a picture that shows the change:
So it’s modulo ap instead of modulo p.
On a wiki with a lot of proofs for the little theorem of Fermat they start with a so called ‘simplification’. The simplification says that you must pick the number a between 0 and p. So if you have an odd prime, say a = 113, does the little theorem only make sense for exponents above 113? And can’t we say anything about let’s say the square 113^2?
With the new version of the little theorem we don’t have such problems any longer. Here is a screen shot from the start of that wiki, the upper part shows you the improvement:
Here is a link to that wiki that is interesting anyway.
If you follow that link you can also scroll down to the bottom of the wiki where you can find the notes they used. It is an impressive list of names like Dirichlet, André Weil, Hardy & Wright and so on and so on. All I want to remark is that non of them found counter examples to the last theorem nor did they improve on the little theorem of Fermat. Now I don’t want to be negative on Dirichlet because without his kernel I could never have crafted my modified Dirichlet kernel that is more or less the biggest math result I ever found. But the rest of these people it is just another batch of overpaid non performers. It’s just an opinion so you don’t have to agree with it, but why do so many people get boatloads of money while they contribute not that much?
End of thing 2.
Now we are finally ready to post the main dish in this post: the proof that the subset of four dimensional rational numbers form a field. Math professors always think it is ‘very important’ if something is a field while in my life I was never impressed that much by it. And now I am thinking about it a few weeks more, the less impressed I get by this new field of four dimensional complex numbers. Inside the theory of higher dimensional complex numbers the concept of ‘imitators of i‘ is important: these are higher dimensional numbers that if you square them they have at least some of the properties of the number i from the complex plane. They rotate everything by 90 degrees or even better they actually square to minus one. Well one of the imitators of i in the space of 4D complex number is dependent of the square root of 2. As such it is not a 4D rational complex number. That detail alone severely downsizes my enthousiasm. But anyway, the next pictures are also a repeat of old important knowledge like the eigenvalue functions. Instead of always trying to get the eigenvalues from some 4×4 matrix, with the eigenvalue functions with two fingers in your nose you can pump out the eigenvalues you need fast. This post is six pictures long each size 550×825 pixels. Here we go:
Yes that is the end of this post that like always grew longer than expected. If you haven’t fallen asleep by now, thanks for your attention and don’t forget to hunt the math professors until they are all dead! Well may be that is not a good idea, but never forget they are too stupid to improve on the little Fermat theorem and of course we will hear nothing from that line of the profession…
This year starting in January I found more and more counter examples to the last theorem of Fermat. As a by product when we looked at the stuff on the 4D complex numbers, we found that if we restrict ourselves to the 4D rationales, they were always invertible. And as such they form a field, this is a surprising result because the official knowledge is that the only possible 4D number system are the quaternions from Hamilton. So how this relates to those stupid theorems of Hurwitz and Fröbenius about higher dimensional complex numbers is something I haven’t studied yet. But that Hurwitz thing is based on some quadratic form so likely he missed this new field of 4D complex rationales because the 4D complex numbers are ruled by a 4 dimensional thing namely the fourth power of the first imaginary unit equals minus one: l^4 = -1. Compare that to the complex plane that has all of it’s properties related to that defining equation i^2 = -1.
And because we now have a 4D field I thought like let’s repeat how you find the inverse of a 4D complex rational number. And also prove that we have a field as basic a proof can be. But while writing this post I had to abandon the second thing otherwise this post would grow too long. Of course in the past I have crafted a post for finding the non-invertible 4D complex numbers but in that post I never remarked that rational 4D complex numbers are always invertible. To be honest in the past it has never dawned on me that it was a field, for me this is not extremely important but for the professional math people it is.
When back in Jan of this year I found the first counter example to the last theorem of Fermat I was a bit hesitant to post it because it was so easy to find for me. But now four months further down the time line I only found two examples where other people use some form of my idea’s around those counter examples and both persons have no clue whatsoever that they are looking at a counter example to the last theorem of Fermat. But in a pdf from Gerhard Frey (that is the Frey from the Frey elliptic curve that plays an important role in the proof to the last theorem of Fermat by Andrew Wiles) it was stated as: (X + Y)^p = X^p + Y^p modulo p. That’s all those professionals have, it is of a devastating minimal content but at least it is something that you could classify as a rudimentary counter example to the last theorem of Fermat. It only works when your exponent if precisely that prime number p and it lacks the mathematical beauty that for example we have in expressions like: 12^n = 5^n + 7^n modulo 35.
Anyway this post contains nothing new but there is some value in repeating how to find inverses of higher dimensional complex numbers. All you need is a ton of linear algebra and for that let me finish this intro on a positive note: Without the professional math professors crafting linear algebra in the past, at present day for me it would be much harder to make progress in higher dimensional complex numbers. And it is amazing: Why is linear algebra relatively good while in higher dimensional number systems we only look at a rather weird collection of idea’s? This post is made up of seven pictures each of size 550×800 pixels.
Stupid typo: Z = 1 + l +… so the real part must be one.
Ok ok this post is not loaded with all kinds of deep math results. But if you have a properly functioning brain you will have plenty of paths to explore. And the professional math professors? Well those overpaid weirdo’s will keep on neglecting the good side of math and that is important too: That behavior validates they are overpaid weirdo’s…
For example the new and improved little theorem of Fermat: The overpaid weirdo’s will neglect it year in year out. That’s the way it is, here is once more a manifestation of the new and improved little theorem of Fermat:
Let’s leave it with that. Thanks for your attention.
A couple of days ago all of a sudden while riding my bicycle I calculated what the so called directional derivative is for 3D & 4D complex numbers. And it is a cute calculation but I decided not to write a post about it. After all rather likely I had done stuff like that many years ago.
Anyway a day later I came across a few Youtube video’s about the directional derivative and all those two guys came up with was an inner product of the gradient and a vector. Ok ok that is not wrong or so, but that is only the case for scalar valued functions on say 3D space. A scalar field as physics people would say it. The first video was from the Kahn academy and the guy from 3Blue1Brown has been working over there lately. It is amazing that just one guy can lift such a channel up in a significant manner. The second video was from some professional math professor who went on talking a full 2.5 hour about the directional derivative of just a scalar field. I could not stand it; how can you talk so long about something that is so easy to explain? Now I do not blame that math professor, may be he was working in the USA and had to teach first year math students. Now in the USA fresh students are horrible at math because in the USA the education before the universities is relatively retarded.
Furthermore I tried to remember when I should have done the directional derivative. I could not remember it and in order to get rid of my annoyance I decided to write a small post about it. Within two hours I was finished resulting in four pictures of the usual 550×775 pixel size. So when I work hard I can produce say 3 to 4 pictures in two hours of time. I did not know that because most of the time I do not work that fast or hard. After all this is supposed to be a hobby so most of my writing is done in a relaxed way without any hurry. I have to say that may be I should have taken a bit more time at the end where the so called Cauchy-Riemann equations come into play. I only gave the example for the identiy function and after that jumped to the case of a general function. May be for the majority of professional math professors that is way to fast, but hey just the simple 3D complex numbers are ‘way to fast’ for those turtles in the last two centuries…
Anyway, here is the short post of only 4 pictures:
Should I have made the explanation longer? After all so often during the last years I have explained that the usual derivative f'(X) is found by differentiating into the direction of the real numbers. At some point in time I have the right to stop explaining that 1 + 1 = 2.
Also I found a better video from the Kahn academy that starts with a formal definition of the directional derivative:
At last let me remark that this stuff easily works for vector valued functions because in the above limit you only have to subtract two vectors and that is always allowed in any vector space. And only if you hang in a suitable multiplication like the complex multiplication of 3D or 4D real space you can tweak it like in the form of picture number 4 above.
That was all I had for you today, this is post number 166 already so I am wondering if this website is may be becoming too big? If people find something, can they find what they are searching for or do they get lost in the woods? So see you in another post, take care of yourself & till the next post.
For a couple of years I have a few pdf files in my possession written by other people about the subject of higher dimensional complex and circular numbers. In the post we will take a look at the work of Shlomo Jacobi, the pdf is not written by him because Shlomo passed away before it was finished. It is about the 3D complex numbers so it is about the main subject of this website.
Link used: http://search.arxiv.org:8081/paper.jsp?r=1509.01459&qid=1603841443251ler_nCnN_1477984027&qs=Shlomo+Jacobi&in=math
Weirdly enough if you search for ‘3D hypercomplex number’ the above pdf does not pop up at all at the preprint archive. But via his name (Shlomo Jacobi) I could find it back. Over the years I have found three other people who have written about complex numbers beyond the 2D complex plane. I consider the work of Mr. Jacobi to be the best so I start with that one. So now we are with four; four people who have looked at stuff like 3D complex numbers. One thing is directly curious: None of them is a math professional, not even a high school teacher or something like that. I think that when you are a professional math professor and you start investigating higher dimensional complex numbers; you colleagues will laugh about it because ‘they do not exist’. And in that manner it are the universities themselves that ensure they are stupid and they stay stupid. There are some theorems out there that say a 3D complex field is not possible. That is easy to check, but the math professionals make the mistake that they think 3D complex numbers are not possible. But no, the 2-4-8 theorem of say Hurwitz say only a field is not possible or it says the extension of 2D to 3D is not possible. That’s all true but it never says 3D complex numbers are not possible…
Because Shlomo Jacobi passed away an unknown part of the pdf is written by someone else. So for me it is impossible to estimate what was found by Shlomo but is left out of the pdf. For example Shlomo did find the Cauchy-Riemann equations for the 3D complex numbers but it is only in an epilogue at the end of the pdf.
The content of the pdf can be used for a basic introduction into the 3D complex numbers. It’s content is more or less the ‘algebra approach’ to 3D complex numbers while I directly and instantly went into the ‘analysis approach’ bcause I do not like algebra that much. The pdf contains all the basic stuff: definition of a 3D complex number, the inverse, the matrix representation and stuff he names ‘invariant spaces’. Invariant spaces are the two sets of 3D complex numbers that make up all the non-invertible numbers. Mr. Jacobi understands the concept of divisors of zero (a typical algebra thing that I do like) and he correctly indentifies them in his system of ‘novel hypercomplex numbers’. There is a rudimentary approach towards analysis found in the pdf; Mr. Jacobi defines three power series named sin1, sin2 and sin3 . I remember I looked into stuff like that myself and somewhere on this website it must be filed under ‘curves of grace’.
A detail that is a bit strange is the next: Mr. Jacobi found the exponential circle too. He litarally names it ‘exponential circle’ just like I do. And circles always have a center, they have a midpoint and guess how he names that center? It is the number alpha…
Because Mr. Jacobi found the exponential circle I applaud him long and hard and because he named it’s center the number alpha, at the end I included a more or less new Euler identity based on a very simple property of the important number alpha: If you square alpha it does not change. Just like the square of 1 is 1 and the square of 0 is 0. Actually ‘new’ identity is about five years old, but in the science of math that is a fresh result.
The content of this post is seven pictures long, please read the pdf first and I hope that the mathematical parts of your brain have fun digesting it all. Most pictures are of the standard size of 550×775 pixels.
Yes all you need is that alpha is it’s own square.
Ok ok, may be you need to turn this into exponential circles first in order to craft the proof that a human brain could understand. And I am rolling from laughter from one side of the room to the other side; how likely is it that professional math professors will find just one exponential circle let alone higher dimensional curves?
I have to laugh hard; that is a very unlikely thing.
End of this post, see you around & see if I can get the above stuff online.
This post took me a long time to write, not that it was so very difficult or so but lately I am learning that graphics program named GIMP. And that absorbs a lot of time and because I am only sitting behind my computer a few hours a day, doing GIMP goes at the expense of writing math…
I always make my pictures with an old graphics program named Picture Publisher 10. It is so old that on most windows 7 and windows 10 it does not run but it has all kinds of features that even the modern expensive graphics programs simply still don’t have. Silently I was hoping that I could use GIMP for my math texts and yes that could be done but in that case I have to use old background pictures forever. Or I have to craft a ‘new style’ for making the background in the math pictures that can last at least one decade.
But let’s not nag at what GIMP cannot do, if you install just one large addon you have about 500 filters extra and my old program PP10 comes from an era when the word ‘addon’ was not a word used ever. Before we jump to the math, let me show you a nice picture you can make with the tiling filter inside GIMP. It is about my total bicycle distance since I bought this bicycle computer, it says 77 thousand km so the Tour the France racers can suck a tip on that:
Just one tile already looks nice.And this is how four of these tiles look.
Ok, let us look at the math of this post. This is part 22 in the introduction to the 4D complex numbers. The 4D complex numbers have three imaginary units, l, l^2 and l^3. And the stuff that makes it ‘complex’ is the fact that l^4 = -1, you can compare that to the complex plane where the square of the imaginary unit equals -1.
On the complex plane, if you know what the logarithm of i is, you can use that to find the exponential circle also known as the complex exponential. This is what the number tau always is in all kinds of spaces: It is always the logarithm of the first imaginary unit that has a determinant of +1. In this post we will calculate the eigenvalues of this important number tau. That will be done with two methods. In the first method we simply use the eigenvalue functions, plug in the number tau and voila: out come the four eigenvalues. In the second method we first calculate the four eigenvalues of the imaginary unit l and ‘simply’ take the logarithm of those four eigenvalues.
It is not much of a secret that my style of work is rather sloppy, I never order my work in theorems, lemma’s or corrolaries. It is not only that such an approach if too much a straight jacket for me, it also frees me from a lot of planning. I simply take some subject, like in this case the eigenvalues of the number tau and start working on explaining that. While writing that out there always comes more stuff around that I could include yes or no. In this post what came around was that only after writing down the four eigenvalues I realized that you can use them to prove that the exponential curve (the 4D complex exponential) has a determinant of 1 for all points on that curve. That was an important result or an important idea so I included it because that makes proving that the determinant is 1 much more easy.
Now a few posts back with that video from that German physics guy Alexander Unzicker I said that he (and of course all other physics professionals) could always use the 4D complex exponential curve for the ‘phase shifts’ that those physics people always do. But for doing such 4D ‘phase shifts’ or unitary transformations in general, you need of course some kind of proof that determinant values are always +1. Well Alexander, likely you will never read this post but below you can find that very proof.
The previous post was from the end of August and now I think about it: Have I done so little math during the last four weeks? Yes there were no results simply left out, it was only penning down these eigenvalues of tau and the idea you can use these eigenvalues for proving the 4D exponential curve always has a determinant of 1. It is amazing that GIMP can hinder the creation of fresh math… 😉
The math pictures are seven in number, all in the usual size of 550×775 pixels. I hope you like it and see you in the next post.
So these are the four eigenvalues of the number tau and based on that the four eigenvalues of the 4D complex exponential for a values of time.
That’s it for this post. See you in a future post.
Added on 27 Sept 2020: This proceeds the two pictures made with GIMP that started this post. I just made the whole stuff on a cube (actually it is a beam because the starting picture is not a square). It is amazing how good such filters in GIMP are:
Ah, finally it is finished. This work grew longer than expected but with a bit of hindsight that is also logical: for example I spell out in detail once more how to find the eigenvalue functions for a arbitrary number X. After all that is an important detail so it is worth repeating. But I skipped the proces of diagonalization because we do not need it in this post. Yet if you teach math and the time has come to do the complex number stuff, you could show the students how to diagonalize the complex multiplication for numbers from the complex plane. Most of the time students only diagonalize just one matrix with some numbers in it and that’s all, they never diagonalize an entire family of matrices. So that is why that would be useful, on the other hand the eigenvalues for a number z from the complex plane is z itself and it’s conjugate… And say for yourself: diagonalizing a number z so that later you must multiply the eigenvalues (also z) is very useless, as a matter of fact it is hard to find anything that is more useless… And once you have explained that diag stuff is usefull and utterly un-usefull at the same time, you can point to the live of the average math professor: also utterly useless…
Of course in higher dimensions the proces of diagonalization is very handy because it gives you for example a way of calculating the logarithm of higher dimensional numbers. And that way can be used in any dimension while all other methods for finding a logarithm get more and more difficult (as far as I know).
In this post I also worked out in detail what the eigenvalues of non-invertible numbers are; the non zero numbers with a determinant of zero have at least one eigenvalue being zero. I calcualted the eigenvalues for the numbers tau and alpha for both the complex and circular 3D multiplication.
This is post number 150 so all in all on average I write just about 30 posts a year. That is a cost of about 2€ per post… 😉 Luckily this hobby of 3D complex numbers is a rather cheap hobby while at the same time it keeps the mind sharp. A disadvantage is that if it takes me just 5 or 10 minutes to do some calculations with pencil and paper, it often takes 5 to 10 hours before it is turned into a post that is more or less readable for other people… And that is something I value highly; so often you come across sloppy explanations that are not carfully thought through. I don’t like that.
Originally I prepared 10 pictures to write the post on but I had so much text that I started expanding those pictures and in the end I made an 11-th picture to get it all on. So I just expanded those pictures to make the text fit more or less precise of most of them have weird sizes. May be it is better to just stick to the size of 550×775 pixels and just make more of them if needed and not this chaotic expansion on the fly. Ok, here we go:
I expect that when you made it this far, you already know what the Cauchy/Schwarz inequality is. But in case you never heard of it, please try to understand that beautiful but very simple inequality. Here is a wiki: Cauchy-Schwarz Inequality. Link used: https://brilliant.org/wiki/cauchy-schwarz-inequality/
Ok, this is more or less what I had to say on the subject of crafting a norm from eigenvalues. Don´t forget in the complex plane the square of the norm is also the product of the eigenvalues of a complex number z. So for centuries the math professors are already doing this although I do not think they are aware that they use a product of eigenvalues. For them likely it is just some stuff that is ´Just like Pythagoras´.
