All in all I am not super satisfied with this post because the math result is not that deep. Ok ok the 4D complex numbers also contain non-invertible numbers, say P and Q, and these are divisors of zero. That means PQ = 0 while both P and Q are non-zero. And just like we did in the case of 3D circular and complex numbers because of the simple property PQ = 0 all mixed terms in (P + Q)^n become 0 and as such: (P + Q)^n = P^n + Q^n.
In the space of 4D complex numbers an important feature of the determinant det(Z) of a 4D complex number Z is that it is non-negative. As such there is not a clear defined layer between the part of the number space where the determinant is positive versus the negative part. During the writing of this post it dawned on me that Gaussian integers in the 4D complex space always have a non-zero determinant. As such the inverse of such a Gaussian exists although often this is not a Gaussian integer just like the inverse of say the number 5 is not an integer. A completely unexpected finding is that the 4D complex fractions form a field…
That made me laugh because the professional math professors always rejected higher dimensional complex numbers because they are not a field. For some strange reason math professors always accept or embrace stuff that forms a field while they go bonkers & beserk when some set or group or ring is not a field. This is a strange behavior because the counter examples that I found against Fermat his last theorem are only there because 3D and 4D numbers are not a field: there are always non zero numbers that you cannot invert. As such a lot of math professors are often busy to make so called field extensions of the rational numbers. And oh oh oh that is just soo important and our perfumed princes ride high on that kind of stuff. And now those nasty 4D complex numbers from those unemployed plebs form a field too… I had to smile softly because 150 years have gone since the last 4D field was discovered, that is known as the quaternions, and now there is that 4D field of rationals that are embedded into something the cheap plebs name ‘4D complex numbers’? How shall the professional math professors react on this because it is at the root of their own behavior over decades & centuries of time?
Do not worry my dear reader: They will stay the overpaid perfumed princes as they are. Field or no field, perfumed princes are not known to act as adult people.
After having said that, this post is only five pictures long all of the ususal size of 550×775 pixels. For myself speaking I like the situation on the 3D numbers more because there you can easily craft an infinite amount of counter examples against the last theorem of Fermat. Ok, here we go:
Yes I have to smile softly: all this hysteria from overpaid math professors about stuff being a field or not. And now we are likely into a situation where the 4D complex numbers are not a field but the space of 4D complex rationals is a field…
Will the math professors act as adults? Of course not. Ok, let’s end this post because you just like me will always have other things to do in the short time that we have on this pale blue dot known as planet earth. Till updates.
I am rather satisfied with the approach of doing the same stuff on the diverse complex spaces. In this case the 2D complex plane and the 3D & 4D complex number systems. By doing it this way it is right in your face: a lot of stuff from the complex plane can easily be copied to higher dimensional complex numbers. Without doubt if you would ask a professional math professor about 3D or higher dimensional complex numbers likely you get a giant batagalization process to swallow; 3D complex numbers are so far fetched and/or exotic that it falls outside the realm of standard mathematics. “Otherwise we would have used them since centuries and we don’t”. Or words of similar phrasing that dimishes any possible importance.
But I have done the directional derivative, the factorization of the Laplacian with Wirtinger derivatives and now we are going to do the total differential precisely as you should expect from an expansion of the century old complex plane. There is nothing exotic or ‘weird’ about it, the only thing that is weird are the professional math professors. But I have given up upon those people years ago, so why talk about them?
In the day to day practice it is a common convention to use so called straight d‘s to denote differentiation if you have only one variable. Like in a real valued function f(x) on the real line, you can write df/dx for the derivative of such a function. If there are more then one variable the convention is to use those curly d’s to denote it is partial differentiation with respect to a particular variable. So for example on the complex plane the complex variable z = x + iy and as such df/dz is the accepted notation while for differentiation with respect to x and y you are supposed to write it with the curly d notation. This practice is only there when it comes to differentiation, the opposite thing is integration and there only straight d‘s are used. If in the complex plane you are integrating with respect to the real component x you are supposed to use the dx notation and not the curly stuff. Well I thought I had all of the notation stuff perfectly figured out, oh oh how ultrasmart I was… Am I writing down the stuff for the 4D complex numbers and I came across the odd expression of dd. I hope it does not confuse you, in the 4D complex number system I always write the four dimensional numbers as Z = a + bl + cl^2 + dl^3 (the fourth power of the imaginary unit l must be -1, that is l^4 = -1, because that defines the behavior of the 4D complex numbers) so inside Z there is a real variable denoted as d. I hope this lifts the possible confusion when you read dd…
More on the common convention: In the post on the factorization of the Laplacian with Wirtinger derivatives I said nothing about it. But in case you never heard about the Wirtinger stuff and looked it up in some wiki’s or whatever what, Wirtinger derivatives are often denoted with the curly d‘s so why is that? That is because Wirtinger derivatives are often used in the study of multi-variable complex analysis. And once more that is just standard common convention: only if there is one variable you can use a straight d. If there are more variable you are supposed to write it with the curly version…
At last I want to remark that the post on the factorization of the Laplacian got a bit long: in the end I needed 15 pictures to publish the text and I worried a bit that it was just too long for the attention span of the average human. In the present years there is just so much stuff to follow, for most people it is a strange thing to concentrate on a piece of math for let’s say three hours. But learning new math is not an easy thing: in your brain all kind of new connections need to be formed and beside a few hours of time that also needs sleep to consolidate those new formed connections. Learning math is not a thing of just spending half an hour, often you need days or weeks or even longer.
This post is seven pictures long, have fun reading it and if you get to tired and need a bit of sleep please notice that is only natural: the newly formed connetions in your brain need a good night sleep.
Here we go with the seven pictures:
Yes, that’s it for this post. Sleep well and think well & see you in the next post. (And oh oh oh a professional math professor for the first time in his or her life they calculate the square Z^2 of a four dimensional complex number; how many hours of sleep they need to recover from that expericence?) See ya in the next post.
Originally I wanted to make an oversight of all ways the so called Dirac quantization condition is represented. That is why in the beginning of this post below you can find some stuff on the Dirac equation and the four solutions that come with that equation. Anyway, Paul Dirac once managed to factorize the Laplacian operator, that was needed because the Laplacian is part of the Schrödinger equation that gives the desired wave functions in quantum mechanics. Well I had done that too once upon a time in a long long past and I remembered that the outcome was highly surprising. As a matter of fact I consider this one of the deeper secrets of the higher dimensional complex numbers. Now I use a so called Wirtinger derivative; for example on the space of 3D complex numbers you take the partial derivatives into the x, y and z direction and from those three partial derivatives you make the derivative. And once you have that, if you feed it a function you simply get the derivative of such a function.
Now such a Wirtinger derivative also has a conjugate and the surprising result is that if you multiply such a Wirtinger derivative against it’s conjugate you always get either the Laplacian or in the case of the 3D complex numbers you get the Laplacian multiplied by the famous number alpha.
That is a surprising result because if you multiply an ordinary 3D number X against it’s conjugate you get the equation of a sphere and a cone like thing. But if you do it with parital differential operators you can always rewrite it into pure Laplacians so there the cones and spheres are the same things…
In the past I only had it done on the space of 3D numbers so I checked it for the 4D complex numbers and in about 10 minutes of time I found out it also works on the space of 4D complex numbers. So I started writing this post and since I wanted to build it slowly up from 2D to 4D complex numbers it grew longer than expected. All in all this post is 15 pictures long and given the fact that people at present day do not have those long timespan of attention anymore, may be it is too long. I too have this fault, if you hang out on the preprint archive there is just so much material that often after only five minutes of reading you already go to another article. If the article is lucky, at best it gets saved to my hard disk and if the article has more luck in some future date I will read it again. For example in the year 2015 I saved an article that gave an oversight about the Dirac quantization condition and only now in 2020 I looked at it again…
The structure of this post is utterly simple: On every complex space (2D, 3D and 4D) I just give three examples. The examples are named example 1, 2 and not surprising I hope, example 3. These example are the same, only the underlying space of complex numbers varies. In each example number 1 I define the Wirtinger derivative, in example 2 I take the conjugate while in the third example on each space I multiply these two operators and rewrite the stuff into Laplacians. The reason this post is 15 pictures long lies in the fact that the more dimensions you have in your complex numbers the longer the calculations get. So it goes from rather short in the complex plane (the 2D complex numbers) to rather lengthy in the space of 4D complex numbers.
At last I would like to remark that those four simultanious solutions to the Dirac equation it once more shouts at your face: electrons carry magnetic charge and they are ot magnetic dipoles! All that stuff like the Pauli matrices where Dirac did build his stuff upon is sheer difficult nonsense: the interaction of electron spin with a magnetic field does not go that way. The only reason people in the 21-th century think it has some merits is because it is so complicated and people just loose oversight and do not see that it is bogus shit from the beginning till the end. Just like the math professors that neatly keep themselves stupid by not willing to talk about 3D complex numbers. Well we live in a free world and there are no laws against being stupid I just guess.
Enough of the blah blah blah, below are the 15 pictures. And in case you have never ever heard about a thing known as the Wirtinger derivative, try to understand it and may be come back in five or ten years so you can learn a bit more… As usual all pictures are 550×775 pixels in size.
Oh oh the human mind and learning new things. If a human brain learns new things like Cauchy-Riemann equations or the above factoriztion of the Laplacian, a lot of chages happen in the brain tissue. And it makes you tired and you need to sleep… And when you wake up, a lot of people look at their phone and may be it says: Wanna see those new pictures of Miley Cyrus showing her titties? And all your new learned things turn into insignificance because in the morning what is more important compared to Miley her titties?
Ok my dear reader, you are at the end of this post. See you in the next post.
Man oh man, the previous post was from 12 Nov so time flies like crazy. Originally I wanted to write a post on a thing you can look up for yourself: the Dirac quantization condition. I have an old pdf about that and it says that it was related to the exponential circle on the complex plane. Although the pdf is from the preprint archive, it is badly written and contains a ton of typo’s and on top of it: the way the Dirac quantization is formulated is nowhere to be found back on the entire internet. In the exponent of the exponential circle there is iqg where q represents an elementary electric charge and g is the magnetic monopole charge according to Paul Dirac. Needless to say I was freaked out by this because I know a lot about exponential curves but all in all the pdf is written & composed so badly I decided not to use it.
After all when I say that electrons carry magnetic charge and do not have bipolar magnetic spin, the majority of professional physics professors will consider this a very good joke. And if I come along with a pdf with plenty of typo’s the professional professors will view that as a validation that I am the one who has cognitive problems and of course they are the fundamental wisecracks when it comes to understandig electron spin. Our Pauli and Dirac matrices are superior math, in the timespan of a hundred years nothing has come close to it they will say.
Here is a screen shot of what freaked me out:
Furthermore I was surprised that the so called professional physics professors have studied stuff like ‘dyons’. So not only a Dirac magnetic monopole (without an electric charge but only a magnetic charge), a dyon is a theoretical particle that has both electric and magnetic charge. But hey Reinko, isn’t that what you think of the electron? There are two kinds of electrons, all electrons have the same electric charge but the magnetic charge comes in two variants. There are so many problems with the idea that electrons are magnetic dipoles, but the profs if you give them a fat salary will talk nonsense like they are a banker in the year 2007.
So I decided to skip the whole Dirac quantization stuff and instead focus a bit on factorizing the Laplacian differential operator. I the past I have written about that a little bit, so why not throw in a Google search because after all I am so superior that without doubt my results will be found on page 1 of such a Google search! In reality it was all ‘Dirac this’ and ‘Dirac that’ when it comes to factorization of the Laplacian on page 1 of the Google search. So I understood the physics professors have a serious blockade in their brains because this Dirac factorization is only based on some weird matrices that anti-commute. These are the Pauli and Dirac matrices and it is cute math but has zero relation to physical reality like the electron pairs that keep your body together.
No more of the Dirac nonsense! I sat down and wrote the factorization of the Laplacian for 4D complex numbers on a sheet of paper. Let me skip all this nonsense of Dirac and those professional physics professors and bring some clarity into the factorization of the Laplacian. It took at most 10 minutes of time, it is just one sheet of paper with the factorization. I hope this is readable:
So that is what I have been doing since 12 April, since the last post on this website. I have worked my way through the 2D complex plane, the 3D complex numbers and finally I will write down what did cost me only 10 minutes of time a few weeks ago…
In a few days the post wil be ready, may be this week. If not next week & in the meantime you are invited to think about eletrons and why it is not possible that they are magnetic dipoles.
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.
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:
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:
Finally I have some time left to update this website. I would like to proceed with another part into the introduction to the four dimensional complex numbers. The previous part 20 dates back to 02 Feb 2019 and that too was about this structure of the non-invertible numbers in four dimensional complex space.
When I was reading back a few of my own old writings like part 20 from 02 Feb it struck me that those non-invertible numbers are all just linear combinations of the so called imitators of i. Imitators of i live always in dimensions higher than 2 and they mimic the behaviour of the number i from the complex plane. For example in the spaces of the 3D complex and circular numbers those imitators are not capable of squaring to minus one but they do a pretty good job at rotating stuff by 90 degrees if you multiply by them. In the 4D complex space there are two of such imitators and they do square up to minus one. Of course this is related to the fact you can find two copies of the 2D complex plane in the 4D complex space. So in that regard the 4D complex numbers are a bit different compared to the 4D quaternions that exist of 3 copies of the complex plane (but those do not commute and as such you cannot differentiate or integrate stuff).
Another interesting detail is that in the 4D complex number system the set on non-invertible numbers consists of just two lines that are perpendicular to each other. That is very different from the 3D situation where the set of non-invertibles is always a plane combined with a perpendicular line through zero. The reason that in 4D complex space the set is so small lies of course in the matrix representation and the determinant. On the 4D complex space the determinant is non-negative, just like the determinant is non-negative on the complex plane. Every 4D complex number has four eigenvalues and they come in conjugate pairs, so the product of these four eigenvalues gives the determinant hence the determinant cannot be a negative real number.
And say for yourself: aren’t the eigenvalue functions a very handy thing? If you want to find the eigenvalues of let’s say the 4D complex number Z = 1 + 2l + 3l^2 + 4l^3, that is often a horrible mathematical exercise. But once you have these four eigenvalue functions, you simply plug in any Z and voila: there are your 4 eigenvalues. This post is seven pictures long, as usual in the 550×775 pixel size.
Ok, that was it for this post. Till updates my dear reader.
I found an old video (what is ‘old’, it is from Jan 2019) and I decided to hang it in the website because it has such a beautiful introduction. The title of the video is The Secret of the Seventh Row. Seldom you see such a perfect introduction and I hope you will be intrigued too when you for the first time see the secret of the seventh row…
Now before I started brewing beer I often made wine. That was a hobby that started when I was a student. In the past it was much more easy to buy fruit juice that was more or less unprocessed, like 100% grape juice for 50 cents a liter. And with some extra sugar and of course yeast in a relatively short time you have your fresh batch of 20 liters wine. And somewhere from the back of my mind it came floating above that I had seen such irregularities arising from wine bottles if you stack them horizontal. But I never knew it had a solution like shown in the video.
Video title: The secret of the 7th row – visually explained
The next video is from Alexander Unzicker, the vid is only five minutes long. First I want to remark that I like Alexander a lot because he more or less tries to attack the entire standard model of physics. That not only is a giant task but you also must have some alternative that is better. For example when I talk or write about electrons not being magnetic dipoles, I never end in some shouting match but I just apply logic.
Let me apply some logic: In the Stern Gerlach experiment a beam of silver atoms is split in two by an inhomeogenous magnetic field. The magnetic field is stronger at one side and weaker at the other. One of the beams goes to the stronger side while the other goes to the weaker side of the applied magnetic field. But the logical consequence of this is that the stream silver atoms going to the weaker side gains potential energy. This is not logical. If you go outside and you throw a few stones horizontal, they always will fall to the earth and there is the lowest potential energy. The stones never fly up and accelerate until they are in space. In order to gain the logical point it is enough to assume that electrons are magnetic monopoles and that is what makes one half of the beam of silver atoms go to the weaker side of the applied magnetic field. If the electrons come in two varieties, either monopole north or monopole south, both streams do what the rest of nature does: striving for the lowest energy state.
Talking about energy states: Did you know that the brain of math professors is just always in the lowest energy state possible?
But back to the video: Alexander is always stating that often when progress is made in physics, all in all things become more easy to understand. That also goes for electrons, all that stuff about electrons being magnetic dipoles is just very hard to understand; why do they gain potential energy?
In his video Alexander gives a bad space as example where a so called three sphere is located. On the quaternions you cannot differentiate nor integrate, they are handy when it comes to rotations but that’s more or less all there is. So Alexander I don’t think you will make much progress in physics if you start to study the quaternions. And by the way don’t all physics people get exited when they can talk about ‘phase shifts’? They use it all the time and explain a wide variety of things with it. I lately observed Sabine Hossenfelder explaining the downbreak of quantum super positions into the pure ground states (the decoherence) as done by a bunch of phase shifts that make all probabilities of super positions go to zero. Well, the 4D complex numbers have a so called exponential curve and voila; with that thing you can phase shift your stuff anyway you want…
Video title: Simplicity in Physics and How I became a Mathematician
Yesterday I started working on the next post. It is all not extremely difficult but ha ha ha may be I over estimate my average reader. After all it is about the non-invertible numbers in the space of four dimensional complex numbers. The stuff that physics and math professors could not find for centuries… So you will never hear people like Alexander Unzicker talk about stuff like that, they only talk in easy to understand common places like the quaternions. And when I come along with my period of now about 18 years completely jobless, of course I understand the high lords of all the universities have more important things to do. All these professors are just soooooo important, they truly cannot react on social slime that is unemployed for decades. I understand that, but I also understand that if such high ranked people try to advance physics with the study of quaternions, the likelihood of success is infinetisimal small…
Anyway, here is a teaser picture for an easy to understand problem: if two squares are equal, say A^2 = B^2, does that always mean that either A = B or that A = -B?
In another development for decades I always avoided portraits and photo’s of myself on the internet at all costs. Of course after 911 that was the most wise strategy: you stay online but nobody know how you look. But over the years this strategy has completely eroded, if for example I just take a walk at some silly beach about 30 km away people clearly recognize me. So I more or less surrender, likely I will still try to prevent my head being on some glossy and contacts with journalists in general will also be avoided for decades to come. But in the present times why not post a selfportrait with a mask?
The upper half of the picture below is modified in the ‘The Scream’ style and the lower half is modified with something known as ‘vertical lines’.
Is the glass half full or half empty? You can argue that it is half full because the so called ‘pure tones’ as introduced in the previous post work perfectly for making a four dimensional Fourier series based on the 4D complex numbers. The glass is half empty because I started this Fourier stuff more or less in order to get some real world applications done, but 1 dimensional signals like a sound fragment do not reconstruct properly.
Why do they not reconstruct properly? Well often you need to take the sum or the difference of a 4D complex number Z and it’s conjugate that I write as Z* (because I cannot do ‘overline’ in this text mode). But the sum or the difference of such 4D numbers removes only the real part or the second imaginary part. The first and third imaginary parts stay in this sum or difference, this stuff is what makes the reconstruction of a signal s(t) going wrong.
Yet I was not crazy, as far as the reconstruction works it does it more or less as expected only you get only half of the signal reconstructed. That is not that worse but the garbage that enters the reconstruction is what makes this kind of making a 4D Fourier series something that will never have any practical benefit. But again does that mean the glass is half empty?
I remember that a long time ago in something like 1991 or 1992 I had found the product of a 3D complex number X and it’s conjugate X*. My naive idea was that this should only give the unit sphere in 3D space, but this product that also two imaginary components that I considered garbage at the time. Back in the time, it was just before the internet era, I could not know that this ‘garbage’ was actually the equation of a cone. And if you intersect this cone with the unit sphere in the space of 3D complex numbers, you get the 3D exponential circle. So it wasn’t garbage, it was the main prize in 3D complex number theory. Back in the time in 1991 it was stuped from me to expect the 3D complex numbers would behave ‘just like’ the two dimensional complex numbers from the complex plane. May be in this year 2020 I am making the same mistake again by expecting a 4D kind of Fourier series must behave ‘the same’ as those defined on the real line (the original Joseph Fourier proposal) and the more advanced version from the complex plane.
With the 4D Dirichlet kernel just like with the 2D Dirichlet kernel from the complex plane, you must take the difference of a number and it’s conjugate. In the complex plane this makes the real component zero and this difference is just an imaginary number. The 2D Dirichlet kernel is the quotient of two such imaginary numbers and as such it is always a real number. For the 4D Dirichlet kernel stuff is not that easy but for me it was surprising that you can show relatively easy the 4D Dirichlet kernel has to be a ‘self conjugate’ number. That means Z = Z* (on the complex plane when you have a number z such that z = z* it means it is a real number).
This post is 8 pictures long, 7 of them have the standard size of 55×775 pixels but I had to make one picture both a bit more broad and higher in order to get the math fitting in it. Ok, let’s upload the math pictures with the stuff around the 4D Dirichlet kernel.
Ok, credits have to go to where that is deserved. I remember that back in the time like in 1990 I found it relatively hard to calculate the 2D Dirichlet kernel. It took me over 15 or 20 minutes but again: that was before the internet era. Yet at present day I was all so simple and why was that? That is because there is a nice Youtube video doing the easy stuff, it is from ‘Flameable math’:
May be at the end I can say the glass is half full because now this reconstruction stuff does not work properly, luckily I do not have to construct the Fejer kernel for 4D complex numbers… Ok, let’s call it a day and let me end this post.
I want to start with a bit of caution: In this post you can find the definition of the 4D Fourier series. It looks a lot like the definition as on the complex plane. But I still did not prove all those convergence questions. And I also do not remember very much from the time when I had that stuff as a student (that was about 30 years ago). So I don’t know if I will be able to make such proofs about convergence and what kind of functions you can use to make a Fourier series from etc etc.
Yet in this post I define a set of possible signals that I name ‘pure tones’ and these clearly have a 4D Fourier expansion because by definition they have a finite number of non-zero Fourier coefficients. Of course when you only have a finite amount of non-zero coefficients, you don’t have any kind of convergence problem. So for the time the convergence problems are avoided.
In this post, number 154 already, I hope I demenstrated enough that the basis functions used in the definition are all perpendicular to each other. After all that was a nasty hurdle we met when it was tried with just the four coordinate functions of the 4D exponential curve as our basis vectors. So we do not meet that problem again using the exponential curve as a whole. If I denote the exponential curve as f(t), the basis functions we use are basically f(nt) where n is a whole number. Just like in the previous posts I always use the notation f(t) when the period is related to the dimension and g(t) when the period is different. Here I use of course a period of two pi because that is convenient and it makes the coordinate functions more easy to write out: the first one is now cos(t)cos(2t) and the other three are just time lags of the first one. But if you want to write g(t) as an exponential, because of the period it now looks a bit more difficult compared to just e to the power tau times t.
For myself speaking I have no idea at all if crafting a Fourier series like this has any benefits of using just the definition as on the complex plane. After all I only have more or less basic knowledge about the use of Fourier series, so I just don’t know if it is ‘better’ in some regards and ‘worse’ in others.
At last without doubt under my readers there will be a significant fraction that wonder if those 4D complex number system is not some silly form of just the complex plane? After all if that 4D space is based on some imaginary unit l with the property that now the fourth power l^4 = -1, how can that be different from the complex plane? The answer to that lies in the logarithm of the first imaginary unit l. If this 4D space was just some silly extension of the complex plane, this log of l should be nicely bound to i pi/4 where i is of course the imaginary unit from the complex plane. But log(l) is the famous number tau because with that you can make the exponential curve f(t) = e^(tau t). Basically the main insight is that i pi/4 makes the complex exponential go round with a period of four because i^4 = 1 and the 4D number tau makes the exponential curve go round with a period of 8 because l^8 = 1.
This post is six pictures long, all 550×775 pixels in size.
As usual I skipped a lot of stuff. For example, how did Fourier do it? After all at the time all this stuff with inner products was poorly developed or understood. That alone would be a cute post to write about. Yet the line of reasoning offered by Joseph Fourier was truly brilliant. In case you are lazy or you want to avoid Google tracking you, here is a link to that cute symbolab stuff: symbolab.com Link used: https://www.symbolab.com/solver/fourier-series-calculator
Ok, that is what I had to say for this tiny math update.