# Part 21: More on the structue of non-invertible 4D complex numbers.

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.

# Two videos & a short intro to the next post on 4D complex numbers.

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’.

Ok, that was it for this post.

# The 4D Dirichlet kernel related to the 4D Fourier series.

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.

# Definition of the 4D Fourier series.

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

Ok, that is what I had to say for this tiny math update.

# The last details before we finally can go to the new way of 4D Fourier series.

This evening I brewed a fresh batch of wort, it is now cooling and tomorrow when it is at room temperature it can go into the fermenter bottles. Everything is relaxed around here, in the last week in my city only four cases of COVID-19 into the hospital.

Orginally I planned to define the 4D way of taking a Fourier series in this post, but while writing this post I realized to would become too long. So this post is seven pictures long (all 550×775 pixels) therefore I hope it does not cross the length of your attention span. For myself speaking always when I read some stuff that is ‘too long’ I start scrolling till the end and as such you often miss a lot of important details.

An important feature of exponential curves is that you have those de Moivre identities that come along with all those exponential curves. In this post I did not prove the 4D version of a de Moivre identity for the 4D complex numbers, but I give some numerical evdence. It has to be remarked that when I wrote the 20 posts around the basics of the 4D complex number system, I did not include the de Moivre identity. So that is more or less an ommision. On the other hand it is of course much more important to be able to find the exponential curve in the 4D space that is the basic material needed to write down such a de Moivre identity.

I categorize these posts about the 4D Fourier series also under ‘integration’. Not that I have many fantastic insights about integration but the reason for this category is much more down to earth: You have to perform an integration for every Fourier coefficient you calculate. In case you missed it: this year I finally wrote that post about how to define integration in the 3D complex numbers. Use the search function of this website in order to find that post in case you are interested.

Ok, that is what I had to say. Let us go to the beef of this post number 153 and that is of course the seven pictures that are all hungry for your attention. Here are the seven pictures:

There is an online Fourier series applet as it is named in these years. In the past we did not have applets but only computer programs. Anyway it is important those applets are there so I want to give a big thumbs up to the people that maintain that website! Here is a link to the Fourier thing:

Fourier series calculator. Link used: https://www.symbolab.com/solver/fourier-series-calculator

At last I want to point at the importance of such free websites. It is very good if a society has enough of this kind of ‘free stuff’. For example when in the year 2012 I picked up my study of the 3D complex numbers for me it was very important that there were free online applets for the logarithm of a matrix. Without such a free website I would have taken me many more years to find those exponential circles & curves. Or I would not have found them at all because after all my biggest breakthrough was when I did read that numerical evidence from log j on the 3D complex numbers: why are the two imaginary components equal I was just wondering? Later I found how crucial that was: only if the imaginary components are equal the eigenvalues are purely imaginary in the complex plane.

Ok, that’s it for this post. In the next post I will finally give that definition and after that I do not have a clue. I still do not have any good proof for convergence of these 4D Fourier series so we’ll see. Till updates.

# New way of Fourier series using the 4D complex numbers.

Warning: This post contains stuff that is not correct! Yet I decided to post it anyway so you can see that crafting math is also just keep on working until you have it right. The reason it does not work in the post below is that the basis functions I use are not all perpendicular to each other. And if you want to reconstruct a function or a signal s(t) with the basis functions as mentioned below, you will get weird overlap and the end result will not be correct.

Ok, that was a bit frustrating. But all in all I don’t have much reputation damage because more or less instantly I found another way of crafting the 4D Fourier series and that seems to work perfectly. So compared to the professional math professors who at one point in time accepted the quaternions and together with that stupid theorem of Frobenius concluded that 3D or 4D complex numbers are not possible, at least I don’t look that retarded. Sometimes I can be stupid too but at least it does not last for over one century. And may be that is also the reason that professional math professors absolutely do not want to talk about my work on 3D & higher dimensional number systems; admitting that you have been stupid for over one century is of course not an easy thing to do. And given the fact I am now unemployed for 17 or 18 years, rather likely the professional professors would rather be eating dog shit compared to speaking out my name… Once more we observe that in this world there is never a shortage of idiots.

In this post I use the coordinate functions of the exponential curve in the 4D complex numbers but I changed the period to 2 pi instead of a period of 8 that comes along with taking the log of the first imaginary unit. I also would like to mention that I use the so called modified Dirichlet kernel and because that kernel originates from Fourier analysis you must not get confused by the name ‘Dirichlet kernel’. The modified kernel is important (anyway for me) because it spits out all those coordinate functions for making exponential circles and curves in all dimensions possible. While if memory serves, the use of the Dirichlet kernel inside Fourier analysis is for using it in proofs of convergence. But may be I remember that completely wrong, after all it was about 30 years ago that I studied the Fourier stuff for the last time. The last two weeks were pleasant from the mathematical point of view, all that old Fourier stuff that somewhere still lingers around in my brain. But so much is gone, what is that Gibbs overshoot? Is that when a male math professor has his yearly orgasm? And what was the Parcival identiy? I don’t have a clue whatsoever.
This post is 7 pictures long, four are of 550×775 pixels and I had to enlarge the other three to a size of 550×850 pixels. So it is not a mess like the previous post where I just enlarged the pictures on the fly until all that text was there. Here it is:

Likely all those basis functions have this problem, if you take the inner product of an arbitrary basis function against the same basis function with three times the speed, it is not zero. And as such it is not perpendicular…

For people who have never heard of inner product spaces done with functions I found a cute pdf where a lot of the basics are explained.

Inner product spaces.

I would like to be the 4D Fourier stuff done in a correct manner in the next post but sifting through what I wrote on 4D complex numbers I realized I never wrote about a de Moivre identiy for the 4D exponential curve. May be I will publish that in a separate post, may be not.
Anyway, have a good time and see you in the next update.

# Cute search engine results found & intro 4D Fourier series.

About a week ago I started investigating how you could craft a Fourier series using the coordinate functions of the 4D exponential curve. The usual way the series of Joseph Fournier are done is with the sine and cosine that are also the building blocks of the exponential circle on the complex plane. So I needed to look up my own work on the 4D complex numbers because in the beginning of 2019 I stopped writing posts about them and after such a long time not every detail is fresh in your brain of course.
Anyway I did a Google search on 4D complex numbers and to my surprise this website popped up above where the quaternions were ranked. I was ranked number one. That was a great victory of course, it means that people are actually reading this stuff… In one of the screen shots below you see the quaternions once more topping my 4D complex numbers but from day to day it seems that Google is shuffeling the top results a bit so the search results look a bit more dynamic on a day to day basis.

This year I didn’t look at the search engine stuff at all, we still have that corona stuff going on and beside that why look at such boring stuff if I can do math instead? But I could not resist and went to the Microsoft Bing search engine. For years they never ranked this website on page 1 if you searched for ‘3D complex numbers’. But all those years if you looked into the picture search of the Bing search engine a giant fraction of the pictures was from this website. That was very strange, how can you return so much pictures from my website while never mentioning me at any significant position in the rankings of the html files? Ok ok most people say Bing is an inferior search engine compared to the Google search engine and as such not many people use the bing thing.

So once more and for the first time in this year 2020 I searched for ‘3D complex numbers’ on the Microsoft search engine. To my surprise instead of being burried down deep on say page 10, at Microsoft they had seen the light. Here is a screen shot:

In the next screen shot you see the html listing of Google when you search for 4D complex numbers. Today when I made the screen shot I was not ranked at no 1 but for some strange reason that did not make me cry like a baby in distress.

And the last screen shot is about the Google thing for pictures when you search for four dimensional complex numbers. Luckily there is no competition but does that mean the rest of humanity is stupid as hell?

Of course not, it only means no one is interested in crafting 4D complex numbers for themselves. Professional math professors don’t want to talk about 4D complex numbers in public, so why are my internet search engine rankings that high? It might be that it is read by non math professors and that more or less explains the high rankings…

Ok, it is now 22 June and I finally wrote down what the new way of Fourier series is using the 4D exponential curve. Writing of the next post is almost finsihed and I think I am going to do it just like Joseph Fourier did. That is without any proof at al for the most important things…

Anyway, in the nexgt picture you see the Fourier series in a 4D style:

It could ber handy to look at the end of an old post from 01 Nov 2019, there I show you how you can use the modified Dirichlet kernels for finding parametrizations of the exponential circles & curves in 2, 3 and 4 dimensions. If it is possible to craft a 4D Fourier series (again this is only postulated so there is no proof at this date) you surely must try to understand the 4D modified Diriclet kernel…

# Two parametrizations for 3D exponential circles.

It is about high time I post the solution in parametrization form of those five equations from 03 Oct 2019. That is almost 2 months back and oh how ashamed am I for my laziness… But for me math is a hobby, an important hobby but a hobby anyway. So other hobby’s are allowed to interfere with my little math hobby.

This post is 10 pictures long and at the end there is a horrible bad video from the Youtube channel Seeker. Begin this week I crossed that video with an intriguing title; Could These Numbers Unravel New Dimensions in Space? I was just curious but it is that Cohl Furey stuff again. It is an attempt to explain particle physics via complex number, quaternions and octonions… What do they have in common? These number systems are always fields that means all non-zero numbers have an inverse. Why the professional math professors find that so important is unknown to me, it is more like they have nothing else in the toolbox. If you are interested you can find the Cohl Furey video’s on Youtube.

In this post I too write about things that are common in the complex plane, complex and circular 3D numbers and 4D complex numbers. You can use the modified Dirichlet kernels as the building blocks for all possible exponential circles or in the case with 4D complex numbers: the exponential curve (in 4D space the curve is in a 3D hyper plane).

But I also wanted to show you the original cosine solution that I found years ago. To this day it is still amazing that the cosine can pull it off; that the cosine can be a building block for a 3D exponential circle. Next year it will be three decades ago when I found the 3D complex numbers and got interested in them. At present day you can wonder why there is never a healthy response from the math communuty. It is all very logical: if there is no healthy response that means the math community in itself cannot be healthy. It is just a community of perfumed princes and that’s it.

After so much blah blah it is high time to go to the ten pictures:

So from the complex plane in two dimensions to 4D complex space; a binding element is how you can use the modified Dirichlet kernels and their time lags to construct these very interesting parametrizations. Of course there is much more that binds those spaces together; the matrix representations are all very similar, just like the eigen values and eigen vectors. But above 2D it is never a field. And again why the professional math professors have this weird fixation on fields is completely unknown to me.
At last, here is that wonderful video that will make your toes curl

End of this post and thanks for your attention!

# Part 20: On the structure of non-invertible 4D complex numbers.

In general it is rather hard to find non-invertible 4D complex numbers because the determinant is non-negative everywhere. Just try it yourself, write down just one 4D complex number that is non-zero and not invertible.

That is not an easy task, after some time you will find some but do you have all?

But if you understand the concept of the eigenvalues that every 4D complex number Z has, it is easy to understand that if a 4D complex number is non-invertible at least one of the eigenvalues must be zero.

In previous posts we already unearthed the four eigenvalue functions that return the four eigenvalues each 4D complex number has.

In this post we will try to find where these eigenvalue functions are zero.

Since eigenvalue functions come in pairs whenever possible in the case of 4D complex numbers we only have two pairs of those eigenvalue functions.

Let’s stop the talking and just post the twelve pictures that make up part number 20 into the basics of the four dimensional complex numbers.

Ok, that was it for part 20 in the series that covers the basics of 4D complex numbers.

See you around my dear reader.

# A teaser question: Can you prove this inequality?

Recently I am working on part 20 to the basics of the 4D complex numbers. Ok ok if you need 20 parts to explain ‘the basics’ how basic is it you can ask yourself.

You can argue long and short on this: are fresh Cauchy integral formula’s really ‘basic stuff’? I don’t know how a democratic vote among professional math professors would fall down.

Anyway, an important property of the determinant of 4D complex numbers is the fact that the determinant is always non-negavite. At least it is zero and at those points in space we have found a non-invertible number.

In part 20 on the basics to 4D complex numbers we will look when the eigenvalues of 4D complex numbers vanish; at those points the stuff is non-invertible & that is what we will be hunting on part 20.

In the next picture you see a difficult to understand inequality & the teaser question is:
Can you prove this inequality via math methods that do not use 4D complex number theory at all?

If so, you should definitely pop up a second pint of perfect beer on a late Friday evening.

Ok, that was it. Till updates in part 20 where we try to find all non-invertible 4D complex numbers in a not too difficult way.