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.
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… Here is the link:
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´.
Oh oh, this is one of those posts where I only calculate in the 3D circular numbers while I classify it as 3D complex numbers. In the past when I made those categories on this website I did not want to have too many categories so that is why I only have 3D complex numbers as a category.
All in all this post (number 146 already) is not extremely important because over the years I have given many proofs that the parametrization for the exponential circle indeed fulfills all those equations like the sphere-cone equation of the fact the determinant is always one. On the other hand, if you have an important mathematical object like the exponential circles, it is always good to have as many proofs as possible. Just like there are many proofs for the theorem of Pythagoras, it would be strange if we only had one proof and nobody cares about more proofs to that theorem that more or less the central to a giant mountain of math.
What do I mean with ‘first principles’? Very simple: that is the summation formula for the exponent of a linear operator or the matrix exponential if you want. In this post I use a somehow slightly different number tau; I use a number tau that gives a period of 2 pi for the exponential circle. The reason is simple: that makes the long calculation much more readable.
Another thing I want to mention is that the long calculation is nine lines long. For myself when I read the works of other people I do not like it if calculations go on and on and on. I always try to avoid too long calculations or I just don’t write posts about them. Almost nobody reads the stuff it it’s too long and gets too complicated so most of the time I simply skip that. Beside that there is always 0% feedback from the mathematical community, so although I always year in year out try to keep it so simple that even math professors can understand it, nothing happens. Just nothing, so after all those years it is not much of a miracle I don’t want to engage with these overpaid weirdo’s at all. Likely if you are born stupid you will die stupid & I have nothing to do with that. Mathematics is not a science that is capable of cleaning itself up, the weirdo’s keep on hanging to their fantastic quaternions and their retarded ideas of what numbers & complex numbers are. Too much money and too much academic titles have not lead to a situation where the science of math is capable of cleaning itself when needed.
Enough of the blah blah blah, after all the physics professors have the same with their electron spin: where is your experimental proof that the electron is a magnetic dipole? For over five years nothing happens except a lot of weird stuff like quantum computers based on electron spin…
This post is five pictures long, for me it was cute to see how those three cosine functions slowly rise from the start of the long calculation. Also of importance is to notice that I had to use the simple formula for cos(a + b) = cos(a)cos(b) – sin(a)sin(b) that comes from the exponential circle in the complex plane. Just once more showing that 3D complex & circular numbers are indeed emerging from the 2D complex plane. Not that the math professional will react, but anyway…
Let’s go to the five pictures:
Again, this is not a ´very important´ post. Given all those results and proofs from the past it is logical such a long calculation has to exist. It´s relevance lies in the fact you simply cannot have enough proofs for the calculation of parametrizations of the 3D exponential circle.
Let me leave it with that. See you in the next post.
A lot of math professionals rather likely still think that 3D complex numbers do not exist, may be for reasons like there are non-invertible numbers or whatever what other reason they have. This post more or less proved such views are nonsense; for example a lot of math on the 2D complex plane does not rely on the fact it is a field (and as such only division by zero is forbidden).
But on the 3D complex and circular number spaces indeed it brings some complications if you have non-invertible numbers in the function you want to integrate over a particular curve. And I have to say that problem could be solved by using the special properties that those numbers have. In this post I only show some examples with the non-invertible number alpha (alpha is the midpoint of the 3D exponential circles and all multiples of alpha are also non-invertible so the line through 0 and alpha are all not invertible).
For me writing this was a good distraction away from all that negative news we have day in day, all those countries reporting daily death toll can make you a bit depressed… So when I am through with the daily news I always do some other stuff like calculating a few of such integrals. That is a very good antidote against all that bad news. After all there is not much gained if you constantly think about things you cannot change at all.
This post is relatively long; at first I crafted 12 pictures but it soon turned out that was not enough. So while filling the 12 pictures with the math and the text I expanded some of the pictures so they could contain more math & text. That was not enough and in the end I had to craft two more background pictures. All in all it is 14 pictures long, that is a record length for this website.
If in your own mathematical life you have performed contour integration in the complex plane, you must be able to understand how this works in the 3D spaces. And for those who have done the thing known as u-substitution on the real line: it is just like that but now this u thing is the parametrization of a path. All that stuff below with gamma in it is either the path or the parametrization of that path. Please remark that you must use the complex or the circular multiplication on 3D, just like integrating over a contour in the complex plane uses the 2D complex multiplication.
In case if you are not familiar with the number alpha that is found at the center of the exponential circle, use the search function of this website and for example look up ‘seven properties of the number alpha’.
I hope I have removed all faults, typo’s etc so that later I do not have to repair the math because that is always cumbersome. Here we go: 14 pictures long so this is hard to grasp in detail in just a few hours. But it is beautiful math & that is why I do this. For me math is a lovely hobby.
Enough of the blah blah blah, here we go:
Ok, let´s first hit the button ´Publish´ and see what will happen… It looks all right but a day after first publication I realized there was some missing text. It turned out I had to rename picture number 2 and now every thing was like it was planned.
Later I will flea through the rest of the text, if needed I will post more addenda. For the time being that was it so till addendums or till the next post.
Ok, the math text is finally written. It took a long time but all in all I am very satisfied with the result. It will be a long post, I estimate about 12 pictures long and that is more or less a record length on this website. I have finished only two pictures and I will take my time to make the other ones because my mouse does not work properly. When I click with the computer mouse, very often that acts as a double click and that makes making pictures a laborsome task because of all the errors that double clicking gives. And when I have to repeat a series of clicks three or four times before it is ok, it will take some time. May be I should buy a new mouse,
Anyway to make a long story short: For years I stayed more or less away from crafting math about integration because it is hard to find a definition that would work always. My favorite way of using Riemann sums could not work always because of the existence of non-invertible numbers in the 3D spaces. And that gave some mathematical fear in my small human mind because path independence came with that way of Riemann summation. All in all it is beautiful math to think about: For example if in 3D you use a primitive to integrate over a closed loop, is it always zero?
So only the first two pictures are posted and I have no idea when all other pictures are finished. Here we go:
It is about time for a small update! Despite all that COVID-19 stuff going round, for myself after all those years I finally tried to put integration on higher dimensional number systems on a more solid footing. All those years I just refrained from it because you cannot use my favorite Riemann sum approach because of the non-invertible numbers we have in 3D or even higher dimensions. But now I am trying to finally make some progress and stop avoiding this subject, I find it is utterly beautiful. It has an amazing array of subtle details involved when you have some non-invertible numbers in your integration stuff. I have no idea when I have finished this rather important detail in my cute theory of higher dimensional complex & circular numbers, so let time be time & in the meantime only post a teaser picture about that lovely integration stuff. In the first lines you see a very familiar integral, likely you have done such calculations in the complex plane. In the case of 3D circular and complex numbers you must (of course) use the multiplication on 3D space to make it all work. Basically you are evaluating (or calculating) three integrals at the same time, just like on the complex plane where you are evaluating two integrals at the same time in your calculation. If you work with a pencil & paper, make sure you have enough paper because all those 3 integrals also have 3 terms in it so your calculations can become quite long… Here we go:
Ok, let me end this update now. Till updates and for some strange reason you must wash your hands while the proper authorities never point to 3D complex numbers… Till updates.
All in all it was a nice day today. Brewing is completed and tomorrow the wort can go into the fermentation bottles and the wonderful process of fermenting can take place. For those of you that also like to brew: A couple of months back I found a cute video explaining that you can also brew beer without cooking it. And I was like seeing water burning or I was like a professional math professor understanding 3D complex numbers for the very first time in their life… Anyway if you are interested search for ‘Raw ale no boil brewing’ on Youtube. It is of interest because if you brew without boiling, only after that you understand what you usually cook away in things that might taste good (or bad).
But let’s go to this post: It is about probability amplitudes as they are used in quantum physics where all those kind of amplitudes are multiplied against their conjugate and that gives a real positive number known as the probability. If you write it in polar coordinates on the complex plane, it is easy to see that those probability amplitudes can have all kinds of phases (the argument of the complex plane number). So for that to work on 3D complex or circular numbers, it would be great if you can write it more or less like the polar coordinates as in the complex plane. And that is easy to do in 3D space: Once you have found and also understand the exponential circles, it is evident that all numbers on those exponential cones are some real multiple of a number from the exponential circle.
As such the numbers found on the exponential cone can be written just like the polar stuff from the complex plane, also now the r as used in polar coordinates can also be negative. That is a strange result because for millions of years we were always indoctrinated by a positive r … 😉
Another important difference with the complex plane lies in the fact that the complex plane is closed under addition. That is obvious, but it is also obvious that on a cone it is very different. Most of the time if you add up two numbers you are either inside or outside the cone. But probability amplitudes are always multiplied against their conjugate and added up only later, so we can still use the exponential cone for things like that. I don’t see that ship stranding, so let’s do it.
I also want to remark I am using the so called ‘pull back map’ once more. The professional professors also have a pull back map but that is a very different thing compared to what I use. So don’t be confused by that: the way I use it is to fix higher dimensional exponential circles (and curves) on the exponential circle in the complex plane. (This for fine tuning the period in time and stuff like that, or for understanding why the numbers are what they are: WTF that square root of 3 in it???
This post is 7 pictures long, most are the usual size of 550×775 pixels. At last I want to remark that for myself speaking I do not know if there is any benefit in trying this kind of use of 3D complex and circular numbers. It is funny to think about positive and negative values for r like for example in electron spin or a wave function for the electron pair. But I just do not know if this add any value or that you can use the complex plane only and miss nothing of all you could have learned.
Ok, here we go:
Ok, that was it for this post. Till updates my dear reader.
First a household message: In about two weeks time this website should go to new very fast servers. In order for that to work properly I have to do all kinds of things that I have never done before. Stuff like updating PHP. Ok, that does not sound too difficult but as always the work explodes because first I have to backup everything. And before I can backup everyting I need a new ftp account. The only luck is I still have a running ftp client on my own computer…
In case this website is gone in two weeks, somewhere I got lost in the woods. And there is no hurry: this math website is just a hobby of me. An important hobby because it is a bit of exercise for the brain… End of the household message.
What is the yeast of this post? Historically the de Moivre identity (or theorem) predates the very first exponential circle on the complex plane. If you use the exponential circle, a proof of the de Moivre identity becomes very very easy. In this short post we will use the 3D exponential circle for circular numbers. Two posts back I showed you a possible parametrization via those 3 cosine expressions, in this post we use those parametrizations to formulate a 3D de Moivre identiy. Because we already have an exponential circle, we do not need to give a rigid math proof for this identity. Once you have and exponential cricle, stuff like that comes for free along with it…
As usual I skipped a lot of things while writing this post. For example I skipped using those modified Dirichlet kernels. I skipped giving the 4D de moivre identity for the 4D complex numbers. All in all I was satisfied to cram this all in a very short post; only three pictures long! In case you are still reading this while having no clue whatsoever what a de Moivre identiy is, here is some stuff from brilliant.org: De Moivre’s theorem Http stuff in the link: https://brilliant.org/wiki/de-moivres-theorem/
Ok, only three pictures long. Here we go:
That was it for this post. If I don’t change plans, in the next post we will look at the 3D exponential cone because on that cone you can do all those quantum probability calculations just like in the 2D complex plane. But before that I have to go though that horrible PHP update…
So see you in the next post or let’s split indefinitely and end this stupid website for no reason at all… 😉
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