This is the shortest post ever written on this website.
I found one of those video’s where the Fourier series is explained as the summation of a bunch of circles. Likely when you visit a website like this one, you already know how to craft a Fourier series of some real valued function on a finite domain.
You can enjoy a perfect visualization of that in the video below:
Only one small screen shot from the video:
Oh oh, the word count counter says 80+ words. Let me stop typing silly words because that would destroy my goal of the ‘shortest post ever’. Till updates.
It took me longer than expected to craft this update. That is also the nature of the subject; you can view and do math with Cauchy integral representation in many ways. In the end I settled on doing it just for polynomials of finite degree and even more simple: these polynomials are real valued on the real line. (So they have only real coefficients and after that are extended to the space of 4D complex numbers).
In another development, last week we had the yearly circus of Noble prizes and definitely the most cute thing ever was those evolving protein molecules. Because if you can use stuff like the e-coli bacteria you can indeed try if you can (forcefully) evolve the proteins they make… That was like WOW. Later I observed an interview with that chemistry Nobel prize winner and she stated that when she began she was told ‘gentlemen don’t do this kind of thing’.
So she neglected that ‘gentlemen stuff’ and just went on with it. That is a wise thing because if you only do what all those middle age men tell you to do you will find yourself in the very same hole as they are in…
The physics prize was also interesting, for myself speaking I was glad we did not observe those physics men totally not understanding electron spin but with the usual flair of total arrogance keep on talking about spin up and spin down.
You can also turn that spin nonsense upside down: If elementary particles only carry monopole electrical charge than why should electrons be bipolar when it comes to magnetism? That Gauss law of magnetism is only a thing for macroscopic things, there is no experimental proof it holds for quantum particles…
But let’s talk math because this update is not about what I think of electron spin. This is the second Cauchy integral representation I crafted in my life. Now the last years I produced a whole lot of math, my main file is now about 600 pages long. But only that very first Cauchy integral representation is something that I printed out on a beautiful glossy paper of size A0. That first Cauchy integral representation was on the space of 3D numbers and there life is hard: The number tau has determinant zero and as such it is not invertible. But I was able to complexify the 3D circular numbers and it was stunning to understand the number tau in that complexification of the 3D circular numbers. Just stunning…
Therefore I took so much time in trying to find an easy class of functions on the space of 4D complex numbers. I settled for easy to understand polynomials, after all any polynomial gives the same value everywhere if you write them as a Taylor series.
Since this property of polynomials is widely spread I can safely say this in this part 14 of the basics to the 4D complex numbers we have the next Theorem:
THEOREM: The math will do the talking.
PROOF: Just read the next 12 pictures. QED.
As usual all pictures are 550 x 775 pixels in size. I also use a thing I name ‘the heart of the Cauchy integral’, that is not a widely known thing so take your time so that the mathematical parts of your brain can digest it…
I truly hope the math in this update was shallow enough so you can use it in your own path of the math that you like to explore.
End of this post, may be in Part 15 we will finally do a bit more about the diagonalization of 4D complex numbers because that is also a universal way of finding those numbers tau in the different dimensions like the 17D circular numbers & all those other spaces.
This is Part 9 in the basics to the complex 4D numbers. In this post we will check numerically that the 4D exponential curve has it’s values on the 4D unit sphere intersected with a 4D cone that includes all coordinate axes. In 3D space the sphere-cone equations ensure the solution is 1 dimensional like a curve should be. In 4D space the sphere-cone equations are not enough, there is at least one missing equation and those missing equations can be found in the determinant of a matrix representation M(Z) for a 4D complex number Z.
But we haven’t done any determinant stuff yet (because you also need a factorization of the 4D determinant in four variables and that is not a trivial task). So this post does not contain numerical evidence that the determinant is always one on the entire exponential curve.
If you want to compare this post to the same stuff in the complex plane:
In the complex plane the sphere-cone equation is given by x^2 + y^1 = 1 (that is the unit circle) and if you read this you probably know that f(t) = e^{it} = cos t + i sin t.
You can numerically check this by adding the squares of the sine and cosine for all t in one period and that is all we do in this post. Only it is in 4D space and not in the two dimensional complex plane…
This post is seven pictures long (all of the usual size 550 x 775).
All graphs in this post are made with the applets as found on:
Like promised in this post I will show you in the greatest detail possible how to find those rather difficult looking four coordinate functions.
I had thought about crafting these four coordinate functions before but the method I had in mind was rather labor some so I balked a bit at that. Not that I am lazy but I also had to work on the basics for the 4D complex numbers like in the last posts…
So one day I decided to look into the specific details of what I name ‘imitators of the number i’ and I was very surprised by their behavior. As a matter of fact these imitators imitate i soo good that you can make exponential circles of them.
And I wrote down the two exponential circles, I looked at them and realized you can factorize the 4D exponential curve with it and as such you will get the four coordinate functions…
That was all, at some point in time on some day I just decided to look at the imitators of the number i from the complex plane and within 5 at most 10 minutes I found a perfect way of calculating these four coordinate functions.
It always amazes me that often a particular calculation takes a short amount of time, like 10 or 20 minutes, and after that you always need hours and hours until you have a nice set of pictures explaining the calculation…
Anyway, this post is five pictures long and as such it contains also Part 6 and 7 of the Basics to the 4D complex numbers.
I hope that in the long run it will be the result in this post that will make 4D complex numbers acceptable to the main stream mathematical community.
But may be once more I am only fooling myself with that, after all back in the year 1991 I was only thinking stuff like ‘If you show them the 3D Cauchy-Riemann equations, they will jump in the air from joy’.
They (the math professors) never jumped from joy, no significant change in brain activity was ever observed by me. So when I write ‘in the long run’ as above, may be I should more think like a geological timescale…
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But let’s not complain because once you understand the factorization, it is so beautiful that it is hard to feel angry or whatever what.
Here are the five pictures:
Ok, that is how you calculate the four coordinate functions.
Lately I have been working on the next post about the basics of the 4D complex numbers. You simply need those basics like matrix representations because later on when you throw in some 4D Cauchy-Riemann equations, it is very handy to have a good matrix representation for the stuff involved.
The next post covering the basics had five parts, let’s not dive in all kinds of math details right now but go straight to part five with the four coordinate functions of the 4D exponential curve:
These four coordinate functions are also time lags of each other.
Judge: But you were caught red handed placing the number tau in a Schrödinger equation while you do not qualify for being a member of the most bright and enlightened persons in our society: The PHYSISCS PROFESSORS.
Reinko: But Judge I can explain, it was that evil guy that Gerard ‘t Hooft who did it. I can prove that because it is on video.
Judge: Yes you already told that into the statements you made after arrest by the police. So we took the freedom and ask Mr. Gerard ‘t Hooft himself about the evil you have done with molesting the Schödinger equation. Mr. ‘t Hoofd said it had to be Hermitian and although I do not know what that means he said that by using anti-Hermitian matrices you, Reinko Venema, you are nothing more as some sadistic pedophile piece of shit.
Reinko: But judge, it is not Hermitian, that is only a trick. You see if you multiply it by the number 1 like 1 = – i squared you see it is not Hermitian.
Judge: Do you think we get complex analysis in law school? We don’t, we asked some experts and all agreed that Gerard is right and you are wrong and right now rewarded by your own evil deeds to 75 years in prison in a maximum security facility.
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After this somewhat strange introduction I repeat I was innocent. I was just looking at a video of a guy that is just like me old and boring.
And that guy, Gerard ‘t Hooft, was able to give me three nice punches in the face.
That is what this post is about; Three punches in the face as delivered by Gerard.
It is the very first time I observe professional physics professors using the number tau while claiming the stuff has to be Hermitian to make any sense.
I was devastated because in my little world of mathematics it had to be anti Hermitian so at a first glimpse it looks like a simple shootout between Gerard and me: Only one can be right…
Let me first show you the Youtube video where right at the start Gerard succeeds to bring my small sack of human brain tissue into an exited state and after that I am rewarded with finding the number tau into the famous Schrödinger equation.
Let me also temper the enthousiasm a little bit because at present date 26 Feb in the year 2018 I only know of one example where three quantum states are rotated into each other:
That is the transport of the color charges as it is found on the quarks inside the proton and neutron…
Here is the video, after that the nine pictures that make up the mathematical core of this new post:
Gerard ‘t Hooft – How Quantum Mechanics Modifies the Space-Time of a Black Hole (QM90)
Let me spare you a discussion on the entire video but only look at what you can find on the very introduction as shown above because all of the three punches at my face are already found there.
Here are the nine pictures for this new post:
For readers who have found themselves lost on what a Hermitian matrix is, here is a wiki:
And for readers who have found themselves lost on finding an ‘analytic handle’ about how to calculate matrices like in picture 09, a good starter would be about the calculation of the 7D number tau:
All details will be in the next post but I succeeded into using matrix diagonalization in order to find this seven dimensional number tau.
For people who do not understand what a number tau is, this is always the logarithm of an imaginary unit. Think for example at the complex plane and her imaginary unit i. The number tau for the complex plane is log i = i pi/2.
The problem with finding numbers tau becomes increasingly difficult as the number of dimensions rise. I remember back in the year 2015 just staring at all those matrices popping up using internet applets like the next one:
Matrix logarithm calculator (it uses the de Pade approximation)
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm
Yet back in the year 2015 I was riding on my noble iron horse (a cheap bicycle) through the swamps surrounding the village of Haren and suddenly I had a good idea. Coming home I tried the idea of matrix diagonalization out in 3 dimensions and it worked.
Now I think that most readers who visit this website are familiar with the concept of finding a diagonal matrix D containing all eigenvalues of a given matrix M. Once you have the eigenvalues you can calculate the eigenvectors and as such craft your matrix C containing all eigenvectors.
You can write the stuff as next: D = C^-1 M C.
Suppose you don’t know what M is but I give you the matrices D, C and the inverse of C. Can you find the matrix M?
Yes that is a beerwalk, all you have to do is calculate M = C D C^-1 and you are good to go.
But with the logarithm comes a whole lot of subtle things for making the right choice for the eigenvalues that you place inside the diagonal matrix D. It turns out you only get the desired result if you use arguments in the complex plane between minus and plus pi.
This is caused by the fact that you always need to make a cut in the complex plane if you want to work with the complex logarithm; but it is a bit surprising that only the cut where you leave out all real negative numbers (and zero of course) makes the calculation go perfect and in all other cases it ends in utter and total disaster.
In the next three pictures I show you some screen shots with numerical values of matrix representations and the logarithm of those matrix representations.
The goal is to find mathematical expressions for the observed numerical values that are calculated via the above mentioned de Pade approximation. We don’t want only numerical approximations but also catch the stuff in a mathematical formulation.
At the end of the third picture you see the end result.
So it took some time to find this result, I wasted an entire week using the wrong cut in the complex plane. And that was stupid because I had forgotten my own idea when riding my noble iron horse through the Harener swamps…
The result for the seven dimensional number tau (circular version) as calculated in the next post is a blue print for any dimension although I will never write stuff down like in a general dimension setting because that is so boring to read.
In part this post picks up where I left the stuff of the missing equations back in the year 2015. The missing equations are found inside the determinant equation; for this to succeed we must factorize determinant of the matrix representations of higher dimensional numbers. A well known result from linear algebra is that the determinant is also the product of the eigen values; so we need to craft the eigen value functions that for every X in our higher dimensional number space give the eigen values.
These eigenvalue functions are also the discrete Fourier transform of our beloved higher dimensional numbers and these functions come in conjugate pairs. Such a pair form two factors of the determinant and if we multiply them we can get rid of all complex coefficients from the complex plane.
A rather surprising result is the fact that if we subtract a cone equation from a sphere equation we get a cylinder…
This post is also a way of viewing the exponential circles and curves as an intersection of all kinds of geometric objects like the unit sphere, (hyper) cones, (hyper) planes and (hyper cylinders. Usually I represent it all as some analysis but you can take a very geometric approach too.
I have no idea if the shape of the higher dimensional curves is studied as a geometrical object; I suspect this is not the case since the use of complex numbers outside the complex plane is very seldom observed. The professionals just want their tiny fishing bowl (the complex plane) and declare it an Olympic swimming pool…
Well, let it be because these people will never change.
All in all this post is 20 pictures long (size 550 x 775) so it is a relatively long read.
The pictures of the graphs were all made with an applet named Animated drawing, here is a link and there you can find it under ´Online calculators and function plotters´±
The above thing should give identical zero for all x.
An important feature of exponential curves in spaces with an odd number of dimensions is that they all are inside a hyperplane. The hyperplane says the sum of the coordinates is always 1. If you cut and past the next sum of the five coordinate functions you see that you always get one for all x:
At last the link to the original update from 2015 where I found the missing equations for the first time. But all I knew they were hidden inside the determinant. A few weeks ago I decided to take a better look and the result is this post.
Recently I am working on a relatively long post where I try to take a much more geometric approach to finding exponential circles and exponential curves. That post is also going forward from a few years back when I was searching for the so called missing equations.
The problem of the missing equations does not arise until you start working in five dimensions or higher; the equations as generated by the sphere-cone equations are just not enough to end up with a one dimensional curve.
Back in the time I simply took a few weeks until I had found the answer: The missing equations can be found inside the determinant!
For example if you have a 17 by 17 matrix in 17 variables (the so called matrix representations of 17-dimensional number systems), all you have to do is factorize this determinant and from those factors you can craft the extra needed equations.
Weirdly enough you find a hyper plane and a bunch of hyper-cylinders.
So in the next post I try to show you how you can have a very geometric approach to finding the higher dimensional exponential curves as the intersection of a sphere, a hyperplane, a bunch of cones and a bunch of cylinders.
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In my old notes I found a mysterious looking line of squares of cosines with their time lags. That was from before I solved the problem and this ‘solution’ has all kinds of faults in it.
That is this small post; it is about something that does not work.
It is just 3 pictures long (550 x 775 pixels):
It was just over two years back I wrote that long update on the other website about the missing equations, I was glad I took those weeks to solve this problem because it is crucial for the development of general higher dimensional theory on this detail.
I hope next week I am ready with the new long post, after that I will likely pick up magnetism again because I finally found out what the professionals mean when talking about ´inverted V-s’; that means there is also an electric field accelerating the particles in the aurora’s of earth and Jupiter.
Tiny problem for the professionals: At the Jupiter site, regardless of inverted V-s yes or no, the plasma particles get accelerated anyway…
So that looks like one more victory for Reinko Venema and one more silence from the professional professors.
This post has many goals, for example in the previous post I talked about a ‘very rudimentary Fourier transform’. In this post it is a bit less rudimentary, a bit more satisfying definition is given but still I did not research all kinds of stuff like the existence of an inverse & lot’s more basic stuff.
For myself speaking I consider this ‘new Fourier transform’ more as some exotic bird that, if capable to fly a few meters, will only draw applause from specialists in Fourier analysis.
So for myself speaking I am far more happy we need a more advanced number tau and the mathematical miracles you can do with it in three dimensions.
Therefore I included two examples of exponential curves that go through the plus and minus of all three basis vectors in 3D space, after all this is one of my most remarkable math results…
In this post I also show you how to use the calculus of ‘opposite points, in three dimensions it works like a bullet train but the higher the dimensions become the harder it is to frame it in simple but efficient calculus ways like using opposite points on exponential circles.
Another thing to remark is that an exponential circle is always a circle; it is flat in the 2D sense and has a fixed radius to some center. When this is not the case I always use the words exponential curve…
This post is nine pictures long, I truly hope you learn a bit from it.
You really do not need to grasp each and every detail, but it is not unwise to understand that what I name the numbers tau are higher dimensional versions of the number i from the complex plane.
Ok, here we go:
In these nine pictures I forget to remark you can also craft a new Cauchy formula for the representation of analytic functions. For myself speaking this was far more important compared to a new way of Fourier transform.
You still need that more advanced version of tau…
Can´t get enough of this stuff?
Ten more pictures dating back to 2014 at the next link:
