Sesamath 6 correction bit
Therefore I also included the four coordinate functions of the exponential 4D curve that we looked at in the previous post. All math loving folks are invited to find the four coordinate functions for themselves, in the next post we will go through all details.
And once you understand the details that say the 4D exponential curve is just a product of two exponential circles as found inside our 4D complex numbers, that will convince you much much more about the existence of our freshly unearthed 4D complex numbers. Of course the mathematical community will do once more in what they are best: After having said that, this post is partitioned into five parts and is 10 pictures long.
It is relatively basic and in case that for example you have never looked at matrix representations of complex numbers of any dimension, please give it a good thought. Because in my file I also encountered a few of those professional math professors that were rather surprised by just how a 3 by 3 matrix looks for 3D complex numbers. How can you find that they asked, but it is fucking elementary linear algebra and sometimes I think these people do not understand what is in their own curriculum….
And may be I am coming a bit too hard on the professional math professors. After all they must give lectures, they must attend meetings where all kinds of important stuff has to be discussed until everybody is exhausted, they must be available for students with the questions and problems they have, they must do this and must do that. At the end of the day, or at the end of the working week, how much hours could they do in free thinking?
Not that much I just guess…. 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.
This tau in any higher dimensional number system or a differential algebra in case you precious snowflake can only handle the complex plane and the quaternions is always important to find. Informally said, the number tau is the logarithm of the very first imaginary component that has a determinant of 1. For example on the complex plane we have only 1 imaginary component usually denoted as i. Complex numbers can also be written as 2 by 2 matrices and as such the matrix representation of i has a determinant of 1.
Anytime they talk about a phase shift they always use this in the context of multiplication in the complex plane by some number from the unit circle in the complex plane.
In this post, for the very first time after being extremely hesitant in using dimensions that are not a prime number, we go to 4D real space. Remark that 4 is not a prime number because it has a prime factorization of 2 times 2. Why is that making me hesitant? That is simple to explain: If you can find the number i from the complex plane into my freshly crafted 4D complex number system, it could very well be this breaks down to only the complex plane.
In that case you have made a fake generalization of the 2D complex numbers. So I have always been very hesitant but I have overcome this hesitation a little bit in the last weeks because it is almost impossible using the complex plane only to calculate the number tau in the four dimensional complex space…. May be in a future post we can look a bit deeper in this danger; if also Cauchy-Riemann equations are satisfied in four real variables, that would bring a bit more courage to further study of the 4D complex number system.
After the introduction blah blah words I can say the 4D tau looks very beautiful. That alone brings some piece of mind. I avoided all mathematical rigor, no ant fucking but just use numerical results and turn them into analytical stuff. That is justified by the fact that Gerard is a physics professor and as we know from experience math rigor is not very high on the list or priorities over there….
That is forgiven of course because the human brain and putting mathematical rigor on the first place is the perfect way of making no progress at all.
In other sciences math should be used as a tool coming from a toolbox of reliable math tools. This post is seven pictures long, all are by pixels in size except for the last one that I had to make a little bit longer because otherwise you could not see that cute baby tau in the 4D complex space. It is very very hard to stay inside the complex plane, of course the use of 4 by 4 matrices is also forbidden, and still find this result….
I am still hesitant about using dimensions that are not prime numbers, but this is a first result that is not bad. I can prove that because it is on video. Yes you already told that into the statements you made after arrest by the police. So we took the freedom and ask Mr.
But judge, it is not Hermitian, that is only a trick. Do you think we get complex analysis in law school? 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. 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 also temper the enthousiasm a little bit because at present date 26 Feb in the year 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:. 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. An important calculation of the 7D number tau circular version.
I really took the time to compose this post; basically it is not extremely difficult to understand. Everybody who once has done matrix diagonalization and is still familiar with the diverse concepts and ideas around that can understand what we are doing here. It is the fact that it is seven dimensional that makes it hard to write down the calculations in a transparent manner. I think I have succeeded in that detail of transparency because at the end we have to multiply three of those large seven by seven matrices with each other and mostly that is asking for loosing oversight.
Luckily one of those matrices is a diagonal matrix and with a tiny trick we can avoid the bulk of the matrix calculations by calculating the conjugate of the number tau.
Just like in the complex plane where the conjugate of the number i equals -i, for tau goes the same. Basically the numbers tau are always the logarithm of the first imaginary component. But check if the determinant is one because you can use the tau to craft an exponential curve that will go through all basis vectors with determinant one. This post is 10 pictures long size x , in the beginning I use an applet for the numerical calculation of the matrix representation of the first imaginary unit in 7D space, here is the link:.
Matrix logarithm calculator http: Two years back in after I found the five dimensional numbers tau every now and then I typed in a higher dimensional imaginary unit and after that only staring at the screen of the computer: How to find those numbers as the log applet says….
The method as shown here can be applied in all dimensions and you now have a standard way of crafting exponential curves in all spaces you want. This method together with the modified Dirichlet kernels that provide always a parametrization of the exponential curve form a complete description. That makes or breaks this method, if done wrong you end up with a giant pile of nonsense….
Have fun reading it and if this is your first time you encounter those matrices with all these roots of unity in them, take your time and once more: If you have never seen a matrix like that it is very hard to understand this post in only one reading….
I am glad all that staring to those numerical values is over and we have the onset of analytical understanding of how they are in terms of the angle 2 pi over 7. The result is far from trivial; with the three or five dimensional case you can use other ways but the higher the dimension becomes the harder it gets.
This method that strongly relies on finding the correct diagonal matrix only becomes more difficult because the size of the matrices grows. So only the execution of the calculation becomes more cumbersome, the basic idea stays the same. I have no idea what the next post is going to be, may be a bit of magnetism because a few days back I got some good idea in explaining the behaviour of solar plasma included all those giant rings that shoot up and land in another spot of the sun.
And we also have those results from the Juno mission to Jupiter where the electrons also come from Jupiter itself without the guidance of electrical fields. 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 problem with finding numbers tau becomes increasingly difficult as the number of dimensions rise. I remember back in the year 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: Yet back in the year 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. Integral calculus done with matrix diagonalization. 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: Can you find the matrix M? 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. 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 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. For example if you have a 17 by 17 matrix in 17 variables the so called matrix representations of dimensional number systems , all you have to do is factorize this determinant and from those factors you can craft the extra needed equations.
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. In my old notes I found a mysterious looking line of squares of cosines with their time lags. That is this small post; it is about something that does not work. It is just 3 pictures long x 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. 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.
Basically tau-calculus is very easy to understand:. This is easy to understand but I could only do this in three and five dimensional space, seven or eleven dimensional space? It is now 2. Anyway back in the time it was a great victory to find the exponential curves in 5D space. May be some people with insect like minds think that the Euler identity is the greatest formula ever found but let me tell you: The more of those exponential circles and curves you find, the more boring the complex plane becomes….
In the months after Jan I wanted to understand the behavior of the coordinate functions that come along those two exponential curves. But the problem kept elusive until I realized I had one more round of internet applets waiting for me. I had to feed this internet applet for the log of a matrix over 50 matrices and write down the answers it gave me; that was a lot of work because every matrix had 25 entries.
I still remember sitting at my table and do the drawing and when finished it was so fast that I realized the next: This is a Dirichlet kernel. And the way I used it, it was so more simple to write down this kernel and if you think how those exponential curves in higher dimensional spaces work with their starting coordinate function and all the time lags that follow… This finding will forever be in the top ten of most perfect math found by your writer Reinko Venema….
Ok, what is in this new post? Nine pictures of size x pixels containing: Hope you can learn a bit from it, do not worry if you do not understand all details because even compared to the 2D complex plane or the 3D complex numbers this website is about: You need a good applet for the log of a matrix representation if you want, for example, crack the open problem in tau calculus for the 7D complex numbers: My original update on the other website about the 5D number systems from Jan If you are more interested in those kind of weird looking integrals suddenly easy to solve if you use a proper combination of geometry and analysis, the update from July upon the missing equations is also worth a visit: This is what I more or less had to say, have a nice life or try to get one.
See you in the next post. Via the so called tau calculus I was able to achieve results in 3 and 5 dimensional number systems and I really had no hope in making more progress in that way because it gets so extremely hardcore that all hope was lost. Yet about two years ago I discovered a very neat, clean and very beautiful formula that is strongly related to the Dirichlet kernel known from Fourier analysis. This modified kernel is your basic coordinate function, depending on the dimension of the space you are working in you make some time lags and voila: There is your parametrization of your higher dimensional exponential periodic curve only in 2D and 3D space it is a circle.
Not often do I mention other mathematicians, but I would like to mention the name of Floris Takens and without knowing how Floris thought about taking a sample of a time series and after that craft time lags on that, rather likely I would not have found this suburb and very beautiful math…. But factoring the Laplacian requires understanding 3D Wirtinger derivatives so likely I will show you how I found the very first modified Dirichlet kernel.
This update contains six jpg pictures each about x pixels and two old fashioned animated gif pictures. I tried to keep the math as simple as possible and by doing that I learned some nice lessons myself… Here we go:. Here is an animated gif of how this coordinate function looks when you combine it with the two time lags for the y t and z t coordinates.
Does it surprise you that you get a flat circle? If it does not surprise you, you do not understand how much math is missing in our human world…. And there is that kind of typo that is a disaster in people trying to understand what this stuff actually means. I am sorry I messed things up, but it is like picture six on the previous update on the length of complex numbers that must have been a root cause of much disunderstanding. In the next picture I show you the correction: Why I was that stupid a few weeks ago is non of your business, but by now stuff is repaired and we no longer have this horrible mistake around any longer….
It was in the Spring of when I was walking in a nearby park when it suddenly dawned on me that this exponential process that ran through the basis vectors 1, 0, 0 , 0, 1, 0 and the z-axis unit vector 0, 0, 1 was periodic. It could not be anything else because I was capable of calculating the logarithm of the first imaginary unit j.
I remember at first I just did not have a clue it would be a circle, I even had vague fantasies like may be it is a vibrating string where all those string physics professors talk about. It dates back to 30 May and I used this picture as a joke about how professional math professors look in my fantasy world. Within a week I found that the 3D complex periodic curve was in fact a circle. So I had to laugh hard about my own joke once more because if I had known the 3D periodic thing would also be a circle I would have made the joke very different… Because one way or the other this picture now also represented me.
You know this last week I am a bit puzzled by what the next post should be, in December I conducted a good investigation into the roots of unity related to the two exponential circles and because every body knows roots of unity it would a nice started for this website. The song of omega reloaded http: At the time I was amazed with all the things you can do with the eigen values of the imaginary components j and j squared. From diagonalization to the roots of unity, my theory got definitely air born.
Later in January I found a new Cauchy integral formula actually two just like I found two sets of roots of unity each for the admissable forms of 3D multiplication.
Also in Jan I cracked the problem of 5D complex numbers. Now, almost 3 years later finally stuff on Google take off, for example if this day I start searching for the phrase 3dcomplexnumbers it returns back three results from this new website.
Just look at the next screen shot picture:. This post is 10 pictures long size x , in the beginning I use an applet for the numerical calculation of the matrix representation of the first imaginary unit in 7D space, here is the link: How to find those numbers as the log applet says… The method as shown here can be applied in all dimensions and you now have a standard way of crafting exponential curves in all spaces you want. That makes or breaks this method, if done wrong you end up with a giant pile of nonsense… Have fun reading it and if this is your first time you encounter those matrices with all these roots of unity in them, take your time and once more: If you have never seen a matrix like that it is very hard to understand this post in only one reading… I am glad all that staring to those numerical values is over and we have the onset of analytical understanding of how they are in terms of the angle 2 pi over 7.
I remember back in the year just staring at all those matrices popping up using internet applets like the next one: I even wrote a post about it on 23 Nov At the end of the third picture you see the end result. 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.
Ok, see you around my dear reader. 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.
All in all this post is 20 pictures long size x so it is a relatively long read. If you cut and past the next sum of the five coordinate functions you see that you always get one for all x: Weirdly enough you find a hyper plane and a bunch of hyper-cylinders. It is just 3 pictures long x pixels: Basically tau-calculus is very easy to understand: Craft analytical expressions that you cannot solve but use internet applets for numerical answers.
Use an applet to calculate the logarithm of your basis vector once you have put it on a matrix representation. Start thinking long and hard until you have solved the math analysis… This is easy to understand but I could only do this in three and five dimensional space, seven or eleven dimensional space?
The more of those exponential circles and curves you find, the more boring the complex plane becomes… In the months after Jan I wanted to understand the behavior of the coordinate functions that come along those two exponential curves.