Category Archives: Exponential curves

A few numerical results related to the 4D sphere-cone equations using the four coordinate functions of the 4D exponential curve.

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:

WIMS https://wims.sesamath.net/wims.cgi?lang=en

For the two graphs from above look for ‘animated drawing’ choose the 2D explicit curves option. There you must use the variable x instead of time t.

Here is the stuff you can place in for the sphere equation:

(cos(pi*x/2)*cos(pi*x/4))^2 +
(0.707107*(cos(pi*x/4)*sin(pi*x/2) + sin(pi*x/4)*sin(pi*x/2)))^2 +
(-cos(pi*x/2)*sin(pi*x/4))^2 +
(0.707107*(cos(pi*x/4)*sin(pi*x/2) – sin(pi*x/4)*sin(pi*x/2)))^2

If you just ‘cut & paste’ it should work fine…
That should save you some typo’s along the way

Ok, that is what I had to say on this numerical detail.

Calculation of the four coordinate functions for the 4D exponential curve (complex version).

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.

Till updates.

The basics of 4D complex numbers.

In the previous post on 4D complex numbers I went a little bit philosophical with asking if these form of crafting a 4D number system is not some advanced way of fooling yourself because your new 4D thing is just a complex plane in disguise…

And I said let’s first craft the Cauchy-Riemann equations for the 4D complex numbers, that might bring a little bit more courage and making us a little bit less hesitant against accepting the 4D complex numbers.

In this post we also do the CR equations and indeed they say that for functions like f(Z) = Z^2 you can find a derivative f'(Z) = 2Z. So from the viewpoint of differentiation and integration we are in a far better spot compared to the four dimensional quaternions from Hamilton. But the fact that the CR equations can be crafted is because the 4D complex numbers commute, that is XY = YX. And on the quaternions you cannot differentiate properly because they do not commute.

So crafting Cauchy-Riemann equations can be done, but it does not solve the problem of may be you are fooling yourself in a complicated manner. 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: ignore all things Reinko Venema related, look the other way, ask for more funding and so on and so on. In my life and life experiences not one university person has ever made a positive difference, all those people are only occupied with how important they are and that’s it. Being mathematical creative is not very high on the list of priorities over there, only conform to a relatively low standard of ‘common talk’ is acceptable behavior…

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…

Ok, here are the 10 pictures covering the basic details of 4D complex numbers:

 

 

 

 

 

 

 

 

Ok, that was the math for this post.

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…

Let’s leave it with that, see you in the next post.

A nice teaser picture about 4D coordinate functions of the 4D exponential curve (complex version).

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.

This new baby number tau keeps on looking cute…

Let me leave it with that, till updates.

Calculation of the 4D complex number tau.

It is about high time for a new post, now some time ago I proposed looking at those old classical equations like the heat and wave equation and compare that to the Schrödinger equation. But I spilled some food on my notes and threw it away, anyway everybody can look it up for themselves; what often is referred to as the Schrödinger equation looks much more like the heat equation and not like the classical wave equation…

Why this is I don’t know.

This post is a continuation from the 26 Feb post that I wrote after viewing a video from Gerard ‘t Hooft. At the end of the 26 Feb post I showed you the numerical values for the  logarithm of the 4D number tau. 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.
And it is a well known result that log i = i pi/2, implicit the physics professors use that every day of every year. 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.

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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.

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This post is seven pictures long, all are 550 by 775 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.

Here we go:

Just take your time and look at this ultra cute number tau.

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.

End of this post.

I am innocent, I did not do it. I just found the numbers tau in the Schrödinger equation your honour…

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.

__________

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:

Hermitian matrix
https://en.wikipedia.org/wiki/Hermitian_matrix

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:

An important calculation of the 7D number tau (circular version).

That’s it, till updates.

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 550 x 775), 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://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

Two years back in 2015 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.
Ok ok those modified Dirichlet kernels always have period pi while this way to calculation the log of the first imaginary unit is always related to the dimension (recall that the 7D first imaginary unit l has the property l^7 while for the complex multiplication in 7D space we have l^7 = -1), but it is very easy to fix the Dirichlet kernels to the proper period in the time domain you want.

The most difficult part of this post is in understanding the subtle choice for the eigenvalues of tau = log l, or better; choosing the eigenvalues of the matrix representation involved. 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: take your time.
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.

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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. But in the preprint archive I still cannot find only one work about it, that might be logical because often people do not write about stuff they don’t understand…

Ok, that was it. I hope you liked it & see ya around.

Intro to the calculation of the seven dimensional number tau (circular version).

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.

I even wrote a post about it on 23 Nov 2015:

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: 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.

Ok, see you around my dear reader.

On a way to find more equations so that the 1D existence of exponential curves in all possible dimensions is assured.

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´±

https://wims.sesamath.net/wims.cgi

For example you can cut and paste the next five dimensional equations that represents a hypercone going through all the coordinate axis:

((1/5)*sin(5*x)/sin(x))*((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5)) +
((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5))*((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5)) +
((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5))*((1/5)*sin(5*(x-pi/5))/sin(x-pi/5)) +
((1/5)*sin(5*(x-pi/5))/sin(x-pi/5))*((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5)) +
((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5))*((1/5)*sin(5*x)/sin(x))

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:

((1/5)*sin(5*x)/sin(x)) +
((1/5)*sin(5*(x-pi/5))/sin(x-pi/5)) +
((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5)) +
((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5)) +
((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5))

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.

From 14 July 2015: The missing equations.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#14July2015

Ok, that is what I had to say. Till updates.

On an old idea that did not work…

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.

Here is a link to that old update±

From 14 July 2015: The missing equations.

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.

See ya around!