Category Archives: 3D complex numbers

The Cayley-Hamilton theorem neglected for 25 years?

April 4, 20193D complex numbers, Matrix representation, UncategorizedReinkoVenema

That is strange, if you don’t know the Cayley-Hamilton theorem; it is the finding that every square matrix A, if you calculate the characteristic polynomial for the matrix A it is always zero. At first this is a very surprising result, but it is easy to prove. It’s importance lies in the fact that in this way you can always break down higher powers of the matrix A in lower powers. In the study of higher dimensional complex and circular numbers we do this all the time. If in 3D space I say that the third power of the imaginary component is minus one, j^3 = -1, we only write the third power as a multiple of the zero’th power…

In this post I will give two simple proofs of the Cayley-Hamilton theorem and although in my brain this is just a one line proof, if you write it down it always gets longer than anticipated.

At the end I show you an old video from the year 1986 from the London Mathematical Society where it is claimed that the CH theorem was neglected for 25 years. Now Hamilton is also famous for having sought the 3D complex numbers for about a full decade before he gave up. And I still do not understand why Hamilton tried this for so long but likely he wanted to include the imaginary unit i from the complex plane in it and that is impossible. Or may be he wanted a 3D complex number system that is also a field (in a field all elements or numbers that are non-zero have an inverse, in algebra wordings; there are no divisors of zero). A 3D field is also impossible and in this post I included a small proof for that.

Furthermore in this post at some point may be you read the words ‘total incompetents’ and ‘local university’. You must not view that as some emotional wording, on the contrary it is a cold clinical description of how math goes over there. So you must not think I am some kind of frustrated person, for me it is enough that I know how for example to craft a 3D complex number system. If they don’t want to do that, be my guest. After all this is a free country and we also have this concept of ‘academic freedom’ where the high shot math professors can do what they want.

And what is this ‘academic freedom’ anyway? If for example unpaired electrons are never magnetically neutral but electron pairs always are magnetically neutral, can the physical reality be that electrons are magnetic dipoles? Of course not, that is a crazy idea to begin with. But 97 years of academic freedom since the Stern-Gerlach experiment have never ever brought any meaningful understanding of the magnetic properties of the electron. If it acts as a magnetic charge and you say it is not a charge it is easy to understand how you can fool yourself for about one century of time.

This post is seven pictures long although the last picture is empty.
The two proofs of the Cayley/Hamilton theorem is how I would prove such a thing but good theorems always have many proofs. All pictures are of the size 550×775 pixels.

Why is the seventh picture without math?

Here is the old video from 1986 where it is claimed the Cayley-Hamilton theorem was neglected for about 25 years. Oh oh oh what a deep crime. But the human mind is not made to produce or understand math, so in my view 25 years is a short period of time if in the good old days math professors were equally smart as the present day math professors. The title of the video is The Rise and Fall of Matrices.

Matrices saved my life from crazy math professors.

Ok let me leave it with that an not post a link to the top wiki on the Cayley-Hamilton theorem where all kinds of interesting proofs are given. Till updates my dear reader.

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

February 26, 20183D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanicsReinkoVenema

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).

December 6, 20173D complex numbers, 5D complex numbers, Exponential curves, Matrix representationReinkoVenema

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.

__________

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).

December 1, 20173D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, UncategorizedReinkoVenema

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.

November 8, 20173D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanicsReinkoVenema

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.

Some corrections and an addendum + a new way of taking Fourier transforms.

June 1, 20173D complex numbers, Corrections, Exponential circle, Exponential curvesReinkoVenema

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:

From 18 Jan 2014: Cauchy integrals
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#18Jan2014

A link to the Nov 2016 post on 2D split complex numbers that contains the disinformation about the sum of the coordinate functions:

The second hybrid: a 4D mix of the complex and the circular plane.

End of this post, likely the next post is about prime numbers and how to demolish the internet security we think we have using huge prime numbers…

So see you around!

Some very rudimentary Fourier stuff + a surprising way to do a particular integral.

May 22, 20173D complex numbers, Exponential circleReinkoVenema

Lately I was looking at some video’s about Fourier analysis and it dawned on me I had never tried if the coordinate functions of my precious exponential circles were ‘perpendicular’ to each other.

Now any person with a healthy brain would say: Of course they are perpendicular because the coordinate functions live on perpendicular coordinate axis but that is not what I mean:
Two functions as, for example, defined on the real line can also have an inner product. Often this is denoted as <f, g> and it is the integral over some domain of the product of the two functions f and g.

That is a meaningful way of generalizing the inner product of two vectors; this generalization allows you to view functions as vectors inside some vector space equipped with an inner product.

Anyway I think that most readers who are reading post number 62 on this website are familiar with definitions of inner product spaces that allow for functions to be viewed as vectors.

__________

So I was looking at those video’s and I was highly critical if my coordinate functions would be perpendicular in the sense of such generalized inner product spaces. But indeed they are also perpendicular in this sense. Yet a bit more investigations soon gave the result you cannot build a completely new kind of Fourier analysis from this stuff.

Ok ok a few years back I already arrived at that insight because otherwise in previous posts you would have found stuff relating to that…

Generalized inner product spaces are often named Hilbert spaces, a horrible name of course because attaching a name like Hilbert to stuff you can also give the name Generalized Inner Product Space brings zero wisdom at the scene.
It is only an attempt to turn the science of math into a religion where the prophets like Mr. Hilbert are given special treatment over the followers who’s names soon will be forgotten after they die.

More on Hilbert spaces: Hilbert space
https://en.wikipedia.org/wiki/Hilbert_space

__________

This update is 11 pictures long, all size 550 by 775 pixels.
I kept the math as simple as possible and all three dimensional numbers used are the circular ones. So you even do not have to worry about j^3 = -1…

Have fun reading it:

I hope you like the alternative way of calculating that integral in picture number 11. It shows that 3D complex and circular numbers are simply an extension of mainstream math, it is not weird stuff like the surreal numbers that decade in decade out have zero applications.

May be in the next post I am going to show you a weakness in the RSA encryption system.
Or may be we are going to do something very different like posting a correction on a previous post.

Let´s wait and see, till updates.

A more or less perfect visualization of the Riemann zeta function observed.

December 18, 20163D complex numbersReinkoVenema

It has been a long time since my last update and that is caused by some stupid medical condition I still have and in my native language it is known as a ‘peesschede ontsteking’.
In practice this means I must do all typing on my computer keyboard with my left hand because in the evening I still cannot use my right hand.

Let me spare you the details but the long durance of the pain could even date back to the time when I was a dumb 15 year old with a broken wrist not seeking medical help.

So for the time being no long updates on perfect new hybrid number systems, it takes too much pain to write those long math stories down. So I retreat and just post a link to what is a very good Youtube video on the Riemann zeta function and it’s continuation into it’s analytic continuation.

Here is the video from the 3Blue1Browne guy:

Visualizing the Riemann zeta function and it’s analytic continuation

Nice vid isn’t it?

Last year on 26 March 2015 I wrote an update on where to find the zero’s of the Riemann zeta function in the 3D complex number system. I still consider this being an important publication although that human garbage known as the ‘professional math professors‘ said nothing all these months, I still think it is worth the trouble and try to post a new link to it:

From 26 March 2015: Zeta on the critical strip (3D version only).
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#26March2015

May be it is best to leave this update with that;

Zero point zero point zero point zero reaction of so called ‘professional math professors’ upon finding the zero’s of the Riemann zeta function in dimensions above 2.

Once an overpaid imbecile, always an overpaid imbecile.
Let’s leave it with that.

__________

Update from 19 Dec: I did not include yesterday a more easy to understand analytic continuation that I wrote myself this year; it is the analytic continuation of the geometric series and as such I am debunking the stuff some of the children of a lesser God seem to think:

1 + 2 + 4 + 8 + 16 + ….. = -1.

Nottingham professors from math and physics seem to think that

1 + 2 + 3 + 4 + 5 + ….. = -1/12.

This is also nonsense and there are many ways to prove this is not the case but inside theoretical physics this is actually used: that is the process of renormalization. Every time professional physics professors encounter an infinity in their calculations it is not that they say ‘Something must be wrong with our theory’. No if they encounter stuff like 1 + 2 + 3 + 4 + etc, they replace it by -1/12.

It works pretty well in order to get rid of those singularities they say.

Anyway here is the link to what I had to say on that subject:

From 15 April 2016: Debunking the Euler evaluation of zeta at minus one.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#15April2016

You can find the analytic continuation of the geometric series in the fifth picture.

Let me close this extra update with the Youtube video from those weird weird Nottingham professors that started it all:

ASTOUNDING: 1 + 2 + 3 + 4 + … = -1/12

And indeed if it were true it would be very very astounding.
What for me is TRULY ASTOUNDING is that the very professors you see doing their show is that they think the harmonic series is divergent. The harmonic series is also the zeta function evaluated at 1:

1 + 1/2 + 1/3 + 1/4 + … = infinity.

So the Nottingham professors think that the harmonic series is divergent (that is correct of course) while the sum of all integers is convergent to be -1/12.

Welcome to the world of 21-th century science. Till updates.

The pull back map applied to the coordinate functions of the 3D exponential circle.

November 3, 20163D complex numbers, Exponential circle, Quantum mechanicsReinkoVenema

In this post, number 50 by the way, I am trying to use as elementary math as possible in order to use the pull back map from the 3D circular number system to the complex plane.

With this the pull back map and the 3D circular number system are treated so basic that with only high school math and a crash course in the complex plane students can understand what I am doing.

So for reading this post number 50, what do you need in mathematical knowledge?
1) Understand how to write cos(a + b) and sin(a + b) in terms of cos a and sin b.
2) Understanding of e to the power it in terms of cos t and isin t.
3) Understanding of the roots of unity as found inside the complex plane, in particular being able to calculate all three roots of unity when we take the third root of the number 1.

That’s all, so basically all first year students in math, physics and chemistry could understand this post at the end of their first year on a local university.

__________

The words above are only one reason to write this post; to be honest for me it took a long time to write down for the first time the coordinate functions for the 3D exponential circle.

And I never did give much solid proof for that these coordinate functions have indeed the properties as described. It all more or less came out of the sleeve as some kind of monkey trick.

Therefore for myself speaking, this post giving the results in it also serves as a proof that indeed there is only one class of coordinate functions that do the job. They can only differ in the period in time they need to go around, if you leave that out the triple of coordinate functions becomes unique.

All in all the goals of this post number 50 are:

1) To do the pull back of an exponential circle as simple as possible while
2) In doing so give some more proof that was skipped years ago.

__________

This update is seven pictures long, each 550 x 775 pixels in size.
Hit the road Jack:

I think I have nothing more to say, so see you around my dear reader in post number 51.

Till updates.

Derivation of the number tau for the circular 3D number system.

October 22, 20163D complex numbers, Exponential circle, Matrix representationReinkoVenema

There are lot’s of reasons for this update; one reason is that the actual calculation is mega über ultra cool. Another important reason is that this collection of plain imitation of how the value for the number i in the complex plane was found serves as a proof in itself that this way of crafting 3D complex and circular numbers is the only way it works.

Don’t forget that on the scale of things the Irish guy Hamilton tried for about a decade to find the 3D numbers but he failed. Yet Halmilton was not some lightweight, the present foundation of Quantum Mechanics via the use of the Hamilton operator is done so via the work of Hamilton…
Wether the professional math professors like it or not; that is the scale of things.

During the writing of this post I also got lucky because I found a very cute formula related to the so called Borwein-Borwein function. I have no clue whatsoever if it has any relevance to my own work on this website but because it is so cute I just had to post it too…

Furthermore I used two completely different numerical applets, one for integration and the other for evaluating the log of a matrix, only to show you that these kind of extensions of the complex plane to three dimensional space is the way to go and all other approaches based on X^2 = -1 fail for the full 100%.

__________

This post is ten pictures long, size 550 x 775 pixels.

At the end I will make a few more remarks and give you enough links for further use in case you want to know more about this subject. Have fun reading it.

The applet for the logarithm of a matrix can be found in this nice collection of linear algebra applets:

Linear algebra
http://calculator.vhex.net/function-index/linear-algebra

In this update you might think that via the pull back principle you observed some proof for the value of the integrals we derived, but an important detail is missing:
In 3D space the exponential circle should be run at a constant speed.
As a matter of fact this speed is the length of the number tau, you can find more insight on that in the theorem named ‘To shrink or to grow that is the question’ at:

On the length of the product of two 3D numbers.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#06May2016

A bit more hardcore is my second proof of the value of the integrals as derived in this post. On 15 Nov 2015 I published the second proof that I found while riding on my bicycle through the swamps near a local village named Haren. It is kinda subtle but you can use matrix diagonalization to get the correct answer.
The reaction from the ‘professional community of math professors’ was the usual: Zero point zero reaction. These people live in a world so far away from me: overpaid and ultra stupid…

Integral calculus done with matrix diagonalization.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#21Nov2015

A link to the online encyclopedia of integer sequences is the next link.
Remark that by writing the stuff as on-line instead of online reflects the fact this website must be from the stone age of the internet. That is why it can have this strange knowledge…

The On-Line Encyclopedia of Integer Sequences
https://oeis.org/ (Just fill in 1, 2, 0, 9, 9, 5, 7 in order to land on my lucky day).

The last link is one of those pages that try to explain as why 3D complex numbers cannot exist, the content of this page is 100% math crap written by a person with 0% math in his brain. But it lands very high in the Google ranking if you make a search for ‘3D complex numbers’.
So there must be many people out there thinking this nonsense is actually true…

N-DIMENSIONAL COMPLEX NUMBERS.
http://www.alenspage.net/ComplexNumbers.htm

Ok, this is what I had to say. Let me close this post, hit the button ‘update website’ and pop up a fresh beer… Till updates.

3Dcomplexnumbers

Math as you have never seen before…

Category Archives: 3D complex numbers

The Cayley-Hamilton theorem neglected for 25 years?

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

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

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

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

Some corrections and an addendum + a new way of taking Fourier transforms.

Some very rudimentary Fourier stuff + a surprising way to do a particular integral.

A more or less perfect visualization of the Riemann zeta function observed.

The pull back map applied to the coordinate functions of the 3D exponential circle.

Derivation of the number tau for the circular 3D number system.