Category Archives: Matrix representation

How I found the first modified Dirichlet kernel in 5D numbers.

July 7, 20163D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanicsReinkoVenema

Back in Jan 2014 I was able to solve the circular and complex 5D numbers systems using the so called ‘tau-calculus’. Basically tau-calculus is very easy to understand:

Find a basis vector (from an imaginary number) that has determinant +1.
Craft analytical expressions that you cannot solve but use internet applets for numerical answers. Or, equivalent:
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? It is now 2.5 years later and I still have no clue whatsoever.

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 2014 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.

Originally I planned for ‘digesting’ 10 matrices a day so it would be a five day project, but when I finally started somewhere in June 2014 after two days I was ready to sketch my strongly desired coordinate function of the very first coordinate in 5D space.

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…

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Ok, what is in this new post?
Nine pictures of size 550 x 775 pixels containing:
Basic definitions upon 5D complex and circular number &
Tau calculus for the exponential curve &
Explaining how I found the graph of the first modified Dirichlet kernel &
A small quantum physics example related to probability amplitudes &
Some cute integrals that are easy to crack now.

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:
The 5D realm is a space on it’s own!

Have fun reading & thinking upon it:

A few links of interest are the next:

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:
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

My original update on the other website about the 5D number systems from Jan 2014:
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#30Jan2014

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 2015 upon the missing equations is also worth a visit:
The missing equations.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#14July2015

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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.

The new 3D coordinate system; [a, r, t] coordinates.

May 24, 20163D complex numbers, Exponential circle, Matrix representation, Quantum mechanicsReinkoVenema

About a full week late all is now finished. Also there was that problem of the hacked webpage; the third update on the Schrödinger equation was loaded and loaded with all kinds of weird comments. But if you did read those comments you could see the overwhelming majority was not written by humans but were posted by bots.

Anyway in the end even advertisements for online sales of viagra and stuff surfaced so I took an evening to see if this could be repaired. For the time being it is not possible to post comments although before it was also impossible but that was due to an unknown technical fault…

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In this update we dive into the [a, r, t] coordinate system and this is very similar to using polar coordinates in the complex plane only in 3D we are moving a half-cone along a line and as such sweep through the entire 3D space. In short it goes like this:

With a we control how far the cone is moved from the origin,
with r we denote the distance from a particular point X to the top of the cone and finally,
with t we figure out how much we need to rotate stuff using an applet for the log of a matrix.

I also give once more the coordinate functions of the exponential circle so that you are able to find coordinates on the main cone (that has the origin as it’s top) for yourself.

Without prove I also give a new de Moivre formula, the original de Moivre formula is from about 50 years before Euler so after 3 centuries there is a bit of moving forward on that detail.

I also give a rather strange looking sum of squares of cosine functions that add up to 1.5; the three cosine functions all differ by 120 degrees or 2pi/3 if you want.

It is important to remark I did my best to keep it as simple as possible so I also concentrate on a few worked examples so it is not just theory but also how to find these new coordinates. In relation to other posts like the Schrödinger equations posts, I think this new coordinate system is definitely of interest to people from quantum physics and chemistry that try to calculate those atomic and molecular orbitals.

This update is relatively long: 14 pictures of size 550 x 775 pixels and for the first time I use unshrunk jpg pictures because year in year out the bandwidth of internet connections is still rising fast so why remove fine detail from jpg pictures any longer?

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I hope you learn something new, here are the pics:

Ok, that was it. For myself speaking I more or less expect everything to be the same:
Professional math professors from around the globe will keep on talking about their own research and just how important it is to use complex numbers from the complex plane.

A few useful links:

De Moivre’s formula
https://en.wikipedia.org/wiki/De_Moivre’s_formula

And a good applet for the logarithm of matrix representations is also handy:

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

And last but not least is a proof for the latest formula above: the circular multiplication on the [a, r, t] coordinates can be found on the other website:

From 06 May 2016 : On the length of the product of two 3D numbers.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#06May2016

Let’s leave it with that. Till updates.

An important correction on a very very stupid typo…

May 19, 20165D complex numbers, Corrections, Exponential circle, Matrix representationReinkoVenema

Let’s not do difficult: there are typo’s and there are typo’s.

The one kind is for every body directly to understand ‘this is a typo’ and the brain pops up what should be the correct thing standing there.

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: in that matrix it now reads log r instead of only r.

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…

Ok, let’s leave it with that. Till updates.

Third post on the Schrödinger wave equation using 3D complex numbers for atomic & molecular orbitals.

April 13, 20163D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanicsReinkoVenema

This update is 10 pictures long, the pictures are sized 550 by 775 pixels.
This update covers more or less everything, but I still have to explain how you find the six coordinate functions the poeple will need in order to see if these kind of complex numbers give the same result as ordinary complex numbers from the complex plane.

For those that cannot wait: In the post from 03 April I posted a teaser picture with the coordinate functions in 3D, if you multiply this against the e to the power i pi alpha thing in this update you have the six coordinate functions…

Ok ok you neatly have to write them out, but basically it is all there.

At first I was thinking it would be hard to get different results using these higher dimensional complex numbers, but when talking about atomic and molecular orbitals it might be more subtle than it looks. At the end I will post a video where some physics guy shows all kinds of orbitals related to hydrogen but his stuff is different from the pictures we observe in chemistry.
He explains this by saying that the people from chemistry always take a super-position of two wave-blobs and as such it gets oriented along the y-axis say.
If you would take super-positions of my 3D complex numbers you will get very similar results. look at the drawing in the one before last picture:
Take a super-position of an exponential circle and it’s conjugate and observe it must have the same behavior as 2D numbers from the complex plane.

(In that drawing your eys is supposed to be along the line through zero and alpha, so zero is right behind the center of the shown circle…)

Enough of the bla bla bla, here are the 10 pictures:

Click on the picture to get a larger version of the drawing:

Now finding these atomic & molecular orbitals is very hard, for simple atoms like hydrogen it is doable but what about uranium or some nice protein with only 3693 atoms in it?

All that kind of stuff falls under what we name n-body problems and for n above 3 it seems impossible to find exact analytical solutions.

There is a nice video out there explaining a bit more on the topic of finding the shapes of atomic & molecular orbitals. It is from Brant Carlson and has the title Hydrogen atom wavefunctions:

Ok, that was it for today. Till updates.

A new type of Cauchy integral formula.

February 28, 20163D complex numbers, Exponential circle, Exponential curves, Matrix representationReinkoVenema

Yesterday I wrote a new post on the Schrödinger equation using 3D complex numbers but before I post that let’s go a bit more hardcore with a brand new Cauchy integral formula.
Actually it is not that brand new because on 18 Jan 2014 I posted it on the other website.

Now in a normal world a brand new Cauchy integral would be greeted with a lot of joy and plenty of discussion, yet that has not happened by now. Once more we observe that among professional math professors there is a severe problem concerning the so called ‘competence question’.
Or may be it is better to frame this into a lack of competence; if you have that you are also not able to judge new results properly and this is what we observe year in year out.

But I have to admit it is a relatively hardcore update, it is 10 pages long and I remember clearly it was fun to write because I wanted to prove the Cauchy formula in this way for a long time.

Source: http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#18Jan2014

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Now a person that is not 100% insane might wonder how the hell you calculate the determinant of a six by six matrix because in parctice that is an awful amount of work. But I used an internet applet and as such got a numerical value like about 106,821 and within a few seconds I recognized this as being pi to the power of six divided by nine.

Once back in the year 1992 I came across that number and it was kinda weird to observe that in 2014 it was still floating around in my brain. Sometimes I wonder if I am the crazy one and the math professors are the ones with healthy brains…;)

Ok, till updates my dear reader.

The Cone Theorem.

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

On the other website I just posted 12 pages about the cone theorem. This theorem states that cones with a central axis the line through 0 and the number alpha and with their top in 0, undergo a fixed rotation when multiplied by one of the imaginary numbers like j or j^2.

You can find that on page four covering stuff posted this year.

It is important to remark I got the idea to study this particular detail because of the article in the preprint archive from Shlomo Jacobi. Now this Shlomo guy seems to be dead so I have to be a bit cautious. Let’s say these 12 pages are the way should study stuff like this & don’t forget I got the idea from this Jacobi guy while the professional math professors as usual contribute nothing.

In the next teaser picture you see how it works, while calculating some inner product you get this equation and if you fill in some allowed number for the control c you get the desired cone.

These cones are online easily made with an applet named Polyray. The great advantage of this applet is that you can fill in implicit equations so you are not bonded by some explicit stuff like

z = bla bla formulae in x and y.

You can click on the picture to land on the new update (open in a new window):

In another development I posted a few more reasons as why electrons are magnetic monopoles in the magnetic page on the other website. Now lately some folks from MIT have run six simulations of nuclear plasma and the results nicely confirm my insights in the behavior of nuclear plasma.

The MIT folks thought that in a nuclear fusion reactor you could simply neglect the contributions from the electrons because their mass is so small compared to the mass of protons and higher isotopes of atomic hydrogen. But ha ha ha, when electrons are magnetic monopoles such thinking is shallow & hollow. Anyway to make a long story short: the simulations point to a magnetic monopole electron.

Problem is I do not know how they model the plasma in detail, don’t forget the weirdo’s from the universities think electrons are magnetic dipoles and if you think that how can you make a reliable model of plasma anyway???

Here is the link around magnetic monopole stuff:
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff.htm#17Feb2016

Enough of the bla bla bla, may be in the next post on this website I am going to dive into stuff related to the Schrödinger equation. Or something else like thousands and thousands of new and previously unknown trigoniometric identities…

We’ll see, till updates.

Seven properties of the number alpha.

February 9, 20163D complex numbers, Laplacian, Matrix representationReinkoVenema

The number alpha is one of my best finds in the field of mathematics. In all kinds of strange ways it connects very different parts of math to one another, for example when it comes to partial differential equations the number alpha plays a crucial role in transforming this of a pile of difficult stuff into something that lives in only one dimension.

You can also use the number alpha for perpendicular projections, you can use it for this and you can use it for that.

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Now in the previous post I told you I would write out some of the elementary properties of the number alpha, but when I finished it the thing was about 5 A4 size pages long and that would be about 10 pictures on this new website.

That would be a bit too long and also I had written nothing in the page for 2016 on the other website. So I decided to hang those five A4 pages in the old website and you get a few teaser pictures on this new website.

Here are the three teaser pictures, click on any to land on the alpha update:

The applet I used is a very helpfull tool, you can find it here:
http://calculator.vhex.net/calculator/linear-algebra/matrix-exponential-using-the-pade-approximation

Ok, that was it. Till updates and do not forget to floss your brain a bit every now and then…

Correction on the 08 Dec 2015 post; there are two typo’s…

January 17, 20163D complex numbers, Corrections, Matrix representationReinkoVenema

It is not a big deal because every person who understands a bit about how matrix representations work sees instantly these must be two typo’s.

But recently about once a week I am scanning how this new website is doing in search engines like Google. And I am very satisfied, every post can now pop up as a separate search result and for example on pictures to my surprise the next picture popped up as pic number six if you search for ‘3d complex numbers’.
This is the version with the two typo’s in it:

And here is the corrected version:

So it is not a big deal but if a search result ends that high it is not unwise to correct it.
And to be honest, I know for years that you can craft let’s say 15 dimensional complex numbers from 3D and 5D complex numbers.
But to be honest, I had never done it until the December update from last year.

And I have learned some stuff too, only if you dive into those technical details like how those basis vectors are actually related you appreciate it so much more.
You know the nicest thing about higher dimensional complex numbers is very simple: I know for sure I am about one of the first humans to hang around in those spaces.
Beside the mathematical beauty the stuff has, it has also that old stuff like discovering new lands that is basically baked into the human genome.

Ok, enough of the phylosofical bla bla. Till updates.

Matrix representations and how to craft them.

January 1, 20163D complex numbers, Matrix representationReinkoVenema

Here it is still 01 jan 2016 so a happy new year.
In this update with five pictures with the standard size of 550 by 550 pixels we are going to look at how to craft matrix representations for higher dimensional complex numbers.

It is all rather basic stuff.

Here we go with post number 1 in the year 2016:

Yeah yeah, every point of this graph represents a 3D complex number that if you craft the matrix representation of it, it is a unitary matrix.

So the next time you see a physics professional professor writing stuff like SU(3) you instantly know you are dealing with some form of idiot life…

Cauchy-Riemann equations for the complex plane and for 3D complex numbers.

December 12, 20153D complex numbers, CR equations, Matrix representationReinkoVenema

In itself the name of ‘Cauchy-Riemann equations’ is a terrible way of naming these equations because it says nothing about why they are important.

It would be better to name the stuff involved like ‘Chain rule equations for partial derivatives’ because if that would be the case you would understand why these equations are worth your precious time anyway…

This update is 8 pictures of size 550 by 550 pixels or about 5 pages of A4 size if crafted in the A4 size format.
Now why are CR equations important?
Very simple: You can find the derivative of a function just like on the real number system or in the complex plane. That is why CR equations are the basic food for understanding higher dimensional complex number systems…

Enough of the bla bla, here are the 8 pictures: