Category Archives: Matrix representation

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

3D complex numbers, Exponential circle, Matrix representation, Quantum mechanics

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:

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0024_23May2016_the_art_coordinates14Ok, 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…

5D complex numbers, Corrections, Exponential circle, Matrix representation

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…

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

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

3D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanics

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:

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

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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:

0021=13April2016=hydrogen_orbitals

Ok, that was it for today. Till updates.

A new type of Cauchy integral formula.

3D complex numbers, Exponential circle, Exponential curves, Matrix representation

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.

0014=27Feb2016=Cauchy_integrals

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.

3D complex numbers, Exponential circle, Matrix representation

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):

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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.

3D complex numbers, Laplacian, Matrix representation

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:

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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…

3D complex numbers, Corrections, Matrix representation

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:

0002=02Dec2015=teaser_6D_complex_numbersAnd here is the corrected version:

0009=02Dec2015=teaser_6D_complex_numbers_corrected

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.

3D complex numbers, Matrix representation

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:

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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.

3D complex numbers, CR equations, Matrix representation

 

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:

 

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This stuff is basic stuff so it should be hanging out on this new website.

Till update my dear reader.

Eight pages on 6D numbers (containing 3D and 2D complex numbers).

3D complex numbers, Matrix representation

The day before yesterday I finished an 8 page long update on six dimensional complex numbers from the viewpoint of inclusion and extension.

So basically I show that the complex plane and the 3D complex numbers are included in the 6D space while the other way around you can make a 6D space starting with the 2D and 3D complex spaces.
Link:
Inclusion and extension of complex spaces
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#06Dec2015

0002=02Dec2015=teaser_6D_complex_numbers

Well well, I still have a long learning curve to do on this new website because you can also make pictures shown at their real size…

Ok, now the previous website has gotten so much attention it would be time to post a bit more stuff in here. After all after just a few weeks and almost no content at all weirdly enough it already ranks relavtively high in the search engine stuff.

Another problem to be solved is that the comment section does not work and even the categories are disfunctional… Very likely this wordpress theme is developed by somebody that has no math insight at all. Why can I make categories while they do not work???
Beat’s me. Anyway, till updates.