Category Archives: 3D complex numbers

Curl, curl and more curl.

3D complex numbers, CR equations, Exponential circle, Matrix representation

This post contains seven examples of the differential operator named curl. The motivation for this update lies in the fact that after my humble opinion inside the set of equations known as the Maxwell equations there is a tiny fault: Rather likely electrons carry at least a net magnetic charge.
And because they carry net magnetic charge they not only accelerated by electrical fields but also magnetic field.

If you have an old television set with one of those fancy tubes that contain one electron cannon for a black & white television and three electron cannons for a color television. With the help of a stack of these strong neodymium magnets it is easy to give experimental proof that the electrons indeed get accelerated… Why in the course of over one hundred years not one of the professional physics professors has done this is unknown to me; may be it is the separation of ‘theoretical professors’ versus ‘experimental ones’ a reason for this omission. May be it is the use of dogma (unproven things that live inside a belief system, in this case the belief system that magnetic monopoles do not exist).

__________

Anyway the Maxwell equations contain a lot of the curl operator, that is not needed per sé but it makes the formulae look sleek and short. Originally Maxwell had a set of like 20 equations or so while at present day only four remain. But if you see those four equations for the first time it is very impressive, only over time you get used to it.

This post is 13 pictures long, size 550 x 775 pixels.
I start with examples that are as simple as possible and very slowly bring in a bit more abstraction.
Therefore I hope it is very readable, have fun with it!

29sept2016-curl-operator01

29sept2016-curl-operator02

 

29sept2016-curl-operator03

 

29sept2016-curl-operator04

 

29sept2016-curl-operator05

29sept2016-curl-operator06

 

29sept2016-curl-operator07

29sept2016-curl-operator08

29sept2016-curl-operator09

29sept2016-curl-operator10

29sept2016-curl-operator11

29sept2016-curl-operator12

29sept2016-curl-operator13

As far as I can remember, in the first year I opened these investigations into higher dimensional number systems again I calculated the curl for the complex multiplication in 3D space. In this post we only looked at the circular version of stuff.

But I can´t find it back so I cannot place a hyperlink to it.

Anyway, here is a nice wiki with the curl expressed as an integral (often much harder to calculate but nice to observe it can also be done that way):

Curl (mathematics)
https://en.wikipedia.org/wiki/Curl_(mathematics)

And because this post was motivated by all that curl in the Maxwell equations, I tried to find back when I originally started writing about electrons having magnetic charge instead of being magnetic dipoles like they are tiny bar magnets. It was 29 April that inside the math pages I found the first update on that. Here it is:

From 29 April 2014 : Do electrons have spin?
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#29April2014

Ok, that was it. Till updates my dear reader.

 

Teaser picture for a new update on differential equations.

3D complex numbers, CR equations

I am going to make this update on the other website so in about one week you can check it out on page 4 of the 3D complex numbers. I admit I have been lazy the last month but now I have stopped smoking over three years my health is still improving so I am more exploring the environment with my bike while I can do that again…

Ha! At my worst about four years back I could only walk 100 to 125 steps and after that I needed to pause 3 to 4 minutes because I got camps in one of my legs. So it finally dawned on me I had to stop this nicotine addiction because the next phase would be a wheelchair combined with an oxygen mask and a tank of high pressured liquid oxygen on the back of my wheelchair.

Looking back I am glad I got so ill because without it I would never have managed to stop smoking those ridiculous amounts of cigarettes day in day out.

Sorry to bother you with my past health problems, this new teaser picture is rather funny and I hope also intriguing: I have crafted three coordinate functions x(t), y(t) and z(t) and if you differentiate them with respect to the time t you get a 3D square of the vector (x(t), y(t), z(t)).

So check it out for yourself; take the derivative of the three coordinate functions and see if you can get the three equations as on the bottom of the teaser picture…

The new update (on the other website) is ten pages long meaning it is 10 pictures of size 550 x 1100. Click on the teaser picture to land on the new update:

03-Sept-2016-teaser-pic-for-differential-equationsFor me it is just so cute: If you differentiate to time you get the square of the position you are in the 3D complex number system.

For use in the science of physics I do not think it is that important because real physical problems never rely on the coordinate system you use, but you never know…
For use in the science of math it is also not important because professional math professors still have not developed the cognitive capabilities of understanding 3D complex numbers.

I also made a teaser picture for use on the other website, it is the same solution to the simple differential equation as above but this time I solved it inside the complex plane. Of course I could not use that on this website as the first teaser picture given the face we more or less always try to focus on the 3D complex & circular number systems…

07-sept-2016-teaser-new-diff-eq-techniques

Ok, end of this post.
And life, as usual life will go on.

Till updates.

Wirtinger derivatives and the factorization of the Laplacian.

3D complex numbers, CR equations, Laplacian, Quantum mechanics

This post could have many titles, for example ‘Factorization of the Laplacian using second order Cauchy-Riemann equations’ would also cover what we will read in the next seven pictures.

The calculation as shown below is, as far as I am concerned, definitely in the top ten of results relating to all things 3D complex numbers. Only when I stumbled on this a few years back I finally understood the importance of the so called sphere-cone equation.

The calculation below is basically what you do when writing out the sphere-cone equation only now it is not with variables like x, y and z but with the partial differential operators with respect to x, y and z. In simplifying the expressions we get I use so called second order Cauchy-Riemann equations, if you understand the standard CR equations these second order equations are relatively easy to digest.

Have fun reading it.

05Aug2016_factorization_of_the_Laplacian01

05Aug2016_factorization_of_the_Laplacian02

05Aug2016_factorization_of_the_Laplacian03

05Aug2016_factorization_of_the_Laplacian04

05aug2016_factorization_of_the_laplacian05 05aug2016_factorization_of_the_laplacian06

 

05Aug2016_factorization_of_the_Laplacian07This post is also categorized under Quantum Mechanics, the reason for that is that the wave equation contains the Laplacian operator and the more you know about that rather abstract thing the better it is in my view.

I would like to close with a link to a wiki on Wirtinger derivatives, originally they come from theory with several complex variables. That explains why in the wiki the Wirtinger derivatives are written as partial derivatives while above we can use the straight d´s for our differential of f.

Here is the wiki: Wirtinger derivatives
https://en.wikipedia.org/wiki/Wirtinger_derivatives

Till updates.

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

3D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanics

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:

  1. Find a basis vector (from an imaginary number) that has determinant +1.
  2. Craft analytical expressions that you cannot solve but use internet applets for numerical answers. Or, equivalent:
  3. Use an applet to calculate the logarithm of your basis vector once you have put it on a matrix representation.
  4. 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…

__________

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:

0027_07July2016_modified_Dirichlet_kernel01

0027_07July2016_modified_Dirichlet_kernel02

0027_07July2016_modified_Dirichlet_kernel03

0027_07July2016_modified_Dirichlet_kernel04

0027_07July2016_modified_Dirichlet_kernel05

0027_07July2016_modified_Dirichlet_kernel06

0027_07July2016_modified_Dirichlet_kernel07

0027_07July2016_modified_Dirichlet_kernel08

0027_07July2016_modified_Dirichlet_kernel09

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

__________

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.

Modified Dirichlet kernels for low dimensions.

3D complex numbers, 5D complex numbers, Exponential circle, Exponential curves

What the hydrogen bomb is in the average nuclear arsenal, that is what modified Dirichlet kernels are for higher dimensional complex & circular number systems.
Via the so called tau calculus I was able to achieve results in 3 and 5 dimensional number systems and I really had no hope in making more progress in that way because it gets so extremely hardcore that all hope was lost.

Yet about two years ago I discovered a very neat, clean and very beautiful formula that is strongly related to the Dirichlet kernel known from Fourier analysis. The formula I found was a dressed down version of the original Dirichlet kernel therefore I named it ‘modified Dirichlet kernel’.

This modified kernel is your basic coordinate function, depending on the dimension of the space you are working in you make some time lags and voila: There is your parametrization of your higher dimensional exponential (periodic) curve (only in 2D and 3D space it is a circle).

For myself speaking: this result of finding the modified Dirichlet kernel is for sure in my own top 10 list of most important results found. Not often do I mention other mathematicians, but I would like to mention the name of Floris Takens and without knowing how Floris thought about taking a sample of a time series and after that craft time lags on that, rather likely I would not have found this suburb and very beautiful math…

__________

I haven’t decided what the next post will be about.
It could be stuff like:

  • How I found the first modified Dirichlet kernel, or
  • Wirtinger derivatives for 3D number systems, or
  • Wow man, can you factorize the Laplacian operator form Quantum Mechanics???

But factoring the Laplacian requires understanding 3D Wirtinger derivatives so likely I will show you how I found the very first modified Dirichlet kernel.

__________

This update contains six jpg pictures each about 550 x 775 pixels and two old fashioned animated gif pictures. I tried to keep the math as simple as possible and by doing that I learned some nice lessons myself… Here we go:

0026_20June2016_Modified_Dirichlet_kernels01

 

This is the animated gif using z = 0 in 3D while this picture is showing the Euler exponential circle:

17-06-2016_exp_circle_on_C0026_20June2016_Modified_Dirichlet_kernels02

 

0026_20June2016_Modified_Dirichlet_kernels03

Here is an animated gif of how this coordinate function looks when you combine it with the two time lags for the y(t) and z(t) coordinates. Does it surprise you that you get a flat circle?
If it does not surprise you, you do not understand how much math is missing in our human world…

17-06-2016_exp_circle_on_3D_circular

0026_20June2016_Modified_Dirichlet_kernels04

0026_20June2016_Modified_Dirichlet_kernels05

0026_20June2016_Modified_Dirichlet_kernels06

In case you are interested in the ‘time lag’ idea as Floris used it, here is a nice Youtube video that gives a perfect explanation. If you apply this time lag idea for example is a 17 dimensional real vector space you get a 17D exponential curve with all of it’s magnificent properties…

Takens’ theorem in action for the Lorenz chaotic attractor

Yes, end of this post. See ya around & have a nice life or try to get one.

Three centuries after de Moivre finally some new baby steps.

3D complex numbers, Exponential circle

I am a little bit late with this post but after the previous post I took some time to enjoy it because not every day you can craft a brand new coordinate system…

Also I was a bit in doubt about this post, shall I finally start with those modified Ditichlet kernels or work out a simple de Moivre example in 3D? It became the latter so likely in the next post I will do some first things with modified Dirichlet kernels.

This post is just 4 pictures long (size 550 x 775 pixels) and to be honest it contains no serious math whatsoever. I was only driven by curiosity about how difficult a simple example of the new 3D versions of it would be.
Now it is not 100% trivial but it is also not a very deep result, at best you can say it is a bit technical because you constantly have to apply those sum and difference formulae for the cos and sine functions.

The real deep math work was crafting those exponential circles in 3D in the first place and later finding the coordinate functions belonging to that. That was the deep math because once you have those, the new de Moivre formulae are a piece of cake (make sure it is gluten free!).

__________
In this update I also would like to make an advertisement for a long update I started about one year ago on the other website; all in all it is 37 pictures long (size 550 x 1100) and it took about 3 weeks to write it. It has the title The Missing Equations because with those modified Dirichlet kernel I knew I had solved a terrible hard problem but the higher in the dimensions I got the more missing equations I had for my wonderful solution… Here is the link:

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

__________

After this small advertisement here are the four pictures of this update:

0025_05June2016_de_Moivre_baby_steps01

0025_05June2016_de_Moivre_baby_steps02

0025_05June2016_de_Moivre_baby_steps03

0025_05June2016_de_Moivre_baby_steps04

Once more: This is not a deep mathematical result or so.
I was just curious of how difficult it would be to get one of the most simple 3D new de Moivre formula results using only the 3 century old stuff from about 50 years before Euler entered the scene…

The most important lesson you can learn from this is that instead of focusing on all kinds of details like n = 2 and only the real part and so on is a waste of time and energy.
After all using the exponential circle is what brings peace to the heart; it is simple, it covers all powers at the same time and you have all coordinate functions at once…

__________

Now you have to wait a few minutes more because I would like to pop up a fresh home brew and after that I will hit the ‘publish post’ button…

Thanks for the waiting 😉 Now I will hit the publish button and see you next time around!

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…

__________

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?

__________

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

0024_23May2016_the_art_coordinates01

0024_23May2016_the_art_coordinates02

0024_23May2016_the_art_coordinates03

0024_23May2016_the_art_coordinates04

0024_23May2016_the_art_coordinates05

0024_23May2016_the_art_coordinates06

0024_23May2016_the_art_coordinates07

0024_23May2016_the_art_coordinates08

0024_23May2016_the_art_coordinates09

0024_23May2016_the_art_coordinates10

0024_23May2016_the_art_coordinates11

0024_23May2016_the_art_coordinates12

 

0024_23May2016_the_art_coordinates13

 

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.

Things have grown emotional and chaotic. But there is a glimmer of hope…

3D complex numbers, Exponential circle

Oh oh all of a sudden all kinds of things that are highly emotional and non-mathematical arise. Stuff like a close but young family member still being suicidal and a young neighbor that was arrested by the local police and is now in the intensive care of the local psychic hospital.

And to finish it off, all of a sudden after waiting far too many years I decided to press some child neglectance/abuse charges related to the boy in the intensive care.
And, just a detail: If you have a very severe spychotic behavior like the boy in the IC of course this could have a pure biological foundation.
But if your mommy is a full blown psychopath this does not do much good either…

Yet I have to say that although this stuff is highly emotional I think I like it.
After all these years finally there could be a small chance of fundamental change for those kids suffering from the psychopath mommy…

We will wait and see.

__________

On the math front I have not been sitting on my hands but stuff is not finished by far.
So the new coordinate system has about one week of delay.
On the naming of the coordinate system it is just about thinking about the name for a new baby and I am more and more leaning to the name that is also a functional description:

[a, r, t] coordinates.

Why not? The three letters a, r and t are how you need to calculate them in that very order.
And the word art stands for stuff that is supposed to be beautiful but also make you think a bit more. As the days go by the less I am in favor of naming this new coordinate system with names like:

  1. Cone coordinate system, or
  2. Conial coordinates, or
  3. Shifted cone coordinate system, or
  4. More stupid names.

No, why not choose [a, r, t] coordinates as the name for this new coordinate system?

On the other website I have posted the first update that studies how the length of two 3D circular numbers change when you multiply them. So given the reactangular coordinates (a, b, c) of A and (x, y, z) of X, what is the length of the product AX?

Here is the link:

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

Ok let me end this update, till the next post.

A brand new coordinate system for 3D complex and circular numbers under development.

3D complex numbers, Exponential circle

With great success I was able to kill my frustration. There are many ways to combat heavy upcomming frustration, for example you could go running against the wind until you are so heavily exhausted that all frustration is gone.

But this time I did it differently: Within a timespan of at most three minutes I finally wrote down that calculation that I avoided for so long, for so many years. And within this small time frame suddenly I was bombarded by sphere and cone equations telling me that story I should have discovered so many years ago.

Within this unloading of a huge amount of frustration I discovered how the length changes when you multiply two 3D cokmplex numbers, say X and Y.
All these years I never understood properly what drove the length of the product XY.

And this result eased my mind, I was no longer frustrated that all those incompletent people get boatloads of money every month while I live in relative poverty compared to those saleries.
And I went to work and I discovered an amazingly strange but also very easy to understand completely new coordinate system for 3D space.

It is not for any ordinary 3D vector space, you must equip it with either the complex or circular multiplication, but it is so beautiful that I hope it will survive in the long run.

I have not decided on a name yet, for the time being I name it ‘special coordinates’. Since cones play such a major role the name ‘cone coordinates’ might be the right thing to do.
But there are already Conical Coordinates yet they act like the common point of 3 non parallel planes; it is the intersection of two cones with a sphere…

__________

All in all I will post the stuff on length preserving/shrinkage/extra growth on the other website in page 4 on the higher dimensional complex numbers.

And on this website I will post an entry with this new coordinate system

So that’s the planning for the time being, 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:

0021=13April2016=third_Schrodinger_post01

0021=13April2016=third_Schrodinger_post02

0021=13April2016=third_Schrodinger_post03

0021=13April2016=third_Schrodinger_post04

0021=13April2016=third_Schrodinger_post05

0021=13April2016=third_Schrodinger_post06

0021=13April2016=third_Schrodinger_post07

0021=13April2016=third_Schrodinger_post08

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

0021=13April2016=third_Schrodinger_post09

0021=13April2016=third_Schrodinger_post10

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.