The stuff is more or less finished, I only have to turn it into a series of pictures so tomorrow or the day after I will post the third post on the Schrödinger equation using higher dimensional complex numbers.
Now on the other website I posted a teaser picture and since we are against cruel discrimination of peace loving websites why not post it here too?
So that is our post for today: Just a teaser picture:
Yeah yeah, once more we observe mathematical perfection.
Till updates.
Ok ok I made a relatively big blunder when sending the people from quantum mechanics using the Schödinger equation into the direction of the 3D complex numbers.
Because as a matter of fact, you cannot solve the factual Schrödinger equation in the 3D complex number system because there is no famous i to be found with the property that i^2 = -1.
You need a more advanced number system and I will explain that in detail in an upcoming post number three on the Schrödinger wave equation.
__________
In the meantime, it has not fallen on deaf ears on my side that after I posted the first Schrödinger post and I later did an internet search, suddenly with the search phrase ‘3Dcomplexnumbers’ I suddenly ended on number 1, 2 and 3.
So like expected it drew a lot of attention.
Therefore I would like to give a kind of present to the physical community because also not fallen on deaf ears, as far as I observe it physics people always try to use a product integral when they can.
Product integrals were my first serious mathematical invention, I found them while I was still trying to get my first year exam al the local university. Math professors almost never use product integrals because they are to stupid for that but physics people often put it in product integral representation.
How the history of that detail is I do not know, may be Paul Dirac had a bit to do with it…
Anyway, some time ago I wrote a pdf with the title
A tribute to Euler. Title: Ten styles for product integrals and product differentiation.
http://kinkytshirts.nl/pdfs/Product_integrals.pdf
__________
So far for the gift or the present, if you want to use other number systems beside the complex plane for depicting atomic orbitals or whatever you want to do with it, you also must know more about exponential circles and curves.
Physics people use so called ‘phase shift’ all of the time, but that is only multiplying some stuff with e to the power it or so. This is the sandpit exponential curve for toddlers.
In the world of the grown ups we have all kinds of other exponential circles and curves and guess what? I have a pdf for you with another 10 pieces of exponential circles & curves:
An overview of exponential circles and curves.
http://kinkytshirts.nl/pdfs/10_exponential_circles_and_curves.pdf
A possible way of parametrization of 3D exponential circles is given in the next picture and understanding this stuff is important when it comes to the third post related to the Schrödinger equation:
This is the end of this intro to the third Schrödinger post.
Have a nice life or try to get one.
Till updates.
A few posts back I wrote a bit about the Schrödinger wave equation related to calculating atomic and molecular orbitals for electrons using 3D complex numbers.
What I said was basically correct but also an over-simplification of the situation.
The problem is very very basic: in the 3D number system, let it be complex or circular, you just cannot solve and equation like $X^2 = -1$.
Hence the number i from the complex plane with i^2 = -1 just does not live in 3D real space.
So using alternative number systems outside the complex plane is not a straightforward thing to do, yet in principle all higher dimensional complex numbers should give the same results.
If not there would be a very basic problem inside the wave equation from quantum mechanics and I am not aware of any faults in that detail of the quantum theory.
Here are two pictures that serve as an addendum on the previous post on the Schrödinger equation:
Now if you are reading this it is very likely that at least once in your life you have seen a solution to the Schrödinger wave equation like the ‘particle in a box’. And that is not a 3D box but the one dimensional box or just an interval of the real line.
Solving the Schródinger stuff for atomic and molecular orbitals is a very different kind of game; these are always many particle systems where every particle influences the system and the entire system influence the individual particles.
Mathematically speaking it is a nightmare; analytical solutions are not possible they say.
It can only be solved numerically…
__________
But keep on dreaming, after all they also say decade in decade out that electrons are magnetic dipoles. There is no experimental proof for that only theoretical bla bla bla.
Let’s leave it with that. Till updates.
It was in the Spring of 2013 when I was walking in a nearby park when it suddenly dawned on me that this exponential process that ran through the basis vectors (1, 0, 0), (0, 1, 0) and the z-axis unit vector (0, 0, 1) was periodic.
It could not be anything else because I was capable of calculating the logarithm of the first imaginary unit j.
I remember at first I just did not have a clue it would be a circle, I even had vague fantasies like may be it is a vibrating string where all those string physics professors talk about.
Now this evening I was just Googleing around a little bit when I came across this picture again:
It dates back to 30 May 2013 and I used this picture as a joke about how professional math professors look in my fantasy world. Within a week I found that the 3D complex periodic curve was in fact a circle.
So I had to laugh hard about my own joke once more because if I had known the 3D periodic thing would also be a circle I would have made the joke very different… Because one way or the other this picture now also represented me.
__________
You know this last week I am a bit puzzled by what the next post should be, in December 2013 I conducted a good investigation into the roots of unity related to the two exponential circles and because every body knows roots of unity it would a nice started for this website.
On the other website you can find it at the 05 Jan 2014 entry:
The song of omega reloaded
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#05Jan2013
At the time I was amazed with all the things you can do with the eigen values of the imaginary components j and j squared. From diagonalization to the roots of unity, my theory got definitely air born.
Later in January 2014 I found a new Cauchy integral formula (actually two just like I found two sets of roots of unity each for the admissable forms of 3D multiplication). Also in Jan 2014 I cracked the problem of 5D complex numbers.
By all standards, as far as I can see it; the two months Dec & Jan in that time were the most productive ever.
__________
Now, almost 3 years later finally stuff on Google take off, for example if this day I start searching for the phrase 3dcomplexnumbers it returns back three results from this new website.
Just look at the next screen shot picture:
So after waiting all these years, finally I begins to look as if stuff starts getting air born on some bigger scale than before.
Ok, end of this update. As usual till updates!
The numerical use of three dimensional complex numbers is almost the same as the situation on the complex plane. This is caused by the simple fact that only on the main cone that includes the three coordinate axes, we have that if you multiply a number X by it’s conjugate, the result is a real number.
In the complex plane this is valid for all numbers in the plane but in higher dimensional complex number systems the situation is different; you must always pick numbers from that main cone where also the exponential circle lives (in 3D) or exponential curves (in higher dimensions).
This update is 5 pictures long, size pics = 550 by 775.
__________
One hour later:
Shit! There is a serious problem with uploading the pictures, they get uploaded but they are not visible… So you must wait at least one day longer because I do not understand the problem at hand…
Till updates.
__________
Problem with the jgp pictures is solved; according to my webhost provider it was caused by the name Schrödinger because that contains an o with two dots: ö.
My computer can handle filenames with ö so for me they looked normal but the server that hosts this website cannot deal with these kinds of symbols…
Anyway after a few days here are the pictures:
Well I am happy this strange problem of invisible pictures has been solved. Till updates.
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
__________
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.
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.
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.
__________
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…
The zero’s of the Riemann zeta function is one the things I will never be able to find because I hate it to write computer code. Always my original enthousiasm fades away quickly and after some time I simply stop working in that direction and foces on things that I like more.
Just like a few posts back when I finally decided to skip the stupid Mandelbrot fractal in three dimenstions. Computers are nice things to build but programming has never been my cup of tea let alone my pint of beer. (I am also a hobby brewer, it is a great hobby and it saves you a lot of money. The more you brew the more money you save…).
Ok in this update we are going to take a look at imitators of the number i from the complex plane. I think that most readers here already know that multiplication by i rotates everything 90 degrees. In 3D space we have similar things but not all higher dimensional number spaces contain the number i from the complex plane. In that case we must use substitutes like what I name the ‘imitators of i‘.
This update is seven pictures long, each picture is 550 by 550 pixels:
In the next post we will flea through the elementary properties of the number alpha, look at the dynamics on the line through zero and alpha (just like on the real line) and so on and so on.
Till updates.
Without doubt the Laplacian is a very important differential operator. It plays a major role in for example the classical wave equation and also the Schrödinger wave equation from quantum mechanics.
Now scroll a bit back until you find the post on the Cauchy-Riemann equations, at the end I used the phrase ‘Cauchy-Riemann equations chain rule style’ and this is how we can crack in a very easy way how the Laplacian operates on functions that obey the CR equations on 3D complex numbers.
I have hundreds and hundreds of pages of math stuff on the 3D complex number system and very often I use the number alpha. This number alpha is so important, not only in 3D, that it is worth to post a few posts on them.
For the time being, I just conducted a simple Google search on the phrase ‘3d complex numbers’ in the search detail for pictures. And every time this old teaser picture from the other website pops up:
At the end you see that (1, -1, 1), well that is three times alpha.
It is a nice exercise to prove that the square of alpha equals alpha.
So alpha is in the same category as for example numbers like 0 and 1 because if you square those you also get the original number back in return.
After all one squared equals one and zero squared equals zero.
End of this update, till updates.