Category Archives: Exponential circle

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

May 15, 20163D complex numbers, Exponential circleReinkoVenema

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.

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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:

Cone coordinate system, or
Conial coordinates, or
Shifted cone coordinate system, or
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.

May 3, 20163D complex numbers, Exponential circleReinkoVenema

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…

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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.

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.

Teaser picture for the third post on the Schrödinger wave equation.

April 11, 20163D complex numbers, Exponential circle, Quantum mechanicsReinkoVenema

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.

Some good math for the physics community.

April 3, 20163D complex numbers, Exponential circle, Exponential curves, Quantum mechanicsReinkoVenema

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.

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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

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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.

When did I find the first exponential circle in 3D space?

March 25, 20163D complex numbers, 5D complex numbers, Exponential circleReinkoVenema

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.

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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.

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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!

Atomic orbitals, the Schrödinger wave equation and 3D complex numbers.

February 29, 20163D complex numbers, Exponential circle, Laplacian, Quantum mechanicsReinkoVenema

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.

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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.

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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.

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.

And now we are with three…

January 19, 20163D complex numbers, CR equations, Exponential circleReinkoVenema

In Oct 1990 (estimated) I found the 3D complex numbers, a few years back I discovered that a guy named Dennis Morris has found them too. Dennis even wrote a book about it, this book was published twice and I was able to get a hand on the second publication.

For normal humans the book from Dennis is a good starter, for me it has the depth of a bird bath.

We do not complain, today on the preprint archive I found the next pdf file:

On a novel 3D hypercomplex number system
http://arxiv.org/pdf/1509.01459v1.pdf

And this work has not the depth of a bird bath, it is much more the depth of a human bathtub.
There ar some dumb typo’s, for example table 1 contains a very stupid error so that has to be corrected.

The writer of the preprint article goes under the name of Shlomo Jacobi and since his residence was Israel we might jump to the conclusion he was a Jew. Let religion be no problem because after all the Muslims were once far ahead of the Western powers but because one of their religious leaders declared math as being from the devil, Muslims find themselves at the receiving end of military powers for about one thousand years…

Now back to our Jewish pdf file: Table 1 should be corrected and I, Reinko Venema, I give them a big applause because if you scroll down to page 39 you observe they have found the 3D exponential circle too!

Well I have found all exponential curves in all possible dimensions, so I am very pleased to invite the Jewish mathematical community upon further investigations into this math detail.

End of this update, till updates.