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):

0013=22Feb2016=teaser_picture_cone_theorem

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:

0012=09Feb2016=teaser_pics_alpha_properties01

0012=09Feb2016=teaser_pics_alpha_properties02

0012=09Feb2016=teaser_pics_alpha_properties03

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…

Imitators of the complex number i and how this relates to the zero’s of the Riemann zeta function in 3D.

3D complex numbers

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:

0011=04Feb2016=imitators_of_i01

0011=04Feb2016=imitators_of_i02

0011=04Feb2016=imitators_of_i03 0011=04Feb2016=imitators_of_i04

0011=04Feb2016=imitators_of_i05

0011=04Feb2016=imitators_of_i06

0011=04Feb2016=imitators_of_i07

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.

Calculating the Laplacian using the Cauchy-Riemann equations.

3D complex numbers, CR equations, Laplacian

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:

0010=30Jan2016=Laplacian_for_3D_stuffAt 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.

Math muscle, does it exist?

3D complex numbers

Two days ago I found that article from a Israeli citizen on the preprint archive. It is well known that the Jewish part of the Israeli society have a very high Nobel prize to capita citizen ratio.

Very likely they have the most Nobel prizes per capita citizen of our small planet…
There are a lot of reasons for this, for example Israelis think about the food they eat. (I mean try to eat some of that weird McDonnalds food and make a math exam later, good luck with it.)

But this post is not about why the Jewish society has so many Nobel prizes, it is about showing off my math muscles. Math muscles? Do they exist?

To my amazement the guy Shlomo Jacobi even investigated the alternating sums as shown below.
He understood the importance of the stuff involved, but was likely not capable of finding the explicit formulaes…

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So now we are comparing math muscle:
The next picture is what the Jacobi guy brings up:

0010=21Jan2016=math_muscle01

Please remark this is just a power series any math student can write down, what is the solution?
And from an update known as the Curves of Grace, I found the explicit stuff.
I recycled this picture from Google search pictures because that saved me a bit of time:

0010=21Jan2016=math_muscle02

Remark you must replace the x by the time variable t in the Taylor series…
It is a typo because in the past I wrote them with x while now I needed them it had to be in time t.

So ok ok, since I want this to be a nice new website it took about 10 minutes to find a corrected version of this. In the meantime I observed how much work I have done over the last 8 months.
Here is the corrected version:

0010=21Jan2016=math_muscle03

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So between me and the Israeli math departments from the universities they have over there, this is just a bit of showing off my math muscles…

Source links: preprint archive of the Jacobi guy: http://arxiv.org/abs/1509.01459
My work on the curves of grace: http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#18April2015

Till updates my dear reader.

And now we are with three…

3D complex numbers, CR equations, Exponential circle

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.

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.

The 3D Mandelbrot set. Part two.

Uncategorized

With my previous post I thought that I am leaving behind a lot of readers into the woods. But the goal of this new website is making easy to digest updates.

Therefore just three pictures explaining a bit more about the 3D Mandelbrot set.
I wish the computer code writing folks a lot of luck; if you are the first that
has a view on the 3D Mandelbrot set you have won the race.

By the way; out there is something known as the Mandelbulb or so. It looks nice but believe it has nothing to do with the 3D Mandelbrot set. This Mandelbulb thing is not based upon a reliable way of multiplication in three dimensions, the results they found are nice looking but it is just not based on a rigid way of 3D multiplication…

Anyway, here are three pictures all sized 550 by 550 pixels that should shine a bit more light on the minds of those that write the code:

0008=14Jan2016=3D_Mandelbrot_fractal01

0008=14Jan2016=3D_Mandelbrot_fractal02

0008=14Jan2016=3D_Mandelbrot_fractal03

On a video channel named Numberphile from Google they have a good explanation about how to color the pixels according to the Mandelbrot scheme. Here is the link:
The Mandelbrot Set – Numberphile

In all dimensions where complex numbers can be defined, you can do this…
Till updates, think well and live well.

Short stuff on the 3D Mandelbrot fractal.

3D complex numbers

About a year ago I decided to take about two years to make it to the 3D Mandelbrot set. So I tried to learn one of those modern programming languages like C++, I did build a new computer because on my old system C++ would not run. And so on and so on.

Decades ago I tried to learn a computer programing language known as Basic. When I found out how those kind of programming languages evaluted an integral, I almost had to vomit.
In those long lost years I already developed a fundamental dislike against programming.

Now I am 52 years of age and it is still the same; me writing computer code is not a happy thing to do. So I killed the project of being the first person on this planet to view the 3D Mandelbrot set using the 3D complex or circular multiplication…

I never made it beyond what is in C++ a ConsoleApllication; you get your output in an old fashioned DOS screen and no graphics at all. And how to embed this into a thing you can actually fly through, I have given up on that.

So I did not write much code, but the results had all you expected it should have: Strong sensitivity to initial conditions and so on and so on.

Well here is the kernel of the 3D Mandelbrot set for the circular multiplication.
Circular simply means we are using 3D circular numbers X = x + yj + zj^2 where j^3 = 1.

In this kernel we have to use so called ‘dummy variables’ because computers are so stupid you cannot tell them how to calculate the next round of variables This despite in the year 2016 most desktops have multiple cores, your programming language still uses the old von Neuman principles.

Here is the kernel with the dummy variables written as capital X, Y and Z while we only want to know how the x, y and z evolve over the iterations… :

int i = 1;
float x = 0f;
float y = 0f;
float z = 0f;
float X = 0f;
float Y = 0f;
float Z = 0f;

while ((i < 80)&(x*x + y*y + z*z < 1600))
{
i = i + 1;
X = x;
Y = y;
Z = z;
x = X * X + 2 * Y * Z + C0;
y = 2 * X * Y + Z * Z + C1;
z = Y * Y + 2 * X * Z + C2;
}

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0007=14Jan2016=Mandelbrot_in_3DOh oh my dear Mandelbrot baby, now I have thrown you into the river I will never be the first human to observe your intrinsic details. Let it be, let it be because for the rest of my life I can still hate that stupid computer code writing.

Till updates.