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:

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:

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:

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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.

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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Oh 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.

This is a very short update from a guy that likely goes under the name of Alen.
Now for years Alan has a very weird page hanging out there and he tries to describe the 3D complex numbers as they should be in his view…

Very very likely this is the approach Hmilton took for 12 to 15 years; it leads to all kinds of horrible difficulties, technical disasters and an end result useless to all people on the globe.

Now you must not jump to the conclusion that I hold this Alen person for some idiot. It is rumored that at any given time there are about 100 thousand professinal math workers out there and I do not mean high school teacher but people from the universities and stuff like that.

All these people cannot find higher dimensional complex number systems themselves either…
Ok, enough of the bla bla bla en prepare yourself to dive in the blurry and fuzzy mindset of how people thought 3D complex numbers must be (it is always with something squared that is minus one…):

N-DIMENSIONAL COMPLEX NUMBERS.
http://www.alenspage.net/ComplexNumbers.htm

It seems that the irish math guy sir Hamilton has sought three dimensional complex numbers for a staggering long period of 15 years. After 15 years he found the four dimensional quaternions, that is nice stuff but for differentiation and integration the quaternions are about the biggest disaster there is around.

Therefore, beside linear transformations, there is no functional analysis on the quaternions.

In this post I originally wanted to give you some link to some stuff that was labeled under the name ‘Alan’s pages’ but I cannot find them back. Alan did everything wrong just like Hamilton did:
Basically they start with the comlex and try this to extend to 3D space; that stuff is guaranteed to fail hard.
Anyway, my first try was a surface named the Riemann surface of the logarithm although at the time I found it I did not know what Riemann surfaces were. This update is five pictures long each 550 by 550 pixels. Have fun reading it.

In another development I wrote reason number 12 as why electrons have to be magnetic monopoles, later this week I will hang this into the other website on the page on magnetic stuff. It is about the plasma they use in nuclear fusion reactors, the plasma is not stable and the physics professors do not understand why this is.
Well these torus shaped fusion reactors constantly accelerate the plasma particles until they get relativety effects, the basic concept of torus shaped reactors is basically what is wrong with it…

Anyway, till updates.

When learning about higher dimensional complex numbers, one of the things you must first understand that the complex plane as known for centuries simply does not live in three dimensional space. If you look at it from a historical perspective people always tried to start with the two dimensional complex plane and tried to expand that into three dimensions.

It does not work that way; for example you can make 21-dimensional numbers by using 3-dimensional and 7-dimensional numbers. The dimension nicely breaks down via the prime number theorem (every natural number can be uniquely written as the factors of prime numbers).

Since 2 is not a divisor of 3, it is impossible to find the complex plane in a three dimensional world…

Now years ago I proved that for the complex multiplication the number i does not exist, yet now I started this new website why not give the same proof for the circular multiplication?

Click on the picture below to read that exiting proof… 😉
(On my browser I first have to click on the picture and after that enlarge it to get the readable stuff.)

Picture size is 550 pixels wide by 1650 pixels high, I tried to write it in such a way that advanced high school folks could understand it.

Till updates.