# Calculation of the four coordinate functions for the 4D exponential curve (complex version).

Like promised in this post I will show you in the greatest detail possible how to find those rather difficult looking four coordinate functions.

I had thought about crafting these four coordinate functions before but the method I had in mind was rather labor some so I balked a bit at that. Not that I am lazy but I also had to work on the basics for the 4D complex numbers like in the last posts…

So one day I decided to look into the specific details of what I name ‘imitators of the number i’ and I was very surprised by their behavior. As a matter of fact these imitators imitate i soo good that you can make exponential circles of them.
And I wrote down the two exponential circles, I looked at them and realized you can factorize the 4D exponential curve with it and as such you will get the four coordinate functions…

That was all, at some point in time on some day I just decided to look at the imitators of the number i from the complex plane and within 5 at most 10 minutes I found a perfect way of calculating these four coordinate functions.

It always amazes me that often a particular calculation takes a short amount of time, like 10 or 20 minutes, and after that you always need hours and hours until you have a nice set of pictures explaining the calculation…

Anyway, this post is five pictures long and as such it contains also Part 6 and 7 of the Basics to the 4D complex numbers.

I hope that in the long run it will be the result in this post that will make 4D complex numbers acceptable to the main stream mathematical community.
But may be once more I am only fooling myself with that, after all back in the year 1991 I was only thinking stuff like ‘If you show them the 3D Cauchy-Riemann equations, they will jump in the air from joy’.
They (the math professors) never jumped from joy, no significant change in brain activity was ever observed by me. So when I write ‘in the long run’ as above, may be I should more think like a geological timescale…
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But let’s not complain because once you understand the factorization, it is so beautiful that it is hard to feel angry or whatever what.
Here are the five pictures:

Ok, that is how you calculate the four coordinate functions.

# The basics of 4D complex numbers.

In the previous post on 4D complex numbers I went a little bit philosophical with asking if these form of crafting a 4D number system is not some advanced way of fooling yourself because your new 4D thing is just a complex plane in disguise…

And I said let’s first craft the Cauchy-Riemann equations for the 4D complex numbers, that might bring a little bit more courage and making us a little bit less hesitant against accepting the 4D complex numbers.

In this post we also do the CR equations and indeed they say that for functions like f(Z) = Z^2 you can find a derivative f'(Z) = 2Z. So from the viewpoint of differentiation and integration we are in a far better spot compared to the four dimensional quaternions from Hamilton. But the fact that the CR equations can be crafted is because the 4D complex numbers commute, that is XY = YX. And on the quaternions you cannot differentiate properly because they do not commute.

So crafting Cauchy-Riemann equations can be done, but it does not solve the problem of may be you are fooling yourself in a complicated manner. Therefore I also included the four coordinate functions of the exponential 4D curve that we looked at in the previous post.

All math loving folks are invited to find the four coordinate functions for themselves, in the next post we will go through all details. And once you understand the details that say the 4D exponential curve is just a product of two exponential circles as found inside our 4D complex numbers, that will convince you much much more about the existence of our freshly unearthed 4D complex numbers.

Of course the mathematical community will do once more in what they are best: ignore all things Reinko Venema related, look the other way, ask for more funding and so on and so on. In my life and life experiences not one university person has ever made a positive difference, all those people are only occupied with how important they are and that’s it. Being mathematical creative is not very high on the list of priorities over there, only conform to a relatively low standard of ‘common talk’ is acceptable behavior…

After having said that, this post is partitioned into five parts and is 10 pictures long. It is relatively basic and in case that for example you have never looked at matrix representations of complex numbers of any dimension, please give it a good thought.

Because in my file I also encountered a few of those professional math professors that were rather surprised by just how a 3 by 3 matrix looks for 3D complex numbers. How can you find that they asked, but it is fucking elementary linear algebra and sometimes I think these people do not understand what is in their own curriculum…

Ok, here are the 10 pictures covering the basic details of 4D complex numbers:

Ok, that was the math for this post.

And may be I am coming a bit too hard on the professional math professors. After all they must give lectures, they must attend meetings where all kinds of important stuff has to be discussed until everybody is exhausted, they must be available for students with the questions and problems they have, they must do this and must do that.

At the end of the day, or at the end of the working week, how much hours could they do in free thinking? Not that much I just guess…

Let’s leave it with that, see you in the next post.

# A nice teaser picture about 4D coordinate functions of the 4D exponential curve (complex version).

Lately I have been working on the next post about the basics of the 4D complex numbers. You simply need those basics like matrix representations because later on when you throw in some 4D Cauchy-Riemann equations, it is very handy to have a good matrix representation for the stuff involved.

The next post covering the basics had five parts, let’s not dive in all kinds of math details right now but go straight to part five with the four coordinate functions of the 4D exponential curve:

These four coordinate functions are also time lags of each other.

This new baby number tau keeps on looking cute…

Let me leave it with that, till updates.