Category Archives: Exponential circle

Probability amplitudes on the 3D exponential cones (circular and complex version).

All in all it was a nice day today. Brewing is completed and tomorrow the wort can go into the fermentation bottles and the wonderful process of fermenting can take place. For those of you that also like to brew: A couple of months back I found a cute video explaining that you can also brew beer without cooking it. And I was like seeing water burning or I was like a professional math professor understanding 3D complex numbers for the very first time in their life… Anyway if you are interested search for ‘Raw ale no boil brewing’ on Youtube. It is of interest because if you brew without boiling, only after that you understand what you usually cook away in things that might taste good (or bad).

But let’s go to this post: It is about probability amplitudes as they are used in quantum physics where all those kind of amplitudes are multiplied against their conjugate and that gives a real positive number known as the probability. If you write it in polar coordinates on the complex plane, it is easy to see that those probability amplitudes can have all kinds of phases (the argument of the complex plane number). So for that to work on 3D complex or circular numbers, it would be great if you can write it more or less like the polar coordinates as in the complex plane. And that is easy to do in 3D space: Once you have found and also understand the exponential circles, it is evident that all numbers on those exponential cones are some real multiple of a number from the exponential circle.

As such the numbers found on the exponential cone can be written just like the polar stuff from the complex plane, also now the r as used in polar coordinates can also be negative. That is a strange result because for millions of years we were always indoctrinated by a positive r … ­čśë

Another important difference with the complex plane lies in the fact that the complex plane is closed under addition. That is obvious, but it is also obvious that on a cone it is very different. Most of the time if you add up two numbers you are either inside or outside the cone. But probability amplitudes are always multiplied against their conjugate and added up only later, so we can still use the exponential cone for things like that. I don’t see that ship stranding, so let’s do it.

I also want to remark I am using the so called ‘pull back map’ once more. The professional professors also have a pull back map but that is a very different thing compared to what I use. So don’t be confused by that: the way I use it is to fix higher dimensional exponential circles (and curves) on the exponential circle in the complex plane. (This for fine tuning the period in time and stuff like that, or for understanding why the numbers are what they are: WTF that square root of 3 in it???

This post is 7 pictures long, most are the usual size of 550×775 pixels. At last I want to remark that for myself speaking I do not know if there is any benefit in trying this kind of use of 3D complex and circular numbers. It is funny to think about positive and negative values for r like for example in electron spin or a wave function for the electron pair. But I just do not know if this add any value or that you can use the complex plane only and miss nothing of all you could have learned.

Ok, here we go:

Ok, that was it for this post. Till updates my dear reader.

A new de Moivre identity.

First a household message: In about two weeks time this website should go to new very fast servers. In order for that to work properly I have to do all kinds of things that I have never done before. Stuff like updating PHP. Ok, that does not sound too difficult but as always the work explodes because first I have to backup everything. And before I can backup everyting I need a new ftp account. The only luck is I still have a running ftp client on my own computer…

In case this website is gone in two weeks, somewhere I got lost in the woods. And there is no hurry: this math website is just a hobby of me. An important hobby because it is a bit of exercise for the brain…
End of the household message.

What is the yeast of this post? Historically the de Moivre identity (or theorem) predates the very first exponential circle on the complex plane. If you use the exponential circle, a proof of the de Moivre identity becomes very very easy. In this short post we will use the 3D exponential circle for circular numbers. Two posts back I showed you a possible parametrization via those 3 cosine expressions, in this post we use those parametrizations to formulate a 3D de Moivre identiy.
Because we already have an exponential circle, we do not need to give a rigid math proof for this identity. Once you have and exponential cricle, stuff like that comes for free along with it…

As usual I skipped a lot of things while writing this post. For example I skipped using those modified Dirichlet kernels. I skipped giving the 4D de moivre identity for the 4D complex numbers. All in all I was satisfied to cram this all in a very short post; only three pictures long!
In case you are still reading this while having no clue whatsoever what a de Moivre identiy is, here is some stuff from brilliant.org:
De Moivre’s theorem
Http stuff in the link: https://brilliant.org/wiki/de-moivres-theorem/

Ok, only three pictures long. Here we go:

That was it for this post. If I don’t change plans, in the next post we will look at the 3D exponential cone because on that cone you can do all those quantum probability calculations just like in the 2D complex plane. But before that I have to go though that horrible PHP update…

So see you in the next post or let’s split indefinitely and end this stupid website for no reason at all… ­čśë

Two parametrizations for 3D exponential circles.

It is about high time I post the solution in parametrization form of those five equations from 03 Oct 2019. That is almost 2 months back and oh how ashamed am I for my laziness… But for me math is a hobby, an important hobby but a hobby anyway. So other hobby’s are allowed to interfere with my little math hobby.

This post is 10 pictures long and at the end there is a horrible bad video from the Youtube channel Seeker. Begin this week I crossed that video with an intriguing title; Could These Numbers Unravel New Dimensions in Space? I was just curious but it is that Cohl Furey stuff again. It is an attempt to explain particle physics via complex number, quaternions and octonions… What do they have in common? These number systems are always fields that means all non-zero numbers have an inverse. Why the professional math professors find that so important is unknown to me, it is more like they have nothing else in the toolbox. If you are interested you can find the Cohl Furey video’s on Youtube.

In this post I too write about things that are common in the complex plane, complex and circular 3D numbers and 4D complex numbers. You can use the modified Dirichlet kernels as the building blocks for all possible exponential circles or in the case with 4D complex numbers: the exponential curve (in 4D space the curve is in a 3D hyper plane).

But I also wanted to show you the original cosine solution that I found years ago. To this day it is still amazing that the cosine can pull it off; that the cosine can be a building block for a 3D exponential circle. Next year it will be three decades ago when I found the 3D complex numbers and got interested in them. At present day you can wonder why there is never a healthy response from the math communuty. It is all very logical: if there is no healthy response that means the math community in itself cannot be healthy. It is just a community of perfumed princes and that’s it.

After so much blah blah it is high time to go to the ten pictures:

So from the complex plane in two dimensions to 4D complex space; a binding element is how you can use the modified Dirichlet kernels and their time lags to construct these very interesting parametrizations. Of course there is much more that binds those spaces together; the matrix representations are all very similar, just like the eigen values and eigen vectors. But above 2D it is never a field. And again why the professional math professors have this weird fixation on fields is completely unknown to me.
At last, here is that wonderful video that will make your toes curl

End of this post and thanks for your attention!

Teaser picture for the next post.

After a lot of rainy days it was perfect weather today for the time of the year. It has been 3 weeks already since the last post and it is not that I have been doing nothing but the next post still isn’t finished. I told you that we would be looking at a parametrization that solves all 5 equations from the last post. So let me give you the parametrization in the teaser picture below. I also included the parametrization based on the modified Dirichlet kernels, by all standards the discovery of those modified kernels was one of the biggest discoveries in my study of higher dimensional number systems. To be precise: I found the first modified Dirichlet kernel years ago when I studied the 5D complex space.

In the last post I may have sounded a bit emotional but that is not the case. I am more or less one 100% through with the behavior of the so called math professors. They are incompetent to the bone and although that is not an emotional thing, it is that coward behavior that I do not like in those people. No, if it is highly overpaid, utterly incompetent and on top of that day in day out a coward, better show them the middle finger.

After having said that (I wasn’t expecting an invitation anyway) let’s look at the teaser picture because it is amazing stuff. I remember when I wrote down the parametrization for the very first time. At the time I did not know if the cosine thing would work because say for yourself: if you have a periodic function and you make two time lags of it, how likely is it they will form a flat circle in 3D space? But the cosine together with the two time lags does the trick because it is not hard to prove the parametrization lies in the plane with x + y + z = 1.

Ok, here is the cute parametrization for the 3D exponential circle:

The cosine & the modified Dirichlet kernel parametrizations

I think next week everything is ready so likely I can finally upload the next post. So thanks for your attention and till updates.

The sphere-cone equation in a matrix notation.

It is about time for a new post on 3D numbers, circular and complex. In this post I write the sphere-cone equation in a matrix notation so see the previous post on conjugates if you feel confused. The sphere-cone equation gives us two equations, as the name suggests these are a sphere and a cone and on the intersection we find the famous exponential circle.

Beside the sphere-cone equation I also demand that the determinant equals 1, now we have three equations and every intersection of those 3 equations has as it’s solution the exponential circle. Can it become more crazy? Yes because it is possible to factorize the third degree determinant into a linear and a quadratic factor. Those factors must also be 1 and now we have five equations! And since you can pick 10 pairs out of five, we now have 10 ways of solving for the intersection where the exponential circle lives…

It is strange that after all these years it is still easy to find 10 video’s where so called ‘professional math professors’ sing their praise upon the exponential circle in the complex plane. They really go beserk over the fact that e to the power it gives the cosine and sine thing. And after all those years still silent, yeah yeah those hero’s really deserve the title of honorable shithole… It is honorable because they often have relatively large salaries and they are shitholes because of their brave behavior when it comes to 3D complex numbers. Bah, I am getting a bad taste in my mouth when I think about the behavior of professional math professors. Let me stop writing about that low form of life.

This post is 8 pictures long. May be, I have not decided yet, is the next post about parametrizations of the exponential circle. In these 8 pictures I work out the case for the circular multiplication, that is the case where the imaginary unit j behaves like j^3 = 1. At the end I only give the 3D complex version of the matrix form of the sphere-cone equation and the rest you are supposed to do yourself.

Ok, again do not confuse this with quadratic forms. A matrix equation as written above has a real and two imaginary components while quadratic forms are often just real valued.

Let┬┤s try to upload this stuff. See you in the next post.

But are these quadratic forms?

This is a lazy easy going summer post, it does not have much mathematical depth. Let’s say the depth of a bird bath. But with most posts I write you also need a lot of knowledge about what was in previous posts and for the average person coming along that is often too time consuming… So we keep it simple today; quadratic forms on 3D space.

If you have had one or two courses of linear algebra you likely have encountered quadratic forms. They are often denoted as Q(X) where the X is a column matrix and the quadratic form is defined as Q(X) = XT A X. Here XT is the transponent of X so that would be a matrix row. As you might guess, the X column matrix contains the variables while the constant square matrix A is the source of coefficients in the quadratic form Q(X).  In most literature it is told the matrix A is symmetric, of course there is no reason at all for that; any square matrix will do. On the other hand it is easy to see or to show that if a square matrix is anti-symmetric the corresponding quadratic form will always be zero everywhere.

In this post we will take matrices that are always the matrix representation of 3D complex & circular numbers. Matrix representations are a complete category on this website so if you don’t know them you must look that up first. (Oh oh, here I go again: this was supposed to be easy but now the average reader must first try to understand matrix representations of higher dimensional multiplications…)

Compared to the previous update on the likely failure of all fusion reactors this post is far less dramatic. If in the future I am right and we will never have fusion power, that will be the difference between life and death of hundreds of millions of people in the long run… So in order to be a bit less depressing let’s lift the spirits by a lightweight new post on quadratic forms! Why not enjoy life as long as it lasts?

Ok, the actual post is seven pictures long, all in the usual size of 550×775 pixels.













As you see the math is only bird bath deep.

I have to admit that for me the use of the number alpha was important because that is at the center of the exponential circles in the 3D complex and circular spaces. So I have a legitimate reason to post this also under the category ┬┤exponential circle┬┤. And from the non-bird bath deep math, that is the big math ocean that is very deep, I like to classify as much posts under that category ┬┤exponential circles┬┤.

Ok, let┬┤s leave it with that and try to upload this post. Till updates my dear reader.

New roots of unity (the 3D complex ones) & The rain theorem.

I just finished brewing the 100-th batch of a beer under the names Dark Matter and Spin 1/2 beer. All in all that is an amazing amount of beer; in the past I brewed 35 liter per batch but now it is 40 liter per batch so all in all an amount of something between 3500 and 4000 liters… So ein prosit my dear reader.

The ‘new’ roots of unity aren’t that new, this post is a re-editing of something I posted on 05 Jan 2014 on the other website. Later that year I started this website. Actually these roots of unity are just over five years old. In mathematical terms that is still very young so in that sense they are still new.

Recall the roots of unity in the complex plane are solutions to z^n = 1 and as such these roots are found on the exponential circle (the complex exponential) in the complex plane. As you have found the exponential circle or exponential curve in some space, from that you can always make new roots of unity. That is hardly a mathematical achievement because it is so simple to do once you have found your exponential circle or curve. But in the diverse spaces these new roots of unity behave very different, for example in this post we will add them all up but unlike in the complex plane they do not add up to zero. That is caused by the fact that in the complex 3D space the number alpha is at the center of the exponential circle, as such if we add n roots of unity in 3D space the result is n times alpha. Last year we studied the space of 4D complex numbers and if you would craft new roots of unity in that space it will behave much more like those in the complex plane because in the 4D complex numbers we have 0 as the center of the exponential curve. (For dimensions above 3 the exponential curve always lies in a hyperplane so it can never be a circle.) It always amazes me that you have all those physics people who study string theory but as far as I know never use exponential curves…

Life is beautiful, because how can you do string theory without math like that? But in physics almost everything is beautiful, for example if they explain the outcome of the Stern-Gerlach experiment always 50% of unpaired electrons align with the applied magnetic field and the other 50% for some mysterious reason do the anti-align thing. And if one hour later the same physics professor explains how a permanent magnet can attract some piece of iron, all of a sudden 100% of the unpaired electrons align and all that talk of 50/50 suddenly is not observed… Life inside the science of physics is wonderful; all you have to do is a bit of blah blah blah and if people complain this is not logical at all you simply say: Quantum physics is such that if you think you understand it, you don’t understand it… How wonderful is the life of physics professors; talk some blah blah blah and if people complain you blame them for ‘not understanding quantum things’. For sure that is a beautiful form of life.

But enough of the talking, somewhere in the next seven pictures I did forget to insert a graph of the determinant. Yet I showed you the structure of the non-invertible numbers so often, I think I post it with that fault included. After all why should life be perfect? If life would be perfect you would have no way of improvement and likely that is the moment you die: no more possibility of improvement. As usual the pictures are 550×775 pixels but I had to make the first one a tiny bit longer. Good luck with digesting it & have a bit of fun in the process.

End of the pictures.

For myself speaking it was fun to read my own two proof for the rain theorem again after five years. Please do not forget that new roots of unity on other spaces can be very different in behavior, after all they are always part of the exponential circle or curve in that space so they will derive their math properties from that. Till updates.

Part 19: Four integrals defining the 4D complex number tau.

It is a bit late but a happy new year anyway! In this post we will do a classic from the complex plane: calculation of the log of the first imaginary unit.

On the complex plane this is log i and on the complex 4D space this is log l .

Because this number is so important I have given it a separate name a long long time ago: These are the numbers tau in the diverse dimensions. In the complex plane it has no special name and it simply is i times pi/2.

On the real line it is pretty standard to define the log functions as the integral of the inverse 1/x. After all the derivative of log x on the real line is 1/x and as such you simply define the log to be the integral of the derivative…

On the complex plane you can do the same but depending of how your path goes around zero you can get different answers. Also in the complex plane (and other higher dimensional number systems) the log is ‘multi valued’. That is a reflection of the fact we can find exponential periodic functions also known as the exponential circles and curves.

The integrals in this part number 19 on the basics of 4D complex numbers are very hard to crack. I know of no way to find primitives and to crack them that way. May be that is possible, may be it is not, I just do not know. But because I developed the method of matrix diagonals for finding expressions for the value of those difficult looking integrals, more or less in an implicit manner we give the right valuations to those four integrals.

With the word ‘implicit’ I simply mean we skip the whole thing of caculating the number tau via matrix diagonalization. We only calculate what those integrals actually are in terms of a half circle with coordinates cos t and sin t.

This post is 8 pictures long in the usual size of 550 by 775 pixels (I had to enlarge the latest picture a little bit). I hope it is not loaded with typo’s any more and you have a more or less clean mathematical experience:

End of this post.

Just a short video on the Fourier stuff.

This is the shortest post ever written on this website.

I found one of those video’s where the Fourier series is explained as the summation of a bunch of circles. Likely when you visit a website like this one, you already know how to craft a Fourier series of some real valued function on a finite domain.

You can enjoy a perfect visualization of that in the video below:

Only one small screen shot from the video:

Oh oh, the word count counter says 80+ words. Let me stop typing silly words because that would destroy my goal of the ‘shortest post ever’. Till updates.

Part 14: The Cauchy integral representation for the 4D complex numbers.

It took me longer than expected to craft this update. That is also the nature of the subject; you can view and do math with Cauchy integral representation in many ways. In the end I settled on doing it just for polynomials of finite degree and even more simple: these polynomials are real valued on the real line. (So they have only real coefficients and after that are extended to the space of 4D complex numbers).

In another development, last week we had the yearly circus of Noble prizes and definitely the most cute thing ever was those evolving protein molecules. Because if you can use stuff like the e-coli bacteria you can indeed try if you can (forcefully) evolve the proteins they make… That was like WOW. Later I observed an interview with that chemistry Nobel prize winner and she stated that when she began she was told ‘gentlemen don’t do this kind of thing’.

So she neglected that ‘gentlemen stuff’ and just went on with it. That is a wise thing because if you only do what all those middle age men tell you to do you will find yourself in the very same hole as they are in…
The physics prize was also interesting, for myself speaking I was glad we did not observe those physics men totally not understanding electron spin but with the usual flair of total arrogance keep on talking about spin up and spin down.

You can also turn that spin nonsense upside down: If elementary particles only carry monopole electrical charge than why should electrons be bipolar when it comes to magnetism? That Gauss law of magnetism is only a thing for macroscopic things, there is no experimental proof it holds for quantum particles…

But let’s talk math because this update is not about what I think of electron spin. This is the second Cauchy integral representation I crafted in my life. Now the last years I produced a whole lot of math, my main file is now about 600 pages long. But only that very first Cauchy integral representation is something that I printed out on a beautiful glossy paper of size A0. That first Cauchy integral representation was on the space of 3D numbers and there life is hard: The number tau has determinant zero and as such it is not invertible. But I was able to complexify the 3D circular numbers and it was stunning to understand the number tau in that complexification of the 3D circular numbers. Just stunning…

Therefore I took so much time in trying to find an easy class of functions on the space of 4D complex numbers. I settled for easy to understand polynomials, after all any polynomial gives the same value everywhere if you write them as a Taylor series.
Since this property of polynomials is widely spread I can safely say this in this part 14 of the basics to the 4D complex numbers we have the next Theorem:
THEOREM: The math will do the talking.
PROOF: Just read the next 12 pictures. QED.

As usual all pictures are 550 x 775 pixels in size. I also use a thing I name ‘the heart of the Cauchy integral’, that is not a widely known thing so take your time so that the mathematical parts of your brain can digest it…

I truly hope the math in this update was shallow enough so you can use it in your own path of the math that you like to explore.

End of this post, may be in Part 15 we will finally do a bit more about the diagonalization of 4D complex numbers because that is also a universal way of finding those numbers tau in the different dimensions like the 17D circular numbers & all those other spaces.

Have a nice life or try to get one.