A lot of math professionals rather likely still think that 3D complex numbers do not exist, may be for reasons like there are non-invertible numbers or whatever what other reason they have. This post more or less proved such views are nonsense; for example a lot of math on the 2D complex plane does not rely on the fact it is a field (and as such only division by zero is forbidden).
But on the 3D complex and circular number spaces indeed it brings some complications if you have non-invertible numbers in the function you want to integrate over a particular curve. And I have to say that problem could be solved by using the special properties that those numbers have. In this post I only show some examples with the non-invertible number alpha (alpha is the midpoint of the 3D exponential circles and all multiples of alpha are also non-invertible so the line through 0 and alpha are all not invertible).
For me writing this was a good distraction away from all that negative news we have day in day, all those countries reporting daily death toll can make you a bit depressed… So when I am through with the daily news I always do some other stuff like calculating a few of such integrals. That is a very good antidote against all that bad news. After all there is not much gained if you constantly think about things you cannot change at all.
This post is relatively long; at first I crafted 12 pictures but it soon turned out that was not enough. So while filling the 12 pictures with the math and the text I expanded some of the pictures so they could contain more math & text. That was not enough and in the end I had to craft two more background pictures. All in all it is 14 pictures long, that is a record length for this website.
If in your own mathematical life you have performed contour integration in the complex plane, you must be able to understand how this works in the 3D spaces. And for those who have done the thing known as u-substitution on the real line: it is just like that but now this u thing is the parametrization of a path. All that stuff below with gamma in it is either the path or the parametrization of that path. Please remark that you must use the complex or the circular multiplication on 3D, just like integrating over a contour in the complex plane uses the 2D complex multiplication.
In case if you are not familiar with the number alpha that is found at the center of the exponential circle, use the search function of this website and for example look up ‘seven properties of the number alpha’.
I hope I have removed all faults, typo’s etc so that later I do not have to repair the math because that is always cumbersome. Here we go: 14 pictures long so this is hard to grasp in detail in just a few hours. But it is beautiful math & that is why I do this. For me math is a lovely hobby.
Enough of the blah blah blah, here we go:
Ok, let´s first hit the button ´Publish´ and see what will happen… It looks all right but a day after first publication I realized there was some missing text. Example two was not ‘cut & pasted’ in the above pictures. So that was my fault, sorry if it made you confused. Thus we need one more picture containing example 2:
Later I will flea through the rest of the text, if needed I will post more addenda. For the time being that was it so till addendums or till the next post.
Ok, the math text is finally written. It took a long time but all in all I am very satisfied with the result. It will be a long post, I estimate about 12 pictures long and that is more or less a record length on this website. I have finished only two pictures and I will take my time to make the other ones because my mouse does not work properly. When I click with the computer mouse, very often that acts as a double click and that makes making pictures a laborsome task because of all the errors that double clicking gives. And when I have to repeat a series of clicks three or four times before it is ok, it will take some time. May be I should buy a new mouse,
Anyway to make a long story short: For years I stayed more or less away from crafting math about integration because it is hard to find a definition that would work always. My favorite way of using Riemann sums could not work always because of the existence of non-invertible numbers in the 3D spaces. And that gave some mathematical fear in my small human mind because path independence came with that way of Riemann summation. All in all it is beautiful math to think about: For example if in 3D you use a primitive to integrate over a closed loop, is it always zero?
So only the first two pictures are posted and I have no idea when all other pictures are finished. Here we go:
It is about time for a small update! Despite all that COVID-19 stuff going round, for myself after all those years I finally tried to put integration on higher dimensional number systems on a more solid footing. All those years I just refrained from it because you cannot use my favorite Riemann sum approach because of the non-invertible numbers we have in 3D or even higher dimensions. But now I am trying to finally make some progress and stop avoiding this subject, I find it is utterly beautiful. It has an amazing array of subtle details involved when you have some non-invertible numbers in your integration stuff. I have no idea when I have finished this rather important detail in my cute theory of higher dimensional complex & circular numbers, so let time be time & in the meantime only post a teaser picture about that lovely integration stuff. In the first lines you see a very familiar integral, likely you have done such calculations in the complex plane. In the case of 3D circular and complex numbers you must (of course) use the multiplication on 3D space to make it all work. Basically you are evaluating (or calculating) three integrals at the same time, just like on the complex plane where you are evaluating two integrals at the same time in your calculation. If you work with a pencil & paper, make sure you have enough paper because all those 3 integrals also have 3 terms in it so your calculations can become quite long… Here we go:
Ok, let me end this update now. Till updates and for some strange reason you must wash your hands while the proper authorities never point to 3D complex numbers… Till updates.
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 … 😉
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.
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… 😉
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
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:
I think next week everything is ready so likely I can finally upload the next post. So thanks for your attention and till updates.
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.
This is another lightweight easy going summer update. It is about matrix representations and how to find the conjugate of a 3D complex or circular number. I use the case of the complex plane of 2D conplex numbers to show that conjugation is not some silly reflection just always but rather simple will always be the upper row of a proper matrix representation. As a matter of fact it is so easy to understand that even the biggest idiots on this planet could understand it if they wanted. Of course math professors don’t want to understand 3D numbers so also this new school year nothing will happen on that front…
Did you know that math professors study the periodic system? Yes they do, anyway in my home country the Netherlands they do because every year they get a pay rise and that pay rise is called a periodic. And as such they study the periodic system deep and hard…
I classified this post only under the categories ‘3D complex numbers’ and ‘matrix representations’ and left all stuff related to exponential circles out. Yet the exponential circle stuff is interesting; after reading this post try to find out if the numbers alpha (the midpoint of the exponential circles) are symmetrix (yes). And the two numbers tau (the log of the first imaginary unit on the circular and complex 3D space) are anti-symmetrix (yes).
This post is just over 7 pictures long. As the background picture I used the one I crafted for the general theorem of Pythagoras. (Never read that one? Use the search funtion for this website please!) All pictures are of the usual size namely 550×775 pixels.
Ok, that was it for this update. Although it is so very simple (for years I did not want to write of just two simple planes that contain all the self-conjugate numbers) but why make it always so difficult? Come on it is summer time and in the summer almost all things are more important than math. For example goalkeeper cat is far more important compared to those stupid 3D numbers. So finally I repost a video about a cat and that makes me very similar to about 3 billion other people.
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.
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.