A detailed sketch of the full theorem of Pythagoras (that matrix version).Part 1 of 2.

Matrix representation, Pythagoras stuff

For me it was a strange experience to construct a square matrix and if you take the determinant of the thing the lower dimensional volume of some other thing comes out. In this post I will calculate the length of a 3D vector using the determinant of a 3×3 matrix.
Now why was this a strange experience? Well from day one in the lectures on linear algebra you are thought that the determinant of say a 3×3 matrix always returns the 3D volume of the parallelepiped that is spanned by the three columns of that matrix.
But it is all relative easy to understand: Suppose I have some vector A in three dimensional space. I put this vector in the first column of a 3×3 matrix. After I add two more columns that are both perpendicular to each other and to A. After that I normalize the two added columns to one. And if I now take the determinant you get the length of the first column.
That calculation is actually an example below in the pictures.

Well you can argue that this is a horrible complicated way to calculate the length of a vector but it works with all parallelepiped in all dimensions. You can always make a non-square matrix, say an nxd matrix with n rows and d columns square. Such a nxd matrix can always be viewed as some parallelepiped if it doesn’t have too many columns. So d must never exceed n because that is not a parallelepiped thing.

Orthogonal matrices. In linear algebra the name orthogonal matrix is a bit strange. It is more then the columns being orthogonal to each other; the columns must also be normalized. Ok ok there are reasons that inside linear algebra it is named orthogonal because if you name it an orthonormal matrix it now more clear that norms must be one but then it is more vague that the columns are perpendicular. So an orthogonalnormalized matrix would be a better name but in itself that is a very strange word and people might thing weird things about math people.
Anyway loose from the historical development of linear algebra, I would like to introduce the concept of perpendicular matrices where the columns are not normalized but perpendicular to each other. In that post we will always have some non-square matrix A and we add perpendicular columns until we have a square matrix.

Another thing I would like to remark is that I always prefer to give some good examples and try not to be too technical. So I give a detailed example of a five dimensional vector and how to make a 5×5 matrix from that who’s determinant is the length of our starting 5D vector.
I hope that is much more readable compared to some highly technical writing that is hard to read in the first place and the key idea’s are hard to find because it is all so hardcore.

This small series of posts on the Pythagoras stuff was motivated by a pdf from Charles Frohman, you can find downloads in previous posts, and he proves what he names the ‘full’ theorem of Pythagoras via calculating the determinant of A^tA (here A^t represents the transpose of A) in terms of a bunch of minors of A.
A disadvantage of my method is that it is not that transparant as why we end up with that bunch of minors of A. On the other hand the adding of perpendicular columns is just so cute from the mathematical point of view that it is good to compose this post about it.

The post is eight pictures long so after a few days of thinking you can start to understand why this expansion of columns is say ‘mathematical beautiful’ where of course I will not define what ‘math beauty’ is because beauty is not a mathmatical thing. Here we go:

With ‘outer product’ I mean the 3D cross product.

Inside linear algebra you could also name this the theorem of the marching minors. But hey it is time to split and see you in the next post.

A visualization of the so called ‘full’ theorem of Pythagoras + a worked example in 4D space.

Pythagoras stuff

A few posts back I showed you that pdf written by Charles Frohman where he shows a bit of the diverse variants of the more general theorem of Pythagoras there is. At school you mostly learn only about the theorem for a triangle or a line segment and it never goes to anything else. But there is so much more, in the second half of this post I show you three vectors in 4D space that span a parallelepiped that is three dimensional. From the volume of such a thing you can also craft some form of Pythagorean theorem; that paralellepiped can be projected in four different ways and the squares of the four volumes you get equals the square of the original parallelepiped.
I would like to remark I hate that word ‘paralellepiped’, if you like me often work without any spell correction this is always a horrible word…;)

Now my son came just walking by, he reads the title of my post and he remarks: It sounds so negative or sarcastic this ‘full theorem’. And no no no I absolutely do not mean this in any form of negative way. On the contrary I reconmend you should at least download Charles his pdf because after all, it was better compared to what I wrote on the Pythagoras subject about 10 years ago.

But back to this post: What Charles names the full theorem of Pythagoras is likely that difficult looking matrix expression and from the outside it looks like you are in a complicated space if you want to prove that thing. The key observation is that all those minor matrices are actually projections of that n x k matrix you started with. So that is what the first part of this post is about.

The second part is about a weird thing I have heard more than once during my long lost student years: Outside the outer product of two vectors we have nothing in say 4D space that gives a vector perpendicular to what you started with. I always assumed they were joking because it is rather logical that if you want a unique one-dimensional normal vector in say 4D space, you must have a 3D dimensional thing to start with. That is what we will do in the second part: Given a triple of vectors (ABC) in four dimensional space, we construct a normal vector M to it. With this normal vector, after normalization of course, it gives now a handy way to calculate the volume of any of those paralellepiped things that hang out there.

Ok, lets go: Six pictures long but easily readable I hope.

All that is left is trying to find back that link to Charles his pdf.

Charles_Frohman_Full_Pythagorean_TheoremDownload

That was it for this post. I hope you liked it, I surely liked the way you can calculate those paralellapipidedted kind of things. Thx for your attention and see you in the next post.

A simple theorem on the zero’s of polynomials on the space of 3D complex numbers.

3D complex numbers

In this post we look in detail at a very simple yet important polynomial namely

p(X) = X (X – 1).

Why does it have four zero’s in the space of 3D complex numbers? Well if you solve for the zero’s of p so try to solve p(X) = 0, that is you are looking for all numbers such that X^2 = X.
These numbers are their own square, on the real line or on the complex plane there are only two numbers that are their own square namely 0 and 1.
On the space of 3D complex numbers we also have an exponential circle and the midpoint of that circle is the famous number alpha. It is a cakewalk to calculate that alpha is it’s own square just like (1 – alpha).

This post is four pictures long in the size 550×825 pixels so it is not such a long read this time. In case you are not familiar with this number alpha, use the search function on this website and search for the post “Seven properties of the number alpha”. Of course since it is math you will also need a few days time of thinking the stuff out, after all the human brain is not very good at mathematics…
Well have fun reading it.

The last crazy calculation shows that a polynomial in it’s factor representation is not unique. Those zero’s at zeta one and zeta two are clearly different from 0 and 1 but they give rise to the same polynomial.

At last I want to remark that unlike on the complex plane there is no clean cut way to tell how many zero’s a given polymial will have. On the complex plane it is standard knowledge that an n-degree polynomial always has n roots (although these roots can all have the same value). But on the complex 3D numbers it is more like the situation on the real line. On the real line the polynomial p(x) = x^2 + 1 has no solution just like it has no solution on the space of 3D complex numbers.
That was it for this post, thanks for your attention.

Is this the most simple proof for a more general version of the theorem of Pythagoras? The inner product proof.

Pythagoras stuff

Last week I started thinking a bit about that second example from the pdf of Charles Frohman; the part where he projected a parallelogram on the three coordinate planes. And he gave a short calculation or proof that the sum of squares of the three projected areas is the square of the area of the original object.

In my own pdf I did a similar calculation in three dimensional space but that was with a pyramid or a simplex if you want. You can view that as three projections too although at the time I just calculated the areas and direved that 3D version of Pythagoras.

Within the hour I had a proof that was so amazingly simple that at first I laid it away to wait for another day or for the box for old paper to be recycled. But later I realized you can do this simple proof in all dimensions so although utterly simple it has absolutely some value.
The biggest disadvantage of proving more general versions of the theorem of Pythagoras and say use things like a simplex is that it soon becomes rather technical. And that makes it hard to read, those math formula’s become long can complex and it becomes harder to write it out in a transparant manner. After all you need the technicalities of your math object (say a simplex or a parallelogram) in order to show something is true for that mathematical object or shape.

The very simple proof just skips that all: It works for all shapes as long as they are flat. So it does not matter if in three dimensional real space you do these projections for a triangle, a square, a circle, a circle with an elleptical hole in it and so on and so on. So to focus the mind you can think of 3D space with some plane in it and on that plane is some kind of shape with a finite two dimensional area. If you project that on the three coordinate planes, that is the xy-plane, the yz and xz-plane, it has that Pythagoras kind of relation between the four areas.

I only wrote down the 3D version but you can do this in all dimensions. The only thing you must take in account is that you make your projections along just one coordinate axis. So in the seven dimensional real space you will have 7 of these projections that each are 6 dimensional…

This post is four pictures long, I did not include a picture explaining what those angles alpha and theta are inside one rectangular triangle. Shame on me for being lazy. Have fun reading it.

So all in all we can conclude the next: You can have any shape with a finite area and as long as it is flat it fits in a plane. And if that plane gets projeted on the three coordinate planes, the projected shapes will always obey the three dimensional theorem of Pythagoras.

Ok, thanks for your attention and although this inner product kind of proof is utterly simple, it still has some cute value to it.

Two pdf’s on more general versions of the theorem of Pythagoras.

Pythagoras stuff

A few months back I found a very good text on more general versions of the good old Pythagorean theorem. Since in the beginning of this text the author Charles Frohman did the same easy to understand calculations as I did a long time ago I more or less trust the entire document. But I did not check the end with those exterior calculations, I don’t know why but I dislike stuff like the wedge product.
The second pdf is from myself, likely I wrote it in 2012 because a proof of a more general version of the theorem of Pythagoras was the first math text I wrote again after many years. After that at the end of 2012 I began my investigations into the three dimensional complex numbers again and as such this website was needed in 2015.

Anyway I selected 3 details from these two pdf’s that I consider beautiful math ideas where of course I skip a definition of what ‘beautiful’ is. After all the property ‘mathematically beautiful’ is not a mathematical object but more a feeling in your brain.

Let me start with four pictures where I look into those 3 selected details, after that I will hang the two pdf texts into this post.

Below follow a few screenshots from the pdf’s:

The first pdf is from Charles Frohman. May be you must download it first before you can read it, I should gain more experience with this because the pdf format is such a hyper modern development…;)
The first text is from 2010:

Charles_Frohman_Full_Pythagorean_TheoremDownload

At last my old text from 2012:

Reinko_Venema_Pythagoras_on_simplicesDownload

(Later I saw there were some old notes at the end of my old pdf, you can neglect that, it has nothing to do with the Pythagoras stuff.)

There is little use in comparing these texts, I only wanted to make a proof that uses natural induction so I could prove the theorem in all dimensions given the fact we have a proof (many proofs infact) for the theorem of Pythagoras with a rectangular triangle. Charles his text is more broader and the main piece is the proof for that determinant version of the theorem of Pythagoras.

At last a remark about the second detail of mathematical beauty: Charles gave the example of a parallellogram where the square of the area equals the sum of squares of the three projections on the three coordinate planes. I think you can take any shape, a square of a circle it does not matter. It only matters it is a flat thing in 3D space. After I found that within the hour I had a proof for the general setting of this problem in higher dimensional real space, may be this is for some future post.

For the time being let us split and go our own ways & thanks for the attention of reading this.

Two videos on electrons and the still missing magnetic monopole.

Magnetism

The first video is very simple, a bit on the high school plus level, but it is well made. The reason I post is that it has a very good explanation as why electrons are viewed as point particles. I had never heard of this explanation and it goes more or less like this:

If the electron had some kind of hard kernel, in that case if you shoot them fast enough into each other they will bounce differently.

This is based on the assumption that at low energies two colliding electrons will not touch each other. It seems that this kind of behaviour keeps on going on at high collision energies.
Another detail that is interesting are questions about the size of an electron. In this video a number like smaller then 10 to the minus 18 cm is mentioned. Since I think that if it is true that electrons are ‘tiny magnets’ so they are bipolar in the magnetic sense, they cannot be accelerated by magnetic fields in a significant manner.

I assume this is the diameter.

The physics professors think that a non constant magnetic field can accelerate an electron. Non constant can mean it varies over time, varies over space or both. If you apply a magnetic field to such a bipolar electrons, say if the north pole of the electron is repelled by that, the other side of the electron must feel an attractive force. The difference should account for the acceleration of the electron.
Lets do an easy calculation: Using the radius being this 10^-18 cm or 10^-20m, the density of an electron is about 2.2 times 10^29 kg per cubic meter of ‘electron stuff’. Suppose we have a ball shaped electron with a volume of one cubic meter thus it has a mass of 2.2 times 10^29 kg, it’s radius is about 60 cm.
So the diameter of our superlarge electron is about 120 cm and it has this rediculous huge mass. Do you think you can accelerate this thing with a magnetic field that has some nonzero gradient?

There are so many problems with the model of the electron being a magnetic dipole. Why should electrons ‘anti align’ themselves with an applied magnetic field? That is strange because they gain potential energy with that. That is just as strange and crazy as the next example:
You have a bunch of stones, one by one you grab them and hold them still in place. You let them loose. Some fall to the ground, the others fly up.
This never happens because nature has this tendency to lower the potential energy.
Another problem is that it is known that the electron pair is magnetically neutral. The ‘explanation’ is that the two electrons have opposite spin and ‘therefore’ cancel each other out. That is a stupid explanation because if it is true that the electron is a bipolar magnetic thing it should be magnetically neutral to begin with.

The second video is from Brian Keating, Brian is an experimental physics guy. This is one of those ‘Where are the magnetic monopoles’ videos that people who like to demenstrate they are dumb post on Youtube and the likes. It makes me wonder: What the hell are they doing with our taxpayer money? The concept of a magnetic monopole is just plain fucking stupid; it is a particle with no electric charge but only one of the two possible magnetic charges.
Why is this fucking stupid? Just look at the electron: if their fairy tales are true, the electron is an electric monopole and a magnetic dipole. If I would look to some dual version of an electron and have drunk lots of beer I would propose a particle that is an electric dipole but also a magnetic monopole.

You never hear those physics people talk about that, it is always that stupid talk of where are the magnetic monopoles or if there is just one magnetic monopole in every galaxy it is ok. What I consider the weirdest thing that if you advertise the electron as a magnetic dipole, should you not give a tiny bit of experimental validation for this? But no, Brian has no time for such considerations.

I should have included the text ‘This is fucking stupid’.

So where are all the magnetic monopoles? If my view on magnetism is correct they are in every electron pair that holds your body together.

May be in your body or your eyes?

Ok, lets leave this nonsense behind. Don’t forget people like this might be infuential but they are too stupid to understand only the smallest part of say three dimensional complex numbers.
End of this post.

Comparing the two sphere-cone equations.

3D complex numbers, Exponential circle, Matrix representation

This channel is of course not meant for political statements but this fucking war is a fucking distraction from doing math. While writing this post in small pieces I was constantly dissatisfied with the level of math (too simple, done too often in the past etc). But when I was finished and read it all over, all in all it was not bad. It is a short oversight of how to find shere-cone equations and once more how to find a conjugate.
And once more: The math professors are doing it wrong when it comes to finding the conjugate for 32 years now & the clock keeps on ticking. On the one hand this is remarkable because if you do internet searches a lot of people understand that the Jacobian matrix should be the matrix representation for the derivative of a complex valued function in say three dimensional space. So that goes good, but when it comes to taking the conjugate for some strage reason they all keep on doing it wrong wrong and wrong again so they will never find serious math when it comes to number systems outside the complex plane or for that matter the quaternions.

The setup of this post is as next:
1) Explaining (once more) how to find the conjugate.
2) Calculating the two sphere-cone equations.
3) The solution of these S-C equations is the exponential circle that is,
4) parametrisized by three so called coordinate functions that we
5) substitute into both S-C equations in order to get
6) just one equation.

Basically this says that the complex and circular multiplication on our beloved three dimensional space are ‘very similar’. Just like that old problem of solving X^2 = -1 is impossible in these spaces while the cubic problem of X^3 = -1 has only trivial solutions like basis vectors. That too is ‘very similar’ behaviour.

Anyway this post is six pictures long.

That was it for this post on my beloved three dimensional complex numbers.

Addendum to the previous post: The new de Moivre identity for the 3D circular numbers + 2 videos.

3D complex numbers, Exponential circle, Matrix representation

I know I know I have published stuff like this before and over again. But that was also years ago and now I do it again it is still not boring to me. After all the professional math professors still are not capable of finding those beautiful exponential circles and curves simply because they all imitate each other. And they imitate each other with how to use and find a so called conjugate. And if you use the conjugate only as some form of ‘flipping a number into the real axis’ all your calculation will turn into garbage. Anyway by sheer coincidence I came across two videos of math folks doing it all wrong. One of the videos is even about the 3D circular numbers although that guy names them triplex numbers.

You can do a lot with exponential circles and curves. A very basic thing is making new de Moivre identities. From a historical point of view these are important because the original de Moivre identity predates the first exponential circle from Euler by about 50 years. In that sense new de Moivre identities are very seldom so you might expect some interest of the professional math community…

Come on, give me a break, professional math professors do a lot of stuff but paying attention to new de Moivre identities is not among what they do. But that is well known so lets move on to the four pictures of our update. After that I will show you the two video’s.

Let us proceed with the two video’s. Below you see a picture from the first video that is about 3D circular numbers and of course the conjugate is done wrong because math folks can only do that detail wrong:

Below you can see the video:

By all standards the above video is very good. Ok the conjugate is not correct and may be the logarithm is handled very sloppy because a good log is also a way to craft exponential circles. But hey: after 30 years I have learned not to complain that much…

The next video is from Michael Penn. He has lots of videos out and if you watch them you might think there is nothing wrong with that guy. And yes most of the time there is nothing wrong with him until he starts doing all kinds of algebra’s and of course doing the conjugate thing wrong. Michael is doing only two dimensional albebra’s in the next video but if you deviate from the complex plane very soon you must use the conjugate as it is supposed to be: The upper row of the matrix representation.

Here a screen shot with the content of the crimes commited:

Most of his other video’s are better, but his knowledge is just a reflection of what professional math professors think about conjugation. It is always just a flip in the real axis.

Here is his vid:

Ok, that was it for this appendix to the previous post.

Once more: The sphere-cone equation.

3D complex numbers, Exponential circle, Matrix representation

It is past midnight, this evening I brewed hopefully a lovely beer. It is late so let me keep the intro short. The last time I often lack stuff for new posts because most of the theory of 3D complex and circular numbers has been posted in this collection of 200+ posts. And you cannot keep it repeating over and over again, if all those years in the past the math professionals did just nothing, why would they change their behaviour in the future? Beside that I do not want have anything to do with them any more, it is and stays a collection of overpaid weirdo’s and there is nothing that can change that.
On the other hand one of the most famous expressions in math is and stays the exponential circle in the complex plane.
That stuff like e^it = cos t + isin t is what makes many hearts beat a tiny bit faster. So when someone comes along stating that he found an exponential circle in spaces like 3D complex numbers, you might expect some kind of attention. But no, once more the math professionals prove they are not very professional. Whatever happens over there I do not know. May be they think because they could not find this in about 350 years no one can so it must all be faulty. For me it was a big disappointment to get discriminated so much, on the other hand it validates that math professors just are not scientists. Ok they have their salary, their social standing, their list of publications and so on and so on. But putting lickstick on a pig does not make it a shining beauty, it stays a pig. So a math professor can have his or her prized title of professor, that does not make such a person a scientist of course. At best they show some form of imitating how a scientist should behave but again does such behaviour make these people scientists?
Anyway a couple of days back at the end of a long day I typed in a search phrase in a website with the cute name duckduckgo.com. Sometimes I check if websites like that track this very website and I just searched for “3D complex numbers”. The first picture that emerged was indeed from this website and it was from the year 2017. I looked at it and yes deep in my brain it said I had seen it before but what was it about? Well it was the product of two coordinate functions of the exponential circle in 3D. It is a very cute graph, you can compare it to say the product of the sine and cosine function in the complex plane.
So I want to avoid repeating all that has been written in the past of this website but why not one more post about the 3D exponential circles?

In the end I decided to show you how likely one of those deeply incompetent “professional” math professors would handle the concept of conjugation. Of course one hundred % of these idiots and imbeciles would do it as “This is just a flip in the real axis or in the x-axis” and totally spoil the shere-cone equation and only find weird garbage that indeed better cannot be published. After all our overpaid idiots still haven’t found the 3D complex numbers, I am still living on my tax payer unemployment benefit and life, well life will go on. But it is not only math, with physics there are similar problems and they all boil down to that often an idiot does not realize he or she is an idiot.

But let’s post the six pictures, may I will add an addendum in a few days, may be not. Here we go:

Isn’t that a cute graph or not?

Ok, may be in will write one more appendix about how these kind of coordinate functions of exponential circles give rise to also new de Moivre identities. That is of interest because the original de Moivre identity predates the Euler exponential circle by about 50 years.

Yet once more: Likely there is just nothing that will wake up the branch of overpaid weirdo’s known as the math professors…
So for today & late at night that was it.
Thanks for your attention.

I found a long pdf about micro magnetism in nano tubes.

Magnetism

It is no secret that I think electrons are not “tiny magnets” having two magnetic poles but that electrons are magnetic monopoles just like they are electric monopoles. Viewing electrons as small tiny magnetis leads to all kinds of logical contradictions. For example a permanent magnet is always explained as a thing where all electron spins of unpaired electrons align and as such together they build that macroscopic magnetic field as you know from stuff like a bar magnet. But in chemistry an important binding element in molecules is the electron pair. Yet now there is something like the Pauli exclusion principle and the two electrons must have opposite spin. End example.
So in a permanent magnet the electrons must align in order to be attractive to each other while in chemistry the opposite must happen. My dear reader this is not logical. Also, why do we find only electron pairs? Well if you look at it as there are two kinds of electrons with both a magnetic charge either ‘north pole’ or a ‘ south pole’ charge, that explains why we only observe electron pairs. If the ‘tiny magnet’ model was true, we should observe all kinds of electron configurations like 5 electrons in a circle or whatever you can make with tiny magnets.
What I self consider a strange thing is that people from the physics community never ever themselves say that all their views on magnetism are often not logical. Are they really that stupid or do they self censor in order not to look stupid?

Anyway five years back in the year 2017 I was studying a new way of making computer memory by IBM: so called racetrack memory in nano wires. I was highly puzzled by that because one of the main researchers said that you cannot move the domain walls of magnetic domains with magnetic fields. You could move the domains themselves but not the walls and I was as puzzled as can be. Yet that same day I found a possible answer: the magnetic domains of say iron can be moved by magnetic fields because they have a surplus of a particular kind of electrons. So two magnetic domains separated by a domain wall must have opposite magnetic charges. In the next picture you get the idea of what IBM tried to do:

It was a cute idea but IBM had to give up on it because they did not use insights that are logical but kept on hanging to the tiny magnet model.

So in the long pdf that is squarely based on the official version of electron spin (the tiny magnet model) has all kinds of flaws in it. For example in the next picture that all does not pan out because those small arrows are not there in reality if electrons carry magnetic charge just like they carry electric charge:

And life, well life will go on…

Ok for me it is an experiment to try include a pdf file, if it fails I will hang this pdf in the pdf directory of the other website and link to that file.

Lets give it a try:

Magnetic_domain_thesis_STANO_2017_diffusionDownload

I leave it this way and do not try to make the pdf visible. After all if you are interested in stuff like this you must download it anyway because it is a few hundred pages long. And it is a funny read so now and then, for example yesterday I came across a section where they took the outer product of two (vector) electron spins and I just wonder WHY?

Ok, let me push the button named Publish and say salut to my readers.