It is cucumber time; I am lazy to the bone and just chilling out…

Often when I am out I try to do a bit of math while riding my noble iron horse known as that old bicycle. The disadvantage of doing math on your bike is that one the one hand you cannot go very towards complicated stuff where you need pencil and paper but on the other hand you can get deep by getting some good idea’s.

And only when you get home and you have access to pencil & paper you can check if the stuff can be written out and see how your idea’s survive in the battle for attention from your brain.

After the previous post about magnetism I was only thinking ‘Why not do some pure 3D complex number stuff again’? But the math well is a bit dry lately when it comes to 3D complex numbers. May be this has a bit to do with the total and utter silence from the so called ‘professional math people’ who excel in staying silent…

But a few times it crossed my mind to do that mind boggling factorization of the Laplacian once more; if I would make a top 10 or top 25 list of the most strange results found this factorization of the Laplacian would end very high.
Yet when I check my own website, all that has to be said was already said about one year ago; on 05 August 2016 I posted the next seven pictures long post upon the factorization of the Laplacian using so called Wirtinger derivatives.

It still is a good read I think:
Wirtinger derivatives and the factorization of the Laplacian.

Wirtinger derivatives and the factorization of the Laplacian.

So there was little use in writing that stuff out again when there is, as usual, never ever any signal from the ‘professionals’ who rather likely are busy spending their too large salaries on stuff they think is important…

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In another development I came across the latest video from the Mathologer, it is very interesting because he claims that the famous Euler identity is not from Euler at all.
But Mr. Mathologer comes up with what is one of the famous Euler stuff, anyway a long long time ago it was one of the details that made Euler famous was finding what the sum of squared reciprocals was: 1/1^2 + 1/2^2 + 1/3^2 + ….

Over 25 years back I did the same calculations as the Mathologer invites you to so let me share the video with you. At first it looks a bit difficult but all you need to do is think about how to write out those infinite products as sums and after that you apply the age old trick of equalling the left and right side of the equation.

Here is the vid:

Euler’s real identity NOT e to the i pi = -1

May be in a future post we will be diving a bit deeper into this because Mr. Mathologer has nice news upon who found what but he skips all that stuff like how to write the entire functions from the complex plane as (infinite) products.
Furthermore he does not explain as why the given infinite product would be valid anyway…

Ok, may be in a next post I will be diving a bit deeper in all those kinds of infinite products.
Or may be it will be something completely different, anyway till updates.

Update from 22 August 2017:

By sheer accident while I was only watching a video about why there is such a break between higher math and higher physics, I came across some weird stuff from a guy named Edward Witten.
And the talk was about so called Seiberg-Witten monopoles, so my interest was aroused because I cannot allow plagiarism of course.

Anyway it turns out that Mr. Witten and his Seiberg pal talk about massless monopoles without laughing. The concept of a massless monopole is so idiot that normal people with just a tiny bit of self respect would never talk about that.

Anyway to make some long story short, Mr. Witten is also Mr. String Theory. You know that kind of theory that is impossible to validate in physical experiments so it is the opposite of what I do because if electrons carry magnetic charge it could be found in more and more experiments…
But the Witten guy wrote about Dirac operators and once more my interest was aroused and I looked it up: Dirac operators are differential operators D and if you square them you get the Laplacian…..

Here is a short wiki about the stuff involved:

Dirac operator
https://en.wikipedia.org/wiki/Dirac_operator

Basically when you try to find operators D that square to the Laplacian it is more like ‘operator problem looking for a fitting math space’ while in my above factorization of the Laplacian it is a math space (3D complex and circular numbers) that want a factorization.
In the wiki you also observe in example number 4 that Clifford algebras are named a possible candidates, that is true but a few remarks are at their place.
That is the content of the next two small pictures:

Ok, this wasn’t how I more or less planned the next update but when idiots come along talking about massless monopoles beside having deep fun I also have the right to expose the names of the idiots in question…

Let’s leave it with that, till updates my dear reader.

More on the failure of IBM’s racetrack memory.

Just over one week ago I posted reason number 48 as why electrons cannot be magnetic dipoles over on the other website, it is about the failure of IBM in crafting a new kind of fast memory. They failed because they treat electron spin like it is a vector while it makes much more sense that electron spin is one of two possible magnetic charges.

Here is the post from the other website:

Reason 48: The failure of IBM’s racetrack memory.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff02.htm#24July2017

It took me relatively long to find where the stuff all went wrong, at first I spend over a week every evening trying to find some stuff on the preprint archive and although there are some explanations found over there, because the writers of those articles are professional physics people they do not understand electron spin.

Also they DO NOT WANT TO UNDERSTAND ELECTRON SPIN because if you view electron spin as a magnetic dipole you end up in a gigantic ocean of nonsense, for example in the science of chemistry very often the electron pair plays a major role in the binding of all kinds of molecules. But if electrons were magnetic dipoles there would be no reason at all to limit the number of electrons to two; you would get all kinds of weird constellations of triplet electrons or whatever you can make with dipole bar magnets…

But if electrons carry two different magnetic charges suddenly it makes a whole lot of sense that we only observe electron pairs; the magnetism is neutral in an electron pair while the repelling electrical charge ensures no larger configurations beyond the pair formation are found. With a magnetic dipole you just would not observe this kind of behaviour…

Now back to IBM’s racetrack memory: All the time I did not understand how the IBM research folks did write the electron spin domains on the racetrack memory; electrons behave very much like cats:
It is easy to chase a cat into a tree but very hard to convince the cat it should leave the tree and come down to earth again…

With electrons you have the same: the two magnetic charges on the electron have a slightly different energy level, when an electron falls from the highest energy level to the lower one we observe the famous 21 cm wavelength photon. It is a well known fact from astronomy that interstellar and intergalactic hydrogen atoms only very very seldom have their electron fall from the highest magnetic energy level to the lowest energy level. This is what I name the ‘combed up universe’; even in intergalactic space most electrons are in the highest energy state because there are plenty of photons flying around to keep them ‘combed up’ when it comes to energy levels…

Back in the year 2004 IBM patented so called racetrack memory; the goal was to leave the 2D structure we have in present day computer hardware and use nano wires to go 3D and as such exploit three dimensional architecture of future computer hardware. The racetrack memory is made from nano wires, those nano wires contain lots of magnetic domains but contrary to the magnetic domains you find in, for example, iron these domains contain only one spin state.

According to IBM researchers all spin states are in the direction of the nano wire (from that you can understand they think electron spin is a vector, the vector represents the magnetic bipolar nature of the electrons according to IBM researchers).

In the next picture you see a boatload of information; the red and blue colour represents of course the two magnetic spin states of the electron. As you see on inspection they can inject blue electrons from the left and red electrons from the right.
If the IBM researchers inject red or blue electrons they can shift the entire column of electrons in the nano wire, according to IBM fellow Stuart Parkin the borders between the magnetic domains get transported…

Of course back in the year 2004 IBM thought they had hit the jackpot because if you neatly follow the standard model of physics where electrons are always having two magnetic poles you will always have that such borders are North pole against North pole (or South against South pole) and as such these borders should be extremely fragile…

And IBM thought they could transport those fragile things at high speeds, if true they would earn not billions but trillions over the long run of a patent.

Yet in my theory of magnetism, if it is true there are two magnetic charges the borders between red and blue magnetic domains are the most strong structures into the nano wire anyway so it is logical they keep intact while the electron column is transported…

Here you see why it is important to keep an open mind on electrons spin because if you follow the standard model companies like IBM cannot make technical progress.

For myself speaking I did not understand how to write information to the nano wire; I was thinking they did it with electro-magnetic radiation because any photon with a wavelength below 21 cm could bring an electron from the lowest energy state to the highest energy state…
But how to go from high energy to low energy is like talking to a cat high in a tree…

And no matter how much articles I did read on the preprint archive, nowhere an answer was to be found…
In the next picture you see how IBM visualizes how a small red region from the nano wire turns into blue: IT IS THAT WRITING WIRE BELOW WITH RED ELECTRONS IN IT!!!!!

Picture source:

Now we have two clashing versions on electron spin:

  1. The standard model version where electrons are magnetic dipoles says: The red writing wire cannot change the red domain in the nano wire because all red electron spins point into the same direction. And if you add more bar magnets perfectly aligned that only makes the red state stronger. Versus:
  2. If electrons carry one of two magnetic charges and we use the principle that like charges repel, the red electrons on the writing wire repel the red electrons in the nano wire into the blue neighbouring blue domains. At the same time blue electrons will flow to the red region.

So if my view on electron spin is true, in that case the simple act of writing information to such nano wires destroys the information in the surrounding magnetic domains.

And that my dear reader is something that the professional physics people still do not want to acknowledge until this present day of 03 August 2017.

By the way, next winter it is about the fourth or even the fifth year I am explaining as why electrons cannot be magnetic dipoles. Those people, the so called ‘professionals’ will keep on hanging to their silly beliefs around electron spin for a much longer time.

Let’s leave it with that.

 

Destroying Internet Security Part Two.

Another misleading title, but it is fun to write it down so why not?

In this post (8 pictures long) we have two parts:
Part 1: The relation between the modulo row’s and the modular arithmetic groups Z/jZ.
Part 2: A proposal (or schematic outline) of an important part of the algorithm that brings you from one stratum to the other.

I think this is my last post on this subject of modulo row’s.

Lately websites using RSA encryption methods (that is why we look at large prime numbers made of two factors N = p*q) have gone from a 1048 bit long key to 2096 binary digits long keys. The idea is that it makes life just so much more safe; but the important part of the algorithm for transport over the strata is remarkable resilient towards such moves…
Furthermore, doubling the length of the encryption keys (squaring the size so to say) will in general also increase the size of the Jente basin as found just before the largest prime factor q.

I do not claim to know a lot about encryption, but as far as I know there is zero point zero use of idea’s like the Jente basin. People use a lot of so called ‘trial division’ but even that is not a real division but mostly just taking N modulo something.
For example; want to know if the number 73 is a factor of N?
They simply calculate N mod 73 and if the outcome is 0 they say that 73 is a factor of N, otherwise it is not. The use of idea’s like a Jente basin so you can scrap a lot of trial numbers in the region you are in is, as far as I know, not used.

To be honest, I also do not know how they factor large hundreds of decimal digits long numbers anyway; so it might very well be they use similar idea’s. But if that were the case why is everybody else only talking about taking a huge number of trial divisions without any strategy behind that?

The numbering of the pictures is a continuation of the previous post.
Here are the 8 pictures, have fun reading it.

At this point in time the so called quantum computers are going from the lab to the field, anyway a lot of people claim this. But since after my humble opinion electrons have one electric charge and one of two possible magnetic charges, it will be a long long time before we have a working quantum computer based on electron spin.
Just like IBM with their racetrack technology for 3D memory; the idea is ok but at IBM too they think electron spin a like a vector and not like a charge. And voila; year in year out you never hear from it again…

Ok, for the time being this is what I had to say. See you in the next post.

Let’s Destroy Internet Security!!!

Ok ok, I admit instantly that the title of this post is way over the top but for once I allow myself a catchy title that has only limited resemblance to what this post is about. In this post, if I write the word computer I always mean a classical computer so not a quantum version of it.

In the previous post there is a video in from the ‘Infinite Series’ that serves as an introduction to the Shor algorithm; if this algorithm could be implemented into a quantum computer that would likely break internet security for a short while. Beside the fact that large prime numbers are used in standard classical encryption, it can also be done with elliptic curves.

This post is about the principle of Jente, with a bit of luck you can find factors of large numbers using the principle of Jente. Counter intuitively the largest (prime) factor will be the easiest to find.
Now how did Jente find the principle of Jente?
Back in the time, end 1997 or begin 1998, we lived in a house without a garden and since I still smoked a lot of tobacco I always had a window open in my working room. Since this work room was next to the entry of the house, very often when the door to my room opened papers would fly from my desk because of wind going through the room.

There was this cute baby crawling around and one day she brought me back a piece of paper that had flown off my desk. And on that piece of paper was a little cute formula that read
m_{j+1}  = m_j – d_j. So that is how this got the name the principle of Jente.

Lately Jente turned 21 years of age, she now lives temporary in Australia, and I decided to write this old stuff down as a kind of present for her. The principle of Jente is extremely easy to understand, but as far as I know mathematical reality this principle has not been exhausted very much by the entire math community over centuries of time.

What is missing in this post is a way to converge fast with high speed to one of the factors of one of those huge composite numbers the software engineers use for internet security. My gut feeling says that it should not be that hard but until now I have never found it. It might very well be that inside things like Diophante equations somewhere the solution to this problem of fast finding the largest prime factor is solved without the person who has done that being aware of it…

I tried to keep this post as short as possible so I scrapped a whole lot of stuff but it is still 15 pictures long (picture size as usual 550 x 775 pixels). A feature that I like very much is that I am using so called Harry Potter beans in order to explain as why the Jente principle works. I feel a bit proud on that because it is so simple you could explain that to elementary pupils in their highest years.

For myself speaking I also like this approach to finding prime factors because it is so different from all other ways, yet it has that underlying undeniable thing in it named the Jente principle. The most important detail in this post is the table with the diagonals in it.
If you understand that table and, for example, you can find another algorithm for quantum computers that solves that problem, you have found an alternative to the Shor algorithm…

Have fun reading it, take your time because it is not meant to be grasped in five minutes or so.

 

I hope you understand the fundamental problem still open after almost two decades:

You start with some number j, calculate m_j = N mod j and d_j = N div j.
Having these, the Jente principle guarantees you can find (j + k) mod N for all k > 0.

But, how oh how, do you converge towards a solution of
m_{j+k} = 0 mod (j+k) ?????

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The Shor algorithm: In the world of quantum computing we have the theoretical side where people just write down all kinds of elaborate scheme’s like the Shor algorithm and just as easy they throw in a lot of Hadamard gates that supposedly will bring a giant bunch of quantum bits into super position.

On the other hand you have people that actually try to build quantum computers.

As far as I know stuff, there is no way of bringing a lot of qbits into a nice super position or, for that matter, entangle them into a good initialization state in order to run your quantum software.

More info:

Hadamard transform
https://en.wikipedia.org/wiki/Hadamard_transform

Shor’s famous algorithm: Shor’s algorithm
https://en.wikipedia.org/wiki/Shor%27s_algorithm

Elliptic Curve Cryptography: a gentle introduction
http://andrea.corbellini.name/2015/05/17/elliptic-curve-cryptography-a-gentle-introduction/

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Ok, that was it. Don’f forget to pop open a few beers. Don’t believe all that nonsense that doctors are telling you like drinking less = good.
As far as I know reality, all people in my social environment that drink far too little beer always get killed in extremely violent events… 😉

Till updates.

Destroying Internet Security using the Jente principle, a teaser introduction.

A few months back suddenly there was a new video channel about math and it goes under the cute name Infinite Series. About two months back the channel posted a way to destroy internet security if you could only find that factorization of two giant prime numbers.

Most of present internet security hangs around the difficulty of observing a giant number N of, let’s say, one hundred digits and our incapability to factorize large numbers like that into their prime factor numbers.

Of course, since the Infinity Channel is USA based, it is completely impossible that fresh math will come from that space. Here is the video and indeed only ancient math is around:

How to Break Cryptography | Infinite Series

The idea’s as expressed in the video are very interesting, but is just does not use the Jente principle that ensures you can find weakness in the integers surrounding the prime numbers that make up the factorization of the stuff you want to encrypt.

In the next two pictures you see that a prime number is extremely weak in avoiding detection using the Jente principle if you are close enough to that prime number.

And if a prime number is detected, in principle you could break down the security of the communication channel.

 

Let’s leave it with that, after all talking about a basin around a prime number that shouts out ‘the prime number is here’ is one hundred percent outlandish to those overpaid USA math professors…

End of this teaser post, I hope I have some more next week so see you around!

Some corrections and an addendum + a new way of taking Fourier transforms.

This post has many goals, for example in the previous post I talked about a ‘very rudimentary Fourier transform’. In this post it is a bit less rudimentary, a bit more satisfying definition is given but still I did not research all kinds of stuff like the existence of an inverse & lot’s more basic stuff.

For myself speaking I consider this ‘new Fourier transform’ more as some exotic bird that, if capable to fly a few meters, will only draw applause from specialists in Fourier analysis.
So for myself speaking I am far more happy we need a more advanced number tau and the mathematical miracles you can do with it in three dimensions.

Therefore I included two examples of exponential curves that go through the plus and minus of all three basis vectors in 3D space, after all this is one of my most remarkable math results…

In this post I also show you how to use the calculus of ‘opposite points, in three dimensions it works like a bullet train but the higher the dimensions become the harder it is to frame it in simple but efficient calculus ways like using opposite points on exponential circles.

Another thing to remark is that an exponential circle is always a circle; it is flat in the 2D sense and has a fixed radius to some center. When this is not the case I always use the words exponential curve

This post is nine pictures long, I truly hope you learn a bit from it.
You really do not need to grasp each and every detail, but it is not unwise to understand that what I name the numbers tau are higher dimensional versions of the number i from the complex plane.

Ok, here we go:

In these nine pictures I forget to remark you can also craft a new Cauchy formula for the representation of analytic functions. For myself speaking this was far more important compared to a new way of Fourier transform.

You still need that more advanced version of tau…

Can´t get enough of this stuff?
Ten more pictures dating back to 2014 at the next link:

From 18 Jan 2014: Cauchy integrals
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#18Jan2014

A link to the Nov 2016 post on 2D split complex numbers that contains the disinformation about the sum of the coordinate functions:

The second hybrid: a 4D mix of the complex and the circular plane.

End of this post, likely the next post is about prime numbers and how to demolish the internet security we think we have using huge prime numbers…

So see you around!

Some very rudimentary Fourier stuff + a surprising way to do a particular integral.

Lately I was looking at some video’s about Fourier analysis and it dawned on me I had never tried if the coordinate functions of my precious exponential circles were ‘perpendicular’ to each other.

Now any person with a healthy brain would say: Of course they are perpendicular because the coordinate functions live on perpendicular coordinate axis but that is not what I mean:
Two functions as, for example, defined on the real line can also have an inner product. Often this is denoted as <f, g> and it is the integral over some domain of the product of the two functions f and g.

That is a meaningful way of generalizing the inner product of two vectors; this generalization allows you to view functions as vectors inside some vector space equipped with an inner product.

Anyway I think that most readers who are reading post number 62 on this website are familiar with definitions of inner product spaces that allow for functions to be viewed as vectors.

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So I was looking at those video’s and I was highly critical if my coordinate functions would be perpendicular in the sense of such generalized inner product spaces. But indeed they are also perpendicular in this sense. Yet a bit more investigations soon gave the result you cannot build a completely new kind of Fourier analysis from this stuff.

Ok ok a few years back I already arrived at that insight because otherwise in previous posts you would have found stuff relating to that…

Generalized inner product spaces are often named Hilbert spaces, a horrible name of course because attaching a name like Hilbert to stuff you can also give the name Generalized Inner Product Space brings zero wisdom at the scene.
It is only an attempt to turn the science of math into a religion where the prophets like Mr. Hilbert are given special treatment over the followers who’s names soon will be forgotten after they die.

More on Hilbert spaces: Hilbert space
https://en.wikipedia.org/wiki/Hilbert_space

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This update is 11 pictures long, all size 550 by 775 pixels.
I kept the math as simple as possible and all three dimensional numbers used are the circular ones. So you even do not have to worry about j^3 = -1…

Have fun reading it:

 

 

 


 

 


 

I hope you like the alternative way of calculating that integral in picture number 11. It shows that 3D complex and circular numbers are simply an extension of mainstream math, it is not weird stuff like the surreal numbers that decade in decade out have zero applications.

May be in the next post I am going to show you a weakness in the RSA encryption system.
Or may be we are going to do something very different like posting a correction on a previous post.

Let´s wait and see, till updates.

New magnetic update + some pictures related to the post on the general theorem of Pythagoras.

I know I know I have not posted very much lately. There was plenty of material to craft new posts from but I skipped easily writing 10 posts or so because I am also wondering as why the so called ‘professional professors’ never make a move.

For the math professors this is 100% logical: If you are math stupid to the bone, you will never understand 3D complex number systems. But why the physics professors do not react in any way is completely unknown to me. Ok ok, for example their explanation of how permanent magnets work is very very strange. They formulate it often this way:

In a permanent magnet all spins align themselves.

That is a very stupid explanation and if you meet a physics professor and you whisper softly ‘quantum computer’ they start talking about atoms and electrons that can be in two places at the same time and that electrons can be in a so called super position of being spin up and spin down at the same time…

So in a permanent magnet the electrons and their spins are glued into place permanently while if those people need more funding all of a sudden even atoms can be in two places at the same time…

It is important to stress that the professional physics professors have a one 100% lousy explanation about how permanent magnets work. They completely miss the important fact permanent magnets have their magnetism because of the place of unpaired electrons inside the inner shells of those magnets… Here is my 04 Jan 2017 explanation of it on this very website:

How permanent magnets work, the official version against what I think of it.

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On 19 March I posted reason number 46 as why electrons cannot be magnetic dipoles on the other website. It is about so called BCS theory that explains super conductivity using so called phonons.

It is a very very very strange theory because just one unpaired electron is capable of distorting a metal lattice at low temperatures to such an extend that another electron is attracted and as such super conductivity emerges…

It is an imbecile kind of theory, in my point of view it is a  basic thing is that the electron pair is magnetically neutral that gives rise to the emerging of super conductivity. And at that point we have the perfect collision with my views on magnetism and the professional view:

The professionals think that the electron is a magnetic dipole because about 150 years ago a guy named Gauss did write down some fancy math explaining flux conservation. I love that kind of math but it just does not go for the electron, furthermore there is zero experimental proof for the electron being a magnetic dipole.

Here is the link that replaces the official BCS theory by a model for super conductivity as I see it:

19 March 2017: Reason 46: BCS theory says electron pairs are bosons…
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff02.htm#19March2017

Don’t forget: It is not a theory but a model.
Yet it should cover all kinds of super conductivity materials from the old school stuff from 1911 by Heike Kamerling Onnes to the present day high temperature super conductors named cuprates.

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After having said the above I am pleased to post five more pictures on the general theorem of Pythagoras. I know that in the last post I said this is the final post but then I was not capable of producing the pictures as shown below.

Here they are, only five pictures with very very simple math in it.

Have fun reading it!

Ok, that was it.

Have a nice life or try to get one.

And in case you are a professor in physics; why not cough up for the first time in you life that the electron is indeed a magnetic dipole???

Why not? A tiny bit of experimental proof would be great.

End of this post, till updates.

The general theorem of Pythagoras (second and final post).

This week I finally did put in the last details of the proof for the general theorem of Pythagoras. Now a long long time ago somewhere like in 1993 or1994 when I found this proof I could only find a very different proof in the official literature.
That proof worked with a matrix, I do not remember how it worked but the important feature is that this proof that used a matrix did not need a special coordinate system.

In the proof that I found I need the origin in the place where all lines, planes, hyper planes etc meet perpendicular so it is pretty natural to use the natural basis in n-dimensional real vector space.
The simplicity of this proof hangs on the construction of a normal vector to a hyper-plane and although I know this result for over two decades once more I was stunned about how easy this normal vector is to find and how easy it is to use the properties of this normal vector in proving the general theorem of Pythagoras.

Because after all; if you are given n + 1 points in n-dimensional space and you must prove something about the convex span of those n + 1 points, most of the time you just scratch your chin a bit, think a bit about it & never make any progress at all…
But using this easy to construct normal vector, instead of a difficult fog you have crystal clear skies over math paradise, what more should a reasonable person want???

In the year 2017 we have a much much better developed internet compared to the times when I originally did find my own version of a proof, but I did not research any of the outlets we have today like, for example, Google books.
If I can find a few good links I will update this post later.

This post is an additional 7 pictures to the previous post, each picture is as usual 550 x 775 pixels.
If you haven’t read the first post containing the first five pictures, please go here.

Once more: The surprising result is how easy to construct this normal vector is…

 

For myself speaking I am a little bit dissatisfied by notations like O with a hat and a + in the exponent, but I could not find a more easy notation so you simply must swallow that:

O hat lives in n-dimensional space while
O hat with the plus in the exponent lives in (n + 1)-dimensional space…

Ok, this is what I had more or less to say. If I can find a few good links I will post these later and if not see you around & try to get a nice life in case you don’t already have such a kind of life.

General Pythagoras theorem part 1: The 3D case.

A long time ago I found a very simple proof for the general theorem Pythagoras. At the time the general public had almost zero access to internet resources and in those long lost years I could not find out if my proof was found yes or no.

As memory serves, Descartes was the one that gave a proof for the 3D version of the Pythagorean theorem… (But I never did read the proof of Descartes.)

Two weeks back I was cleaning out my book closet so I could store more bottles of beer for the ripening process and I came across that old but never perfectly finished proof.

And it entered my mind again because it is fascinating that just by constructing that perfect normal vector, you make it of unit length, calculate a few higher dimensional volumes and voila:
There is you proof of the general theorem of Pythagoras.

In this post we only look at the 3D example for the theorem of Pythagoras. But already here we use a normal vector together with the 2D theorem of Pythagoras in order to prove the result for 3D space.
Basically this is also precisely the way the proof works in all higher dimensions, ok ok the notations and ways of writing the stuff down is a bit more technical but if you understand the proof in this post you will immediately understand how the general proof works.

The general proof is based on the principle of natural induction, likely the reader is familiar with natural or mathematical induction because beside it’s elegance it is also easy to explain to first year students in exact sciences. Basically you prove some stuff for low values of n, say n = 2 or 3 for 2D and 3D space and after that you do the so called ‘induction step’ where you must show that if it holds for a particular value of n, the stuff you want to prove is also true for n + 1.

Here is a wiki on the subject: Mathematical induction
https://en.wikipedia.org/wiki/Mathematical_induction

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This post is five pictures long (size 550 x 775 as usual) so have fun reading it:

23Feb2017_generalized_Pythagoras_theorem01

23Feb2017_generalized_Pythagoras_theorem02

23Feb2017_generalized_Pythagoras_theorem03

23Feb2017_generalized_Pythagoras_theorem04

23Feb2017_generalized_Pythagoras_theorem05

In the last line of the proof it is important to remark that both the length of XY is done with the 2D version of Pythagoras, but the height h of triangle XYZ is also done with the 2D version of Pythagoras. And so you get the 3D version of the famous Pythagoras theorem.

See you in the next post where it is all a bit more abstract and not slammed down to just two or three dimensions. Have a nice life or try to get one.