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The inverse theorem of Pythagoras (part 2).

Somewhere last year I just looked some nice video from the Mathologer about the theorem of Pythagoras. And since I myself have found a proof for the general theorem of Pythagoras in higher dimensions, I was puzzled about what the so called ‘inverse theorem of Pythagoras’ actually was.

Could I do that too in my general proof? And the answer was yes, but when I wrote that old proof of the general theorem of Pythagoras it was just a technical blip not worthwhile mentioning because it was a simple consequence of how those normal vectors work.

Anyway to make a long story short, a few days back I likely had nothing better to do and for some reason I did an internet search for ‘the inverse theorem of Pythagoras’. All I wanted to do is read a bit more about that from other people.

To my surprise my own writing popped up as search result number 3, that was weird because I wanted to read stuff written by other people… Here is a screenshot of the answers as given by the Google search machine:

Ok ok, not bad at search result number 3.

Now why bring this up? Well originally I forgot to post to the video that started my thinking in the first place. It is from the Mathologer and here at 16.00 minutes into his video is where my mind started to drift off:

The video from the Mathologer is here (title Visualizing Pythagoras: ultimate proofs and crazy contortions):

It is a very good video, my compliments.

After so much advertisements for the Mathologer, just a tiny advertisement for what I wrote on the subject of the inverse theorem of Pythagoras on March 20 in the year 2018:

What is the inverse Pythagoras theorem?

Ok, that was it. Till updates.

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.

What is the inverse Pythagoras theorem?

It is already late in the evening, actually it is past midnight so I will keep the text of this post short. It was a nice day today and this evening I brewed the 23-th batch of a beer known as ‘Spin half beer’. (I name it that way because it contains only half of the dark malts I use in the beer known as dark matter…;) so it has nothing to do with electrons).

This is a very basic post about some ‘inverse Pythagoras theorem’ as came flying by in some math video. I was rather surprised that I have not seen it before but there are so many theorems out there using that old fashioned Euclidian geometry that I might have forgetten all about it.

Within 10 minutes I had a good proof for the 2D version of this ‘inverse Pythagoras theorem’. You can find it in the first picture below.

One day later when I was riding a bit around I tried to find the higher dimensional analog of that easy to understand 2D statement or theorem. And as such it crossed my mind the important role a distance number d played in my proof for the general theorem of Pythagoras that acts on simplexes that are the higer dimensional analog of 2D triangles.

Coming home it was easy to write out the details, but for me it was all so simple that does this stuff deserve the title ‘theorem’? Well make up your own mind about that, but if it is not a real complicated theorem it is still a nice and cute result…

This post is six pictures long (all 550×775 pixels beside the last one that needed a bit expansion because the math did not fit properly so that one is 600×775 pixels).

At times it might look difficult but this is only because it is in a general setting when it comes to the number of dimensions, the basic idea’s are all simple things like taking an inner product with a normalized normal vector.

Here are the six pictures:

That is a cute result but for me the normal vector is just as cute but only a bit harder to write out because that part deals with general setting where the dimension n is not fixed.

For the time being is this the end of this post. See you around my dear reader.

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Addendum added on 30 March 2018: In the previous post I forgot to place a link to the proof of the general theorem of Pythagoras as I crafted it once a long time ago.

Before this link I would like to show you once more how to prove the general theorem of Pythagoras for the 3D case using only the 2D theorem.

After all, that is the first basic step in my proof for the general theorem of Pythagoras…

Here are the two addendum pictures outlining how this basic step from the two dimensional plane to the 3D space goes:

Here is the link to the proof of the general theorem of Pythagoras:

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

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

Ok that was it, till updates.

Oops; CERN did not find magnetic monopoles.

It has to be remarked that the physics folks are very persistant to keep on trying to find the so called Dirac monopole. How this has come to be is still a miracle to me. After all if the electron has one electric charge and for the rest it is a magnetic dipole, it would look naturally to look for a particle that is a magnetic monopole and an electric dipole at the same time…

But I have never heard about such an investigation, it is only the Dirac magnetic monople and that’s it.

Here is a quote from sciencenews dot org:

If even a single magnetic monopole were detected, the discovery would rejigger the foundations of physics. The equations governing electricity and magnetism are mirror images of one another, but there’s one major difference between the two phenomena. Protons and electrons carry positive and negative electric charges, respectively, but no known particle has a magnetic charge. A magnetic monopole would be the first, and if one were discovered, electricity and magnetism would finally be on equal footing.

Source:

Magnets with a single pole are still giving physicists the slip
https://www.sciencenews.org/article/magnetic-monopoles-single-pole-physics

Comment on the quote: Because in my view I consider the electrons having one electrical charge and one of two magnetic charges, I think we have a nice equal footing of electricity and magnetism… (End of the comment.)

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Back to CERN and stuff. Last month it came out that the MoEDAL experiment has failed in the sense that no magnetic monopoles were observed. Here is a small screenshot from the preprint archive stuff:

Comment: No idea what these people are talking about when they talk about 68.5 times the electric charge… Are they talking about electric charge or magnetic charge?
(End of comment)

Source of the content of the picture above:

detector in 2.11 fb−1 of 13 TeV proton-proton collisions at the LHC.

https://arxiv.org/pdf/1712.09849.pdf

After a bit of searching I found back this beautiful video, coming from CERN, explaining how to find magnetic monopoles. It is clear they never ever studied the electron.

Yeah yeah my dear average CERN related human; what exactly is a magnetic monopole?

Does it have electric charge too and why should that be?

In my view where the electrons carry both electric and magnetic charge, a magnetic monopole with zero electric charge just does not exist.

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Ok, let me bring this post to an end by observing that at CERN they were not capable in the year 2017 of detecting the magnetic monopole as it should exist following the lines of thinking like Paul Dirac once did.

So that is a good thing because after thinking about four years about magnetism it would be horrible for me to find that at CERN they had a major discovery about magnetic monopoles…

Sorry CERN folks, your failure to find magnetic monopoles your way does not prove that electrons are indeed carrying magnetic charge. It just makes it a little bit more plausible that they do…

So my dear CERN folks, thanks for publishing your failure because for me it is another tiny quantum move into the direction of accepting the electron as it is.

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End of this post.

Intro to the calculation of the seven dimensional number tau (circular version).

All details will be in the next post but I succeeded into using matrix diagonalization in order to find this seven dimensional number tau.
For people who do not understand what a number tau is, this is always the logarithm of an imaginary unit. Think for example at the complex plane and her imaginary unit i. The number tau for the complex plane is log i = i pi/2.

The problem with finding numbers tau becomes increasingly difficult as the number of dimensions rise. I remember back in the year 2015 just staring at all those matrices popping up using internet applets like the next one:

Matrix logarithm calculator (it uses the de Pade approximation)
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

Yet back in the year 2015 I was riding on my noble iron horse (a cheap bicycle) through the swamps surrounding the village of Haren and suddenly I had a good idea. Coming home I tried the idea of matrix diagonalization out in 3 dimensions and it worked.

I even wrote a post about it on 23 Nov 2015:

Integral calculus done with matrix diagonalization.

Now I think that most readers who visit this website are familiar with the concept of finding a diagonal matrix D containing all eigenvalues of a given matrix M. Once you have the eigenvalues you can calculate the eigenvectors and as such craft your matrix C containing all eigenvectors.
You can write the stuff as next: D = C^-1 M C.

Suppose you don’t know what M is but I give you the matrices D, C and the inverse of C. Can you find the matrix M?

Yes that is a beerwalk, all you have to do is calculate M = C D C^-1 and you are good to go.

But with the logarithm comes a whole lot of subtle things for making the right choice for the eigenvalues that you place inside the diagonal matrix D. It turns out you only get the desired result if you use arguments in the complex plane between minus and plus pi.
This is caused by the fact that you always need to make a cut in the complex plane if you want to work with the complex logarithm; but it is a bit surprising that only the cut where you leave out all real negative numbers (and zero of course) makes the calculation go perfect and in all other cases it ends in utter and total disaster.

In the next three pictures I show you some screen shots with numerical values of matrix representations and the logarithm of those matrix representations.

The goal is to find mathematical expressions for the observed numerical values that are calculated via the above mentioned de Pade approximation. We don’t want only numerical approximations but also catch the stuff in a mathematical formulation.

At the end of the third picture you see the end result.

So it took some time to find this result, I wasted an entire week using the wrong cut in the complex plane. And that was stupid because I had forgotten my own idea when riding my noble iron horse through the Harener swamps…

The result for the seven dimensional number tau (circular version) as calculated in the next post is a blue print for any dimension although I will never write stuff down like in a general dimension setting because that is so boring to read.

Ok, see you around my dear reader.

On the derivative and integral of the inverse function.

After all that magnetism stuff it is about time to throw in a tiny bit of simple math around how to find the derivative and primitive (the integral) of the inverse of a function.

In most (introductory) textbooks on calculus you will find a nice way of finding the derivative of the inverse of a given function f(x) defined on the real line. For integration where you need to find the anti-derivative there is also a very elegant way of calculating those, but in my life I have never ever seen it in print on paper in an actual existing book.

Now last week I came across a video where another guy claimed that finding the primitive in this way was completely new but within 60 seconds with the help of the Google search engine you can find this is not the case.

According to a wiki on the subject of integration of the inverse of a function, the first know results date back to 1905. This is a remarkably short time ago and for myself speaking I think that many folks found this way too but for some strange reasons it never popped up to the surface. It is strange to observe that for example the method of the calculation of variations was invented included those fine differential equations that form the way to find for example the path of least action or minimal time but somehow those people never found the way to integrate the inverse of a function…

On the other hand, I have seen it myself that there can easily be a complete vacuum in mathematics; in my first year at the university I invented the so called product integral. Normally when you calculate an integral you can view that as adding up all the area under the graph of a certain function, with a product integral you do the same but you do not add it up but you multiply all stuff.
And in it’s most natural setting you do that with raising a function f(x) to the power dx.
That was my invention but although product integration has been studied for over a century, nobody had ever taken a function to the power dx…

Now enough of the blah blah blah done, this post is four pictures long and the wiki stating this cute formula was found in the year 1905 is the next:

Integral of inverse functions
https://en.wikipedia.org/wiki/Integral_of_inverse_functions

This post is four pictures (550 x 775 pixels), here they are:

So that was it for this post, see ya around my dear reader.

Updated on 16 Oct 2017:
Today I found that video back where some guy made those unsubstantial claims that this result was never ever found in the entire history of mathematics. That is not true but it is strange that the derivative is in every introductory course or book while the integral version is always absent.
We can safely jump to the conclusion that the integral version is not widespread known and this causes authors of those books not to include it.

The video goes under the title:
Rare Integration Strategy – You won’t learn this in Calculus.

So that was it for this update on this post, see ya around my dear reader.

Updated on 25 June 2018:
By sheer coincidence I came across a very nice video today. I remember that I wanted to discuss the situation as described into the video but I also want to keep the writing short at about 500 words.

So I skipped a discussion as where the function f(x) is hard to find but it is better to attack it via the inverse function. I know this sounds a bit vague but in the video you have such a situation.

The video goes under the title:
Integration Problem: Thinking Outside the “Box,” or the Given Region (From Stanford Math Tournament)

It seems to be from the year 2014 so it is refreshing to observe that it is not true that all math departments around the world are only occupied by zombies…

So that’s it for this update on this post.

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.

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!