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

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.

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:

At last my old text from 2012:

(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.

Impending Nobel prize & recycled Pythagoras theorem & it’s ‘inverse’.

Tomorrow is the new Noble prize in physics out, actually it is already past midnight as I type these words so it is actually today. But anyway. I am very curious if this year 2020 the Nobel prize in physics will once more go to what I name those ‘electron idiots’. An electron idiot is a person that just keeps on telling that electrons are magnetic dipoles because of something retarded like the Pauli matrices. May be idiot is a too harsh word, I think that a lot of that kind of behavior or ideas that can’t be true simply stay inside science because people want to belong to a group. In this case if you tell the official wisdom of electron spin you simply show that you belong to the group of physics people. And because people want to belong to a particular group they often show conformistic behavior, when it comes to that there is very little difference between a science like physics or your run of the mill religion.

In this post I would like to share a simple experiment that every body can do, it does not blow off one of your arms it is totally safe, and shows that those Pauli matrices are a very weird pipe dream. Here we go:

The official explanation of the Stren Gerlach experiment always contains the next: If electron spin is measured into a particular direction, say the vertical direction, if later you measure it again in a direction perpendicular on the vertical once more it has 50/50 probability. So if it is measured vertically and say it was spin up, if you after that measure it in say a horzontal manner once more the beam should split according to the 50/50 rule.

Ok, the above sound like highly IQ level based on lots of repeated laboratorium experiments. Or not? And what is a measurement? A measurement is simply the application of a magnetic field and look what the electron does; does it go this way or that way?

Electron pairs are always made up of electrons having opposite spins, in chemistry a pair of equal spins is named a non-bondig or an anti-bonding pair. Chemical bonds based on electron pairs cannot form if the electrons have the same spin.

Now grab a strong magnet, say one of those strong neodymium magnets and place it next to your arm. Quickly turn the magnet 90 degrees or turn your arm 90 degrees, what does happen? Of course ‘nothing happens’ but if electron spin would follow that 50/50 rule, in that case 50% of your electron pairs would become an anti bonding pair. As such your flesh and bones whould fly apart…

Now does that happen? Nope njet & nada. As far as I know it has never been observed that only one electron pair became an anti-bonding pair by a simply change of some applied external magnetic field…

As far as I know the above is the most easy day to day experiment that you can do in order to show that electrons simply do not change spin when a different magnetic field is applied…

I have been saying this for over five years but as usual when it comes to university people there is not much of a response. In that regard physics is just like the science of math: It has lost the self cleaning mechanisms that worked in the past but now in 2020 and further those self cleaning mechanisms do not work anymore. It is just nothing. It is just a bunch of people from blah blah land. So let’s wait & see if one of those ‘electron idiots’ will get the Nobel prize tomorrow.

Luckily I have a brain for myself. I am not claiming I am very smart, ok may be compared to other humans I do well but on the scale of things like understanding the universe I am rather humble. I know 24/7 that a human brain is a low IQ thing, but just like all other monkeys it is the only thing we have.

Very seldom the human brain flares up with a more or less bright idea that simplifies a lot of stuff. A long time ago I wanted to understand the general theorem of Pythagoras, I knew of some kind of proof but I did not understand that proof. It used matrices and indeed the proof worked towards an end conclusion but it was not written down in a transparent way and I just could not grasp what the fundamental idea’s were.

So I made a proof for myself, after all inside math the general theorem of Pythagoras is more or less the most imporatant theorem there is. I found a way to use natural induction. When using natural induction you must first prove that ‘something’ is true for some value for n, say n = 2 for the two dimensional theorem of Pythagoras. You must also prove that if it holds for a particular value of n, it is also true for n + 1. That is a rather powerful way to prove some kind of statement, like the general theorem of Pythagoras, holds for all n that is holds in all dimensions.

I crafted a few pictures about my old work, here they are.

It is from March 2018 when I wrote down the ‘inverse’ theorem of Pythagoras:

And from March 2017 when I wrote the last piece into the general theorem of Pythagoras:

Ok, let me leave it with that and in about 10 hours of time we can observe if another ‘electron idiot’ will win the 2020 Nobel prize in the science of physics. Till a future post my dear reader. Live well and think well.

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.

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.

__________

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.

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

__________

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

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.