More on electrons, the discovery of electron spin and how permanent magnets work.

January 3, 2017MagnetismReinkoVenema

Back in the year 1921, almost one hundred years ago, Herr Stern and Herr Gerlach conducted a very intriguing experiment. They heated up silver until it was a gas and they did send the beam of silver ions through an inhomogeneous magnetic field.
They observed the beam splitting into two streams of silver ions, they thought they had found ‘spatial quantization’…

Here is a picture of a schematic set up of the Stern Gerlach expeirment:

The upper side of the magnetic field is stronger compared to the strength of the bottom field, Stern and Gerlach expected the beam to split in the direction of the gradient of the magnetic field.

At present day we know that a beam of electrons also gets split, when three years ago I did read the results from this experiment I was buffled, baffled and bewildered: it was ok by me that a part of the beam went up towards the strongest parts of the magnetic field.

But why was a part of the beam attracted to the weaker parts of the magnetic field?????
This makes no sense, after all in those years I nicely believed electrons were magnetic dipoles because everybody said so. Let me demonstrate in a gedanken experiment why this behavior of the electrons is very strange if electrons are magnetic dipoles:

Begin Gedanken Experiment:

Let an electron cannon send a beam of electrons into an inhomogeneous magnetic field, if electrons are indeed magnetic dipoles in that case you can view them as little vectors. These vectors can point anywhere, together all vectors from a sphere.

Only one of the vectors of that sphere is in perfect anti-alignment with the magnetic field. If we think of the vectors as pointing from the south to the north pole, only the vector that points perfectly south will have perfect anti-alignment.

All vectors that are not perfectly aligned will be pulled into alignment, so if electrons are magnetic dipoles it is expected that almost all electrons will go to the strongest part of the magnetic field.

End Gedanken Experiment.

Yet in practice about 50% of the electrons go up and the other 50% go down…

And I was just puzzled so much; how can the weaker parts of the magnetic field attract magnetic dipoles??? After one or two days I ran the experiment again in my head but at some point I don’t know why I thought ‘Let’s try a magnetic monopole’.

To my amazement a magnetic monopole did give the results as we know them from the Stern Gerlach experiment. And I just thought by myself ‘Hey hey Reinko, not so fast because electrons are not magnetic monopoles but magnetic dipoles. It is even in the Maxwell equations Reinko so think before you speak’.

But a day later I was walking around in the local park thinking about chemical bonds; if electrons were magnetic monopoles that would also explain why we only have electron pairs in chemistry.

Anyway now after three years I have about 40 reasons as why electrons cannot be magnetic dipoles but on the universities where about 100 thousand ‘professional’ physics professors are deployed there is zero reaction to my insights.
On the contrary; they avoid talking about me like I am having the pest…

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Back to the year 1927 at the Solvay Conference people like Niels Bohr and Wolfgang Pauli argued that for free electrons it would make no sense to do some kind of Stern Gerlach experiment.
Here is a screen shot of a video I will link below to:

So five years after the experiment and four years after publication all those guys like the Einstein / Bohr / Pauli / Schrödinger / Dirac / Heisenberg / Bose complex, none of those men understood the basic nature of the electron:

An electron is a localization of electrical charge and one of the two magnetic charges.

As such there are two types of electrons: a magnetic monopole north and a south variant.
Also known as ‘spin up’ or ‘spin down’.

The next documentary is about one hour long, if you know nothing about electron spin it is a bit much to swallow in one time. But for me it was a true treasure trove, the guy that gives the talk is eighty years old and has given lectures in quantum physics for decades and decades:

The Stern-Gerlach Experiment And The Discovery Of Electron Spin – Sandip Pakvasa [2016]

Ok I see this post is getting a bit too long so a detailed explanation upon how permanent magnets work is skipped to some future date. In the meantime we have only scratched the surface when it comes to the ‘official version of electron spin’ versus my little set of 40 reasons as why electrons cannot be magnetic dipoles. You can find it in my page on magnetics:

A primer on the electrons that are the long sought magnetic monopoles.
Author: Reinko Venema.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff.htm#27Oct2015

It is now 23.27 hours and I have more stuff to do in my life so till updates & how permanent magnets work will be dealt with in a new post. (By the way for me it is completely weird and strange that the professional physics people still do not understand permanent magnets. They think the electrons are glued in place…)

Don’t forget to roast the ‘professional’ physics professors with their crazy ideas about electrons.

See yah around.

Happy new year! + I hope you drank enough beer during the feast while I only post a picture showing math superiority before cracking down on physics professors in the next post…

January 2, 2017Magnetism, UncategorizedReinkoVenema

Once more a happy new year! Luckily the number 2017 is a prime number but let us not talk on 2017-dimensional complex number systems but keep it simple:

In the next post I will explain to you how permanent magnets work in detail, you might think ‘wow man permanent magnets are studied for centuries and longer’ but my point is they had it wrong on important details.

But if you go to a high paid physics professor and you say ‘wow man your ideas upon permanents magnets are based upon electrons being the source of magnetic dipole behavior’, most of the time you get a cold shoulder.

These imbeciles, those professional physics professors they cannot even explain permanent magnets and they only do ‘bla bla bla the Gauss law of magnetism says that more bla bla is the only way forward’.

That kind of behavior is very interesting, why make nonsense to be your basic line of reasoning?

________

I have nothing more to say; in the next post I will explain how permanent magnets work, how they get permanent magnetism and how they can loose it.

For the time being because I am well aware of how arrogant all these physics professors are, I simple post and infinite product that shows how my own brain handles the stuff that flows in:

By the way, I crafted the outcome of this limit to 1/2 because when we talk electrons in the next post they are known as spin half particles. Beside this it is estimated that all professional physics people will react strongly dismissive of the simple fact that electrons cannot be magnetic dipoles…

Come on, this is the year 2017 and there will be no mercy for the physics professors.
Let’s leave it with that.

A more or less perfect visualization of the Riemann zeta function observed.

December 18, 20163D complex numbersReinkoVenema

It has been a long time since my last update and that is caused by some stupid medical condition I still have and in my native language it is known as a ‘peesschede ontsteking’.
In practice this means I must do all typing on my computer keyboard with my left hand because in the evening I still cannot use my right hand.

Let me spare you the details but the long durance of the pain could even date back to the time when I was a dumb 15 year old with a broken wrist not seeking medical help.

So for the time being no long updates on perfect new hybrid number systems, it takes too much pain to write those long math stories down. So I retreat and just post a link to what is a very good Youtube video on the Riemann zeta function and it’s continuation into it’s analytic continuation.

Here is the video from the 3Blue1Browne guy:

Visualizing the Riemann zeta function and it’s analytic continuation

Nice vid isn’t it?

Last year on 26 March 2015 I wrote an update on where to find the zero’s of the Riemann zeta function in the 3D complex number system. I still consider this being an important publication although that human garbage known as the ‘professional math professors‘ said nothing all these months, I still think it is worth the trouble and try to post a new link to it:

From 26 March 2015: Zeta on the critical strip (3D version only).
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#26March2015

May be it is best to leave this update with that;

Zero point zero point zero point zero reaction of so called ‘professional math professors’ upon finding the zero’s of the Riemann zeta function in dimensions above 2.

Once an overpaid imbecile, always an overpaid imbecile.
Let’s leave it with that.

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Update from 19 Dec: I did not include yesterday a more easy to understand analytic continuation that I wrote myself this year; it is the analytic continuation of the geometric series and as such I am debunking the stuff some of the children of a lesser God seem to think:

1 + 2 + 4 + 8 + 16 + ….. = -1.

Nottingham professors from math and physics seem to think that

1 + 2 + 3 + 4 + 5 + ….. = -1/12.

This is also nonsense and there are many ways to prove this is not the case but inside theoretical physics this is actually used: that is the process of renormalization. Every time professional physics professors encounter an infinity in their calculations it is not that they say ‘Something must be wrong with our theory’. No if they encounter stuff like 1 + 2 + 3 + 4 + etc, they replace it by -1/12.

It works pretty well in order to get rid of those singularities they say.

Anyway here is the link to what I had to say on that subject:

From 15 April 2016: Debunking the Euler evaluation of zeta at minus one.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#15April2016

You can find the analytic continuation of the geometric series in the fifth picture.

Let me close this extra update with the Youtube video from those weird weird Nottingham professors that started it all:

ASTOUNDING: 1 + 2 + 3 + 4 + … = -1/12

And indeed if it were true it would be very very astounding.
What for me is TRULY ASTOUNDING is that the very professors you see doing their show is that they think the harmonic series is divergent. The harmonic series is also the zeta function evaluated at 1:

1 + 1/2 + 1/3 + 1/4 + … = infinity.

So the Nottingham professors think that the harmonic series is divergent (that is correct of course) while the sum of all integers is convergent to be -1/12.

Welcome to the world of 21-th century science. Till updates.

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

November 24, 2016Exponential circle, Exponential curvesReinkoVenema

Update from 30 Nov: My health problems persist, my right wrist is still swollen and hot all the time. So after one week it is clear I need to see a doctor…
Anyway I can type text with one hand so here we go:

In this update I talk about the circular plane because I want to use the same language in 2D as in 3D or higher, yet for those living in the mud this stuff is mostly named split complex numbers. There are more names going round: for those people that do not understand what the conjugate of a number is and how to find those, they name it hyperbolic numbers.

This update is about finding the log of the first and only imaginary unit of the circular (also named split or hyperbolic) numbers. This mathematical goal can only be achieved by replacing the real scalars in the circular plane by numbers from the complex plane.
That replacement stuff is known in my household as the Sledgehammer Theorem, this theorem says you can more or less always replace scalars by higher dimensional numbers. But this has to make some sense; for example you have a number from the complex plane like z = a + bi, now if you replace the two real numbers a and b with general numbers from the complex plane you did not gain much. As a matter of fact you gained nothing at all because you are still inside the complex plane and other people will only laugh at you:

That is just like the way Donald Trump will expand the US economy…

For myself speaking I do not understand that a math result as in this post is more or less unknown to the professional math community. How can it be that Euler has all that stuff of finding the God formula while century in century out the math professors make no progress at all?

Every day I am puzzled by this because I am not ultra smart or so, it is only my emotional system is a bit different: I never get scared when hunting down some good math…

__________

Anyway from the mathematical point of view I am proud of this ten picture long update: it is as close as possible to the calculation that unearthed the very first exponential circle. That was the discovery that in the complex plane the log of i is given by i pi over 2.

Need a cold shower by now?

Want to restore your faith in the old masters with their superior use of math?

Try the next video from the Youtube channel, it only uses insights from the circular plane and he runs fast and far: The Lorentz boost inside special relativity:

Split complex numbers and the Lorentz boost.

Let’s leave this update with that, have a good life or try to get one.

Update from 04 Dec 2016: I would like to post the number one wiki when you do an internet search of split complex numbers. (There are all kinds of names going round, but the circular plane is also the split complex number plane for sure.)
As usual all that stuff has the conjugate wrong, but in the next wiki you see more or less the combined wisdom of the math community when it comes to expanding the complex plane to higher dimensions. (It is a dry desert, human brains are not that fit for doing math):

Split-complex number.
https://en.wikipedia.org/wiki/Split-complex_number

Once more: those people have got it wrong about how to find the conjugate and as such you can also find lots of pdf files about circular (or split-complex) numbers that say they are hyperbolic numbers.
The common fault is that they use the conjugate just as if you conjugate an ordinary complex number from the complex plane.

I remember I did that too for a couple of years until it dawned on me that we are only looking at the projections of the determinant; it has nothing to do with lengths, even in the complex plane it is not the norm of the complex number but it’s matrix representation and the determinant.

All stuff you find on this on the internet is nothing but shallow thinking.

End of this update, see yah in the next post.

The pull back map applied to the coordinate functions of the 3D exponential circle.

November 3, 20163D complex numbers, Exponential circle, Quantum mechanicsReinkoVenema

In this post, number 50 by the way, I am trying to use as elementary math as possible in order to use the pull back map from the 3D circular number system to the complex plane.

With this the pull back map and the 3D circular number system are treated so basic that with only high school math and a crash course in the complex plane students can understand what I am doing.

So for reading this post number 50, what do you need in mathematical knowledge?
1) Understand how to write cos(a + b) and sin(a + b) in terms of cos a and sin b.
2) Understanding of e to the power it in terms of cos t and isin t.
3) Understanding of the roots of unity as found inside the complex plane, in particular being able to calculate all three roots of unity when we take the third root of the number 1.

That’s all, so basically all first year students in math, physics and chemistry could understand this post at the end of their first year on a local university.

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The words above are only one reason to write this post; to be honest for me it took a long time to write down for the first time the coordinate functions for the 3D exponential circle.

And I never did give much solid proof for that these coordinate functions have indeed the properties as described. It all more or less came out of the sleeve as some kind of monkey trick.

Therefore for myself speaking, this post giving the results in it also serves as a proof that indeed there is only one class of coordinate functions that do the job. They can only differ in the period in time they need to go around, if you leave that out the triple of coordinate functions becomes unique.

All in all the goals of this post number 50 are:

1) To do the pull back of an exponential circle as simple as possible while
2) In doing so give some more proof that was skipped years ago.

__________

This update is seven pictures long, each 550 x 775 pixels in size.
Hit the road Jack:

I think I have nothing more to say, so see you around my dear reader in post number 51.

Till updates.

More on the pull back map, just a teaser picture and some blah blah blah.

November 2, 2016UncategorizedReinkoVenema

In the previous post we had some stuff on the pull back map but also those links to complicated theorems. Therefore I tried to explain the inner workings of the pull back map that pull higher dimensional complex & circular numbers back to the complex plane in as easy to understand chunks as I could.

In the next post I only use advanced high school math (for my own country that would be the VWO education line, the classes 5 and 6) and for the rest any university student that has followed an elementary crash course on the complex plane.

I am very glad I could find such simple ways to pull back higher dimensional exponential circles and curves back to that goodie good old Euler formula that says stuff like e^it is related to the cosine and the sine functions.

All stuff is boiled down to things you can see in the teaser picture below, no new advanced 20th century math ideas, only using century old well known trigonometric equations and that is all…

Once more: Higher dimensional complex number systems are just there, it is a natural thing like the natural numbers like 1, 2, 3, 4, 5 etc are. Where the complex plane is something like a fish bowl, the higher dimensional complex & circular numbers are a big ocean.
But if you as a so called professional math professor can only swim your circles inside the fish bowl, can you survive the currents in this giant ocean?

No of course you can’t, so good luck with your future life inside the complex plane.

After this blah blah blah (remark the math professors are also extremely smart if you look at how much salary they suck in let alone the ‘research money’ they get to form global research groups that use at best two dimensional complex numbers) it is time for the teaser picture:

At last I would like to remark that the pull back map is on equal footage with the modified Dirichtlet kernels for my individual emotional system; I am glad I am still alive and can find stuff like this.

Till updates.

Derivation of the number tau for the circular 3D number system.

October 22, 20163D complex numbers, Exponential circle, Matrix representationReinkoVenema

There are lot’s of reasons for this update; one reason is that the actual calculation is mega über ultra cool. Another important reason is that this collection of plain imitation of how the value for the number i in the complex plane was found serves as a proof in itself that this way of crafting 3D complex and circular numbers is the only way it works.

Don’t forget that on the scale of things the Irish guy Hamilton tried for about a decade to find the 3D numbers but he failed. Yet Halmilton was not some lightweight, the present foundation of Quantum Mechanics via the use of the Hamilton operator is done so via the work of Hamilton…
Wether the professional math professors like it or not; that is the scale of things.

During the writing of this post I also got lucky because I found a very cute formula related to the so called Borwein-Borwein function. I have no clue whatsoever if it has any relevance to my own work on this website but because it is so cute I just had to post it too…

Furthermore I used two completely different numerical applets, one for integration and the other for evaluating the log of a matrix, only to show you that these kind of extensions of the complex plane to three dimensional space is the way to go and all other approaches based on X^2 = -1 fail for the full 100%.

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This post is ten pictures long, size 550 x 775 pixels.

At the end I will make a few more remarks and give you enough links for further use in case you want to know more about this subject. Have fun reading it.

The applet for the logarithm of a matrix can be found in this nice collection of linear algebra applets:

Linear algebra
http://calculator.vhex.net/function-index/linear-algebra

In this update you might think that via the pull back principle you observed some proof for the value of the integrals we derived, but an important detail is missing:
In 3D space the exponential circle should be run at a constant speed.
As a matter of fact this speed is the length of the number tau, you can find more insight on that in the theorem named ‘To shrink or to grow that is the question’ at:

On the length of the product of two 3D numbers.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#06May2016

A bit more hardcore is my second proof of the value of the integrals as derived in this post. On 15 Nov 2015 I published the second proof that I found while riding on my bicycle through the swamps near a local village named Haren. It is kinda subtle but you can use matrix diagonalization to get the correct answer.
The reaction from the ‘professional community of math professors’ was the usual: Zero point zero reaction. These people live in a world so far away from me: overpaid and ultra stupid…

Integral calculus done with matrix diagonalization.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#21Nov2015

A link to the online encyclopedia of integer sequences is the next link.
Remark that by writing the stuff as on-line instead of online reflects the fact this website must be from the stone age of the internet. That is why it can have this strange knowledge…

The On-Line Encyclopedia of Integer Sequences
https://oeis.org/ (Just fill in 1, 2, 0, 9, 9, 5, 7 in order to land on my lucky day).

The last link is one of those pages that try to explain as why 3D complex numbers cannot exist, the content of this page is 100% math crap written by a person with 0% math in his brain. But it lands very high in the Google ranking if you make a search for ‘3D complex numbers’.
So there must be many people out there thinking this nonsense is actually true…

N-DIMENSIONAL COMPLEX NUMBERS.
http://www.alenspage.net/ComplexNumbers.htm

Ok, this is what I had to say. Let me close this post, hit the button ‘update website’ and pop up a fresh beer… Till updates.

Too little time left so only a second teaser picture on the next post on the details of the 3D tau calculus.

October 20, 2016UncategorizedReinkoVenema

Originally I planned to upload tonight the new post on the integrals related to the number tau for the circular multiplication. But I found this very cute result from some other math professors, I believe these are two brothers Borwein & Borwein.

Beside that I also had more time to spend on a very important hobby: Brewing beer…;)

Four years back when I for the first time derived integrals like this with the cosine and sine stuff to the power three in it, I just had no clue whatsoever how to find analytical stuff for their value. These kind of integrals cannot be solved by throwing in some simple primitive or so.

At present I have two independent proofs for their value.
Back in the time I knew there was some internet website that contains a whole lot of integer sequences so if I could find that I would have at least some analytical clue about that nasty problem. Only a long time later I found that website, but is said ‘we do not know’.
Or ‘unknown integer sequence’ or whatever what.

But yesterday when I tried more or less to get a negative result my luck changed for the better: the website with the integer sequences in it actually returned an answer.

And for my few pounds of human brain tissue the answer was completely crazy.
Therefore I decided to put the result of this Borwein function on top in the teaser picture and my own idea’s at the bottom. Here it is:

I have absolutely no clue as why these two things should be the same, but four years back I had absolutely no clue as what this numerical value like 1.2092 actually meant…

The link to what might be the Borwein & Borwein function

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.
https://oeis.org/A248897

Let’s leave it with that, see ya in the next post.

Three new magnetics updates + Intro to a new post about calculation of the number tau.

October 16, 2016Magnetism, UncategorizedReinkoVenema

On the other website I posted reason number 37, 38 and 39 about why it is impossible for electrons to be magnetic dipoles. Let me give you the links and short descriptions about their content.

05 Oct 2016: Reason 37: Old and new experiments upon the bonkers force.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff.htm#05Oct2016

Once more the importance of repeatable experiments is stressed; my own simple experiments with that old color television is explained once more. Furthermore I am proposing a few other experiments that I cannot do here myself because, for example, they should be done is a space without magnetic or electrical fields.

The thing ‘bonkers force’ is acting along the magnetic field lines and makes electrons (and protons etc) accelerate. So it is perpendicular to the Lorentz force.

10 Oct 2016: Reason 38: The Hendrik Casimir effect and the vacuum catastrophe.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff.htm#10Oct2016

The Nobel prize in physics went this year to three men who studied two dimensional structures of electrons. So with just 50 to 70 minutes of labor I managed to do the same and explain as why the experiment of Hendrik Casimir has a wrong experimental set up because there they use the idea that electrons are magnetic dipoles. En passant using this wrong set up of Hendrik Casimir I can explain the root cause of the so called ‘vacuum catastrophe’.
The theoretical value of the so called zero-point energy of one cubic centimeter of space should be 10 to the power 112 erg of energy, yet at present day the best value found is about 10 to the power -8.

That is off the mark by just a factor of 10 to the power 120…

14 Oct 2016: Reason 39: The acceleration of the solar wind.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff.htm#14Oct2016

This is just one of the many things you cannot explain with electrons and protons being magnetic dipoles; despite gravity and or the influence of electrical fields the solar wind does not go down in speed. The professional physics professors cannot explain this nasty detail because they keep on holding on to the Gauss law for magnetism that says magnetic monopoles do not exist…

For the electron pair the Gauss law is valid but not for loose electrons.
As far as I know the winners of the Nobel prize from this year also believe electrons are magnetic dipoles so the Nobel committee has done a great disservice to the progress in physics.

So from the vacuum catastrophe to the properties of the solar wind: the professional physics professors will not find an explanation century in century out because you must not think that by writing down how stuff likely works they will change their ways.

But, ha ha ha my dear but incompetent and coward physics professors: My experiment with an old television can be repeated by any person and you, you fxckheads, cannot explain it…

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Ok, we proceed with math: The next post will be about how to find the number tau that you must use for crafting exponential circles and curves in dimensions above 3.

In order to focus the mind I would like to repeat a rather famous calculation from the complex plane: the calculation of the logarithm of the imaginary number i.
It is a beautiful calculation and it says that log i = i*pi/2.

Three teaser pictures to ram home to the brains of professional physics professors that I know plenty of complex numbers and that in my view using only 2D complex numbers simply shows what kind of brain matter you folks are made of:

$16oct2016-famous-math-calculation01$

$16oct2016-famous-math-calculation02$

$16oct2016-famous-math-calculation03$

At the closing of this small update I would like to remark that in the next post we are going to try and find logarithm values for imaginary numbers from 3D space.

And if in the future the Nobel committee would select Nobel prize winners that can actually think deeply and not all this shallow stuff, that would be great!

See you around my dear reader, till updates.

Curl, curl and more curl.

September 29, 20163D complex numbers, CR equations, Exponential circle, Matrix representationReinkoVenema

This post contains seven examples of the differential operator named curl. The motivation for this update lies in the fact that after my humble opinion inside the set of equations known as the Maxwell equations there is a tiny fault: Rather likely electrons carry at least a net magnetic charge.
And because they carry net magnetic charge they not only accelerated by electrical fields but also magnetic field.

If you have an old television set with one of those fancy tubes that contain one electron cannon for a black & white television and three electron cannons for a color television. With the help of a stack of these strong neodymium magnets it is easy to give experimental proof that the electrons indeed get accelerated… Why in the course of over one hundred years not one of the professional physics professors has done this is unknown to me; may be it is the separation of ‘theoretical professors’ versus ‘experimental ones’ a reason for this omission. May be it is the use of dogma (unproven things that live inside a belief system, in this case the belief system that magnetic monopoles do not exist).

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Anyway the Maxwell equations contain a lot of the curl operator, that is not needed per sé but it makes the formulae look sleek and short. Originally Maxwell had a set of like 20 equations or so while at present day only four remain. But if you see those four equations for the first time it is very impressive, only over time you get used to it.

This post is 13 pictures long, size 550 x 775 pixels.
I start with examples that are as simple as possible and very slowly bring in a bit more abstraction.
Therefore I hope it is very readable, have fun with it!

As far as I can remember, in the first year I opened these investigations into higher dimensional number systems again I calculated the curl for the complex multiplication in 3D space. In this post we only looked at the circular version of stuff.

But I can´t find it back so I cannot place a hyperlink to it.

Anyway, here is a nice wiki with the curl expressed as an integral (often much harder to calculate but nice to observe it can also be done that way):

Curl (mathematics)
https://en.wikipedia.org/wiki/Curl_(mathematics)

And because this post was motivated by all that curl in the Maxwell equations, I tried to find back when I originally started writing about electrons having magnetic charge instead of being magnetic dipoles like they are tiny bar magnets. It was 29 April that inside the math pages I found the first update on that. Here it is:

From 29 April 2014 : Do electrons have spin?
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#29April2014

Ok, that was it. Till updates my dear reader.

3Dcomplexnumbers

Math as you have never seen before…

More on electrons, the discovery of electron spin and how permanent magnets work.

Happy new year! + I hope you drank enough beer during the feast while I only post a picture showing math superiority before cracking down on physics professors in the next post…

A more or less perfect visualization of the Riemann zeta function observed.

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

The pull back map applied to the coordinate functions of the 3D exponential circle.

More on the pull back map, just a teaser picture and some blah blah blah.

Derivation of the number tau for the circular 3D number system.

Too little time left so only a second teaser picture on the next post on the details of the 3D tau calculus.

Three new magnetics updates + Intro to a new post about calculation of the number tau.

Curl, curl and more curl.