A primer on the three dimensional complex numbers.
Author: Reinko Venema. 

Page 1  Page 2  Page 3 Page 4


Introduction: This file is a collection of the pictures from Feb 2012 until now that are related to the 3D complex numbers.
3D complex numbers are in many ways just like the ordinary complex numbers from the complex plane; only in the 3D case you simply 'create' an 'imaginary number' j who's third power equals minus one.

Recall that in the ordinary complex plane the 'imaginary unit' i has a square of minus one.

Just like ordinary complex numbers (written often as z = x + iy) it looks rather elementary to use rectangular coordinates in real numbers (x, y, z).

Therefore the number one is written as 1 = (1, 0, 0)
the number j is j = (0, 1, 0) and the square of j is in the z-axis direction (0, 0, 1).


So only pictures in this stuff below... 

Remark: Everywhere where in these pictures it says 'see below' you must read that as 'see above' in some other picture. (Because this file is chronological while the homepage of this website is anti-chronological.) 

Updates in this html file not mentioned on the homepage
(these updates are a bit more technical): 

12 May 2013; the number i is not in 3D to be found   
21 May 2013, the wedge product 
30 May 2013, the log for j squared 
13 June 2013, tau calculus 
13 June 2013, making chopped meat of the 1, 2, 4, 8 theorem 
31 July 2013, a 6 by 6 matrix for log j 
08 Sept 2013, 10 properties for the exponential tau 
23 Sept 2013, without comment just 3 circles 
06 Oct 2013, exponential circles  
18 Oct 2013, determinant graphs in 3D + a glimpse of the general theory 
01 Nov 2013, eigenvalues and eigenvectors 
08 Nov 2013, the wrath of Diophantus  
28 Nov 2013, relations with linear control theory 
08 Dec 2013, correction and addendum Fourier transforms 
24 Dec 2013, general 2D stuff  & the song of omega 

Proceed on this 2014 page: 

05 Jan 2014, the song of omega reloaded  
18 Jan 2014, Cauchy integrals 
30 Jan 2014, 5D complex numbers and the golden ratio  
03 March 2014, a simple model for the proton and the neutron   
29 April 2014, do electrons have spin?  
28 May 2014, on the quantumization of magnetism 
02 Aug 2014,
the outer product in 3D 
15 Sept 2014, superposition of quantum states using 3D numbers 
26 Nov 2014, an oversight of 10 exponential circles & curves 
16 Dec 2014, what did Hamilton do wrong in 3D? 

Proceed on this 2015 page 3.   



From 05 Jan 2014: This is the second page in these petite investigations in the higher dimensional complex (and circular) numbers.  

Today we are going to look at the roots of unity as found on the exponential circle in the three dimensional complex case. The three dimensional roots of unity are very different compared to the old and known ones from the complex plane; as we will see this is caused by the fact that the center of the exponential circle is not zero but the number alpha... 

Here is the 3D reloaded stuff, we start with 5 things that are handy to have in your memory/consciousness: 















In this update I skipped the matrices with roots of unity related to the discrete Fourier transform, very likely they are all non-invertible while in the complex plane these are unitary matrices...
So may be in some future update we will look at matrices where the entries are no longer real numbers or complex numbers from the complex plane, but higher dimensional entries... 

That was it for this update, see you around. 


From 18 Jan 2014: Cauchy integrals. 

This update is about double length compared to the size of the usual updates. There is a simple reason for this length: the stuff involved is rather complicated so it takes a lot more trouble to explain everything properly. 

Let' s not waste time and put your brains to work!
Here we go: 





Updated 26 March 2016: I corrected the above matrix for the log of -j, circular multiplication. I am sorry it took me so long to remark the fault there was within. Well this is directly related to the fact my work is never peer checked, that is good for the quantity but at the same time I cannot guarantee it is always flawless... 















In case you have never seen a Cauchy integral, a relative good youtubber is made by doctorphys (the video contains a small fault but all in all it explains the complex plane relatively good): 

S3. Cauchy Integral Formula

Wikipedia has a nice collection that contains both the resolvent and the spectrum mapping theorem:

Holomorphic functional calculus 


Ok, that was it for this time. Till updates.  


From 30 Jan 2014The 5D complex numbers and the golden ratio.  

This is another 10 page long update and I ask myself as why am I doing this?
After two years and zero response from the academical societies I feel like I am throwing pearls to the
swines, so I am considering a second pause of about 20 years (just like the first pause from 1992 to 2012). 


Ok, after having said that, on 22 Feb I replaced a few things because of typo´s, better formulae format and most of all: in 5D these are not exponential circles but exponential curves. 
I am sorry I did not check that detail and that as such the previous updates were not one 100% correct. 

Let us look at the stuff from this improved 22 Feb update: 




















Not related very much to the above, but a Popular Science outlet named the New Scientist has a funny file that says two dimensional complex numbers can be harmful if they do not fit...  

From i to u: Searching for the quantum master bit 

Let' s leave it with that. 


From 03 March 2014: A simple model for the proton and the neutron. 












Further reading: 

The Standard Model 

Till updates. 


From 29 April 2014 : Do electrons have spin? 

For myself speaking I do not believe electrons have spin since they behave a lot like magnetic monopoles.
But standard physics theory says that magnetic monopoles do not exist so the scientists from the field of physics have to come up with ´other stuff´ in order to explain the behavior of so called ´spin 1/2´ particles. 

So be it, but in this update it is a bit more about how I look at the spin stuff.
Here we go: 













My dear reader, what do you think? Do electrons have spin and are they standard magnetic dipoles with a north and a south pole? In the science of chemical things, they say electrons only come in pairs but if they were magnetic dipoles also 4, 6, 8 etc configurations would be allowed. 

Since we only observe pair like behavior logic says that electrons might very well be magnetic monopoles...

Till updates. 

Update from 02 May: At the end of the Feynman lectures on physics you can observe all kinds of interesting thingelings like the original Stern-Gernbach experiment from the year 1922. But for myself speaking I observe the honorable Richard Feynman stating stuff like that it is impossible for the electron spin to be exactly measured into the z-axis direction... 

Again: Standard quantum theory does not say why the z-axis would be special in any way. 

But if you see it as the direction of the alpha axis in 3-dimensional complex numbers, suddenly the stuff makes a lot more sense. After all alpha is a non-invertible number and although I never mentioned it, alpha also plays a crucial role in the wave equation.
Let's leave it with that. 

Update from 08 May: Not much important or complicated but those spin half numbers have a strange property.  


Till updates. 



From 28 May 2014: This update contains no math at all. In this update I first tried to find some counter examples against electrons being magnetic monopoles, yet everything found points into the direction that electrons are magnetic monopoles.
















The 'wrong picture' is from a Youtubber with the title 

From Atoms to X-rays 

The energy splitting picture is from 

Electron Paramagnetic Resonance: Theory 

And I don't remember where I got the Zeeman effect picture from. 


A guy named Paul Callaghan presents a nice series on MRI where all the basics are covered in a 
pleasant and accessible way: 

Introductory NMR & MRI 

Also important is understanding the so called rare earth metals, they are known as the Lanthanide Series in the periodic system of chemical elements. The importance lies in the fact they all have only 2 electrons in their most outer orbital but inside they have lots of unpaired electrons.
If my idea's about electrons having monopole magnetic quantum states is in fact correct, these properties perfectly explain why the rare earth metals have their magnetic properties... 

The Lanthanide Series 


Till updates. 


From 02 Aug 2014: The outer product in 3D. 

This update is five pictures, or equivalent, five A4 size pages long but most stuff is repeating from previous updates. The only new thing is the outer product. 

Does that mean I do not have new things? 

No, I have plenty of new things but lately I arrived at the conclusion that apparently no university and no department of mathematics is waiting for this. There is only this wall of silence and so I wondered as why I should let insult myself by a bunch of incompetent people?  

Not only incompetent from the professional point of view, after all century in century out they could not find the higher dimensional complex numbers; but also that incompetence on the emotional level.
They think staying silent is the best way to behave, will can be just a childish as they are behaving... 

Ok, here we go: 









That was it for this update, Till updates. 


From 15 Sept 2014
The superposition of quantum states using 3D circular numbers from the 3D exponential circle.

Important remark: Since my knowledge of quantum mechanics is rather limited, I decided to use the three colors of quarks as the three states that are supposedly to be in a 'superposition'. 
Of course in practice it is extremely hard to verify if protons and neutrons are in a superposition state yes or no... 

I take no responsibility for the Quantum Mechanics used, I take only responsibility for the math parts used like higher dimensional complex and circular numbers. To be precise: circular numbers from our 3D world... 

End of this remark. 


Six pictures will paint what I had to say on this subject, here we go: 













Embargo information: 
This update contains only math that could easily crafted using previous updates. 
So the embargo of publishing at most 10 to 20% of new results is fully respected. 


Useful links: 

From MIT OpenCourseWare an intro in superposition of quantum states, it's a Youtube video: 

Introduction to superposition 

And as usual I made the picture about the determinant with an applet under the name of Polyray.
It is a French website and it has strongly 'session related' links, that is not an evil thing because they have to manage all those languages, but my http link will not work for you. 

So search for Polyray: 

WWW Interactive Multipupose Server at wims.unice.fr 


End of this update.    


From 26 Nov 2014: An overview of 10 exponential circles & curves; back in the year 1748 a guy named Leonhard Euler found the very first exponential circle. It is the unit circle around the origin in the complex plane. The very first exponential circle goes under a lot of names, but if you write down the two words 'Euler identity' most readers understand you are writing about the unit circle in the complex plane.

As far as I know, during a period of 260 years no other folks have found more exponential circles.
And as the complex spaces have more dimensions, they are no longer circles but become curves that cannot be put inside a simple two dimensional plane. 

The reason for that is very simple, let me give you a simple example like for the five dimensional real vector space: The exponential curve will go through all five basis vectors but it is not possible for a two dimensional plane to go through all five basis vectors...
In the five dimensional space, the exponential curves live inside the hyperplane that passes through all five end-points of the basis vectors.

(Ok ok, that is not 100% true, but for the circular version of multiplication of 5D complex numbers it is. For the complex version in 5D space, sometimes not the positive basis vector but the minus of that basis vector is in the path of the exponential curve.) 

For myself speaking, I consider the finding of all these exponential circles and curves as definitely inside the top 10 of mathematical results found during my entire life. It is not very often that you can improve on an important result in math that is the blood and veins of complex analysis.  


Ok, let's hit the road; Here are 20 pages with the description of 10 exponential circles and curves.
I did my best to avoid clumsy notation and tried to be as transparent as possible, I hope I succeeded more or less in that task. 






































And now you are at the end of this 19 pages long update on 10 new exponential circles and curves. 
Don't forget: The professional math professors did not find any one more in a time span of 260 years while in the last 1.5 years I crafted 10 more of this stuff.  

On page 20 just some pictures where I try to visualize the 3D exponential circle a little bit: 



Usefull links: 

For the matrix exponential and the log of a matrix you can go to: 


The visualization pictures for the 3D exponential circle come from an applet named polyray.
You have to search a little bit because I suspect a direct link does not work.


Till updates.  


From 16 Dec 2014: What did Hamilton do wrong in 3D? 

A lot of the so called 'professional math professors' seem to think that 3-dimensional complex numbers do not exist because for example that Irish guy named Hamilton could not find them.
Yesterday by coincidence I observed the line of reasoning Hamilton followed and it is riddled with all kinds of elementary mistakes. 

In this update I explain what Hamilton did do wrong; the most important detail is the fixation on the Euclidean norm that he thought had to be preserved under multiplication. 
This assumption is plain wrong because you must look at the determinant of the matrix representation of the higher dimensional numbers involved. 

This update is short, just 3 pages.
Here it is: 






Source file: Liberation_Of_Algebra_part1.avi 

Till updates. 



From xx xxx 2015: 

From xx xxx 2015: