Imagining Gaussian fractions

 It all starts with imagination:

The Bible tells us that our imagination is responsible for all human evil:

Gen 6:5  And GOD saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart was only evil continually.

Gen 6:6  And it repented the LORD that he had made man on the earth, and it grieved him at his heart.

Later on the evil is forgiven:

Gen 8:20  And Noah builded an altar unto the LORD; and took of every clean beast, and of every clean fowl, and offered burnt offerings on the altar. 

Gen 8:21  And the LORD smelled a sweet savour; and the LORD said in his heart, I will not again curse the ground any more for man's sake; for the imagination of man's heart is evil from his youth; neither will I again smite any more every thing living, as I have done. 

Gen 8:22  While the earth remaineth, seedtime and harvest, and cold and heat, and summer and winter, and day and night shall not cease.

But not entirely. 

Is our imagination really so evil? What about these thoughts of great minds:

Logic will get you from A to B. Imagination will take you everywhere.

Albert Einstein

Logic will get you from A to B. Imagination will take you everywhere. Share this Quote Albert Einstein
Read more at https://www.brainyquote.com/quotes/george_bernard_shaw_113045

Imagination is the beginning of creation. You imagine what you desire, you will what you imagine and at last you create what you will.

George Bernard Shaw

And so I have imagined Gaussian fractions. These are complex numbers which are complex fractions of two Gaussian integers

z = (m+n i)/(p+q i),

where m and n and  p and q are integers. Then I took those of them which have positive imaginary part. And then plotted so obtained numbers on the complex plane z=x+iy, x horizontal, y vertical. Here is the result

You can download the scalable vector image as pdf from here.

Then I followed my imagination and made the Cayley trnsform z -> (z-i)/(z+i) of these points:

 

You can download the scalable vector image as pdf from here.

Now my imagination tells me to consider the case of the conformal group SU(2,2), where the Shilov boundary of the eight-dimensional complex domain is our (compactified)  four-dimensional space-time of events, and not just a circle of time surrounding the disk as above. This will come step by step in the following posts.

P.S.1 In the plots above I took m,n,p,q to run from -10 to 10, avoiding zeros in the denominator. I have neglected the information contained in the fact that certain fractions appeared more than once, for instance 2/2= 3/3=4/4 etc. I could have ascribed different colors depending on the number of repetitions. I have used Mathematica program to produce these images.

P.S.2 To enhance our imagination we are using at home hyperbaric chamber. Here it is:

Clicking on the image will give you a full resolution photo.

P.S.3.  Marcelo Amaral has sent me the information about the following (Open Acess) paper by Klee Irwin, Marcelo Amaral and David Chester "The Self-Simulation Hypothesis Interpretation of Quantum Mechanics" - the result of work of Quantum Gravity Research Group The paper deals even with  "Unification of Emergentism and Panpsychism" and "Spiritulism". I am looking forward to studying this interesting work.

P.S.4 And in the meantime I have learned about this sad and depressing story: