Follow the clues to find the treasure - Cayley transform

 First there was Light (we omit the Darkness). Then there was Amber and Volta. Then there was Maxwell and electromagnetism. Then there was  Einstein. When creating the Special Theory of Relativity Einstein did not care to follow the clue, and the clue was already there. Einstein (with Minkowski) looked for symmetries of Maxwell equations. They rediscovered and formulated algebraically Lorentz transformations that lead from one uniformly moving reference to another. But in fact Maxwell equations are invariant under a larger group of transformations - the "conformal group" that was allowing not only for a uniform motion, but also for accelerated one.

Suppose now we follow the clue and look for the treasure. What shall we find?

Here are the illustrations for this post. Mathematica code is simple:

g[i_, j_] := {Re[((i + I*j) - I)/((i + I*j) + I)], Im[((i + I*j) - I)/((i + I*j) + I)]};

igauss2 =  ListPlot[Table[g[i, j], {i, -100, 100}, {j, 0, 100}], AspectRatio -> 1, Background -> Black, PlotStyle -> {White, PointSize[0.008]}, PlotRange -> {{-1, 1}, {-1, 1}}, ImageSize -> 570]

igauss2a =  ListPlot[Table[g[i, j], {i, -100, 100}, {j, 0, 100}], AspectRatio -> 1, Background -> Black, PlotStyle -> {White, PointSize[0.008]}, PlotRange -> {{0.75, 1}, {-0.125, 0.125}}, ImageSize -> 570]

igauss2b =  ListPlot[Table[g[i, j], {i, -100, 100}, {j, 0, 100}], AspectRatio -> 1, Background -> Black, PlotStyle -> {White, PointSize[0.008]}, PlotRange -> {{0.95, 1}, {-0.025, 0.025}}, ImageSize -> 570]

Export["igauss2.jpg", igauss2]

Export["igauss2a.jpg", igauss2a]

Export["igauss2b.jpg", igauss2b] 

The explanations will follow soon

Zoom:

More zoom:

Short explanation of the graphics. We are taking a particular transformation of the complex plane into itself. It's name is Cayley transform. It is given by the formula

f(z) = (z-i)/(z+i)

Here z is a complex number z = x+iy, where x,y are coordinates of the point on the plane and i stands for the imaginary unit, the square root of -1.

This map transforms the half-plane of positive y into the interior of the unit circle  When x and y are integers, z=x+iy is called Gaussian integer. Gaussian integers form a regular grid on the plane. After Cayley's transform they form nice  rainbow rays inside the unit circle.

What is the relation of this Cayley transform to space-time, conformal group and physics? This will come later. There is also a puzzle of the strange "black hole" on the right? Where it coms from? What does it mean? 

Good exercise for students - because right now I do not know the answer! (If no one will solve this problem, I will post its solution next Saturday. But of course I will have to solve it first!)

This post will be continued in the coming day or two....

And in the meantime you can also check my animated  "continuous Cayley transform" on Wikimedia 

P.S. Unfortunately, as of 19:50, Thursday, my calculations misbehave. I am not obtaining expected results, which is a dissapointment. Of course I will continue, but at the moment the time frame of obtaining the desired results is highly uncertain. And the rules of life by Jordan Peterson are not helping in this case. It is not excluded that I am trying to obtain something impossible. But in this case I would like to know WHY it is impossible! And I have no idea as to "why?". Sunday 17:48 - problem solved.

P.S.2. In the meantime on my abandoned Polish blog the following discussion is taking place today:

Logical thinking15 December 2022, 20:58
@Arkadiusz Jadczyk 

Do you want to live in the truth, or do you prefer deception by misconceptions, others' and your own?

Arkadiusz Jadczyk15 December 2022, 21:22
@Logically thinking

And more specifically? What is it about?

Logically thinking15 December 2022, 21:44
@Arkadiusz Jadczyk 

About the truth. The simple question is.

Why do you evade answering the most important question that every human being must ask himself?


Arkadiusz Jadczyk16 December 2022, 08:18
@Logically thinking

And more specifically? What is it about?

Logically thinking16 December 2022, 08:55
@Arkadiusz Jadczyk 

About living in truth, in time and in space i.e. always and everywhere, in all circumstances and on all subjects.

Arkadiusz Jadczyk16 December 2022, 09:09
@Logically thinking

Very nice idea. The only thing is that every person's "truth" is different. There is the truth of Zelenski ( if there is one), there is the truth of the EU, there is the truth of Kaczynski and Morawiecki, there is the truth of Biden and there is the truth of Trump and there is the truth of Musk. And every person's truth is different. So "live in the truth," by itself, does not say anything na m. That's why I ask for specifics. Because the devil is hiding in them.

Logically thinking16 December 2022, 10:08
@Arkadiusz Jadczyk 

After all, it's not true that everyone has their own truth. There is only one truth. What you write about are stories, narratives, interpretations. Some of them come closer to the truth, more or less, and some are completely hypocritical.

I believe that there is only one truth, and a person can reach it, can know it. That is why I persevere in my quest to know the truth, defend it and preach it.

Arkadiusz Jadczyk16 December 2022, 12:27
@Logically thinking

As logical thinkers, our process of arriving at the truth is subject to analysis. We note that whoever we would ask - will answer the same way, that they also want to pursue the truth. And yet these truths that different people arrive at are sometimes extremely different. Why? What logically follows from this?

Maybe because in the process of arriving at the truth, each person decides for himself what is important and what is less important. Is our own health important? Is our own happiness important? How important? Is not harming others important? How important? Etc.

We additionally note that our own pursuit of the truth is hampered by the fact that there is a lot of lying around us. The media lies, people lie, we lie to ourselves sometimes. Why do people lie? Because they want to get to their truth, and when lying to others helps them do so, so they do. So, when telling others the truth hinders us from reaching our own personal truth, does it make sense to tell others the truth when it will harm ourselves?

Lots of questions. And a logical person should not leave these questions without scrupulous analysis.

Translated with www.DeepL.com/Translator (free version)

P.S.3. Commenting on the above a Reader kindly sent me a link to the book by Carol Tavris and Elliot Aronson "Mistakes Were Made (but Not by Me)", Publisher: Mariner, Year: 2020, ISBN: 0358329612,9780358329619

I did not read it yet myself (though I am going to), but here is the description of its content:

“Entertaining, illuminating and—when you recognize yourself in the stories it tells—mortifying.” — Wall Street Journal

“Every page sparkles with sharp insight and keen observation. Mistakes were made—but not in this book!” — Daniel Gilbert, author of Stumbling on Happiness

Why is it so hard to say “I made a mistake”—and really believe it?

When we make mistakes, cling to outdated attitudes, or mistreat other people, we must calm the cognitive dissonance that jars our feelings of self-worth. And so, unconsciously, we create fictions that absolve us of responsibility, restoring our belief that we are smart, moral, and right—a belief that often keeps us on a course that is dumb, immoral, and wrong. Backed by years of research, Mistakes Were Made (But Not by Me) offers a fascinating explanation of self-justification—how it works, the damage it can cause, and how we can overcome it. This updated edition features new examples and concludes with an extended discussion of how we can live with dissonance, learn from it, and perhaps, eventually, forgive ourselves.

“A revelatory study of how lovers, lawyers, doctors, politicians—and all of us—pull the wool over our own eyes . . . Reading it, we recognize the behavior of our leaders, our loved ones, and—if we’re honest—ourselves, and some of the more perplexing mysteries of human nature begin to seem a little clearer.” — Francine Prose, O, The Oprah Magazine

The same Reader is curious about my insights after reading this book. Even before reading it the following question occurred to me: Why is it so that even if silently we may recognize our own error, admit it to ourselves, but we never-never-never admit it to another person? Is such a fear evolutionary justified as Jordan Peterson would try to explain it?  From my own experience getting rid of this fear is making us free. On the other hand we should not give freely weapons to those who we perceive as our "enemies". But are we not ever making errors about who is our enemy? Lot of questions here.

P.S4: Melatonin can reduce the rate of severe COVID outcomes, help regulate sleep, promote genomic stability, and protect against neurodegeneration

From my own experience: one has to find by experiments your own optimal dose, and it is not constant in time.

P.S.5 Started reading "Mistakes Were Made (but Not by Me)" mentioned in P.S.3. There, in the introduction, I read:

In 2006, with Iraq sliding into civil war and sixteen American intelligence agencies having issued a report that the occupation of Iraq had increased Islamic radicalism and the risk of terrorism, Bush said to a delegation of conservative columnists, "I've never been more convinced that the decisions I made are the right decisions."1 Of course, Bush had to justify the war his administration pursued in Iraq; he had too much invested in that course of action to do otherwise—thousands of deaths and, according to a conservative estimate from the American Enterprise Institute in 2006, at least a trillion dollars. Accordingly, when he was proved wrong in his original reasons for the war, he found new ones: getting rid of a "very bad guy," fighting terrorists, promoting peace in the Middle East, bringing democracy to Iraq, increasing the security of the United States, and finishing "the task [our troops] gave their lives for." In other words, we must continue the war because we began the war.

And today the story is repeating:

"Congress agreed to provide another $800 million (at least) in defense aid to Ukraine next year via the Ukraine Security Assistance Initiative. The additional $800 million—again, at a minimum—is $500 million more than what President Joe Biden previously asked of Congress for the fund."

BTW: The highlighted sentences (from 2008 edition) were changed in third (2022) edition of the book.

P.S.2a At the end of this discussion I was asked the question if I would sign the petition "A Scientific Dissent from Darvinism". I answered: "No. For adding my name there could bring harm to this idea, which is not without some rationale."  To which a Reader of this blog commented: 

"The idea in itself can and is not without some rationale. Nevertheless, I consider the adoption of creationism in its classical version to be foolishness. Adopting Darwinism, moreover, I also consider foolishness. This is because Darwinism contradicts many discoveries related to, among other things, epigenetics. Nevertheless, it is a certain approximation of reality, as is Newton's theory."