The magic of numbers 3,4,5

 Pikabu the cat knows how to count 0,1,2. Dr. Marek Wolf, the author of "Plato was right", in a response to my previous post "The magic of numbers 0,1,2" (featuring Pikabu ze cat), kindly informed me in a private communication, that there exists empirical evidence that hens (and other birds) can count to three. And indeed, on Researchgate we can easily locate the appropriate paper on this subject. The paper "Any bird and chicken can count to three " by Alexander Yurkin provides us with the following observation:

"It is known that any bird and chicken can count to three (if the chicken incubates 4 eggs or more, and someone  takes an egg, then she sits quietly, if the chicken incubates 3 eggs and someone takes one egg, then she begins to scream, about the fact that her eggs are stolen)."

Newborn chicks can count, at least up to three,
a new study suggests

The paper by Yurkin was inspired, in turn,  by another paper, by Pradeep Mutalik (or Hradeep Mitalik?) in Quanta Magazine, "How Randomness Can Arise From Determinism". Quanta Magazine goes into the Quantum Future of Quantum Waves. We are not there .... yet. So, let us, for a while, remain at the level of cats and birds, and integer numbers.

Reality is all entangled. Everything connects to everything in an acausal synchronistic web of connecting arrows. Every day, while in the hyperbaric chamber, at the pressure of 2.0 atmospheres and oxygen level 90%, I am either meditating or listening to an audio book. Recently I am listening to Dan Brown's novel "Origin".

And so yesterday I listened, in particular, to a fragment containing this piece:

"The Ancient of Days, Langdon thought, squinting through the darkness at Blake’s famous 1794 watercolor etching.(...)
The piece was so futuristic in style that, centuries later, the renowned physicist and atheist Stephen Hawking had selected it as the jacket art for his book God Created the Integers."

2006 edition cover.
2007 edition cover is, for some reason, totally unimpressive.

The paper "How Randomness Can Arise From Determinism", mentioned above, features the Pascal triangle

Puzzle 3: Predicting Individual Behavior

It starts with 1,11,then 121. There are no zeros shown in this triangle. In my previous post, "The magic of numbers 0,1,2", there were lot of zeros. Let us have a look at these 0,1,2 again:

Our attention concentrates on triangular patterns. It is so easy to miss the main point: 

In each row a squared plus b squared gives c squared.

The numbers are big, so you would need a 24 digit calculator (or Wolfram Alpha) to check this statement.  For instance

21x21 + 220x220 = 221x221

etc. 

Each row is a Pythagorean triple. 

How many are there of such triples? How to generate them? 

Everybody knows the famous 3,4,5 and 9+16=25. This is the "Mother Triple". 

But then what would be the "genealogical tree" of these big 0,1,2 triples in the image above? And what does it all have to do with "The Origin" and "Quantum Future"?

To be continued

P.S. 31-08-23 Actually studying:

Andrzej Trautman, Pythagorean Spinors and Penrose Twistors