Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.

Chaotic behavior can be observed in many natural systems, such as the weather. Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.

chaos, in a mathematical context, is produced by regularity - i.e. by recursive algorithms ( you will iterate simple mappings ).

In other words: whenever you search for statistically balanced data ( chaos = highest entropy ), you have to proceed by strictest rule !

Example : Take the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... ( every term is the sum of the two preceeding terms ); the generation is extremely regular (recursive ).

But write now the sequence of binary notations ( 1=1, 2=10, 3=11, 5=101 etc. ):

1110111011000110110101...

This sequence is chaotic ( no regularity whatsoever ): the practical proof:

LZW ( a usual compression algorithm ) cannot compress it !!

So: Chaos is ( mathematically ) generated by extreme regularity !!!

Math and Pyshics are everything, I must give this greater thought.    I will re-post this subject.  Peter, I find math interesting, it is everthing, I know most rules, laws and the minus signs are my worst enemy.

The number of Shannon

a simple proof how deep chess can be

Claude Shannon

Claude Elwood Shannon (1916-2001) was a famous electrical engineer and mathematician, remembered as "the father of information theory". He was fascinated by chess and was the first one to calculate with precision the game tree complexity of chess i.e. the number of possible chess games. He based his calculation on a logical approximation that each game has an average of 40 moves and each move a player chooses between 30 possible moves. That makes a total of 10¹²⁰ possible games. This number is known as the number of Shannon.

To a similar conclusion came Peterson in 1996. An interesting comparison is the estimation of the total numbers of atoms in the universe 10⁸¹ . The number of legal positions in chess according to him, however, is about 10⁵⁰ .

All these calculations will suffer slight changes when we apply new rules to chess, such as the Sofia rule or further estimation of the effect of en-passant. However, the numbers are close enough to show you how deep chess can be.

Other game tree complexities (log game tree):

Tic tac toe 5
Connect Four 21
Othello 58
Chess 120
Backgammon 140
Connect six 140
Go 766

01010111 01100101 00100000 01100001 01110000 01110000 01110010 01100101 01100011 01101001

01100001 01110100 01100101 00100000 01111001 01101111 01110101 01110010 00100000 01110011

01110100 01110010 01101111 01101110 01100111 00100000 01100101 01101101 01101111 01110100

01101001 01101111 01101110 01110011 00100000 00100001 00100001 00100001

it is a nice articles, thank you for sharing.

We appreciate your strong emotions !!!

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