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Monday, January 6, 2014

The evolution from a mathematical perspective

This time I'm going to share a fascinating proposal about a considered a cellular automaton, which is known as Game of Life, or simply Life among experts in Artificial Intelligence and mathematics, which was created by the English mathematician John Horton Conway in 1970.

The game is determined by its initial state, without need of further input.  The game begins with an initial configuration and it is seen how it evolves.

The game is due to the interest of Conway in a problem presented by the mathematician John von Neumann who was trying to find a machine that could build copies of itself. 

Neumann succeeded when he found a mathematician model for a machine of this type with rules applied in a rectangular grid, which is very similar to the so-called Universal Turing machine, which is described as an automaton artifact that was unveiled in 1936 in the Journal Proceedings of the London Mathematical Society and which is basically a device that manipulates symbols on a strip of tape according to a set of rules that can be adapted to simulate the logic of any algorithm which can help to understand the limits of mechanical calculations, which provides advances to complexity theory.

Returning to the Game of Life, Conway found a way to dramatically simplify the notion of von Neumann, using only 4 rules for the implementation of a cellular automaton.

The game was released in October 1970 in the magazine Scientific American column mathematical games of Martin Gardner and opened a line of mathematical research known as the field of cellular automatons because of the analogy of the emergence, transformation and fall of any society of living organisms belonging to well-known simulation games in which the patterns can evolve.

Life allows an example of emergence and self-organization of patterns by what has attracted the interest of multiple fields of science and giving way to studies of emerging complexity or self-organization systems.

Life  is played in an infinite orthogonal network cell square (I recommend widely to visit the link of Wikipedia that I am here referring), each cell is in one of two possible States, alive or dead.

Every cell interacts with its 8 neighbors, which are the horizontal or vertical, diagonal or adjacent cells, with the passage of time, it is possible to observe the following transitions which are the rules of the game:

The initial pattern is considered the seed of the system. The first generation is created through the application of the above rules simultaneously to every cells of seeds that includes births and deaths which can occur at the same time and discreetly called this a tick, which implies that each generation is a pure function of the precedent and the game continues until the last cell dies.

From these simple rules, life has become one of the examples of what is known as emergent complexity and self-organization systems.

Now, let me explain why this made me jump off my seat and go running the stairs to search Google in this respect:
These simple rules allow the understanding of such complex phenomena such as the arrangement of the petals of a rose, or patterns on the skin of a zebra, and it is that in life, science attempts to explain complex patterns, always applying the idea that simple is best, I remember clearly professor Colm Donaldson saying loudly: make it simple, simplicity is more beautiful in science.

So those simple patterns of interaction are the beginning a complex process of interaction that unlike the rest of the games, life part of the standards themselves to make patterns, while in conventional games, developers create multiple game situations that must be met to advance. 

Designed by Paul Rendell 02/April/00, disponible en
Life has served even to analyze patterns of conduct testing different models and watching their interaction. For example the so-called R-Pentomino was the first pattern observed by Conway, which is very stable and thus easy to predict, although 1 103 steps are required for this purpose.

This is why some of the programs designed for Life at the beginning were limited to describe the fate of a pattern of small and specific, however with the development of computers, it is now possible to run more complex patterns.

One of the questions of Conway was to determine if the initial pattern of life could grow indefinitely, or if any system could, so it offered a prize of $50.00 to anyone who could respond to questions. In 1970 a group of MIT led by RW Gosper won the prize with a pattern known as Glider Gun, which emits a new agent every 30 generations indefinitely, by what the pattern grows forever.

As they were more and more patterns, other aspects of the game are defined, for example the speed of light is defined as maximum sustainable speed by any moving object, or the rate of spread given in one step either horizontally, diagonally or vertically by generation. In this regard for both processes are the maximum rate at which information can travel and thus determines the speed of the pattern.

Based on that, the mathematicians have played and created different theorems that are observing with developed patterns.

This is part of a set of ideas that are integrated into the Game Theory, and just one of them is known as Nash equilibrium, which is a concept of the solution of a game and decision-making, which analyzes the strategies employed by the players from the benefit that can be obtained by changing their strategies which creates a principle of stability in the solution exchange during the game and is known as the theory of Nash equilibrium.
Of course, one of the fields that it has enabled more development is known as artificial life that it is defined as a life made by a human mind and not by nature and relates also with the study of non-organic bodies, beyond of the creations of nature, possessing essential properties that allow you to understand it inside an artificial environment created specifically for its development within a programmable machine (usually you can think in) a computer) so that the artificial life (A Life) allows to understand three properties of the nature that are the reproduction, the emergent properties and evolution. 

I am sure you are wondering how does this explain students would ability to change patterns from simple action rules?, well, it seems that many researchers have tried to apply these principles on biology, personally I think that the program of Neuro-modulation environmental assisted which starting with micro-tasks capable of creating patterns of conduct specifies, can be an applied example, but undoubtedly there is much reading to do simple the complexity of learning.


Ashwani, K.  (2013) Cellular automation: A discrete approach for modeling and simulation of artificial life systems. International Journal of Scientific and Research Publications.  3 (10) Available at:

Gardner, M (1970) Mathemathical Games: The fantastic combinations of John Conway’s new solitarie game “life”. Available at:

Gymerek, M. (2010) Conway’s game of life. Available at:

Wikipedia. Conways’s Game of Life. Available at:

 If you want to see very cool patterns, you can visit here:

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