Genetic aspects of the Game of Life.

[Gunnar Johnsson]

By genetic aspects I mean more specifically the study of the ancestor
distributions in various cyclic LIFE patterns.
In the Game of Life each newborn cell has 3 parents . If the pattern
is cyclic with period N it is possible to identify corresponding cells
N generations apart and hence we can define an ancestor relation between

cells in the same generation.
By using two colours initially and a colouring rule that states that
each
newborn cell gets the arithmetical mean colour of its 3 parents one can
study in a visual manner the distribution of 'black genes', i.e each
cell
in any generation shows its share of black ancestry in its current shade

of gray.
This game is now available online in a Javascript implementation:
http://www.math.kth.se/~gunnarj/LIFE/lifegame5.html
(Games of Life in Colour). The mentioned game is called Rainbow Life
Game.
There is also another game ,Black&White Life Game, also known as the
Immigration Game,
in that site.
Some mathematical aspects are also discussed.
The colour development is shown to be of linear algebraic nature
and the governing matrices turn out to be so called stochastic matrices
(each entry in the interval [0,1] and each row sum = 1.)
known from the theory of Markov chains.
The problem of determining the asymptotic colouring for a cyclic pattern

from a given initial black and white pattern is completely solved.
The analysis leads to the proposal of the distinction between primary
and secondary rotor cells (i.e. a refinement of the classical
stator-rotor distinction).
The secondary cells are those that have zero 'genetic impact', i.e. an
initial
black cell in that position produces asymptotically an all white
pattern.
In the glider case one has the following picture (x:primary,
o:secondary):
o
x
x x x

This approach is completely general and applicable on all cellular
automata
where the births are related to at least two parents.
Comments and suggestions are welcome.

Gunnar Johnsson