The tridimensional John Conway's life game with random initial conditions -1.6% of occupied cells- [*Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.6% de cellules occupées-*].

The bidimensional life game
was initially defined by Conway. It uses an empty
square mesh (all vertices are turned **off**).
At time t=0 some vertices are occupied (they are turned **on**): this is the initial
state. To go from the time t to the time t+1, it suffices
to count for each vertex -or "Cell"- C(x,y) the number N of its
neighbours (it is less than or equal to 3^{2}-1=8) and then to possibly change the state of
M according to the following bidimensional automata
rules:
[R1 = Birth] ((C(t).IS.**off**).AND.(N == 3)) ==> C(t+1) **on**
[R2 = Death] ((C(t).IS.**on**).AND.((N < 2).OR.(N > 3))) ==> C(t+1) **off**
[R3] other cases ==> C(t+1)=C(t)

The boundary conditions can be periodical or not.

This process can extended in a tridimensional space. The number N
of neighbours of the vertex -or "Cell"- C(x,y,z) is computed
(it is less than or equal to 3^{3}-1=26) and
the preceding rules can be extended as follows:
[R1 = Birth] ((C(t).IS.**off**).AND.((N >= NB1).AND.(N <= NB2))) ==> C(t+1) **on**
[R2 = Death] ((C(t).IS.**on**).AND.((N < ND1).OR.(N > ND2))) ==> C(t+1) **off**
[R3] other cases ==> C(t+1)=C(t)

The bidimensional and tridimensional processes can be extended one step further using two binary lists
'LD' and 'LA' ("Dead" -**off**- and "Alive" -**on**- respectively):
[R1 = Birth] ((C(t).IS.**off**).AND.(LD[N] == 1)) ==> C(t+1)=**on**
[R2 = Death] ((C(t).IS.**on**).AND.(LA[N] == 1)) ==> C(t+1)=**off**
[R3] other cases ==> C(t+1)=C(t)

("1" means "to change the state" and "0" means "the state is unchanged").

For this picture, the parameters have the following values:
LD="000010001110000011100010000"
LA="111111100011111110000111111"

See some related pictures (including this one):

[More information about the bidimensional John Conway's life game and the *bidimensional extended life game*]

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