The blinker. An example.

The blinker is the smallest oscillator in the Game of Life. It consists of three cells in a row that alternates between horizontal and vertical positions. Suppose that the three blinker cells initially have colour indices represented by the components of the 3-dimensional vector v₀ and that the colours of three cells in the next generation are represented similary by the vector v₁. The transition between the generations (or states) can be pictured in the Game of Life field as follows:

v0[1]  v0[2]  v0[3]  ->

v1[1]   v1[2]   v1[3]

where the colour rule induces the following relations:

v1[1] = ( v0[1] + v0[2] + v0[3] )/3 v1[2] = v0[2] v1[3] = ( v0[1] + v0[2] + v0[3] )/3	or in matrix form:
v1 = T v0 ,          where       T    =

The blinker alternates between vertical and horisontal rows in such a way that the algebraic transition always is described by the same transition matrix, T. This means that the transition from generations 0 to 2 is given by the matrix

A    =     T2    =

.

During this transition the blinker completes a full cycle, since the original position is resumed. Hence, the matrix A is called the cycle matrix for the blinker.

In general, a cyclic pattern with period N has N different transition matrices T_j, j=0, ... ,N-1.   T_j defines the transition between the generations j and j+1. The dimensions of T_j is r_j+1 x r_j, where the numbers r_j and r_j+1 represent the numbers of living cells in the corresponding generations. The cycle matrix A is the product T_N-1T_N-2...T₁T₀.

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