The Math Behind the Motifs

STEP 1:  The zeroth generation

We start with "Generation 0", an initial motif that just has four "live" cells. We will call the white cells "alive" or "on", and the main-color cells "dead" or "off". In the case of the Granny Life project, each cell represents a cluster of two half-double crochet stitches. The outer border will always be white, and is not included in the algorithm.

STEP 2:  Count your neighbors

Each cell has four neighbor cells at its corners. Along the diagonals these neighbor cells might be oriented in different directions, and at the edges we will copy and wrap around. The white image shows various ways that four pink neighbor cells can surround a blue cell. 

In our original "Generation 0" image, each of the center white cells has two alive neighbors, each of the teal cells that touch the white center cells has one alive neighbor, and all of the other teal cells have zero alive neighbors. 

STEP 3:  The next generation

Now we will use the neighbor counts of each cell to decide if that cell will be alive or dead in the next generated motif. The rule we are going to use is the following: 

• Live/white cells will be alive in the next generation only if they have NO alive neighbors.

• Dead/teal cells will be alive in the next generation only if they have JUST ONE alive neighbor.

What this means for our "Generation 0" is the all the white cells with 2 neighbors are going to turn OFF in the next generation, all of the teal cells with 1 neighbor are going to turn ON in the next generation, and all of the other teal cells (which have 0 neighbors) are going to stay OFF. According to our rules, Generation 1 will have eight alive cells, around the original white cells.