A small change of the parity rule gives us new interesting images. In addition to the states of the six nearest neighbors of a cell we count the state of the cell itself too. The cell has state =1 in the next generation if the sum is odd.
Starting with a single active cell (state=1) we now get sometimes 7 active cells. As before, six are at the corners of a regular hexagon and the extra one is at the center. This appears with increasing sizes and at intermediate stages we get fractal self-similar patterns built on units of 7 elements, see the first figure.
This rule does not give concentric rings of hexagons. Instead we get a structure made of six fractal triangles. They resemble somehow the Sierpinski triangle:
Most of this complicated image collapses at the next generation, leaving only 7 cells of state 1. Generally, this modified rule gives more interesting pictures. After collision with the border, there appear structures resembling laces: