Medley of spirals from cyclic cellular automata

Cyclic cellular automata on the integer planar lattice are known to typically evolve through distinct phases ending with minimal periodic terminal states that usually appear as intertwined spirals. Here we explore the diversity of spirals that arise from nonstandard neighborhoods on the integer lattice and from looking at the automata on quasi-crystalline arrangements of cells. We see that phase transitions and development of spirals are almost ubiquitous yet the particular form of the spirals is very dependent upon the particulars of the underlying neighborhoods; in fact the spiral forms echo the neighborhoods. The quasi-crystalline illustrations provide much more subtle echoes in the spiral forms that show artifacts from the non-periodic local symmetry that occurs.