Growth behavior of helical cellular automata.

A helical cellular automata (HCA) model constructed on a two-dimensional grid of cells with a helical structure is presented and the pattern formation of this model studied by numerous computer simulations. It is found that the evolutions of the HCA are sensitive to the circumference of the helix p. With various p, the initial growth of the model generates various patterns ranging from Sierpinski triangle gasket, complex textured pattern, to lateral quasiperiodic structure. A sudden transition from regular fractal to compact pattern occurs near the point where p is equal to a positive integer power of 2. With increasing height of the patterns (increasing growth time), the model also exhibits different growth behaviors in the vertical direction for various p, including the formation of regular periodic patterns and the evolution from initial regular patterns to eventual random structures. Fractal dimension analysis is used to characterize these different evolution processes quantitatively.