Another interesting topic of study is the use of cellular automata to simulate such basic biological structures as the primordial soup in which life first began or simple multi-cellular life forms. Basically, cellular automata have attempted to create artificial life forms through simple algorithmic rules. Fractals have even shown up in such unlikely places as these.Pictures created from 1-D A-lifeOne of the most basic textbook cases of a cellular automata was proposed by Conway in the now popular game of 'Life'. With specified set of rules, Conway attempted to simulate the advent of life on a two-dimensional plane. Each cell is represented by a bit. A 1 represents a life-cell, and a 0 represents an empty-cell. Because this is a two dimensional plane, each cell will have eight neighbours, corresponding to the positions that are 1 unit away in each direction, including the diagonals. Three basic rules apply: an empty-cell with exactly three neighbouring cells will become a life-cell in the next round, and a life-cell with 4 or more neighbours will become an empty cell due to overcrowding, and a life-cell 1 or 0 neighbours dies from isolation. These rules are applied each generation, where each cell is looked at simultaneously to determine whether it should exist in the next round. Even with such simple rules, this behaviour is shown to be quite unpredictable. These provide a simulation of life, because they fulfil all the general requirements: a means for survival, reproduction, and death. There are no fixed initial conditions to the life: the bit-pattern can start out randomly, or with a set pattern: after awhile the . With taking a little thought, though, the set of life-cells would necessarily have fractional dimension, using the box-counting technique: the artificial life can never fill the entirety of the grid because it would die out on the next generation. Actually, the box counting dimension could even provide a measure for the tendency for the life-cells to remain stable: if the dimension is getting too low, then it shows that the life is dying out, and if too high, then overcrowding could occur.
Surprisingly, though, using a one-dimensional cellular automata with simple rules precise fractals can be created. This one has a single rule: a life-cell will exist if the three cells comprising the one to its left, the one to its right, and itself are not all empty or not all filled. If this is started with two adjacent, occupied cells, with each generation displayed on the y-axis, then Sierpinski’s triangle will miraculously form. If the initial conditions are set randomly, then a complicated pattern results, consisting of a myriad of triangles, that could be thought of as an interacting set of Sierpinski’s triangles. This cellular automata also exhibits sensitive dependence on initial conditions. A single bit in a certain place can determine whether the entirety of life on a grid dies out or stays alive potentially forever, as can be seen in my diagrams. In can be hypothesized that other fractal forms can be created with precise rules and initial conditions. Chaos and fractals seem to be appearing everywhere imaginable!