Universe 3/03 - 20 x 20 Field
Starting from an empty field, rule set 3/03 (3 living neighbors stays alive, 0 or 3 live neighbors give birth, all else dies) begins with a Big Bang and then creates an incredible amount of snowflake-like patterns that are symmetrical vertically, horizontally and diagonally. As I was watching them I noticed they didn't seem to repeat nor achieve a constant state. Curious about this, I started thinking about the Universe of 3/03 and the overall topology... what were the limits of the unique iterations and patterns?
Adjusting the Cellular Automaton demo to generate much faster iterations and log a hexadecimal serialization snap-shot of the field, I set out to document every iteration of Universe 3/03 on a 20x20 field starting from an empty field. Given the highly symmetrical patterns, there are only 55 unique bits (a triangular half of one quadrant) that make up a given pattern. That means there are 255 or 36,028,797,018,963,968 possible combinations... at most. Given the rule set, I thought it was highly unlikely the Universe would generate every possible combination. But, the question remained, "How many?"
Given a finite number of patterns possible, and an infinite amount of time, at some point a pattern will be created that will ultimately evolve back around into itself. The sequence will not just simply 'end', since the beginning and first iteration are completely empty and completely full (via the 0 birth rule) and therefor any state that completely fills or completely empties the field will definitely loop. There must be an iteration, Point of Repetition, that causes the sequence of iterations to eternally loop. If you think of the topology of the Universe as a series of iterations strung together in a long rope, the shape of the Universe would resemble a rope with a noose... a section of unique iterations, Prime Iterations, leading to a Point of Repetition that then has a series of Repetitive Iterations joining back into the Point of Repetition. My quest was to find that point and thereby document the entire topology of a 3/03 Universe on a 20x20 field.
This concept is true, as well, if the Point of Repetition is the first iteration, with a 0 length Prime Iterations... creating a circular shape to the universe. In addition, if the Point of Repetition is the last iteration, the loop can be thought of as 0 iterations long and looping into itself.