In this section, we will explore the following questions.
What is Conway’s Game of Life?
What is “sensitive dependence on initial conditions”?
In Section 8.2, we noticed that changing our initial complex number \(z_0\) could result in markedly different Julia sets. That is, the long-term behavior of the iterates of the function \(f(z) = z^2+c\) depended a great deal on the number \(c\text{.}\) This is an example of the pheonomenon known as sensitive dependence on initial conditions.
Another setting in which we can study such dynamics is the Game of Life, a “zero-player game” created by mathematician John Conway, meaning that the initial configuration of the game determines its long term behavior.
Rules of the Game of Life.
The Game of Life is played on an infinite, two-dimensional orthogonal grid of square cells. A cell has two possible states: alive and dead. Every cell interacts with its eight neighbors. The population of cells evolves according to the following rules.
A live cell with fewer than two live neighbors dies in the next generation, as though by underpopulation.
A live cell with two or three live neighbors lives to the next generation.
A live cell with more than three live neighbors dies in the next generation, as though by overpopulation.
A dead cell with exactly three live neighbors becomes a live cell in the next generation, as if by reproduction.
Activity8.3.1.
Examine the example below. Try to guess what the next generation will look like before you play it!
Activity8.3.2.
Examine the example below. Try to guess what the next generation will look like before you play it!
Activity8.3.3.
Examine the example below. Try to guess what the next generation will look like before you play it!
Activity8.3.4.
The examples below are more complex. As such, it may not be feasible to work out future generations. Instead, see what broader dynamics you observe. Do populations always persist? Do they eventually reach a steady state?
Figure8.3.5.Figure8.3.6.Figure8.3.7.
ExercisesExercises
1.
The R-pentomino configuration was found by John Conway in 1969. According to LifeWiki 1
conwaylife.com/wiki/R-pentomino
, it is the most active polyomino with fewer than six cells; it stabilizes in generation 1103, while all others stabilize in at most 10 generations.
