John Conway's Game of Life
In the first couple of lectures we talked about emergence of systems-level properties that don't exist in the constituent elements through interactions between those elements. Thus, even when you know the complete set of rules by which each of the individual elements behave, the result of interactions between them may give rise to surprising phenomena that are often impossible to foresee on the basis of the elementary rules of action alone.
As a concrete example let us consider the 2-dimensional cellular automata Game of Life proposed by John H Conway. The "game" is played on a 2-dimensional square lattice (checkerboard) where each cell could be at any point of time either alive (state 1) or dead (state 0). The evolution of the states of each cell occurs in discrete time-steps according to the following rules:
1. If a cell is alive at a certain time step t, it will remain alive in the next time step if it has 2 or 3 live neighboring cells - else it dies (Death)
2. If a cell is dead at a certain time step t, it becomes alive if it has exactly 3 live neighboring cells - else it remains alive. (Birth)
The first task of the assignment is to program this "game" so that you can see its evolution on the computer. Create a program such that one of the parameters is the initial density of live cell (which are randomly distributed on the lattice) and another parameter is the size of the lattice (total number of cells).
Plot the time-evolution of the live cell density starting with different initial density of live cells for a given lattice size (say 20 by 20).
The next step of the assignment is to see, for a given lattice size (say 20 by 20) and initial density of live cells (say 0.2), what is the distribution of times for which activity continues in the lattice without converging to either time-invariant (static) patterns or trivial period-2 cycles like "blinkers".
Then look at how the distribution changes as you change the system size (to 40 by 40 and then 60 by 60) for different initial density of live cells.
Finally, explore other rules of birth and death (different from the ones Conway proposed) and see what other set of rules can give non-trivial behavior.
You are free to research the web to find out more about the properties
of this cellular automata (but not allowed to copy)...