2.17. Lecture 11: Game of Life¶
Before this class you should review the following:
Have an excellent reading week!
Read Think Complexity, Chapter 6
Before next class you should:
Read Think Complexity, Chapter 7, and answer the following questions:
What is diffusion? What is reaction diffusion?
How does cellular automata relate to diffusion?
What is percolation? What exactly are porous cells?
How would you use cellular automata to simulate physical systems?
What is cross correlation and convolution? How can we use these to define diffusion?
Form a project team and choose a topic
Note taker: Hemanth Sundaresan
2.17.1. Overview¶
Today’s lecture introduced Conway’s Game of Life, a two dimensional cellular automaton used to model complex systems and emergent behavior. The Game of Life demonstrates how simple local rules can produce complex global patterns. It serves as a foundational example of how complexity can appear from decentralized interactions.
Key topics covered:
Understanding cellular automata and Conway’s Game of Life
Cell states and neighbor relationships
Rules governing cell survival, death, and reproduction
Common patterns such as still lifes, oscillators, and spaceships
Conway’s conjecture and long lived patterns
Philosophical interpretations such as realism and instrumentalism
How to implement the Game of Life using computational tools
2.17.2. Conway’s Game of Life¶
The Game of Life is a two dimensional cellular automaton developed by John H. Conway in 1970 and popularized by Martin Gardner in his Scientific American column. It consists of a two dimensional grid of cells where each cell can exist in one of two states:
Alive, represented as 1
Dead, represented as 0
Each cell interacts with its eight surrounding neighbors. This group of neighbors is called the Moore neighborhood.
The grid evolves over time according to the following rules:
If a cell is alive and has 2 or 3 neighbors, it stays alive
If a cell is alive and has fewer than 2 neighbors, it dies due to isolation
If a cell is alive and has more than 3 neighbors, it dies due to overcrowding
If a cell is dead and has exactly 3 neighbors, it becomes alive
Otherwise, dead cells remain dead
These rules define how purely local interactions determine the future state of the system. Despite their simplicity, repeated application of these rules across the grid generates complex and often unpredictable behavior. This illustrates a key principle of complexity science: global patterns can emerge from simple decentralized rules.
2.17.3. Emergence and Common Life Patterns¶
Emergence refers to complex behavior that appears from simple rules and interactions between components of a system.
Although each cell follows only a few simple rules, when many cells interact together, the system as a whole can form:
Stable patterns that do not change
Moving patterns that travel across the grid
Patterns that last for a long time
Complex systems that continue evolving over many generations
No single cell controls the system. Instead, organized structures and behaviors arise naturally from local interactions.
This idea applies to many real world systems such as biology (cell growth and organization), physics (particle interactions), traffic flow, social systems, and computer simulations. In each case, large scale patterns arise from simple local interactions between individual units.
Several common patterns appear frequently in the Game of Life:
Still life patterns: Patterns that remain unchanged over time because each cell has a stable number of neighbors.
Example: Beehive pattern
Fig. 1: A stable pattern called a beehive.
Oscillators: Patterns that repeat after a fixed number of steps. These patterns cycle between different configurations. The number of steps required for a pattern to return to its original configuration is called its period. For example, a pattern with period 2 returns to its initial state every two generations.
Example: Toad oscillator
Fig. 2: An oscillator called a toad.
Spaceships: Patterns that move across the grid while undergoing a periodic cycle of shape changes. Like oscillators, spaceships return to their original configuration after a fixed number of steps, but shifted to a new position on the grid.
Example: Glider
Fig. 3: A spaceship called a glider.
These patterns show how stable, repeating, and moving structures can form from simple rules. They demonstrate that cellular automata can create complex behavior without external control.
2.17.4. Conway’s Conjecture and Behaviors¶
Most initial configurations eventually stabilize or enter oscillatory cycles. However, some patterns persist for a very long time before settling into stability. These are called Methuselahs.
Example of a Methuselah:
R-pentomino: This pattern begins with only five cells but evolves through many generations before stabilizing. It stabilizes after 1103 generations and produces 116 live cells before settling into a collection of stable and oscillating structures. It demonstrates how simple initial conditions can produce complex and extended dynamics.
The existence of long lived patterns led Conway to propose a conjecture. He hypothesized that there was no initial configuration that would produce unbounded growth in the number of live cells. However, this conjecture was later disproven with the discovery of patterns that generate infinite growth.
Examples:
Gosper’s Glider Gun: A stable configuration that periodically produces gliders. As the stream of gliders moves outward indefinitely, the number of live cells grows without bound.
Puffer Trains: Moving patterns that leave behind a trail of persistent debris, resulting in unbounded growth as the pattern moves across the grid.
These discoveries demonstrated that the Game of Life allows sustained and infinite growth under certain conditions, contradicting Conway’s original conjecture.
2.17.5. Realism and Instrumentalism¶
The Game of Life raises philosophical questions about whether patterns and structures are “real.”
Scientific realism: suggests that theoretical entities are real if they exist within scientific models.
Instrumentalism suggests that models are simply tools used to describe and predict behavior, regardless of whether the entities truly exist.
For Example:
A glider is not a physical object, but it behaves like one. It is simply a pattern of cells that moves across the grid. However, we treat it as an object because doing so helps us understand and describe system behavior.
This highlights how models in complexity science are often explanatory tools rather than literal representations of reality.
2.17.6. Implementing the Game of Life¶
The Game of Life can be implemented computationally using arrays and mathematical operations. The grid is represented as a two dimensional array (matrix), where each element represents a cell.
Each cell is represented as:
1 for alive
0 for dead
To compute the next generation:
Count the number of live neighbors for each cell
Apply the Game of Life rules based on the neighbor count
Update the grid to create the next generation
This process is repeated to simulate the system over time.
Neighbor counting is done by checking the eight surrounding cells (Moore neighborhood). The new grid is computed from the old grid and then replaces it for the next step.
There are multiple ways to program this:
Using nested loops
Using matrix operations
Using cross correlation techniques
These implementations allow the simulation to run efficiently. In particular, cross correlation provides a faster and more efficient method because it computes neighbor counts for all cells at once using matrix operations instead of checking each cell individually with loops.
This demonstrates how computational simulation is a powerful tool for studying complex systems. When analytical solutions are difficult or impossible, simulation allows us to observe system behavior and explore emergent phenomena.
2.17.7. Key Takeaways¶
Important concepts from this lecture:
The Game of Life is a cellular automaton
Simple rules can produce complex emergent behavior
Cells interact locally but produce global patterns
Common patterns include still lifes, oscillators, and spaceships
Some configurations demonstrate long term complexity and unbounded growth
Computational simulation is essential for studying complex systems
The Game of Life is a fundamental example of complexity science and demonstrates how structured behavior can emerge from simple, rule based systems.