2.12. 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:

    1. What is diffusion? What is reaction diffusion?

    2. How does cellular automata relate to diffusion?

    3. What is percolation? What exactly are porous cells?

    4. How would you use cellular automata to simulate physical systems?

    5. What is cross correlation and convolution? How can we use these to define diffusion?

  • Form a project team and choose a topic

Note taker: Yuvraj Nag

2.12.1. Quick Facts about Conway:

  • Full Name: John Horton Conway

  • Born: December 26, 1937, in Liverpool, England

  • Field of Study: Mathematics, particularly in game theory, group theory, and cellular automata

  • Academic Positions: Professor at the University of Cambridge and later at Princeton University

  • Other Contributions:

    • Developed the Surreal Numbers system.

    • Created the Doomsday Algorithm for easily calculating the day of the week for any given date.

    • Made significant contributions to knot theory and group theory.

  • Died: April 11, 2020 (age 82) from COVID-19

2.12.2. Introduction to Game of Life (GoL)

The Game of Life (GoL) is a cellular automaton devised by the British mathematician John Horton Conway in 1970 It is a zero-player game, meaning that its evolution is determined by its initial state, requiring no further input. GoL Operates on a 2D grid of live and dead cells. The Live cells represent the active parts of the system, while Dead cells represent the empty or inactive parts. Each cell follows strict deterministic rules for survival, birth, or death.

2.12.3. Why Care??

Now we know what is GoL, but why should we care? I would just say because it is Awesome and interesting since it helps with AI. But with my marks in mind here are some more reasons:

  • Artificial Life (ALife): Simulates self-replicating structures.

  • Theoretical Computer Science: Proving Turing completeness meaning GoL can perform any computation given the right initial conditions.

  • Mathematical and Complexity Research: Demonstrates emerging patterns from simple rules.

2.12.5. Rules of the Game

Each cell interacts with its eight nearest neighbors (north, south, east, west, and diagonals). The next state of a cell depends on the number of live neighbors:

Current State

Number of Live Neighbors

Next State

Live

2-3

Stays Live

Live

0-1, 4-8

Becomes Dead

Dead

3

Becomes Live

Dead

0-2, 4-8

Stays Dead

2.12.6. Common Patterns in GoL:

2.12.6.1. Beehive (Stable)

  • A fixed formation that does not evolve further.

  • Live cells have 2-3 neighbors so all survive.

  • No new cells are born.

2.12.6.2. Toad (Oscillator)

  • Alternates between two states (Period = 2).

  • Forms a looped sequence of births and deaths.

2.12.6.3. Glider (Moving Spaceship)

  • Moves diagonally across the grid.

  • Period = 4, shifts after each cycle.

2.12.6.4. r-Pentomino (Chaotic Evolution)

  • A five-cell initial state forming an “r” shape.

  • Evolves for 1103 steps before stabilizing.

  • Produces a final configuration of oscillators and gliders.

2.12.7. Pattern Libraries

GoL has many professionals and hobbyists create an extensive library of patterns. They can be viewed using the following link:

Conway’s Game of Life Patterns: https://www.conwaylife.com/patterns/

2.12.8. Conway’s Conjecture

In the early development of the Game of Life (GoL), John Conway hypothesized that there are no initial patterns that will NOT stabilize. This conjecture was later disproven when researchers discovered patterns capable of infinite expansion. There are two specific types of self-sustaining patterns that proved Conway’s hypothesis wrong:

  1. Guns - Stable formations that periodically produce spaceships, leading to continuous expansion.

  2. Puffer Trains - Moving patterns that leave live cells behind, resulting in an increasing number of live cells over time.

2.12.9. Scientific Realism vs. Instrumentalism

GoL raises philosophical questions about the nature of patterns and scientific theories. Two contrasting views are:

Scientific Realism Assumes that mathematical and scientific theories describe real entities.

Example: Some theories about biology are expressed in terms of genes.

Instrumentalism Theories are views as useful models whether they are true or not.

Example: A hurricane is just a pattern of air flow, but it is a useful description because it allows us to make predictions and communicate about the weather.

2.12.10. Attributions

[1] Think Complexity by Allen B. Downey.

[2] Lecture Notes: Graham Taylor, University of Guelph

[3] https://www.nytimes.com/2020/04/15/technology/john-horton-conway-dead-coronavirus.html

[4] https://www.math.princeton.edu/news/john-h-conway-1937-2020

[5] https://en.wikipedia.org/wiki/John_Horton_Conway

[6] https://www.cambridge.org/core/journals/mathematical-gazette/article/john-horton-conway-19372020/