Lecture 11: Game of Life
========================

Before this class you should review the following:

.. include:: prep11.txt

Before next class you should:

.. include:: prep12.txt

Note taker: Yuvraj Nag

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

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.


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.

Why so Popular??
----------------
Now we know (hopefully) why we should care for GoL, but why is it so popular? A few reasons are:

- **Simple Rules**: Easy to understand and implement.
- **Complex Behavior**: Simple rules lead to complex emergent behavior.
- **Universality**: GoL is **Turing complete**, meaning it can simulate any computer algorithm just like **Rule 110**.

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**:

.. list-table::
   :widths: 10 10 10
   :header-rows: 1

   * - **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


Common Patterns in GoL:
-----------------------

Beehive (Stable)
^^^^^^^^^^^^^^^^
- A **fixed** formation that does not evolve further.
- **Live cells** have **2-3** neighbors so all survive.
- No new cells **are born**.

Toad (Oscillator)
^^^^^^^^^^^^^^^^^
- Alternates between two states (**Period = 2**).
- Forms a **looped sequence of births and deaths**.

Glider (Moving Spaceship)
^^^^^^^^^^^^^^^^^^^^^^^^^
- Moves **diagonally** across the grid.
- **Period = 4**, shifts after each cycle.

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.

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/

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.

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.


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/